Osaka Central Advanced Mathematical Institute (OCAMI)
Osaka Metropolitan University
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 4 (2022)
doi: 10.24544/ocu.20220721-001

Geometry, Representation Theory and
Quantum Fields
Organized by
Hiroshi Itoyama
Kazunobu Maruyoshi
Jaewon SongMarch 22–25, 2022
Abstract
We report the international workshop “Geometry, Representation Theory and Quantum Fields” held in Nambu Yoichiro Institute of Theoreticaland Experimental Physics (NITEP), Osaka City University, and Osaka City
University Advanced Mathematical Institute (OCAMI).

2020 Mathematics Subject Classiﬁcation.
20C32, 70H06, 81T30, 81T32, 81T35, 81T40
Key words and Phrases.
Integrability, Geometry, Representation theory, Supersymmetry, Quantum ﬁeld
theory, String theory© 2022 OCAMI.
OCAMI. Geometry, Representation Theory and Quantum Fields. OCAMI Reports. Vol. 4, Osaka
Central Advanced Mathematical Institute, Osaka Metropolitan University. 2022, 386pp.
doi: 10.24544/ocu.20220721-001

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OCAMI Reports Vol. 4 (2022)

Preface
Geometry and Representation theory appear in various research areas in quantum ﬁeld theory (QFT). While these have been independently hot topics for several years, it turned out that they play important roles in understanding the nonperturbative behaviors in QFT. Especially in supersymmetric ﬁeld theories, geometry enables us to understand the non-perturbative dynamics of QFT systematicallyin relation with string/M-theory compactiﬁcation. The representation theory of thesymmetry groups (or associated algebras) of QFTs which is directly connected tothe spectra of the theories has been another hot topic in these ﬁelds.
The focus of the international workshop “Geometry, Representation Theory and
Quantum Fields” is on QFT and string theory in these relations with geometry andrepresentation theory. The purpose of this workshop is to gather the experts of theseﬁelds to discuss the recent developments. At the same time, young researchers andgraduate students are welcome to interact them to stimulate their researches. The
workshop covers the topics such as dualities, exact results in supersymmetric gauge
theories, nonperturbative methods in QFT, higher symmetry in QFT, geometricrepresentation theory, vertex operator algebra, and AdS/CFT correspondence.
The workshop was held in a hybrid format. All talks were broadcasted in Zoom
and a small number of domestic participants gathered in Osaka City University
Media Center. 209 people have been registered, and there are roughly 40 - 80 people
participated each talk on zoom. We had 18 talks by the invited speakers including
mathematicians and physicist, as well as 11 short talks by young researchers. Theygave talks focusing on recent developments, which promoted the fruitful discussions.
This will lead to future research projects and a discovery of a new relation.
In this report, we record the abstracts and the slides of the talks.

Geometry, Representation Theory and Quantum Fields

iii

Organizers
Hiroshi Itoyama
Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka
Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: itoyama@omu.ac.jp
Kazunobu Maruyoshi
Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-Kitamachi,Musashino-shi, Tokyo 180-8633 Japan
Email address: maruyoshi@st.seikei.ac.jp
Jaewon SongDepartment of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea
Email address: jaewon.song@kaist.ac.kr

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OCAMI Reports Vol. 4 (2022)

Contents
Shoto Aoki
Chiral fermion on curved domain-wall . . . . . . . . . . . . . . . . .

1Jinbeom Bae
Towards a Classification of Fermionic Rational Conformal Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2Lakshya Bhardwaj
Generalized Symmetries and Anomalies in 3d with Applications to
N=4 SCFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Giulio Bonelli
Spinning Black Holes, 2D CFTs, N = 2 supersymmetric gauge theories & Heun functions . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Sunjin Choi
The Yang-Mills duals of AdS black holes . . . . . . . . . . . . . . . . 51
Zhihao Duan
Resurgence of complex Chern-Simons theory at higher levels . . . . . 58
Abhijit Gadde
A scattering amplitude for massive particles in AdS. . . . . . . . . 68

Dongmin Gang
Exceptional Dehn fillings in 3D-3D correspondence . . . . . . . . . . 69
Dongwook Ghim
Brane Brick Models and the Z 2 -labelled family of toric Calabi-Yau
4-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Philip Glass
Cheshire cat resurgence in quantum field theories . . . . . . . . . . . 110
Sergei Gukov
Fermionic characters, knots, and BPS quivers . . . . . . . . . . . . . 111
Amihay Hanany
Hasse diagrams and Quiver Gauge Theories . . . . . . . . . . . . . . 134

Geometry, Representation Theory and Quantum Fieldsv

Yosuke Imamura
Finite N superconformal index via the AdS/CFT correspondence. . 152

Katsushi Ito
ODE/IM correspondence and WKB periods . . . . . . . . . . . . . . 175
Omar Kidwai
Topological recursion and BPS structures for hypergeometric spectral
curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Si Li

Elliptic chiral homology and quantum master equation. . . . . . . . 215

Sanefumi Moriyama
Duality Cascades and Affine Weyl Groups . . . . . . . . . . . . . . . 236
Sunil Mukhi
Poincare Series, 3d Gravity and Averages of Rational CFT. . . . . 237

Emily Nardoni
Confinement in Adjoint QCD from Seiberg Witten Geometry. . . . 261

Takahiro Nishinaka
The Nekrasov partition function and Schur index of gauged ArgyresDouglas theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Natalie Paquette
Symmetries, universal defects, and holography . . . . . . . . . . . . . 263
Rajath Radhakrishnan
Quantum codes from conformal field theory. . . . . . . . . . . . . . 273

Shu-Heng Shao
Non-invertible Symmetries in Higher Dimensions . . . . . . . . . . . 281
Marcus Sperling
Magnetic quivers and line defects . . . . . . . . . . . . . . . . . . . . 288
Kaiwen Sun
Twisted Elliptic Genera . . . . . . . . . . . . . . . . . . . . . . . . . 302

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OCAMI Reports Vol. 4 (2022)
Yuji Tachikawa
On a rarely-mentioned class of 3d N=4 superconformal field theories
Philsang Yoo
Derived Geometry for Physicists

316. . . . . . . . . . . . . . . . . . . . 347

Gabi Zafrir
Exceptional moduli spaces for exceptional N=3 theories . . . . . . . . 356
Yunqin Zheng
Symmetry Protected Topological Criticality. . . . . . . . . . . . . . 373

Geometry, Representation Theory and Quantum Fields

Chiral fermion on curved domain-wall
Shoto Aoki
Abstract. We consider a massive fermion system having a curved domainwall embedded in a square lattice. In a similar way to the conventional flat
domain-wall fermion, chiral massless modes appear at the domain-wall butthese modes feel ”gravity” through the induced spin connections.

(S. Aoki) Osaka University

1

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OCAMI Reports Vol. 4 (2022)

Towards a Classification of Fermionic Rational
Conformal Field Theories
Jinbeom Bae
Abstract. A fermionic CFT refers to a conformal field theory with primariesof half-integer spin and entails the spin structures. The torus partition functions of each spin structure are associated with the level-two congruence subgroups such as Γθ , Γ0 (2) and Γ0 (2). To explore the landscape of the fermionic
RCFTs, we develop the Modular Linear Diﬀerential Equations (MLDE) forthe level-two congruence subgroups. We show that the partition functions of
the fermionic RCFTs are mapped to that of the bosonic RCFTs, such as the
WZW models, via the generalized Jordan-Wigner transformation. Furthermore, we propose the necessary conditions for promoting fermionic RCFTs to
the supersymmetric RCFTs and discuss the emergent supersymmetry on theedge modes.

(J. Bae) Oxford University

Geometry, Representation Theory and Quantum Fields

Towards a Classification of Fermionic Rational
Conformal Field Theories
Jin-Beom Bae
University of Oxford
Based on the[2010.12392], [2105.02148], [2108.01647], [2112.14130]
works with Zhihao Duan, Kimyeong Lee, Sungjay Lee, Matthieu Sarkis
Geometry, Representation Theory and Quantum Fields

Intro and Motivation

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OCAMI Reports Vol. 4 (2022)

Rational Conformal Field Theory• In this talk, we mainly focus on the torus partition function of two-dimensional
conformal field theory, which is defined asihcc
Z (τ, τ̄ ) = TrH q L0 − 24 q̄ L̄0 − 24 ,q ≡ e 2πiτ ,q̄ ≡ e −2πi τ̄ .

• The torus partition fuction is decomposed in terms of the Virasoro characters
Z (τ, τ̄ ) = χ0 (τ )χ̄0 (τ̄ ) +Xdh,h̄ χh (τ )χ̄h̄ (τ̄ ) +Xdn χ0 (τ )χ̄n (τ̄ ) + (c.c.) .

n

h,h̄

• If the Virasoro primaries are reorganized into a finite number of ‘conformal
characters’, which is associated with an extended chiral algebra,
Z (τ, τ̄ ) =N
X

Mi,j fi (q)f¯j (q),i,j=1then we call such theory as the rational CFT. c and h are rational valued.

Quantum Hall Effect• Consider the two-dimensional electron system that is subjected to the strong
magnetic field B and low temperature T . The Hall conductivity is given bye2
σxy = 2π~
ν. Here ν called the filling fraction.
• The plateaux were observed at integer and fractional filling fraction
such as ν = 1, 2, · · · (integer quantum Hall effect) and ν = 13 , 23 , 15 , · · ·(fractional quantum Hall effect).

[Tsui, Stormer, Gossard]• (The macroscopic perspective)
Chern-Simons theories are known tocapture the response of the quantum Hall ground state to low-energy
perturbations.

Geometry, Representation Theory and Quantum Fields

5

Low-energy Effective Description• Let us consider the abelian Chern-Simons theory with an action
S=

k
4π

Z

d 3 x µνρ Aµ ∂ν Aρ ,where k is the CS level and Aµ is the background electromagnetic field. Then,hJi i = −k
ij Ei ,2πhJ0 i =kB.
2π
2Therefore, the above theory describes the quantum Hall effect with k = e~ν .
• To described the Laughlin states (fractional quantum Hall theory for ν = 1/m),
we introduce an emergent dynamical gauge field aµ . It is associated with the’collective’ behavior of the electrons.


Z
m µνρ
1 µνρ
3
S = d x −  aµ ∂ν aρ +
 Aµ ∂ν aρ .
4π
2π

The Bulk-Boundary Correspondence• The interactions between electrons yield a new state of matter. For instance,
the anyons carry the fractional spin and charge. In the low-energy effective field
theory, the anyons are described by the Wilson lines. For instance, in an abelian
CS theory with even level k, we consider the Wilson line of the formH

Wn (C) = e in C a ,2nwhere the spin of Wn (C) read 2k
mod 1. Since hWk (C)i = 1, n ∼ n + k, namely nruns for 0, ±1, · · · , ±(k/2 − 1), k/2.

• The rational CFTs are realized as the boundary theory of bulk CS theory.
Bulk

Boundary(G )k Chern-Simons Theory

WZW model for (G )kSpin ` of the Wilson line

Weight h = ` of the primary state

One-form symmetry

Center symmetry

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OCAMI Reports Vol. 4 (2022)

Fermionic Rational Conformal Field Theory• Consider the U(1)k CS theory. For odd k, there are 2k distinct Wilson lines
Wn (C) labeled by n = 0, ±1, · · · , ±(k − 1), k. For instance, the theory with k = 3
possesses the anyons of spin 0, 1/6, 1/6, 1/2, 2/3, 2/3. What is the corresponding
object at CFT side?
√• Let us recall the free boson theory compactified on a circle of radius R = k.
2
For even k, the theory becomes RCFT with conformal primaries of h = m
2k(m = −k/2 + 1, · · · , k/2). On the other hand, the ‘reorganization’ does notoccur for odd k. There is no extended chiral algebra.
• For odd k, the manifold depends on the choice of spin structure. And theboundary CFT also has the spin structure. The anyons of the U(1)3 theory aremapped to the conformal primaries of the NS-NS sector partition function of thefirst member of the N = 2 supersymmetric minimal model.
• Extend the bulk-boundary correspondence with the bulk theories defined on the
manifold with the spin structure.

The Goal of Talk
• We aim to classify the fermionic RCFT using the modular linear differential
equation(MLDE) approach. To this end, we develop the MLDE associated with
the Congruence subgroup of level two.
• We explore the space of supersymmetric RCFTs. We first start from thefermionized (G )k WZW models by applying the fermionization.
• Notation :

NS

NS

f
NS
R

NS

R
NS

NS

e
R
R

R

R

Geometry, Representation Theory and Quantum Fields

7

Short Reviews
1. Classification of 2d RCFT via MLDE
2. Fermionization (Jordan-Wigner transformation)

Modular Property of the Conformal Characters• Three conformal characters f0 (τ ), f 21 (τ ), f 161 (τ ) of the Ising model are known tobe transformed under the τ → −1/τ asf0 (−1/τ )1
1
 f 1 (−1/τ )  2
=  1√2f 1 (−1/τ )
2
16

1
1
√
− 2


√ 
f0 (τ )2√ 

− 2   f 12 (τ )  ,0f 1 (τ )16

and they form a modular invariant partition function Z = |f0 |2 + |f 21 |2 + |f 161 |2 .
• Conformal characters are transformed under the modular transformation as,fi (γτ ) =NX

ρij (γ)fj (τ ) ,j=1

where ρij (γ) is an unitary representation ρ : SL(2, Z) → GL(N, C). Such behavior
guarantees the modular invariance of partition function.

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OCAMI Reports Vol. 4 (2022)

Modular Linear Differential Equation• The conformal characters fi (τ ) are regarded as the independent solutions of thefollowing N-th order differential equation [Mathur, Mukhi, Sen]


N−1
X
N
r
D +
ϕr (τ )D fi (τ ) = 0
r =0

where the coefficient function ϕr (τ ) is a modular form of weight 2N − 2r anddkcovariant derivative is defined as D = q dq
− 12
E2 (τ ). The covariant derivative
increases modular weight by two.
• The Wronskian index ` = 0, 2, 3, 4, · · · counts the singularity of ϕr (τ ). (e.g.,` = 0 for ϕr (τ ) = E4 (τ ), E6 (τ ) and ` = 2 for ϕr (τ ) = E6 (τ )/E4 (τ )) The MLDE is
specified by two lables N and `.
• The MLDE provide us with the expressions of the conformal characters. Further
efforts are required to pick up the solutions satisfying positivity, integrality andconsistent Fusion rule algebra.

Exercise : N = 2 RCFT with ` = 02• For ` = 0, the MLDE takes the form of D + µ1 E4 (τ ) fi (τ ) = 0.
 N(N−1)PN−1c• Due to the Valence formula, we have i=0 hi − 24
− 12 = − 6` . Thus,c(c+4)
conformal weight of primary is fixed by h = c+2
12 . Furthermore, µ1 = 144 .
• One can recast the MLDE with λ = (θ2 (q)/θ3 (q))4 .
λ2 (1 − λ)2 ∂λ2 −
c(c + 4)2λ(1 − λ)(2λ − 1)∂λ +(λ(1 − λ) − 1) fi (λ) = 0
3
144

It is invariant under the S : λ → 1 − λ. The solutions are given by the regular
hypergeometric functions.
− c


12
1
4−c
c 4−c
f0 (λ) =λ(1 − λ)
,− ;
;λ ,
2 F1
16
12
4
6

 c+4


12
1
c +8 c +4 c +8
f1 (λ) = N
λ(1 − λ)
,
;
;λ
2 F1
16
12
4
6


• S-matrix is obtained from the monodromy of hypergeometric function.

Geometry, Representation Theory and Quantum Fields

9

Fusion Rule Algebra• The fusion coefficient Nijk counts the number of independent couplings among
three primaries φi , φj , φk . Nijk should have positive integer.
• We use the Verlinde formula to find fusion rule algebra coefficient.
Nijk =X Si` Sj` (S −1 )`k
S0``• Need to find an extend S-matrix to find a consistent fusion rule algebra. For
instance, recall the solution of the second-order MLDE with c = 2.
(1)

(2)

1

1

−e ±π/3−e ∓π/3−e ∓π/3 −e ±π/3

Z = |f0 |2 + 2|f1 |2 = |f0 |2 + |f1 |2 + |f1 |2
The S-matrix is enlarged by
1
√
3

1
1

2
−1

!



1
1 
→√  13
1

Application to the MLDE with N = 2 and ` = 0• We need to solve the MLDE + non-negative integer Fourier coefficients in
q-expansion + Consistent fusion rule coefficients to find the conformal
characters of the RCFTs. [Mukhi, Hampapura]• The following seven solutions are known to constitute the well-defined partition
function of an unitary RCFT.
ch

Identification

Ne

c

h

Identification

Ne

1

1/4

SU(2)1 WZW

2

26/5

3/5

(F4 )1 WZW

2

2

1/3

SU(3)1 WZW

3

6

2/3

(E6 )1 WZW

3

14/5

2/5

(G2 )1 WZW

2

7

3/4

(E7 )1 WZW

2

4

1/2

SO(8)1 WZW

4• The MLDE has been applied for various (N, `) = (2, 2), (2, 4), (2, 6), (3, 0), · · · .
• The classification based on the integrality conjecture has been proposed.
[Kaidi, Lin, Parra-Martinez]

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OCAMI Reports Vol. 4 (2022)

The Jordan-Wigner Transformation[Karch, Tong, Turner][Hsieh, Nakayama, Tachikawa]• A partition function of fermionic theory F is related to the the bosonic theory B
with non-anomalous Z2 symmetry as follows.
 

Z
1 X
ZF [ρ + T ] = g
ZB [s] exp iπ arf[ρ] + arf[ρ + s] + s ∪ T
2 sHere ρ is the spin structure on Σg , s denotes the ZB
2 gauge field and T is theFbackground Z2 gauge field.
• arf[ρ] is called the Arf invariant and is a mod two index. On T 2 , its value for[PP] is one and zero for other spin structures. In particular, e iπarf[ρ] correspond to
the partition function of the Kitaev spin chain on Σg .
• For the bosonic theory B with non-anomalous Z2 symmetry, gauging the Z2
symmetry leads to an orbifold Be = B/Z2 . In this case, the partition function read
 Z

1 X
ZBe[T ] = g
ZB [s] exp iπ s ∪ T
2 sTwisted Partition Functions• Let us write the twisted partition functions ZB [s] with line defect Lg .
hi
Z(1,1) (τ, τ̄ ) ≡ ZB (τ, τ̄ ) = TrHu q L0 −c/24 q̄ L̄0 −c/24 ,hiZ(g ,1) (τ, τ̄ ) ≡ TrHu Lg q L0 −c/24 q̄ L̄0 −c/24 ,hi
Z(1,g ) (τ, τ̄ ) ≡ TrHt q L0 −c/24 q̄ L̄0 −c/24 ,hiZ(g ,g ) (τ, τ̄ ) ≡ TrHt Lg q L0 −c/24 q̄ L̄0 −c/24 .
• We take Lg as the Velinde line Lh that is associated with Z2 symmetry. Lh
preserves left and right chiral algebras separately.
Si,k• Its action on a primary state φk is given by Lh0 φk = S0,kφk .

• In the view of the bulk-boundary correspondence, the Verlinide line of the WZWmodel may have a relation with the Z2 one-form symmetry generator (Wilson line)of the Chern-Simons theory.

Geometry, Representation Theory and Quantum Fields

11

Fermionization• The Jordan-Wigner transformation provide us with the partition function of
the fermionized theory. Each sector partition function of fermionized theory F are
known to have the form of
1
2
1
f
ZFNS =
2
1
ZFR =
2
1
e
ZFR =
2

ZFNS =

Z(1,1) + Z(g ,1) + Z(1,g ) − Z(g ,g ) ,Z(1,1) + Z(g ,1) − Z(1,g ) + Z(g ,g ) ,Z(1,1) − Z(g ,1) + Z(1,g ) + Z(g ,g ) ,Z(1,1) − Z(g ,1) − Z(1,g ) − Z(g ,g ) .

• In terms of the twisted partition functions, the partition function of Z2 orbifold
theory (Be = B/Z2 ) read:
ZBe =


1
Z(1,1) + Z(g ,1) + Z(1,g ) + Z(g ,g )2

Example : Ising model
We note that the Ising model that involves three primaries I, (h = 1/2), andσ(h = 1/16). We denote their conformal characters as f0 (q), f 12 (q), f 161 (q) as
before. Let us choose the Verlinde line Lh= 12 . Using the S-matrix, one can findZ(g ,1) (τ, τ̄ ) = |f0 |2 + |f 21 |2 − |f 161 |2 .
In addition,
Z(1,g ) (τ, τ̄ ) = f 12 (q)f¯0 (q̄) + f0 (q)f¯12 (q̄) + |f 161 (q)|2 ,Z(g ,g ) (τ, τ̄ ) = −f 12 (q)f¯0 (q̄) − f0 (q)f¯12 (q̄) + |f 161 (q)|2After applying the Jordan-Wigner transformation, we have
ZFNS = |f0 (q) + f 12 (q)|2 ,ZFNS = |f0 (q) − f 12 (q)|2 ,f

ZFR = 2|f 161 (q)|2 ,and the above partition functions can be identified with the free fermion theory.
(N.B. GSO projection)

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OCAMI Reports Vol. 4 (2022)

Classification of the Fermionic RCFTs

Congruence Subgroup of Level Two• The partition functions of four spin structures are related by the modulartransformation. The NS-NS sector partition function is invariant under T 2 and S.
T

T
NS

R
NS
S

NS
NS

T
R

R
S

R
S

• The conformal characters of fermionic RCFT involves half-integer exponent,
therefore they are associated with the congruence subgroups.
n
o
1 0
0 1
Γθ = γ ∈ SL2 (Z) |γ ≡ 0 1 or 1 0 mod 2 ,no? ?Γ0 (2) = γ ∈ SL2 (Z) |γ ≡ 0 ? mod 2 ,no? 0Γ0 (2) = γ ∈ SL2 (Z) |γ ≡ ? ? mod 2

Geometry, Representation Theory and Quantum Fields

13

Fermionic Modular Differential Equation• The modular property of the partition functions guarantees that the conformal
characters transformed by themselves, under the congruence subgroup of level
two. Therefore, fi NS (τ ) is regarded as the independent solutions of the following
differential equation.


N−1
X
N
s
D +
ϕr (τ )D fi NS (τ ) = 0
r =0• The coefficient functions ϕr (τ ) are generated by weight 2N − 2r modular formof the congruence subgroup Γθ . We use the fact that the modular forms of eachcongruence subgroup have the following ring structure.
4 s 4rM2k (Γθ ) = h (−ϑ42 )r ϑ4s
4 + (−ϑ2 ) ϑ4 , r ≤ s, r + s = k i,
4s
4s 4r
M2k (Γ0 (2)) = h (−1)r +s (ϑ4r
3 ϑ4 + ϑ3 ϑ4 ), r ≤ s, r + s = k i,
4s
4s 4r
M2k (Γ0 (2)) = h ϑ4r
2 ϑ3 + ϑ2 ϑ3 , r ≤ s, r + s = k i

Fermionic First Order MLDE
• The first-order fermionic MLDE (` = 0) of the NS-NS sector is given byh i NSd1 − 2λ4
4D + µ ϑ4 − ϑ2 f = 0 ⇔+µf NS = 0.
dλλ(1 − λ)The solution of the above differential equation is very simple:
f0NS (λ) =h16 i 12c=λ(1 − λ)

ϑ3 (q)
η(q)c.

It is identical to the NS-sector partition function of 2c-copies of the Majorna-Weyl
free fermions.
• The partition function of other spin structures can be obtained from f0NS (λ), byapplying the modular transformations. More precisely,ff0NS (q) =

ϑ4 (q)
η(q)c,f0R (q) =ϑ2 (q)
η(q)

c
.

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OCAMI Reports Vol. 4 (2022)

Fermionic Second Order MLDE
• Let us move to the fermionic second-order MLDE of the NS-NS sector,
assuming no singularities of the coefficient functions.
hiD2 + µ1 −θ24 (τ ) + θ44 (τ ) D + µ2 θ38 + µ3 E4 f NS (τ ) = 0,
m

h d2
dλ2

+

2(1 + 3µ1 )(1 − 2λ) d4(µ2 + µ3 ) − 4µ3 λ(1 − λ) i NS
f (λ) = 0,+3λ(1 − λ)dλλ2 (1 − λ)2• The analytic solution is expressed in terms of the regular hypergeometricfunction.


c
1
c
c
c
f0NS (λ) = 2 3 λ− 12 (1 − λ)− 12 2 F1 β + , β − ; 1 − 2hNS ; λ = q − 24 (1 + a0 q 2 + a1 q + · · · ),


NS
NS
c
c
c
f1NS (λ) = b0 2 3 −8h λ2h − 12 (1 − λ)− 12 2 F1 β + + 2hNS , β − + 2hNS ; 1 + 2hNS ; λ ,
with

β± =


p
1
2 − 8hNS ± 4 − a1 + 2c − 32hNS + 2a1 hNS − 4chNS + 64(hNS )2 .
4

In addition, we find c and hNS such that satisfy the positivity and integrality.

Classification of the Second-Order fMLDE• With the assumption of a0 = 0 (no free-fermion currents), we find that thefollowing cases satisfy the positivity, integrality.
type

property

central charge c

BPS, IcRh−= 24, a0 = 0

1, 94 , 6, 39
4 , 11, 12

non-BPS, I

c
R
h−
> 24
, hNS 6= 12

7 133 91 102
85
114
10 , 10 , 5 , 5 , 21, 4 , 22, 5

non-BPS, IIcRh−, hNS = 12
> 24

9
11
13
15
2 , 5, 2 , 6, 2 , 7, 2

non-BPS, III

one-parameter family

16cR• If we assume the unitarity bound is saturated (h−= 24), the list of reasonable
solutions are given bytype

property

central charge c

BPS, II

c
R
h−
= 24
, a0 6= 0

3 3
21 45
4 , 2 , 3, 6, 9, 2 , 4 ,12Geometry, Representation Theory and Quantum Fields

15

Identification of the BPS Solutions• The BPS type I solutions are mapped to the WZW models via thefermionization. We note that the following WZW models possess non-anomalous
Z2 symmetry, which is the subgroup of the center symmetry.
chNS

fermionizationchNS

fermionization

1

1/6

U(1)3 WZW model

11

5/6

SU(12)1 WZW model

9/4

1/4

SU(2)6 WZW model

39/4

3/4

Sp(12)1 WZW model6

1/2(SU(2)1 )⊗6 WZW model• The BPS type II solutions with c = 3/4 yields the supersymmetric Lee-Yangmodel. It is the non-unitary N = 1 supersymmetric minimal model. The
solutions with c = 3/2, 3, 6, 9, 21/2 can be understood with the ADE WZW
rk(g)/2 gmodel with level one. More precisely, fi NS = (ϑ3 (τ )/η(τ ))χi (τ ) andg = A1 , A2 , D4 , E6 , E7 .

Third-order fermionic MLDE
We extend the MLDE analysis to the three-character case. We assume that thecunitarity bound is saturated, hR ≥ 24
, and no free-fermion currents. Then, thesolutions satisfying the positivity and integrality are given by(c, h1NS , h2NS )Identification(c, h1NS , h2NS )Identification

3m 1 m c
2 , 2 , 8 , 24

(SO(m)1 )⊗3

1 3
( 15
2 , 2, 4)

SU(4)4 /Z2(2, 16 , 13 )(U(1)3 )⊗2

3 7
( 42
5 , 5 , 10 )(Sp(6)1 )⊗2

3 2
( 18
5 , 10 , 5 )(SU(2)3 )⊗2(10, 23 , 56 )(SU(6)1 )⊗2( 92 , 14 , 12 )(SU(2)6 )⊗2 /Z2

3 3
( 39
2 , 4, 2)(Sp(24)1 )⊗2(5, 13 , 12 )Sp(4)34 11( 66
5 , 5 , 10 )?

(7, 12 , 23 )Sp(6)2(22, 56 , 53 )(SU(12)1 )⊗2 /Z2



Most of the solutions are related to the WZW models (or Z2 orbifold of the WZWmodels) via the fermionization. One exception is the solution with c = 66/5:
there is no bosonic RCFT that is related to this solution with the fermionization.

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Comments on the Supersymmetric
RCFTs

Supersymmetric Rational CFTs
• Examples of the supersymmetric RCFTs:
8
1. Supersymmetric Minimal Models with c = 32 1 − `(`+2)
2. Extremal SCFT c = 12 [Frenkel, Lepowsky, Meurman]
Z NS (q) = q −1/2 + 276q 1/2 + 2048q + 11202q 3/2 + · · ·3. Beuaty and Beast SCFT with c = 24 [Dixon, Ginsparg, Harvey]
Z NS (q) = q −1 + 4096q 1/2 + 98580q + 1228800q 3/2 + · · ·• Let us consider the N = 1 supersymmetric RCFTs. This class involves achiral primary G (z) of conformal weight h = 3/2 satisfying
T (z)G (0) ∼3
1+ ∂G (0),2zz

G (z)G (0) ∼2c
1+ T (0).
33zzNote that the above chiral primary is a descendant of the vacuum.

Geometry, Representation Theory and Quantum Fields

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Necessary Conditions for SUSY
• It is hard to check the susy algebra for general interacting 2d CFTs. Instead, we
consider the common features of the susy RCFTs.
1. The NS sector has spin-3/2 currents as the vacuum descendants and they areinterpreted as the supersymmetry currents.
2. Torus partition fuction of R-R sector becomes constant.
c.
3. The R-sector primaries satisfy the susy unitarity bound hR ≥ 24
4. Torus partition function allows decomposition with the supersymmetric
Virasoro characters.
Proposal : If a given fermionic RCFT satisfy the above conditions, it is regarded
as the supersymmetric RCFT.
5
R• Example : Z6 parafermion theory with c = 5/4 : h−
= 96
6
6
ZFNS
e = ψ6,6 + ψ6,02

6
6
+ ψ2,2+ ψ4,2

6
6
ZFRe = ψ1,1
− ψ5,1

2

6
6+ ψ5,5− ψ5,−1e2

66+ ψ4,4+ ψ4,−22

2

6
6+ ψ5,−3− ψ5,3

2= 12 + 12 + 0

Candidates of the Supersymmetric RCFTs• Fermionization of the following WZW models are satisfy the SUSY conditions.
(Only covers the WZW models with simple groups.)
Type

Gk

A-typeSU(12)1 , SU(4)3 , SU(6)2 , SU(2)6

A0 -type

SU(4)4 /Z2

C-typeSp(4)3 , Sp(6)2 , Sp(12)1

D-typeSO(8)3 , SO(12)2 , SO(24)1

SO(N)3SO(3)3 (= SU(2)6 ), SO(4)3 (= (SU(2)6 )2 ),SO(5)3 (= Sp(4)3 ), SO(6)3 = SU(4)3 , SO(7)3 , · · ·E-type(E7 )2 , (E8 )2

OrbifoldSO(16)2 /Z2 , SU(16)1 /Z2 , SU(8)2 /Z2• BPS solutions of the second and third order fermionic MLDEs.

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Generalized Symmetries and Anomalies in 3dwith Applications to N=4 SCFTs
Lakshya Bhardwaj

Abstract. I will discuss generalized symmetries (1-form, 0-form and 2-group)of 3d gauge theories, along with their t Hooft anomalies. Applying this togood 3d N=4 gauge theories lets us also determine the generalized symmetries
and anomalies involving enhanced Coulomb branch 0-form symmetries in the
IR 3d N=4 SCFTs. The information about these symmetries and anomalies
allows us to determine the precise global forms of 3d mirrors/magnetic quiversof 4d Class S theories and 5d SCFTs. The global form of 0-form symmetrygroups predicted by these magnetic quivers matches results obtained via othermethods in 4d and 5d.

(L. Bhardwaj) Oxford University

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Spinning Black Holes, 2D CFTs, N = 2
supersymmetric gauge theories & Heun functions
Giulio Bonelli
Abstract. The study of perturbation theory around the Kerr black hole is aclassical problem in General Relativity. Due to the high order of symmetry of
the gravitational field and the consequent separation of variables at the linear
order, the problem reduces to the study of linear ordinary second order diﬀerential equations. The resulting equations are of Fuchsian type and therefore,
as already observed long ago by A.M.Polyakov, can be solved exactly in termsof classical irregular Virasoro conformal blocks. By making use of the specificexact expressions of the latter implied by the AGT dual perspective on theconformal field theory, it is possible to explicitly solve the connection problemof the resulting (confluent) Heun equation and give novel exact and explicit
formulas for the grey body factor, quasi-normal modes and Love numbers of
the Kerr black hole.

(G. Bonelli) SISSA

Geometry, Representation Theory and Quantum Fields

Spinning Black Holes
2D CFTsN = 2 supersymmetric gauge theories&
Heun functions
Giulio Bonelli (SISSA & INFN – Trieste)

March 22, 2022

talk @ Geometry, Representation Theory and Quantum Fields

OutlineBased on• G.B., Cristoforo Iossa, Daniel Panea Lichtig, AlessandroTanzini "Exact solution of Kerr black hole perturbations via
CFT2 and instanton counting. Greybody factor, Quasinormal
modes and Love numbers"; e-Print: 2105.04483 [hep-th]
Phys.Rev.D 105 (2022) 4• G.B., Cristoforo Iossa, Daniel Panea Lichtig, AlessandroTanzini "Irregular conformal blocks and connection formulae for
Heun functions", e-Print: 2201.04491 [hep-th]

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Abstract

The study of perturbation theory around the Kerr black hole is a classicalproblem in General Relativity. Due to the high order of symmetry of thegravitational field and the consequent separation of variables at the linear
order, the problem reduces to the study of linear ordinary second orderdifferential equations. The resulting equations are of Fuchsian type andtherefore, as already observed long ago by A.M.Polyakov, can be solved
exactly in terms of classical irregular Virasoro conformal blocks. By making
use of the specific exact expressions of the latter implied by the AGT dual
perspective on the conformal field theory, it is possible to explicitly solve theconnection problem of the resulting (confluent)Heun equation and give novel
exact and explicit formulas for the grey body factor, quasi-normal modes andLove numbers of the Kerr black hole.

OutlineIntroduction
Teukolsky equation and BH perturbation theory
CFT2 and Heun equation
Application to BHs
Conclusions

Geometry, Representation Theory and Quantum Fields

Introduction
I Partial Differential Equations are an important mathematical instrumentin theoretical physics (and in other quantitative sciences)
I Quantum Mechanics, General Relativity, Quantum Field Theory, String

Theory, Statistical Physics...
I The solutions of abstract PDE/ODEs can represent states of a physicalsystem, its reaction after an external probe, relations among observable
properties of the system, etc
I Every new exact solution is an advance, usually a new corner of atheory gets enlighten
I Exact solutions are rare and implied by additional symmetries which

reduce the PDE to an integrable system (by separation of variables)
I Also the study of perturbation theory around exact solutions contains

important information about the physical system (e.g. stability analysisof the object described by the solution, spectrum of stimulated
emission)

Introduction

I Fortunately not always different branches of science advance

independently and separately (this is called interdisciplinarity).
I Sometimes in a branch an open problem gets solved, finding a NEW

different perspective to compute a given physical quantity.
I Potentially this is a NEW solution of a PDE (or more simply of an ODE)I If so, it can be exported to other branches of science where the samedifferential equation arises, describing a maybe completely different
aspect of a different theory (this is called cross-fertilisation)
I This talk is about one of such a fortunate cases...

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Introduction
The theories in the play are three:
I General Relativity in D=4
I Supersymmetric gauge theories with N =2 and D=4
I Conformal Field Theories in D=2

The problems we link are respectively:
I Physical properties of Spinning Black HolesI Computation of BPS partition functions with surface defects
I Correlation functions in Liouville Field Theory

The equations we work on are respectively:
I Einstein Equations linearised around Spinning Black Holes [Teukolsky

equation]
I Renormalisation Group equation for BPS defects [Quantum spectral

curve]I Null states equation at level 2 [Belavin-Polyakov-Zamolodchikov

equation]
These reduce, after the proper specifications are taken, to a concrete set of
linear II order differential equations known as Heun Equations (and its
confluences/reductions).

Teukolsky equation
I Let us focus first on the General Relativity side
I The PDE is the Einstein equationGµν = Rµν −1gµν R + (Λgµν ) = 8πGTµν2

I The particular exact solutions we consider is the (AdS)-Kerr metric (andits generalisations with EM charges)
I This describes the physical spacetime of a spinning Black HoleI (AdS)-Kerr metric is an highly symmetric solution of Einstein equation(enjoys an hidden symmetry given by a conformal Killing-Yano 2-tensor,
generalising the spherical symmetry of the Schwarzschild metric)
I The Einstein equation linearised around the spinning BH metric is

known as Teukolsky equation
I This is important because it describes gravitational waves emitted by

the spinning Black HoleI This approach can be developed for other metric sourced by different

compact objects. (The absence of a Birkoff’s theorem for static
axisymmetric space-times makes the story more complicated than thespherical one.)

Geometry, Representation Theory and Quantum Fields

Teukolsky equation

I After a small perturbation or in a resilient process the BH spacetime isdescribed by an approximate metric g = g BH + ψ. (E.g. Early inspiraland ringdown phases of binary BH)
I Up to higher order corrections (highly interesting and much moredifficult to be exactly studied), the metric perturbation ψ solves the
Teukolsky equation
LinGµν [g BH ](ψ) = 0
where LinGµν [g BH ] is the second order differential operator obtained bylinearising the Einstein tensor around the Black Hole solution.
I Because of the hidden symmetry, Teukolsky equation is completely

separable. [Also geodesic equations and free particle propagation inthe (AdS)-Kerr background are separable].

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Teukolsky equation
I Concretely, the static and cylindrical symmetric BH metric in cylindricalcoordinates (Boyer-Lundquist) (x µ ) = (t, r , φ, θ)BH
ds2 = gµν(θ, r ; M, a)dx µ dx νwhere M is the BH mass and a its angular velocity (more complicatedmore general BH solutions exist with E/M and/or NUT charges andpossibly Λ).
I Because of static and cylindrical symmetry, one can Fourier transformfixing the frequency ω and the angular momentum m of the perturbation
Z
X iφm+itωψ = dω
eψω,m,l (θ, r )m,lI Substituting in the Teukolsky equation one finds that the θ and r

variables dependence of ψ separates asψω,m,l (θ, r ) = Sω,m,l,aω (cos(θ))Rω,m,l (r )where Sω,m,l,aω (u) and Rω,m,l (r ) satisfy two resulting linear second
order ODEs in the corresponding variables.
I There’s an accompanying separation constant problem.

Teukolsky equation=Heun equation
I The differential equations satisfied by both the factors are reduced to

Heun equation or one of its confluences (depending on the specific BH
solution at hand)[Teukolsky]
I This is the linear second order ODEs with four regular singularities



d2
+
dz 2



γ
δ

+
+
z
z −1
z −t



d
αβz − q+dzz(z − 1)(z − t)w(z) = 0 ,where α + β + 1 = γ + δ + . The auxiliary parameter q is related to theasymptotic condition at z ∼ 0 defining the Heun functionq
HeunG (t, q, α, β, γ, δ, z) = 1 + z + O(z 2 ) .
tγ
I HeunG is not exactly solved in math textbooks, but Mathematica

knows it well numerically at arbitrary precision.
I It reduces to the hypergeometric when  = 0 and q = αβt
I For example the confluent Heun is obtained by t → ∞ and rescalingsome parameters. It reads


d 2w
γ
δ
dw
αz − q++++
w = 0.
dz 2
z
z −1
dz
z(z − 1)and is the relevant equation for Kerr BH.

[Teukolsky]

Geometry, Representation Theory and Quantum Fields

37

Heun equation
I Usually GR people uses numerical or approximate hypergeometricsolutions to study BH physics, but one can do better.
I Actually Heun equation and its confluences have been exactly solved inanother branch of theoretical physics.

[Gamayun-Iorgov-Lisovvyy]

I We can therefore import the solution of Heun equations to GR andobtain from it exact properties of BHs.
I Heun functions can be identified with Virasoro classical (irregular)

conformal blocks in CFT2.

[Polyakov]

I [Alday-Gaiotto-Tachikawa] (AGT) correspondence makes these calculable interms of well defined combinatorial formulas, once they get identified
with equivariant Nekrasov partition functions in N = 2 supersymmetricgauge theories in D=4 and SU(2) gauge group.
I In short, equivariant multi-instanton counting solves Heun equations.
I Let’s review all this and then go back to some issues in BH physics.

Heun equation: clarificationsI 1) By exactly solved I mean that one does have a closed form for thecoefficients of local expansions and the connection coefficients. For
hypergeometric 2 F1 (α, β, γ, z) one has at z ∼ 0 the well known
combinatorial formula
X (α)n (β)n nΓ(α + n)zwhere (α)n =2 F1 (α, β, γ, z) =n!(γ)nΓ(n)n≥0and explicit connection coefficients, so it is exactly solved. We will
display analogue (more complicated) formulas for Heun functions which
are therefore exactly solved in the above sense (expansion andconnection).
I 2) Heun equation displays four regular singularities on P1 . The

confluences and the reductions of Heun function correspond tocollisions and parametric rescalings which raise and lower the order ofthe singularities in the equation. (e.g. Confluent Heun, the relevant one
for Kerr BH, has two regular and one irregular of rank 1. Heun equation
has four regular points and corresponds to AdS-Kerr. BPS
Kerr-Newman corresponds to bi-confluent Heun)I 3) To master the solutions of Heun equations for actual applications to

BH, we also need to exactly compute the connection matrices relating
distant local solutions in the z-plane.

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CFT2 and Heun equation
I Conformal Field Theory in D=2 is a QFT with a traceless holomorphic

stress energy tensor with self-OPE
T (z)T (w) ∼2T (w)
c/2++ l.s.t4(z − w)(z − w)2

I Irregular vertex operators of rank r are characterised by r + 1

parameters {Λi }i=0,...,r and the OPE (r = 0 are primaries with ∆ = Λ0 )


Λr
Λr −1
Λ0
+
+ ... +
Φr (z)+ l.s.t
T (z)Φr (w) ∼(z − w)2r +2(z − w)2r +1(z − w)r +2
I Irregular vertices can be obtained by the confluence of r + 1 regular
(i)

(i)

vertices, i.e. primaries, Φ0 of conformal dimension Λ0 as
r
Y

(i)

Φ0 (wi ) ∼wi →z Φr (z) + l.s.t

i=0
(i)

and a proper prescription on the scaling of wi ’s and Λ0 ’s
I Irregular vertices can be reduced by setting some of their parameters Λito zero.

CFT2 and Heun equation
I Reducible Virasoro representations are characterised by null states,

which correspond to primary vertices χr ,s with degenerate conformal
2
dimension ∆r ,s = Q4 − αr2,s (Liouville theory c > 1; αr ,s = − br2 − 2bs , Q = b + b1 andc = 1 + 6Q 2 . ).
I The lowest non trivial null operator is at level 2 and satisfies(b−2 ∂z2 + T (z))χ1,2 (z) = 0 .
I Conformal Ward identities on correlation functions with a degenerate

are the celebrated [BPZ ] equations. Schematically, given a multi-vertex
operator OV (z1 , . . . , zN ) with OPE
T (z)OV (z1 , . . . , zN ) ∼ Vi (z; zi , ∂zi )OV (z1 , . . . , zN ) + reg. as z ∼ zi
one gets the linear PDE
X[b−2 ∂z2 +Vi (z; zi , ∂zi )]Ψ(z) = 0 ,Ψ(z) = hχ2,1 (z)OV (z1 , . . . , zN )ii

I To reduce this PDE to an ODE, one gets rid of the ∂zi in the Vi by ab → 0 limit (semiclassical Liouville) properly rescaling the parameters(a)Λi of the multi-vertex operator O. (e.g. for a primary, Λ0 ∼ b−2 Λ̂0 andso on)[Polyakov 1986]

Geometry, Representation Theory and Quantum Fields

39

CFT2 and Heun equation
I Correlators in Liouville Theory holomorphically factorise in conformal

blocks, that is (for example)<nYΦ(pi ) (zi ) >=

i=0

X

0

1

n−1

Cpq0 p1 Cqq0 p2 . . . Cqqn−2 pn |Fqpn
p0 p1 q0 p2. . . qn−2
n−1 ; z0 , . . . zn
q0 q1

Q

I Holomorphic factorisation is not unique, as it depends on the

successive order of taking OPEs of the vertices.
I Conformal blocks with a degenerate vertex insertion are meromorphic

solutions of the BPZ equation in a given pant decomposition of thepunctured plane C \ {zi }i=0,...n .
I Crossing symmetry (reshuffling the order of the OPEs won’t change theresult) relates different local solutions of the very same differential
equation.
I Therefore, local solutions are related by the composition of linearr
operators determined by recombining the three point functions Cpq.
r
I In Liouville theory, Cpq
are given by the DOZZ [Dorn-Otto-Zamolodchikov2 ]formula as a definite ratio of Γ2 -functions.

CFT2 and Heun equation
To summarise:
I The classical limit of Virasoro conformal blocks gives the local solutionsof the linear II order ODE obtained from BPZ.
I The classical limit of the crossing relations among the conformal blocks

gives the connection formulae for the different local solutions.
I The program works for solutions of II order linear ODEs and theirconfluences by using irregular vertices.
I Before proceeding one needs a detailed classification of irregular

Virasoro conformal blocks and their explicit expression.
Notice that some authors already tried to work out this program at different levels [Zograf-Thaktajan, Matone,
Menotti-Seminara, Hadasz, Piatek, David, Hollands-Kidwai, Castro et al., Motl-Nietzke,...] but we couldn’t findconcrete formulas to apply to BHs computations. For more concrete results on confluent Heun, see[Lisovyy-Naidiuk, Lisovyy-Nagoja-Russilion].



|2

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AGT and Conformal blocks
I The hard part of CFT2 comes now into play. While three point functions

enjoy DOZZ formula, the analytic structure of conformal blocks stayed
obscure for some time.
I It has been solved by the [Alday-Gaiotto-Tachikawa] (AGT) correspondence.
I AGT claims that BPS sectors in N = 2 supersymmetric gauge theories

in D=4 can be characterised by two dimensional integrable QFTs
I In particular, S 4 partition functions of N = 2 supersymmetric quivergauge theories with SU(2) gauge nodes has been shown to coincidewith Liouville multi-vertex correlation functions.
I If the gauge theory has a single SU(2) gauge group and Nf = 4, it has

been shown that
4S
ZSU(2),N({mi }, q) =<
f =4

4
Y

1

eαi φ(zi ) >PLiou

i=1

where q = zz12 zz34 , {mi } ∼ {αi } (up to lin.comb.) and 1 /2 ∼ b−2 .
13 24

I To read the details of this formula one needs to separate the N/Spatches contributions on S 4 and compare with the holomorphic
factorised form of Liouville correlation functions.

AGT and Conformal blocks
I In gauge theory, using equivariant localisation, one has
4

S
ZSU(2),N(R, {mi }, q) =f =4Z

[Pestun]4

Rda |ZSU(2),N(R, {mi }, q, a)|2
f =4

Here
4

R
ZSU(2),N(R, {mi }, q, a) = Z1loop (R, {mi }, a)Zinst (R, {mi }, q, a)f =4

where Z1loop is a given ratio of Γ2 -functions andX
Zinst =
q |Y1 |+|Y2 | R(Y1 , Y2 ; mi , a, R)
Y1 ,Y2 | Y.T.

with R a given rational function in its arguments.

[Nekrasov]

I In CFT

<

4
Y

1

eαi φ(zi ) >PLiou =Z

dαCαα1 α2 Cαα3 ,α4 |F (α1 α2 α α3 α4 |q) |2

i=1

I Comparing, one consistently finds that

|Z1loop |2 = CCandZinst = F.

I therefore Virasoro conformal blocks are calculable in terms of welldefined combinatorial formulas.
I All this is proved, as Vir  H • (Minst )[Nakajima] and Vir irreps [Alba et al.]

Geometry, Representation Theory and Quantum Fields

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CFT2 and Heun equation
I To apply to the computation of conformal blocks with degenerate

insertions, one needs a multi-point generalisation of these prototypicalformulas. It exists and it is proved.
I The insertion of degenerate primaries in gauge theory corresponds toinsertions of surface operators. These can be obtained from quiver
theories in a given ungauging limit.
I The classical limit in Liouville theory is known in the gauge theory as
the Nekrasov-Shatashvili limit, where one considers the theory on a
general Ω-background (equivariant parameters labeling (C∗ )2 action on
C2 ) and reduces the vev of the surface operator Ψ(z)
Ψ(z) ∝ e−  1

1 2(F inst (1 )+2 W(z;1 )+O(22 )),(1 , 2 ) → (, 0)

I To obtain Heun functions as the classical limit of conformal blocks, one

has to normalise and gets (four points at (0, t, 1, ∞) and 0 < |z| < |t|)
the two linear independent solutions of Heun equation from the two
blocks with shifted momenta (there are two channels in the OPE of the
primaries with the degenerate)
lim

c→∞

F({αi }, ±|z/t, t)∝ HeunG± (z)F({αi }|t)with exact expressions for the (easy) proportionality factors. See[B.-Iossa-Panea Lichtig-Tanzini] (A whole class of confluences done explicitly)

CFT2 and Heun equation
I The last piece we need is the solution to the connection problem for

Heun (and confluences)
I This is explicitly provided by the classical limit of the crossing relationsof conformal blocks after normalisation.
I The crossing symmetry relations on degenerate conformal blocks are
obtained by comparing the holomorphic factorization formulas before
and after a global PSL(2, C) transformation γ.
I The resulting formulas come in the schematic formX
F (γ) ({αi }|γ(wi ), γ(z)) = [Jacγ ](wi , z)
M± (b{αi })F({αi ±eiγ b/2}|wi , z)±where eiγ ∈ {0, 1} label the momenta of the vertices crossed by thedegenerate. For a single crossing M± (b{αi }) are the hypergeometric). Forconnection matrices (a ratio of Euler Γ functions Mhyper ∼ ΓΓΓΓmultiple crossings the action of the Mhyper accumulates against multiply
shifted conformal blocks. These do not compose as matrices, but they
will compound (with other crucial factors) the Heun connection after theclassical limit b → 0!).
I For irregular blocks one uses the connection coefficients for the
confluent or reduced hypergeometric (that is Wittaker or Bessel
functions which are the irregular 3 + 1 blocks)

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CFT2 and Heun equation
I After normalising and taking the classical limit one gets a concrete

formula for the connection matrices of Heun functions
I In the regular case these are given byMθθ0 = Mhyper
for simple crossings
θθ 0
X
−θ”κD
Mθθ0 =
Mhyperfor multiple crossings
Mhyperθθ” e
θ”θ 0 . . .
θ”=±

where κD = ln(t)a + 12 aD is a given function of the CFT momenta andmodulus.
I In gauge theory language κD is a combination of the quantum SW

periods.
I Technically, these arise because in the b → 0 limit the multiple shift in

the crossed momenta of the normalised blocks exponentiates leaving
behind κD .
I The intertwining of the hypergeometric connection matrix with theexponential of the quantum periods gives the explicit solution to theconnection problem for Heun equations.

CFT2 and Heun equation

I For the confluent and reduced cases one has to resort to theappropriate connection coefficients (Whittaker and Bessel) to start with,
but the general scheme is the same.
I In [B.,Panea-Lichtig,Iossa,Tanzini] we performed a numerical verification of theclassical conformal blocks vs Heun with Mathematica. We also
checked the above connection formulas via comparison between thehypergeometric approximation and the approximation resulting from thetruncation of the multi-instanton contribution of the quantum periods.
Already the one and two instanton corrections sharply converge to thenumerics in both cases.

Geometry, Representation Theory and Quantum Fields

CFT2 and Painlevé
I There is a precise parallel of this approach to Heun equations whichapplies to the corresponding Painlevé τ -functions.
I Painlevé functions solve ODE (wrt moduli) obtained from the study of

isomonodromic deformations of linear ODEs.
I Heun differential equations are the linear system corresponding toclassical Painlevé functions.
I Like Heun equations, Painlevé equations are ubiquitous in theoreticalphysics (Ising model, matrix models, 2d quantum gravity, etc)
I Since the ’90s [Jimbo-Nagoya-Sakai] they have been identified with c = 1 four

points conformal blocks.
I After AGT multi-instanton counting solves Painlevé equations[Gamayun-Iorgov-Lisovyy, Litvinov-Lukyanov-Nekrasov-Zamolodchikov,
B-Lisovyy-Maruyoshi-Sciarappa-Tanzini]

I The interesting link among the two (c = 1 vs c = ∞) is provided by theso called "blow-up" equations [Nakajima] and a proper limit[Bershtein-Shchechkin], [Grassi-Gu], [Nekrasov] on the s.

BH and Heun equations
I Let’s apply the above to the exact computation of some relevant physics

of the Kerr black hole.
I In this case the Teukolsky equation reduces to the confluent Heun

equation.
I In the radial part, the two regular singularities correspond to the innerand outer horizons and the irregular one to radial ∞.
I The grey body factor is σBH = |T |2 , where T is the transmission

coefficient from the BH horizon to ∞.
I One can normalise the monodromy matrix between two points to be

M ∈ SU(1, 1) and directly read the reflection R and transmission T
coefficients as
 1

R
T
T
where |R|2 + |T |2 = 1
M = R∗
1T∗

T∗

I Therefore

σBH =

1
|M11 |2

where M is the appropriate Heun connection matrix computed after
reversing the dictionary with Teukolsky equation.

43

44

OCAMI Reports Vol. 4 (2022)

BH and Heun equations
Explicitly one finds the exact grey body factor for Kerr BH to beσBH =ω − mΩ
2πTH

×



−2
inst
−iω(r+ −r− ))(−2iω(r+ − r− ))−a−2iMω e
exp ∂F|a =a,a =−aΓ(1 + 2a)Γ(2a)Γ(1 − i ω−mΩ
2πTH
∂a1
1
2

 
 

+ (a → −a)Γ 12 − 2iMω + a Γ 12 − i ω−mΩ
+ 2iMω + a Γ 12 − 2iMω + a
2πT
H



inst



where ∂F
|a1 =a,a2 =−a is the quantum dual period of the associated∂a1
Nekrasov partition function in the NS limit and is computed explicitly.
We positively tested this result against
I ω = 0: reproduce the first order grey body factor formula computedfrom the hypergeometric connection by asymptotic matching.
[Strominger Maldacena]

I WKB approximation of the Teukolsky equation[Carneiro da Cunha et al.]

I direct numerical checks with Mathematica.

BH and Heun equations
I Quasi-normal modes (QNM) are the resonance frequencies of the BH.
I These are the values of the frequency of the perturbations ω matching

purely ingoing boundary condition at the horizon and purely outgoing at
infinity.
I These correspond to set to zero the connection coefficient of the

excluded channel (that is ingoing at the horizon and at ∞)MH∞in in = 0
I Using the explicit form of the confluent Heun connection, one reduces

the above condition to the simpler exact equation
1 − e∂a F = 0 ⇒ ∂a F = 2πin , n ∈ Zquantising the NS dual SW period (F is the quantum SW prepotential in
the NS limit).
I This equation was conjectured (and numerically checked) by

Aminov-Grassi-Hatsuda (see also [Hatsuda, Hatsuda-Kimura]).
I The concrete computation of the QNMs ωn passes through theapplication of the dictionary from the gauge theory to Kerr perturbation
theory.

Geometry, Representation Theory and Quantum Fields

BH and Heun equation
I The separation constant of the Teukolsky equation can be computed

exactly from the requirement of regularity of the spheroidal harmonics:
quantisation of the a period a = ` + 21 (after the dictionary with theangular part of the Teukolsky equation!).
I Another application is the exact computation of the Kerr tidal Love

numbers.
I These are obtained by considering the linear perturbation with purely

ingoing boundary conditions at the horizon.
I This perturbation can be analysed in the intermediate regime (before

radial infinity before oscillations damp it) where the multipolar expansion
holds.
I This corresponds to the regime at which the radial perturbation has

power-like behaviour leading term, that means where the potential isdominated by the quadratic pole.

BH and Heun equation
I Love numbers are obtained as the ratio of the connection coefficientsHψ∼ψinH andH−pw pw
H−pw pw
ψ pw∼ Min > ψ> + Min < ψ<

pw
pw
(where ψ<is the response and ψ>is the probe)

kL =

MH−pw
in <
MH−pw
in >

[explicit formula too long to show]
I We reproduced the static (ω = 0) Love numbers as a well defined limitof the non-static ones (ω 6= 0) which are obtained in the literature via
more complicated regularising approaches (analytic continuation in thespin) [Charalambous-Dubovsky-Ivanov]. We have pursued the computation to
next to leading order in mω ∼ Λ (0 at 1 inst, not computed at 2). [As expected,
they do not develop any real part.]

45

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OCAMI Reports Vol. 4 (2022)

Recap

I We have explicitly shown how to use multi-instanton calculus in D=4

N = 2 supersymmetric gauge theories to concretely solve spinning BH
perturbation theory
I This is done by translating the relevant differential equations in CFT2

and characterise their solutions via AGT correspondence.
I We provide exact expressions for the grey-body factor, quasi-normal

modes, separation constant in Teukolsky equation and tidal Love
numbers for the Kerr BH.
I All our results match the existing literature, pass strong numerical tests

and extend to a larger domain in parameter space (non static
perturbation).

Conclusions

I In principle the Polyakov approach of using the classical limit of Liouvilleconformal blocks to solve linear differential equations extends naturally
to general Fuchsian equations [Zograf-Taktajan] and their confluences. The
solution corresponds to the partition functions in a linear quiver N = 2gauge theories with SU(2) nodes (class S at genus 0) with a surface
operator insertion. Also higher genus is possible.
I We should extend the analysis to the full Heun table. We still missreductions of Heun with singularities of order higher than one. It is not asurprise that these correspond to Argyres-Douglas theories in gauge
theory. These cases are harder also for the parallel Painlevé τ -function
problem.
I Extend to difference equations: qHeun from 5D gauge theories on S 1

and qVirasoro blocks (qPainlevé). The 5D problem is more natural fromthe topological string viewpoint.

Geometry, Representation Theory and Quantum Fields

Conclusions
I Here we DO NOT claim that there is any sort of duality between thegravitational problem and the gauge theory, but just that there are somequantities on the two sides which obey the same II order linear ODE(Heun equation). However, one could try to interpret our results in the
Kerr-CFT correspondence.
I The detailed analysis of gravitational wave emitted by BHs or bydifferent compact objects via exact techniques is potentially a richmethod to obtain a fine structure spectrum to distinguish different
objects from far away and test possible higher order or quantum
corrections to classical GR.
See e.g. [Bianchi-Consoli-Grillo-Morales]
I The full analysis of the AdS-Kerr BH using the Heun connection matrix(fully regular case) gives access to some of its physical parameters.
I Many other problems (e.g. in QM) reduce to Heun equation or one of its

confluences/reductions [Fitziev]. Our results and techniques can beexported to these other situations to possibly obtain interesting new
results.

Thank you!

47

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OCAMI Reports Vol. 4 (2022)

Technical slides:

CFT - CBF
Regular
Confluent
1F
Reduced Confluent
1F
2
Doubly Confluent
1 D1
Reduced Doubly Confluent
1E 1
2
Doubly Reduced Doubly Confluent
1 E1
2

2

F → 1F%&

12

SU(2) Gauge Theory
Nf = 4
Nf = 3
Nf = 2 asymmetric
Nf = 2 symmetric
Nf = 1
Nf = 0F

1 D1&
1E 1 → 1 E 1
2
2
2
%

Heun
HeunG
HeunC
HeunRCHeunDCHeunRDC
HeunDRDC

Geometry, Representation Theory and Quantum Fields

49

Connection formulas: example - the regular blockbefore c → ∞Expand the following at 0 < |z| < |t| < 1
X Zh∆∞ |V1 (1)Vt (t)Φ(z)|∆0 i =

α
2,1

α
t

dα Cα 0θ α Cα α
0

θ=±

0θ

α

z

0

t

2,1
α
α
Cα∞ α1 α F (α∞ α t α0θ α |t,12)

The same correlator can be expanded for z ∼ t and t << 1 after the transformation γ(x) = x−t1−t∆ −∆1 −∆t −∆2,1 −∆0 2h∆∞ |V1 (1)Vt (t)Φ(z)|∆0 i = (1 − t) ∞h∆∞ |V1 (1)V0

=

X Z

α
2,1

α
0

dα Cα tθ αt Cα α

θ=±

t






Φ

t −1

z−t

1

|∆t i =1−tα2,1
∆ −∆1 −∆t −∆2,1 −∆0α
α
Cα∞ α1 α (1 − t) ∞F (α∞ α 0 αtθ αt |

tθ



t
t −1

,

2

t −z

)

t

.

0

2
θ
Comparing the two: eliminate a common Cα∞ α1 α ; express the resulting (CC)θ
LHs = |Mθθ 0 | (CC)LHs using
DOZZ, split in holomorphic and anti-holomorphic getting
α

2,1
α
α
F (α∞ α t α0θ α |t,X=

Mθθ0 (bα0 , bαt ; bα)e

z

0

1

iπ(∆−∆0 −∆2,1 −∆t )

t

)

α2,1
∆ −∆1 −∆t −∆2,1 −∆0αα(1−t) ∞F (α∞ α 0 αtθ0 αt |1

θ 0 =±

t
t −1

,

t −z

)

t

where Mθθ0 are the hypergeometric connection coefficients.

Kerr:Teukolski-CHeun dictionary2ds = −∆ − a2 sin2 θ

!

Σ

2

dt +

Σ

where

2

∆

2

2

∆

d R

+ (s + 1)

dr 2

2

Σ
2

2

sin θ dφ −∆ = r − 2Mr + a = (r − r− )(r − r+ )

Φs = eimφ−iωt

dr dr

∆

"
2

Σ

dt dφ

Sλ,s (θ, aω)Rs (r )K − 2is(r − M)K+

2a sin2 θ(r 2 + a2 − ∆)

2

2

d∆ dR

2

∂x (1 − x )∂x Sλ + (cx) + λ + s −2!

2

Σ = r + a cos θ ,2(r 2 + a2 )2 − ∆a2 sin2 θ2dr +Σdθ +!
− Λλ,s + 4isωr(m + sx)2
1 − x2R = 0,#− 2csx Sλ = 02where x = cos θ, c = aω, K = (r + a )ω − am and Λλ,s = λ + a2 ω 2 − 2amω . We determine the separation
constant λ as the eigenvalue of the angular eq. by imposing regularity at θ = 0, π. Both the above equation reduce
s+1

r −r

to CHeun. For example the radial, by setting z = r −r− and ψ(z) = ∆(r ) 2 R(r ) becomes+−d 2 ψ(z)
dz 2
2

Â

2

a (1 − m ) − Mr+ 4amM ω(M −

√ 2
M

=
0
r
1

Vr (z) =− a2 ) + 4M 2 ω 2 (a2 − 2M 2 ) + 8M 3

√ 2
M

4
X
r i
Âi z

1

z 2 (z − 1)2 i=0− a2 ω 2

am

√ 2
M

− a2 − 2a2 M ω + 2M 2 ω(M −

√ 2
M

+(is)
4(a2 − M 2 )
2

Â

2

+ Vr (z)ψ(z) = 0 where

22

4a λ − 4M λ + (8amM ω + 16a M ω2

2(a2 − M 2 )√− 32M 3 ω 2 ) M 2 − a2 + 4a4 ω 2 − 36a2 M 2 ω 2 + 32M 4 ω 2=
4(a2 − M 2 )
r

Â

2

2

2

2

2

= −λ − 5a ω + 12M ω − 12M ω
Â

r
3

2

2

2

2

2

q

= 8a ω − 8M ω + 8M ωr22

M 2 − a2 + (is)(i − 6ωq

M 2 − a2 + (is)4ω
2

2= 4(M − a )ω ,
4

Â

q

q


−i +
+(is) M 2 − a2 ) − s

M 2 − a2

2

s

2

−
4


√
(2a2 ω − am) M 2 − a2 
a2 − M 2− a2 )

+s

2

50

OCAMI Reports Vol. 4 (2022)

Explicit conformal blocks (regular case)α1
α−2( Q +α1 )( Q +αt )∆−∆t −∆022×
α t;t = t
(1 − t)
α∞ α0

 Y




X |~
Y|
~
~ , α + θα z
~ , α + θα∞
×tzvec α~, Yzhyp α
~, Y
~, Y
t
0
hyp α
1


F

θ=±

~
Y




Y
~, µ =
zhyp α~, YY
αk + µ + b

−1


i−

k =1,2 (i,j)∈Yk~zvec α
~, Y



=

Y

Y

2


+b



1
j−
2




0 0
Q − E αl − αk , Yl , Yk , (i , j )YE (αk − αl , Yk , Yl , (i, j))1(i 0 ,j 0 )∈Ylk ,l=1,2 (i,j)∈Yk
E (α, Y1 , Y2 , (i, j)) = α − b LY ((i, j)) + b AY ((i, j)) + 1 .
21
α2,1 zαα=
F α1∞ α t α0θ α ; t
0
t


Q
Q
z −2( 2 +αt )( 2 +α2,1 )−2( Q +α1 )( Q +α2,1 )+θbα0−2( Q +α1 )( Q −αt )∆−∆t −∆0θ bQ
2
2
2
2
(1−t)
1−
(1−z)
×
=t
z 2
t
  |W
~|






X |~
z
Y|
~ zvec α~ , W~ z~ , α~ , W~ ;α ××t
zvec α
~, Y
~, Y
t
0θ
0θ
bifund α
t
~ ~
−1

Y ,W

×

Y





~ , α + σα∞ z~zhyp α~, Y
1
hyp α~0θ , W , α2,1 + σα0

σ=±



~ , β,~ ;α =~ W
zbifund α~, YtY
k ,l=1,2

×Y(i,j)∈Yk
E (αk − βl , Yk , Wl , (i, j)) −

Q
2


+ αt

Y
(i 0 ,j 0 )∈Wl


Q−E






Q
0 0
βl − αk , Wl , Yk , (i , j ) −+ αt2Geometry, Representation Theory and Quantum Fields

The Yang-Mills duals of AdS black holes
Sunjin Choi

Abstract. We construct large N saddle points of the matrix model for the
N=4 Yang-Mills index dual to the BPS black holes in AdS5 X S5. We findnovel solutions given by areal eigenvalue distributions after slightly reformulating the saddle point problem. We also construct a class of multi-cut saddle
points, showing that they sometimes admit nontrivial filling fractions. As a
byproduct, we find that the Bethe ansatz equation emerges from our saddle
point equation.

(S. Choi) Korea Institute for Advanced Study

51

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OCAMI Reports Vol. 4 (2022)

The Yang-Mills duals of AdS black holes
Sunjin Choi
(Korea Institute for Advanced Study)
Geometry, Representation Theory and Quantum Fields
March 22nd, 2022

This talk is based on- SC, Saebyeok Jeong (Rutgers U.), Seok Kim (SNU) and Eunwoo Lee (SNU)“Exact QFT duals of AdS black holes”
arXiv: 2111.10720 [hep-th]Geometry, Representation Theory and Quantum Fields

53

Microstates of black holes• Quantum properties of black holes are important and interesting subjects of quantum gravity.
- Hawking radiation [74] ~ information paradox- Bekenstein-Hawking entropy [72] ~ statistical understanding of the number of microstates- Quantum chaos ~ SYK model, JT gravity [Shenker, Stanford 13] [Kitaev 14] , …
• AdS/CFT correspondence provides an ideal setting to study the microstates of the black holes.
- Usually difficult to analyze quantitatively due to lack of our understanding about the strong
coupling dynamics.
• Focus on the solvable sector: the BPS sector- Certain operators are protected, i.e. do not receive anomalous dimensions at strong coupling(~ gravity).
- Non-trivial thermodynamic ensemble exists: phase transitions, thermodynamic instabilities, …
𝑍 = 𝑇𝑟[𝑒 −Δ∙𝑅−𝜔∙𝐽 ] (𝑅: R-charge, 𝐽: spin)(σ ∆ − σ 𝜔 = 2𝜋𝑖)

[SC, J.Kim, S.Kim, Nahmgoong 18] [SC, Jeong, S.Kim 21]

Index• Today, we study N =4 𝑈(𝑁) SYM to study the black holes in AdS5 × 𝑆 5 . [Gutowski, Reall 04; Chong, Cvetic, Lu, Pope 05;
Kunduri, Lucietti, Reall 06]• Index of the N =4 SYM on 𝑆 × ℝ: counts BPS operators with (−1)
3𝐹[Romelsberger 05] [Kinney, Maldacena, Minwalla, Raju 05]

𝑚𝑜𝑑 2𝜋𝑖- 𝑅: Cartan charges of 𝑆𝑂(6) R-symmetry ~ electric charges- 𝐽: Cartan charges of 𝑆𝑂(4) rotation symmetry on 𝑆 3 ~ angular momenta- BPS condition: 𝐸 = 𝑅1 + 𝑅2 + 𝑅3 + 𝐽1 + 𝐽2
• Unitary matrix integral representation

Elliptic gamma func. :54

OCAMI Reports Vol. 4 (2022)

Periodic potentials• Consider a toy model with 𝜏-periodic 2-body holomorphic potential (no external potential) :𝑍 ~ ς ‫ 𝑒 𝑎𝑢𝑑 ׬‬− σ 𝑉(𝑢𝑎𝑏 ) where 𝑉 𝑢 + 𝜏 = 𝑉(𝑢) for some 𝜏 ∈ ℂ→ Trivial large 𝑁 saddle point : 𝑢𝑎 =𝐹=−1𝑑‫𝑥 𝜏 𝑉 ׬‬−𝑦𝑑𝑢𝑥 0𝑎
𝜏 ~ 𝜏𝑥 (𝑥 ∈ (0,1))𝑁

1 𝜕𝑉 𝜏 𝑥−𝑦𝜏𝜕𝑦𝑑𝑦 = ‫׬‬01
𝜏

𝑑𝑦 = (𝑉 𝜏𝑥 − 𝜏 − 𝑉 𝜏𝑥 ) = 0• Are there any non-trivial periodicities hidden in our problem?

𝜏

Yes!

Factorized periodic potentials• 𝑺𝑳(𝟑, 𝒁) modular properties of elliptic gamma function (~analogous to 𝑆𝐿 2, 𝑍 modular properties of 𝜗-functions)- “S-trasf.” :,periodicity :• “S-tranformed” integrand :
(𝐼𝑚

𝜎
𝜏

> 0)

?

Geometry, Representation Theory and Quantum Fields

55

Large 𝑁 saddle point…?
• We have found two factorized periodic 2-body potentials (only adjoint matters) :→ Trivial large 𝑁 saddle point: periodic in both directions (𝜎, 𝜏) → uniform parallelogram distribution !
1
1
2
2
𝜕
𝜕𝑉𝜎 𝑢12𝜕𝑉෨𝜏 𝑢12
𝑢 𝑥, 𝑦 = 𝜎𝑥 + 𝜏𝑦,𝐹=−න 𝑑𝑥1 𝑑𝑦1 𝑉𝜎 𝑢12 + 𝑉෨𝜏 𝑢12 = න 𝑑𝑥1 𝑑𝑦1+=01
𝜕𝑢2 −1
𝜎𝜕𝑥
𝜏𝜕𝑦1
1
−
2

2

• Large 𝑁 saddle points of the unitary matrix models are realized as a linear distribution not areal. What’s wrong?

Branch points• Our potentials are made of the logarithm of the elliptic gamma function:
2𝜋𝑖𝑧𝑉෨𝜏 ∋ log Γ ∋ log(1 − 𝑒 𝜏 𝑒 ±2𝜋𝑖𝑢𝑎𝑏𝜏)• Logarithm function has a branch point, which can spoil the periodicity of its argument.

• Branch points inside our ansatz?
- Always exists! Comes from the Haar measure.

“S-trasf.”

56

OCAMI Reports Vol. 4 (2022)

Deformation of the problem• We slightly deform the problem so that the Haar measure singularity can be avoided.
- Molien-Weyl formula: For any permutation invariant function 𝑓(𝑢), we can drop half of the Haar-like measure.

• With 𝜅 = 1/𝜏, order the eigenvalues so that 𝑒

2𝜋𝑖𝑢𝑎𝑏𝜏< 1 for 𝑎 < 𝑏.

→ No branch point coming from the Haar measure inside our ansatz.
• Branch points coming from the other matter contributions?
- Can be evaded in the restricted domain of the chemical potentials (with 𝜎 ↔ 𝜏 version)• Under these constraints, our parallelogram ansatz is the large 𝑁 saddle point!

Bekenstein-Hawking entropy• Resulting large 𝑵 free energy (Δ1 +Δ2 +Δ3 − 𝜔1 − 𝜔2 = ±2𝜋𝑖) [Hosseini, Hristov, Zaffaroni 17] [Agarwal, SC, J.Kim, S.Kim, Nahmgoong 20]→ Legendre transformation yields the Bekenstein-Hawking entropy of the BPS black holes in AdS5 × 𝑆 5 , thus
statistically accounting for their microstates.

=

BH

Geometry, Representation Theory and Quantum Fields

57

Instanton contributions• Physical meaning of our constraints?

• D3-brane instanton contributions to the Euclidean black hole solutions [Aharony, Benini, Mamroud, Milan 21]- When D3-brane wraps 𝑆 3 ⊂ 𝑆 5 and 𝑆1 ⊂ 𝑆 3 ⊂ AdS5 :
𝛿
𝜎

𝛿
𝜏

𝑖𝑘𝑆𝐷3
𝑆𝐷3 = 2𝜋𝑁 𝐼 , 2𝜋𝑁 𝐼 ⟹ 𝑍 ~ σ∞𝑘=0 𝑍𝑘 𝑒- If 𝐼𝑚 𝑆𝐷3 < 0, the Euclidean black hole solutions will be unstable under the condensation of the D3-brane instantons.
- 𝐼𝑚 𝑆𝐷3 > 0 : stability condition of the Euclidean black hole solutions• Our constraints are stronger than 𝐼𝑚 𝑆𝐷3 > 0 suggesting that there are more stability conditions from other instantons.
• One may try to construct (𝜎 + 𝑟, 𝜏 + 𝑠)-parallelogram ansatz with 𝑟, 𝑠 ∈ ℤ to satisfy the above constraints.
- When𝐼𝑚(𝜏)∉ ℚ, one can always find certain 𝑟, 𝑠.
𝐼𝑚(𝜎)- They all give the same entropy identical to the Bekenstein-Hawking entropy of dual black holes.

Concluding remarks• We studied the physics of BPS black holes in AdS5 × 𝑺𝟓 from the matrix model for 4d maximal 𝑼(𝑵) SYM on
𝑆 3 × ℝ.
- We constructed the exact large 𝑵 saddle points dual to the black holes in AdS5, thus statistically accounting fortheir Bekenstein-Hawking entropy.
- Evasion of the branch points gives the stability condition against D3-brane instantons.
• We also found multi-cut saddle points, thus constructed large 𝑁 saddle points corresponding to all known Betheroots.
- They sometimes admit continuous filling fractions (no analogue in Bethe root) → gravity interpretation?
• When 𝜎, 𝜏 are collinear, the branch points become very close to our saddle point configurations.
- Certain operators become very light.
- Relation to the light near-horizon modes on the BPS black holes? → hairy AdS black holes

58

OCAMI Reports Vol. 4 (2022)

Resurgence of complex Chern-Simons theory at
higher levels
Zhihao Duan
Abstract. Three-dimensional complex Chern-Simons theory is deeply related to quantum topology of three manifolds and knot theory. In particular,
the perturbative expansion of its partition function gives rise to factoriallydivergent series, which exhibits interesting resurgent structure and provides
new insights into 3D-3D correspondence. In my talk, I will discuss resurgent
properties of the partition functions on some hyperbolic manifolds at a genericlevel k, extending previous works on the level k = 1 case. This is based on an
upcoming work with Jie Gu.

(Z. Duan) Korea Institute for Advanced StudyGeometry, Representation Theory and Quantum Fields

59

Resurgence in complex Chern-Simons theory
at higher Levels
Zhihao Duan
Korea Institute for Advanced Study

ZD and J. Gu, 2204.XXXX

Geometry, Representation Theory and Quantum Fields, Osaka
March 23, 2022

Motivation• In physics we are interested in quantities like partition functions, correlators, etc.
• The universal tool is perturbation theory: assume the coupling g to be smallf(g) ∼a gn∑ nn• Perturbative series are typically asymptotic

an ∼

n!
An

Factorial growthDiverges everywhere on g plane• ∃ non-perturbative sectors that are exponentially suppressed

e−A/g

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OCAMI Reports Vol. 4 (2022)

Resurgence
Take all sectors into accountDi erent sectors are relatedFirst we need a tool to make sense of asymptotic series
an ∼

n!
An

ff

Borel method: remove n! by hand to construct analytic function

Borel Resummation

ξ plane?
θAlong the ray θ = Arg(g) to make it convergent

Geometry, Representation Theory and Quantum Fields

61

Stokes PhenomenonAA• sΓ+[ f ] and sΓ−[ f ] di er by exponentially small term
an = S(n − b)!
A
A2
1+a1A
+a2A
+⋯
n
A
n−b
(n − b)(n − b − 1)()

aiA coe cients of perturbative series in A-sector• When we further vary the coupling constant, the exponentially small term can be dominantff

ffi

E.g. Airy function describing the intensity of rainbow [Casselman, “the mathematics of rainbows”]

Stokes ConstantsΓ+[ f0] = sΓ−[ f0] + S0,A ⋅ e−A/g ⋅ s[ fA ]⏟perturbation in A-sector

In general we can have SA,B

• Stokes automorphismpackage all sectors into a vector(S[ f1], S[ f2], S[ f3], ⋯)Tcrossing a singularity = automorphism
S[ fA] →∑B

A,BS[ fB],

• Stokes constants often have integrality structure𝔖

𝔖

BPS state degeneracy / topological, moduli space invariantsA,B = δA,B + ∑ SA,B ⋅ e(A−B)/g⋅ EA,B62

OCAMI Reports Vol. 4 (2022)

Goal• Understand the resurgent structure of a “solvable” system• Explore mathematical/physical meaning of the Stokes constantChern-Simons theory• Chern-Simons theory with gauge group G as TQFT
k SCS(A) ={k
2
on the metricTrA ∧ dA + A ∧ A ∧ A No dependence
De ned on M3
∫
4π
3
M3

G = U(1): (fractional) Quantum Hall e ect∫

A e ik SCS = Witten-Reshetikhin-Turaev invariant of M3

fffi

k is quantizedof Wilson loop in representation ℛ⟨ ⟩ = VEV(Colored) Jones polynomial of knot K

K, ℛ

𝒟

Z=

G = SU(2): Topological invariants [Witten, ’89]!

∃ state sum model, skein relation…

Geometry, Representation Theory and Quantum Fields

63

Complex CS theory• G = SL(2,ℂ)• Motivation{

Aa → (Aa, Aa)

Volume conjecture relates knot theory & hyperbolic geometry[Kashaev, ’97]

Categorifcation of WRT invariant, 3d-3d correspondence…

cf. Dongmin Gang & Sergei Gukov’s talkSSL(2,ℂ) =• M3 hyperbolic Z =k − is → 0∫

k + isk − isSCS(A) +
S (A)
2
2 CS

k → (k, s)!

Unconstrained
Quantized

non-abelian

A A e i SSL(2,ℂ)

State sum integral[Dimofte et al, ’09]

1. Decode the “exact” form in terms of summing over perturbation around critical points
2. Critical points are complex at connections, a whole family with actions di er by 2πiℤ
ff

fl

𝒟

Example: 41 knot
• Complement of 41 knot in S 3 is hyperbolic
Non-compact with nite volume = V = 2.029883…

e.g. Thurston’s lecture• For (k = 1, s)
b → 0 limit :x 2 − ix22 dx
Z non−abelian =Φb() e 4πb
∫ℝ+iϵ
2πb
2πb
Φb(x): Faddeev’s quantum dilogarithm b 2 =Φb(x) |b→0 ∼ exp(fi𝒟[Witten, ’10][Gukov, Marino, Putrov, ’16]

U0
+ ⋯)2πib 2

1 − is
1 + is(1 − ex)(1 − e−x) = 1⇓

Saddlepoints

{

2πi
3
2πi
x=−
3

x=

Geometric
Conjugate

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Resurgent structure

Peacock (孔雀 공작) Pattern[Gukov, Marino, Putrov, ’16][Gang, Hatsuda, ’17]
[Garoufalidis, Gu, Marino, ’20 &’21]

ξ plane• Around a saddle point, evaluate Gaussian integral
fG(

b
1
11b 2
697b 4) = 1/4 +++⋯
2πi
3 ( 72 −3 2(72 −3)2
)

I

II

−2V/(2π)fG• Two di erent perturbative series (f = f )C

2V/(2π)

III

IV

G

− 2πi2• Half-circle Stokes automorphism: power series in q = e
α1,1(q) α1,2(q)(α2,1(q) α2,2(q))α1,1 = 1 − 8q − 9q 2 + ⋯Coe cients all integral
Conjecture the full series; Numerically veri ed up to q 4[Garoufalidis, Gu, Marino, ’20]

fi

𝔖

IV→I =

ff

ffi

𝔖

I→II ∘b

Resurgent structureξ planefG• Two di erent perturbative series (f = f )
C

18
-9

G

-8

IV→I =

α1,1(q) α1,2(q)(α2,1(q) α2,2(q))⟵∏=

IV<θ<II= 2 theory[Garoufalidis, Gu, Marino, ’20]

𝔖

𝔖

[Garoufalidis, Gu, Marino, ’21]

fi

fl

𝒩

fi

I

IV

• Turning on the holonomy parameter in the state integral ⇒ re ned index
ff

9

θ

αi,j = Unre ned superconformal index with certain magnetic ux of dual 3d

𝔖

3

3

• Uniquely factorize Stokes automorphism given poles
I→II ∘-39

Geometry, Representation Theory and Quantum Fields

65

Higher Level[ZD, Gu, to appear]• For (k = 2, s)
Znnon−abelian ∝Extra label ∈ {0,1}

∫ℝ+iϵ

ix2

e− 2πb2 (Db(x,0)Db(x, n) − i Db(x,1 − n)Db(x,1))
ib
ib
ib
ib
Db(x, n) = Φb x − − (−1)n⋅ Φ x + + (−1)n(44 ) b(
4
4)• Saddle point equation is the same for both n

⇓

n=0

2πi
3
2πi
− 3

g πi3 = g− 2πi3g
( 2πi3 = g− πi3 )f πi3 = f− 2πi3

(f 2πi3 = f− πi3 )

n=1

ff

1 + e2x + e4x = 0 ⟹ x =[Dimofte, ’11]

• Expansion gives only two di erent perturbative series
for each n

πi
3

− πi3

dx
2πb [Anderson, Kashaev, ’11]

Resurgent structure
f πi3(f 2πi3 = f )f πi3 = 1 +

πi
3

17 3i
432

b2 −

4321 4b +⋯124416

Expand to order b 600( last coe cient ∼ 9.3 × 10518 + 1.3 × 10515i )4Focus on sector I:

2

1. Use Padé approximant to infer the location of singularities
A + ℤπi, A = 0.323...

-1.0

2. Numerically evaluate Borel resummation S[f] =S[ f πi3 ](S[ f 2πi3 ])3. Compare it with suitably chosen basis (H (b)) = MI(q) ⋅ S[f]1
H0(b)Choose b =eπi/6 eπi/6,
to x MI(q) numerically up to q 6
7
6
−43
fi

ffi

∼ 10

+ 10−57i

q 0:

πi

I
0.5

-0.5

-2

πi

1.0

IV

-4

• = Poles of Padé approximant−1 + 1 × 10−7 − 3 × 10−7i 1 − 5 × 10−7 − 1 × 10−6i(−1 + 6 × 10−7 − 2 × 10−6i −1 − 3 × 10−6 − 7 × 10−6i)

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OCAMI Reports Vol. 4 (2022)

Conjecture• Conjecture the whole q series for MI(q). Similarly for all other sectors
Based on holomorphic factorization of state integral
I→II ∘• Half-circle Stokes automorphism

for

IV→I

g πi3

(g 2πi3 = g πi3 )Elements are re ned SCI with magnetic ux of dual 3d
I→II ∘

for

IV→I

f πi3

(f 2πi3 = f πi3 )Elements are re ned SCI with magnetic ux of dual 3d• Conjecture: for (k = N, s)= 2 theory evaluated at 1

= 2 theory evaluated at −1

1. We have N pairs of perturbative series from n ∈ {0,⋯, N − 1} state integrals
2πin
N

fi

fl

fl

fi

fi

𝔖

𝔖

Re ned Conjecture• Turn on the holonomy parameter: adding extra free parameter u to everything• Conjecture: for (k = N, s)
1. We have N pairs of perturbative series from n ∈ {0,⋯, N − 1} state integrals
2. Half-circle Stokes automorphism gives re ned SCI at exp (

2πu
Nb

+

2πin
)
N

fi

appears naturally in
3d-3d correspondence

fi

𝔖

𝔖

𝒩

Checked for N = 2,3,4

𝒩

2. Half-circle Stokes automorphism gives re ned SCI at exp

Geometry, Representation Theory and Quantum Fields

67

Another Example: 52 knot

[ZD, Gu, to appear]• For (k = 2, s)
We nd 6 di erent perturbative series, grouped into 2 vectors

I→II ∘• Repeat the previous process
Numerically checked up to q

f1
f2
f3

g1
g2
g3

IV→I

for

(f1 f2 f3)

Elements are re ned SCI with magnetic ux of dual 3d

4

I→II ∘IV→Ifor= 2 theory evaluated at 1

(g1 g2 g3)T

Elements are re ned SCI with magnetic ux of dual 3d• Conjecture: for (k = N, s)T= 2 theory evaluated at −1

1. We have N pairs of perturbative series from n ∈ {0,⋯, N − 1} state integrals

fi

fl

fl

ff

fi

fi

𝔖

𝔖

Summary• Study and conjecture the resurgence structure of complex CS theory at all levels• Understand the conjecture from 3d-3d correspondence?

• Incorporate abelian at connection?

[Garoufalidis, Gu, Marino, wheeler, ’21]

fl

fi

𝔖

𝔖

𝒩

2πin
N

𝒩

2. Half-circle Stokes automorphism gives re ned SCI at exp

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A scattering amplitude for massive particles in
AdS
Abhijit Gadde

Abstract. What is the local bulk observable that a dual conformal field
theory correlator computes? In the so-called flat space limit, it is the flatspace S-matrix. But away from this limit, we don’t have a clear answer to thisquestion. In this talk, I will define an observable, called the AdS S-matrix,
that is computed using local bulk eﬀective field theory that has the property
that it reduces to the flat space S-matrix in the large radius limit. I will
describe how to conveniently calculate it using Feynman-like rules and relate
it to a more conventional CFT observable namely the Mellin amplitude.

(A. Gadde) Tata Institute of Fundamental Research, Mumbai

Geometry, Representation Theory and Quantum Fields

Exceptional Dehn fillings in 3D-3D
correspondence
Dongmin Gang

Abstract. According to Thurston’s Hyperbolic Dehn filling Theorem, Dehn
fillings of a 1-cusped hyperbolic 3-manifold are always hyperbolic except for
only finitely many exceptional cases. In 3D-3D correspondence, Dehn filling corresponds to gauging of su(2) symmetry and the exceptional Dehn fillings correspond to gauging with small CS levels. Various interesting nonperturbative IR phenomena (such as spontaneous SUSY breaking, generationof mass gap, SUSY enhancement, etc) can emerge under the ’exceptional’
gauging. In this talk, I will propose a systematic way of reading oﬀ the IR
phases of class R theory associated with non-hyperbolic 3-manifold M frombasic topological properties of the 3-manifold.

(Dongmin Gang) Seoul National University

69

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OCAMI Reports Vol. 4 (2022)

Exceptional Dehn Fillings
In
3D-3D correspondence
Dongmin Gang
(Seoul National University)

Work in progress with
Hee-Cheol Kim (POSTECH) and Sunjin Choi (KIAS)

Geometry, Representation Theory and Quantum Fields

Introduction
Extra-dimensions and compactification are common ideas in
String/M-theory
A typical problem is determining the physics of low-dimensional
theory from the mathematical structures on internal space
3D-3D correspondence provides an ideal set-up for the problem

3-manifolds(2+1)D field theory: Rich Geometrical structures: Rich IR dynamics: Well-studied in Mathematics(Chern-Simons, Monopole, Duality)(Classifications): SUSY Localization

Geometry, Representation Theory and Quantum Fields

71

Outline of my talkExceptional Dehn fillings in 3D class R theory
Classification of closed 3-manifolds
IR phases of class R theories from the classification
Conclusion/Future directions

What is Dehn filling?
Let N be a hyperbolic 3 manifold with a torus boundaryRμν = − 2gμν

H1(∂N, ℤ) = {pA + qB : (p, q) ∈ ℤ × ℤ}
ds 2 =

dt 2 + dθ12 + dθ22
(near boundary at t → 0)t2

2D analogy Riemann surface (with a S 1 boundary)

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OCAMI Reports Vol. 4 (2022)

What is Dehn filling?
Let N be a hyperbolic 3 manifold with a torus boundaryRμν = − 2gμν

H1(∂N, ℤ) = {pA + qB : (p, q) ∈ ℤ × ℤ}
ds 2 =

dt 2 + dθ12 + dθ22
(near boundary at t → 0)t2

Dehn filling to NpA+qB : (closed 3 manifold)

NpA+qB :

(

(/ ∼

∪

pA + qB ∼ S 1shrinkable

What is Dehn filling?

NpA+qB :2D analogy

(

∪

(/ ∼

pA + qB ∼ S 1shrinkable

Closing puncture on Riemann surface

(

∪

=

(/ ∼

Geometry, Representation Theory and Quantum Fields

What is Exceptional Dehn filling?

NpA+qB :

(

∪

(/ ∼

1
pA + qB ∼ SshrinkableThurston′s Hyperbolic Dehn Surgery Thm
Except for finitely many (p, q)s, NpA+qB is hyperbolic
Exceptional Dehn filling

Why exceptional Dehn filling interesting?
Hyperbolic N
3ℍ

Non-hyperbolic NpA+qB

S 3, 𝔼3, ℝ × S 2, ℝ × ℍ2,

˜
SL (2,ℝ), Nil and Sol

Thurston's eight geometries

Drastic change of geometrical structure

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OCAMI Reports Vol. 4 (2022)

3D-3D counterpart of Dehn fillingClass R theory[Terashima-Yamazaki; 2011][Dimofte-Gaiotto-Gukov; 2011]

6D A1 (2,0) theory on ℝ1,2 × M

size(M)→0

3D T[M] on ℝ1,2

Top'l twisting : SO(3) × SO(2) → SO(2) = U(1)geo
R

16Qs → 4Qs (generically) 3D𝒩 = 2 SUSY algebra

3D-3D counterpart of Dehn fillingClass R theory[Terashima-Yamazaki; 2011][Dimofte-Gaiotto-Gukov; 2011]

6D A1 (2,0) theory on ℝ1,2 × M

size(M)→0

3D T[M] on ℝ1,2

Top'l twisting : SO(3) × SO(2) → SO(2) = U(1)geo
R

16Qs → 4Qs (generically) 3D𝒩 = 2 SUSY algebra

T[N, A; B]
T[NpA+B] ==′′SO(3)′p′

T[N, A; B]( SU(2) )/ℤ1−form,2pT[N, A; B]( SU(2)pfor even p⊗ U(1)−2)/ℤ1−form,2

for odd p[Gang-Yonekura; 2018]

T[N, A; B] : 3D theory associated with N (and A) [Dimofte-Gaiotto-Gukov; 2011]
Based on an ideal triangulation of N
We choose A ∉ Ker[H1(∂N, ℤ) → H1(N, ℤ2)] and NA = (Lens space)
B ∈ Ker[H1(∂N, ℤ) → H1(N, ℤ2)]

Geometry, Representation Theory and Quantum Fields

75

3D-3D counterpart of Dehn fillingClass R theory[Terashima-Yamazaki; 2011][Dimofte-Gaiotto-Gukov; 2011]

6D A1 (2,0) theory on ℝ1,2 × M
T[N, A; B]

size(M)→0( SU(2) )/ℤ1−form,2

T[N, A; B]
T[NpA+B] ==′′SO(3)′p′

p

T[N, A; B]( SU(2)p

3D T[M] on ℝ1,2

for even p⊗ U(1)−2)/ℤ1−form,2

for odd p[Dimofte,Gaiotto,Gukov; 2011] [Gang,Yonekura; 2018][Pei; 2015]

T[M]here = Tirred[M] is diﬀerent from Tfull[M] in literatures [Gukov,Pei,Putrov,Vafa; 2017](R 2 × S 1 vacua) ≃[Eckhard,Kim,Shafer-Nameki,Willet; 2019]{(Adj)-irreducible SL(2,ℂ) flat connections , for Tirred[M]all SL(2,ℂ) flat connections , for Tfull[M]The precise relation is unclear to me
Little is known about Tfull[N, A; B] with hyperbolic N(Even the existence of a single 3D theory which see all SL(2,C) flat connections)

Why exceptional Dehn filling interesting?
Class R theory under Dehn filling : T[NpA+B] =For suﬃciently large | p | , NpA+B is hyperbolic

T[N, A; B]′′SO(3)′p′

Weak coupling

Nothing interesting/surprising happens during the gauging
T[NpA+B] is a just 𝒩 = 2 SCFT whose CT = CT(T[N, A; B]) + O(1/ | p | )For small | p | , NpA+B is non-hyperbolic
Strong coupling

Non-perturbative eﬀects become important (light monopole operators)
Various IR phases (SUSY broken, SUSY enhancement, mass gap, etc) emerge

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OCAMI Reports Vol. 4 (2022)

Example I
T[(S 3\52)pA+B] = (U(1)− 12 × "SO(3)"p−4 coupled to Φ in (Adj)+1)
p = 1,2,3

SUSY enhancement to N=4 SCFT
Of “rank 0” (w/ no Higgs/Coulomb)

From various BPS ptns.
Believe me

Nothing interesting,cf )p = 0,4

1/2 BPS monopole operators
for “SO(3)”otherwisep = 0,1,2,3,4

Non-hyperbolic(S 3\52)pA+B
Hyperbolic

otherwise

Example II
T[(m006)pA+B] =U(1)− 12 × U(1)− 32 coupled to 3Φs of Q = (0, − 1), (−1,1), (1,1) W = Φ21Φ2Φ3′′SO(3)′p−1
′

p=−2
p=−3

SUSY enhancement to N=4 SCFT of “rank 0”
1/2 BPS monopole operators

p = 1, − 1

Mass gap (TQFT)

cf )SUSY broken

p=0

Nothing interesting,otherwise

Non-hyperbolicp = − 3, − 2, − 1,0,1(m006)pA+B
Hyperbolic

otherwiseGeometry, Representation Theory and Quantum Fields

77

Example II
T[(m006)pA+B] =U(1)− 12 × U(1)− 32 coupled to 3Φs of Q = (0, − 1), (−1,1), (1,1) W = Φ21Φ2Φ3′′SO(3)′p−1
′

p=−2
p=−3

SUSY enhancement to N=4 SCFT of “rank 0”

Main Goal:

1/2 BPS monopole operators

Determine the IR phases of

p = 1, − 1

Mass gap (TQFT)

T[M = NpA+B] with non-hyperbolic M
SUSY broken

p=0

Nothing interesting,otherwisefrom topogical properties of M
cf )Non-hyperbolicp = − 3, − 2, − 1,0,1(m006)pA+B
Hyperbolic

otherwise

Main Goal:
Determine the IR phases ofT[M = NpA+B] with non-hyperbolic M
from topogical properties of M
Strategy

Experiments

SnapPy → Ideal triangulation of N = m003,m004,…
→ Field theory T[NpA+B]→ SUSY partition functions using Localization→ Determine IR phase

Theory(Interpretation using M-theory)→ Compare with topological types of M

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OCAMI Reports Vol. 4 (2022)

Main Goal:
Determine the IR phases ofT[M = NpA+B] with non-hyperbolic M
from topogical properties of M
Strategy

Experiments

SnapPy → Ideal triangulation of N = m003,m004,…
→ Field theory T[NpA+B]→ SUSY partition functions using Localization→ Determine IR phase

Theory(Interpretation using M-theory)→ Compare with topological types of M

Main Goal:
Determine the IR phases ofT[M = NpA+B] with non-hyperbolic M
from topogical properties of M
Strategy

Experiments

SnapPy → Ideal triangulation of N = m003,m004,…
2011]→ Field theory T[NpA+B] [Dimofte,Gaiotto,Gukov;[Gang,Yonekura; 2018]→ SUSY partition functions using Localization→ Determine IR phase

Theory(Interpretation using M-theory)→ Compare with topological types of M
[Dunfield; 2020]"A census of exceptional dehn fillings"
Study exceptional Dehn fillings of 205,822 Ns

Geometry, Representation Theory and Quantum Fields

Separating

IR phases of Class R
T[M = NpA+B] =1) Existence of essential sphere/torus
SL(2,ℂ)(M)
2) Properties of the set χirred3) Unitarity/Non-unitary of TFT[M]

T[N, A; B]′′SO(3)′p′

Nothing interesting,

Graph Manifolds

SUSY broken

Seifert-fiberedspace

1/2 BPS monopole operators
S 2 × S 1 = L(0,1)

Mass gap (TQFT)SUSY enhancement to N=4 “rank 0” SCFT
Need more studies
Finite/real79

Hyp piece In
JSJ decomposition

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0: Unitary: Infinite: Non-unitary: Empty: Finite/Complex

The space of 3-manifolds 1) Existence of essential sphere/torus
Essential sphere : sphere that does not bound 3-ball

𝕋2 ×φ S 1

2D analogy
Essential

Not essential(it bounds 2-ball)S 2 × S 1 = L(0,1)

L(p, q) |p≠0ToroidalReducible

Irreducible(no e. sphere)

Atoroidal(no e. torus)

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The space of 3-manifolds 1) Existence of essential sphere/torus

𝕋2 ×φ S 1

Hyperbolic
Hyperbolic piece In
JSJ decomposition
Graph Manifolds

S 2 × S 1 = L(0,1)

L(p, q) |p≠0

The space of 3-manifolds 1) Existence of essential sphere/torus

Hyperbolic
Hyperbolic piece In
JSJ decomposition
Graph Manifolds
S 2 × S 1 = L(0,1)

SOL manifolds
Seifert-fibered space

S 2((p1, q1), (p2, q2), (p3, q3))
L(p, q)p≠0

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81

Graph Manifolds
SOL manifolds
𝕋2 ×φ S 1 = (𝕋2 × I)/ ∼Torus bundle 𝕋2 ×φ S 1 with | Tr(φ) | > 2(x,0) ∼ (φ(x),1)

Seifert-fibered space

p1 /q1p2 /q2cc

L(p, q) : Lens spacecp4 /q4 = 0

S 2((p1, q1), (p2, q2), (p3, q3))

p3 /q3p/q

Essential Spheres
2D analogy

Essential but
non-separating

Not essential(it bounds 2-ball)Essential and separatingΣg=2 = (Σg=1)♯(Σg=1) (connected sum)

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Essential surfaces but non-separating

S2 × S1S2 × I

𝕋2 ×φ S 1

𝕋2 × Icf ) Σg=2 × S 1 has a separating essential torus

× S1
Separating

The space of 3-manifolds 1) Existence of essential sphere/torus

S 2 × S 1 = L(0,1)

Seifert-fiberedspace

Hyp piece In
JSJ decomposition

Seifert-fiberedspace

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0

Graph Manifolds

S 2 × S 1 = L(0,1)

Graph Manifolds

Hyp piece In
JSJ decomposition

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0: Having S.E. sphere/torus: no S.E. sphere/torus

Geometry, Representation Theory and Quantum Fields

83

Implications in 3D class R theories
3D index[Dimofte,Gaiotto,Gukov; 2011] [Gang,Yonekura; 2018]

Rgeo

IM(q) = Trℋrad(S2)(−1)Rgeoq 2 +j3(SO(3) × SO(2) ∈ SO(5) → SO(2) = U(1)geo)R: it counts 1/4 BPS local operators in T[M]M2 branes wrapped on (separating) essential surface Σ[Gadde,Gukov,Putrov; 2013][Garoufalidis,Hodgson,Hoﬀman,Rubinstein; 2016]

OΣ, OΣ2, … : 1/2 BPS CPOs with Rgeo(OΣ) = − χ(Σ) =⇒ IM(q) diverges due to∞∑

0, torus{−2, sphereχ(Σ)nq − 2 (1 + descendants)

n=0

if there exists (separating) essential torus or sphere inside M

Implications in 3D class R theories
3D index[Dimofte,Gaiotto,Gukov; 2011] [Gang,Yonekura; 2018]

Rgeo

IM(q) = Trℋrad(S2)(−1)Rgeoq 2 +j3(SO(3) × SO(2) ∈ SO(5) → SO(2) = U(1)geo)
R

Diverge (as a q-series)d{d∃Separating Essential sphere or torus

M is of infinite type

Converges (as a q-series) ∃no separating essential sphere/torus
M is of finite type

In T[M = NpA+B] = T[N, A; B]/′′SO(3)′p′,∃U(1)F and 1/2 BPS CPOs (OΣ)n charged under the U(1)Fmonopole operators of SO(3)so that (RIR = Rgeom + νFu(1) of OΣ) ≥

1
2

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OCAMI Reports Vol. 4 (2022)

Separating

The space of 3-manifolds 1) Existence of essential sphere/torus
Hyp piece In
JSJ decomposition

Graph Manifolds

Hyp piece In
JSJ decomposition

Seifert-fiberedspace

SOL manifolds

S 2 × S 1 = L(0,1)

S 2 × S 1 = L(0,1)

Seifert-fibered
space

Graph Manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolic

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0

L(p, q)p≠0: Infinite (having S.E. sphere/torus): Finite (no S.E. sphere/torus)

3D Index as SL(2,C) Chern-Simons ptn
IM(q) = ∫ (gauge) exp ([d𝒜][d𝒜]=i(k + iσ)i(k − iσ)CS[𝒜; M] +CS[𝒜; M]))
2
2

1
B ρ(q 1/2)Bρ(q −1/2)∑
2 ρ∈χ SL(2,ℂ)(M)

4π(q := e σ = e ℏ)

irred

[Dimofte,Gaiotto,Gukov; 2011][Lee,Yamazaki; 2013][Yagi; 2013]

4π
k=0,σ= logℏ(

CS[𝒜; M] :=1
2
Tr 𝒜d𝒜 + 𝒜3
4π ∫M (3 ))SL(2,ℂ)χirred:= set of (adj)-irreducible flat SL(2,ℂ) connections

B ρ(q 1/2) =∫Γ(ρ)

Bρ(q −1/2) =2π[d𝒜]exp − CS[𝒜; M]
( ℏ
)

∫Γ(ρ)

[d𝒜]exp⇒ IM(q = e ℏ)ℏ→02π
CS[𝒜; M]
(ℏ
)

ℏ→0

ℏ→0

exp −
(

exp

2πCS[𝒜ρ, M])ℏ

2πCS[𝒜ρ, M]ℏ()(Holomorphic Blocks)

1
4πi × Im[CS[𝒜ρ]]
exp −+ O(ℏ0)()
2 ρ∈χ∑ℏ
SL(2,ℂ)
irred

Geometry, Representation Theory and Quantum Fields

85

Empty, Finite/Real and Finite/Complex
Rgeo

IM(q) = Trℋrad(S2)(−1)Rgeoq 2 +j3
IM(q = e ℏ)ℏ→0For finite M=0,1
4πi × Im[CS[𝒜ρ]]
exp −+ O(ℏ0)()
2 ρ∈χ∑ℏ
SL(2,ℂ)
irred

SL(2,ℂ)if χirred=∅infinite power series in q 1/2 (ex . 1 + q 1/2 − 3q + …), if ∃ρ with Im[CS[𝒜ρ]] ≠ 0
q-indepent finite number in q → 1 (ex . 1), if Im[CS[𝒜ρ]] = 0 ∀ρ: Empty: Finite/real: Finite/Complex

Separating

The space of 3-manifolds 1) Existence of essential sphere/torus
SL(2,ℂ)(M)
2) Properties of the set χirred

S 2 × S 1 = L(0,1)

Seifert-fiberedspace

Hyp piece In
JSJ decomposition

Seifert-fiberedspace

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0

Graph Manifolds

S 2 × S 1 = L(0,1)

Graph Manifolds

Hyp piece In
JSJ decomposition

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0: Infinite: Finite/Complex: Finite/real: Empty

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OCAMI Reports Vol. 4 (2022)

Implications in 3D class R theories
Rgeo

IM(q) = Trℋrad(S2)(−1)Rgeoq 2 +j3=

0 , if M is empty

infinite power series in q 1/2 (ex . 1 + q 1/2 − 3q + …), if M is finite/complex
q-indepent finite number in q → 1 (ex . 1), if M is finite/real

T[M = NpA+B] = T[N, A; B]/′′SO(3)′p′=

SUSY broken , if M is empty
SCFT (nothing interesting), if M is finite/complex
TQFT (???), if M is finite/real

Implications in 3D class R theories : Finite/Real
IM(q) = Tr

ℋrad(S 2)

(−1)

Rgeo

q

Rgeo
2 +j3

= q-indepent finite number in q → 1 .

It turns out to be IM(q) = 1 (!!)

TQFT? Actually there is a map : M → TFT[M][Cho,Gang,Kim; 2020]

Geometry, Representation Theory and Quantum Fields

87

Implications in 3D class R theories : Finite/RealRgeo

IM(q) = Trℋrad(S2)(−1)Rgeoq 2 +j3= q-indepent finite number in q → 1 .

It turns out to be IM(q) = 1 (!!)

TQFT? Actually there is a map : M → TFT[M][Cho,Gang,Kim; 2020]

For some cases, surprisingly, TFT[M] is non-unitary
T[M] =

mass gap (TQFT), if TFT[M] is unitary{𝒩 = 2 SCFT with U(1)F, if TFT[M] is non-unitary(IM(q, u) ≠ 1 but IM(q, u) |u=1 = 1)

Implications in 3D class R theories : Finite/RealTQFT? Actually there is a map : M → TFT[M][Cho,Gang,Kim; 2020]

For some cases, surprisingly, TFT[M] is non-unitary
T[M] =

mass gap (TQFT), if TFT[M] is unitary{𝒩 = 2 SCFT with U(1)F, if TFT[M] is non-unitaryMore recently, (𝒩 = 4 SCFT of rank 0) ⇒ (non-unitary TQFT)[Gang,Kim,Lee,Shim,Yamzaki; 2021]

So, we naturually conjecture that
T[M] =

mass gap (TQFT), if TFT[M] is unitary{𝒩 = 4 SCFT of rank 0 , if TFTM] is non-unitary

88

OCAMI Reports Vol. 4 (2022)

Implications in 3D class R theories : Finite/RealSo, we naturually conjecture that
T[M] =

mass gap (TQFT), if TFT[M] is unitary{𝒩 = 4 SCFT of rank 0 , if TFTM] is non-unitaryAnd, we naturually divide the finite/real 3-manifolds into
: Unitary

Finite/real: Non-unitary

Finite/Real 3-manifolds∙ S 2 ((p1, q1), (p2, q2), (p3, q3)) =unitary, if qi = ± 1 (mod pi) ∀i = 1,2,3{non-unitary, otherwise∙ 𝕋2 ×φ S 1 is unitary for all φ ∈ SL(2,ℤ)∙ Others? (among toroidal Seifert-fibered spaces)

Geometry, Representation Theory and Quantum Fields

89

Separating

The space of 3-manifolds 1) Existence of essential sphere/torus
SL(2,ℂ)(M)
2) Properties of the set χirred3) Unitarity/Non-unitary of TFT[M]Hyp piece In
JSJ decomposition

Graph Manifolds

Seifert-fiberedspace

SOL manifolds

S 2 × S 1 = L(0,1)

S 2 × S 1 = L(0,1)

Seifert-fibered
space

Graph Manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0

Unitary
Non-unitary: Finite/Complex: Finite/real: Empty

1) Existence of essential sphere/torus
SL(2,ℂ)(M)
2) Properties of the set χirred3) Unitarity/Non-unitary of TFT[M]

T[N, A; B]′′SO(3)′p′

Nothing interesting,

Graph Manifolds

SUSY broken

Seifert-fibered
space

Mass gap (TQFT)SUSY enhancement to N=4 “rank 0” SCFT
Need more studies

S 2 × S 1 = L(0,1)

1/2 BPS monopole operators

Finite/real: Infinite

Separating

IR phases of Class R
T[M = NpA+B] =Hyp piece In
JSJ decomposition

Hyp piece In
JSJ decomposition

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0: Unitary: Infinite: Non-unitary: Empty: Finite/Complex

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OCAMI Reports Vol. 4 (2022)

Example I
T[(S 3\52)pA+B] =(U(1)− 12 × SU(2)p−4 coupled to Φ in (Adj)+1 ) ⊗ U(1)2ℤ1−form2

cf )SUSY enhancement to N=4 SCFT
(w/ no Higgs/Coulomb)

p = 1,2,3

1/2 BPS monopole operators

p = 0,4

Nothing interesting,otherwise

S 2((2,1)(3,1), (11, − 9)), p = 1 aFinite/real/non-unitary
b
{
c
p=3

S 2((2,1)(4,1), (7,5)), p = 2(S 3\52)pA+B

2

S ((3,1)(3,1), (5, − 3)),Graph manifold, p = 0,4

InfiniteHyperbolic , otherwise

Finite/Complex

Example II
T[(m006)pA+B]=

p=−2

SUSY enhancement to N=4 SCFT,
1/2 BPS monopole operators,

p=−3

U(1)− 12 × U(1)− 32 coupled to 3ΦsMass gap (TQFT),

p = 1, − 1+W = Φ21Φ2Φ3

SUSY broken,

p=0′′SO(3)′p−1′

Nothing interesting,cf )S 2((2,1)(3,1), (7, − 5)), p = − 2(m006)pA+B

Graph manifold ,p=−3

S 2((2,1)(2,1), (3,2)),

p=1

S 2((2,1)(3,2), (4, − 3)),L(15,4),p=0

otherwise

Finite/real/non-unitary

InfiniteaFinite/real/Unitarybp = − 1c{

EmptyHyperbolic , otherwise

Finite/Complex

Geometry, Representation Theory and Quantum Fields

91

More examples* . m005,m008 are non-hyperbolic* . (m004) = (S 3\41)

Separating

Summary
T[M = NpA+B] =1) Existence of essential sphere/torus
SL(2,ℂ)(M)
2) Properties of the set χirred3) Unitarity/Non-unitary of TFT[M]

T[N, A; B]′′SO(3)′p′

Nothing interesting,

Graph Manifolds

SUSY broken

Seifert-fibered
space

Mass gap (TQFT)SUSY enhancement to N=4 “rank 0” SCFT
Need more studies
Finite/real

S 2 × S 1 = L(0,1)

1/2 BPS monopole operators

Hyp piece In
JSJ decomposition

SOL manifolds

S 2((p1, q1), (p2, q2), (p3, q3))

hyperbolicL(p, q)p≠0: Unitary: Infinite: Non-unitary: Empty: Finite/Complex

92

OCAMI Reports Vol. 4 (2022)

Future Directions
Refined analysis for infinite type
Infinite :: Real: Complex

Generalization to
Non-hyperbolic manifold with torus boundaries
A, D, E typeClassfications of
3D N=4 SCFT of rank 0
Non-unitary 3D TQFT

3-manifolds

From of Finite/real/non-unitaryThanks for your attention !!!

Geometry, Representation Theory and Quantum Fields

93

Brane Brick Models and the Z 2-labelled family of
toric Calabi-Yau 4-folds
Dongwook Ghim

(D. Ghim) Korea Institute for Advanced Study

94

OCAMI Reports Vol. 4 (2022)

Brane Brick Models andZ2-labelled families of CY 4-folds

Dongwook Ghim
KIAS School of Physics
Mar. 22nd, 2022
Geometry, Representation Theory and Quantum Fields
Osaka City University (via ZOOM)
Based on joint work (arXiv:2203.xxxxx) with S. Franco (CUNY) and R.-K. Seong (UNIST)+ previous works (arXiv:1506.03818, 1609.07144, 1801.00799, etc)

Brane Brick Models andZ2-labelled families of CY 4-folds[1] Construction : Brane Brick Models[2] Tiling-facilitated development: dimensional reduction, orbifold reduction and 3d printing[3] Application : Y p,k family of Calabi-Yau 4-folds

1

Geometry, Representation Theory and Quantum Fields

95

Brane Brick Models
Geometric engineering of 2d (0,2) gauge theories
Low-energy effective theory of D1-branes probing a toric Calabi-Yau 4-fold cone
[Franco, DG, Lee, Seong, Yokoyama 15]
shows (0,2) supersymmetry.

N D1-branes

toric CY4

Toric Calabi-Yau 4-fold = cone over Sasaki-Einstein 7-manifold which enjoys
4 isometries corresponding to U (1)R ⇥ U (1)3 global symmetry of gauge theory.
22d (0,2) gauge theory
SUSY multiplets of 2d (0,2) theories
Following three types of multiplets are allowed in (0,2) gauge theories:[1] chiral multiplet+<latexit sha1_base64="iTXQRpFKqQSwuI1YZGMQBfgpNSg=">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</latexit>

=

+ ✓+

+

i✓+ ✓ D+ ,

ij

D

=0

[2] fermi multiplet<latexit sha1_base64="w7P15Hz0+LP4kvcZ6iHazkCoyZg=">AAACiHicbVHbahsxENVukjZ1b0762JchppDixOwWN5eHQmgTmoc8uFAnActdZrVyLKK9RJotmGW/pf/Ut/5NtbYh1wHBmXPOaEajuNDKUhD88/yV1bVnz9dftF6+ev3mbXtj89zmpRFyKHKdm8sYrdQqk0NSpOVlYSSmsZYX8fW3Rr/4LY1VefaTZoUcp3iVqYkSSI6K2n/4mTMnCF+A6zmKdmEXOE0l4a8ufHeJuk157i5relULqnbccdS9X/uE5wT4zg7wmxKTu3qKNBWoq+O6httBTmAb+GCqIgUfo3Yn6AXzgMcgXIIOW8Ygav/lSS7KVGYkNFo7CoOCxhUaUkLLusVLKwsU13glRw5mmEo7ruaLrOGDYxKY5MadjGDO3q2oMLV2lsbO2YxuH2oN+ZQ2KmlyMK5UVpQkM7FoNCk1UA7Nr0CijBSkZw6gMMrNCmKKBgW5v2u5JYQPn/wYnH/qhXu9/o9+52hvuY519p5tsW0Wsn12xE7ZgA2Z8Na8rtf3PvstP/D3/cOF1feWNe/YvfC//gdGkb9z</latexit>

⇤=

✓+ G

+

i✓+ ✓ D++✓ E,

[3] vector multiplet
Y =i✓+ (DiF01 )+i✓+ ✓ @+

D⇤ = E(i)⇤ij

Quiver notation for
bi-fundamental matters

3

96

OCAMI Reports Vol. 4 (2022)

2d (0,2) gauge theory
Holomorphic interactions compatible with (0,2) SUSY
[1] E-term : allowed deformation in chirality++⇤=✓+ G i✓+ ✓ D+
✓ E,

[Witten 93]<latexit sha1_base64="w7P15Hz0+LP4kvcZ6iHazkCoyZg=">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</latexit>[2] J-term : (derivative of) superpotential
ZX
LJ =d2 xd✓+(⇤a Ja ( i )|✓+ =0 )D⇤ = E(i)

ij<latexit sha1_base64="bM98JKl/p7xo8+JF8BUE66B2DL0=">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</latexit>

h.c.

⇤ij

a

[3] SUSY compatibility condition

Quiver notation for
bi-fundamental matters

4

Toric Calabi-Yau 4-fold
A conical geometry over Sasaki-Einstein 7-manifold
shows singularity at the origin, which is described by toric diagram.

Q1,1,1<latexit sha1_base64="6F5XybECOndXF0ysM8eDhn7x6a0=">AAAB8HicbVDLTgIxFL2DL8QHqEs3jcTEhSEzhqBLEjcuIZGHgZF0Sgca2s6k7ZiQCV/hxoXGuPVz3Pk3FpiFgqdpcnLOvbn3niDmTBvX/XZyG5tb2zv53cLe/sFhsXR03NZRoghtkYhHqhtgTTmTtGWY4bQbK4pFwGknmNzO/c4TVZpF8t5MY+oLPJIsZAQbKz00H1Pv0r7ZoFR2K+4CaJ14GSlDhsag9NUfRiQRVBrCsdY9z42Nn2JlGOF0VugnmsaYTPCI9iyVWFDtp4uFZ+jcKkMURsp+adBC/d2RYqH1VAS2UmAz1qveXPzP6yUmvPFTJuPEUEmWg8KEIxOh+fVoyBQlhk8twUQxuysiY6wwMTajgg3BWz15nbSvKl6tUm1Wy/VaFkceTuEMLsCDa6jDHTSgBQQEPMMrvDnKeXHenY9lac7Jek7gD5zPH0dTj2M=</latexit>

C4<latexit sha1_base64="cLawJzbK5tKJOjFURXX6NYDe4f8=">AAAB83icbVDLSsNAFL3xWeur6tLNYBFclURKdVnoxmUF+4Amlsl00g6dTMI8hBL6G25cKOLWn3Hn3zhps9DWAwOHc+7lnjlhypnSrvvtbGxube/slvbK+weHR8eVk9OuSowktEMSnsh+iBXlTNCOZprTfiopjkNOe+G0lfu9JyoVS8SDnqU0iPFYsIgRrK3k+zHWkzDMWvPH+rBSdWvuAmideAWpQoH2sPLljxJiYio04VipgeemOsiw1IxwOi/7RtEUkyke04GlAsdUBdki8xxdWmWEokTaJzRaqL83MhwrNYtDO5lnVKteLv7nDYyOboOMidRoKsjyUGQ40gnKC0AjJinRfGYJJpLZrIhMsMRE25rKtgRv9cvrpHtd8xq1+n292mwUdZTgHC7gCjy4gSbcQRs6QCCFZ3iFN8c4L86787Ec3XCKnTP4A+fzB9WTkYY=</latexit>

dP3 ⇥ C<latexit sha1_base64="iPaN07QU+qYXn3M4OyVlJ/BIZgI=">AAAB/3icbVDLSsNAFL2pr1pfUcGNm8EiuCqJluqy0I3LCvYBTQiTyaQdOnkwMxFK7MJfceNCEbf+hjv/xknbhbYeGDiccy/3zPFTzqSyrG+jtLa+sblV3q7s7O7tH5iHR12ZZILQDkl4Ivo+lpSzmHYUU5z2U0Fx5HPa88etwu89UCFZEt+rSUrdCA9jFjKClZY88yRoe1fIUSyiEjkRViPfz1tTz6xaNWsGtErsBanCAm3P/HKChGQRjRXhWMqBbaXKzbFQjHA6rTiZpCkmYzykA01jrO+5+Sz/FJ1rJUBhIvSLFZqpvzdyHEk5iXw9WSSUy14h/ucNMhXeuDmL00zRmMwPhRlHKkFFGShgghLFJ5pgIpjOisgIC0yUrqyiS7CXv7xKupc1u1Gr39WrzcaijjKcwhlcgA3X0IRbaEMHCDzCM7zCm/FkvBjvxsd8tGQsdo7hD4zPHy8OlY0=</latexit>

C(++) = (C ⇥ C)/Z2<latexit sha1_base64="yt52iKI7+LxuW3djIwvpQwbsML4=">AAACLnicbZBbSwJBFMdn7WZ22+qxlyEJFMF2RayXQJCgR4O8kCvL7Djq4OyFmdlAlv1EvfRV6iGoiF77GM3qEqYdGPjP75zDOefvBIwKaRhvWmZtfWNzK7ud29nd2z/QD4/awg85Ji3sM593HSQIox5pSSoZ6QacINdhpONMGkm+80C4oL53J6cB6bto5NEhxUgqZOvXlovkGCMWNWI7KpRKxRhewQJcwNCS1CVijhwnIUV4/vu9j+2KreeNsjELuCrMVORBGk1bf7EGPg5d4knMkBA90whkP0JcUsxInLNCQQKEJ2hEekp6SC3Qj2bnxvBMkQEc+lw9T8IZXeyIkCvE1HVUZbKjWM4l8L9cL5TDy35EvSCUxMPzQcOQQenDxDs4oJxgyaZKIMyp2hXiMeIIS+VwTplgLp+8KtqVslkrV2+r+XottSMLTsApKAATXIA6uAFN0AIYPIJn8A4+tCftVfvUvualGS3tOQZ/Qvv+Aco3pyc=</latexit>• Example of toric diagrams (on Z3 lattice; for Calabi-Yau 4-folds)
SL(3,Z) redundancy

5

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97

Starting line : D1 probing C4
Maximally supersymmetric Yang-Mills theory in 2d
D1 branes probing C4 is obtained by dimensional reduction of 10d N=1 SYM.
[Franco, DG, Lee, Seong, Yokoyama 15]

C4<latexit sha1_base64="cLawJzbK5tKJOjFURXX6NYDe4f8=">AAAB83icbVDLSsNAFL3xWeur6tLNYBFclURKdVnoxmUF+4Amlsl00g6dTMI8hBL6G25cKOLWn3Hn3zhps9DWAwOHc+7lnjlhypnSrvvtbGxube/slvbK+weHR8eVk9OuSowktEMSnsh+iBXlTNCOZprTfiopjkNOe+G0lfu9JyoVS8SDnqU0iPFYsIgRrK3k+zHWkzDMWvPH+rBSdWvuAmideAWpQoH2sPLljxJiYio04VipgeemOsiw1IxwOi/7RtEUkyke04GlAsdUBdki8xxdWmWEokTaJzRaqL83MhwrNYtDO5lnVKteLv7nDYyOboOMidRoKsjyUGQ40gnKC0AjJinRfGYJJpLZrIhMsMRE25rKtgRv9cvrpHtd8xq1+n292mwUdZTgHC7gCjy4gSbcQRs6QCCFZ3iFN8c4L86787Ec3XCKnTP4A+fzB9WTkYY=</latexit>

NB

quiverof C4-theory

Four isometries in toric Calabi-Yau 4-foldcorrespond to U (1)R ⇥ U (1)3 global symmetry of gauge theory.

6

Starting line : D1 probing C4
Maximally supersymmetric Yang-Mills theory in 2d[Franco, DG,
Yokoyama 15]
D1 branes probing C4 is obtained by dimensional reductionofLee,
10dSeong,
N=1 SYM.

toric condition on J and E
NB

quiverof C4-theory

Four isometries in toric Calabi-Yau 4-foldcorrespond to U (1)R ⇥ U (1)3 global symmetry of gauge theory.

7

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OCAMI Reports Vol. 4 (2022)

Periodic Quiver
encodes both gauge and global symmetry representation simultaneously.
[Franco, DG, Lee, Seong, Yokoyama 15]
D1 branes YM.
Λ1

Λ2
Λ3

periodic quiverof C4-theory on T3
NB

quiverof C4-theory

Four isometries in toric Calabi-Yau 4-foldcorrespond to U (1)R ⇥ U (1)3 global symmetry of gauge theory.

8

Brane Brick
The graph-dual of periodic quiver is called Brane Brick configuration.
[Franco, DG, Lee, Seong, Yokoyama 15][Franco, Lee, Seong 15]
Λ1

Λ2
Λ3

periodic quiverof C4-theory on T3
NB

Brane Brickof C4-theory on T3

toric diagramof C4 on Z3

In string perspective, it appears naturally in T-dual (IIA) of initialconfiguration.
9

Geometry, Representation Theory and Quantum Fields

99

Wrap-up in the middle

Brane Brickgraph dual

Periodic Quiver

T-duality

Toric diagram

3-torusmesonic
moduli space

Quiver diagram

10

Brane Brick Models andZ2-labelled families of CY 4-folds[1] Construction : Brane Brick Models[2] Tiling-facilitated development: dimensional reduction, orbifold reduction and 3d printing[3] Application : Y p,k family of Calabi-Yau 4-folds

11

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Polytope from polygon
Observation
Many of toric diagrams for CY4 can be drawn by lifting up and down a few
vertices of planar toric diagrams for CY3.

12

2d (0,2) theories from 4d N=1 theories
Observation
Combinatorial data and the Lagrangian of 4d N=1 theories can be employed
in engineering 2d (0,2) brane brick theories.

dimensional reduction

orbifold reduction

3d printing[Franco, DG, Lee, Seong, Yokoyama 15][Franco, Lee, Seong 15][Franco, Hasan 18]

13

Geometry, Representation Theory and Quantum Fields

101

2d (0,2) theories from 4d N=1 theories [1]Dimensional Reduction ó CY3 x C ó lifting a point of toric diagram

lifting a vertex by height 1

N1

N2

14

Brane Tilinge.g. [Franco, Hanany, Kennaway, Vegh, Wecht 05]

4d N=1 gauge theories with (at least) U(1)2 x U(1)R global symmetry arise in
D3 brane probing toric CY 3-folds.

Brane Tilinggraph dual

T-duality

Toric diagram

Periodic Quiver

2-torusmesonic
moduli space

Quiver diagram
N1

N2

102

OCAMI Reports Vol. 4 (2022)

GLSM field (perfect matching)
Perfect matching, or GLSM description,
assigns the collection of chiral fields to a vertex of toric diagram.

N1

W = X12 Y21 Y12 X21<latexit sha1_base64="Z4err4Vvpbwbe9PLOZFDK+eruGI=">AAACMHicbVDLSgMxFM3UV62vUZdugkVwY5kppboRCi50WcG2I+0wZNK0Dc08SDJCGeaT3PgpulFQxK1fYeYh2NYLIeeecy7JPW7IqJCG8aaVVlbX1jfKm5Wt7Z3dPX3/oCuCiGPSwQELuOUiQRj1SUdSyYgVcoI8l5GeO71K9d4D4YIG/p2chcT20NinI4qRVJSjXw88JCcYsbiXwEtoObFZT+C9E9fN7Eo7K+/OflVrTs29jl41akZWcBmYBaiCotqO/jwYBjjyiC8xQ0L0TSOUdoy4pJiRpDKIBAkRnqIx6SvoI48IO84WTuCJYoZwFHB1fAkz9u9EjDwhZp6rnOl6YlFLyf+0fiRHF3ZM/TCSxMf5Q6OIQRnAND04pJxgyWYKIMyp+ivEE8QRlirjigrBXFx5GXTrNbNZa9w2qq1mEUcZHIFjcApMcA5a4Aa0QQdg8AhewDv40J60V+1T+8qtJa2YOQRzpX3/AIzJpjE=</latexit>

N2

X12 X21 Y12 Y21

e.g. [Franco, Hanany, Kennaway, Vegh, Wecht 05]

16

2d (0,2) theories from 4d N=1 theories [1]Dimensional Reduction ó CY3 x C ó lifting a point of toric diagram
lifting a vertex by height 1

adding z-direction in theplanar periodic quiver17

Geometry, Representation Theory and Quantum Fields

103

Dimensional Reduction
Rule
Extension of planar periodic quiver into z-direction is done by following procedures

18

Dimensional Reduction
Rule
Extension of planar periodic quiver into z-direction is done by following procedures

N1

N2

19

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OCAMI Reports Vol. 4 (2022)

2d (0,2) theories from 4d N=1 theories [2]Orbifold Reduction ó (CY3xC)/Z ó lifting / dragging down a pt of toric diagram

lifting a vertex by height 2

lifting and dragging downa vertex by height ±1
20

Orbifold Reduction
Rule
Extension of block periodic quiver along z-direction is done by following procedures

21

Geometry, Representation Theory and Quantum Fields

105

Orbifold Reduction
Rule
Extension of block periodic quiver along z-direction is done by following procedures22

2d (0,2) theories from 4d N=1 theories [3]Accumulating new vertices on top of (or below) vertices in planar toric diagram[Franco, Hasan 18]

23

106

OCAMI Reports Vol. 4 (2022)

3d printing of 2d (0,2) theories
Rule[Franco, Hasan 18]

Accumulation of several “building block” periodic quivers along z-direction

24

Brane Brick Models andZ2-labelled families of CY 4-folds[1] Construction : Brane Brick Models[2] Tiling-facilitated development: dimensional reduction, orbifold reduction and 3d printing[3] Application : Y p,k family of Calabi-Yau 4-folds

25

Geometry, Representation Theory and Quantum Fields

107

Toric Calabi-Yau 4-fold
There are two families of toric Calabi-Yau 4-folds, labelled by two integers; p and k.
[Martelli, Sparks 08]<latexit sha1_base64="zTsGH1mhCQ05M8UYTucDjqnS+L0=">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</latexit>

E( 1, 1, k)<latexit sha1_base64="AlC0qVt1g13N8gVKTlkZxZTbwFY=">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</latexit>C ( 1, 0, k)<latexit sha1_base64="pvsTkgXfLF/nsilMr3/NrDtH7j4=">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</latexit>

P (0, 0, p)<latexit sha1_base64="pvsTkgXfLF/nsilMr3/NrDtH7j4=">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</latexit>

P (0, 0, p)<latexit sha1_base64="pnpuwSU4mMdqWiTFZQqp9L73Ca0=">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</latexit>

D (0, 1, k)<latexit sha1_base64="vZMh+vOg5a6h3isX4KZDuhQk3Xo=">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</latexit>

B (0, 1, 0)<latexit sha1_base64="vZMh+vOg5a6h3isX4KZDuhQk3Xo=">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</latexit>

B (0, 1, 0)<latexit sha1_base64="kvaPtBU/rTmhjoNMPnzF9Op2YEc=">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</latexit>

A (1, 0, 0)<latexit sha1_base64="kvaPtBU/rTmhjoNMPnzF9Op2YEc=">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</latexit>

A (1, 0, 0)

Y p,k (P1 ⇥ P1 )Y p,k (P2 )<latexit sha1_base64="yjGdwCA2GWq8Cc91ultBBnrToQA=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0WoIiWRUl0W3LisYB/SpGUynbRDJ5MwMxFKyD+48VfcuFDErRt3/o2TNotaPXDhcM693HuPFzEqlWV9G4WV1bX1jeJmaWt7Z3fP3D9oyzAWmLRwyELR9ZAkjHLSUlQx0o0EQYHHSMebXGd+54EISUN+p6YRcQM04tSnGCktDcyz+34SnU9SWHECpMaelzTTvg0dRQMi4aJ2OjDLVtWaAf4ldk7KIEdzYH45wxDHAeEKMyRlz7Yi5SZIKIoZSUtOLEmE8ASNSE9TjvRKN5n9lMITrQyhHwpdXMGZujiRoEDKaeDpzuxIuexl4n9eL1b+lZtQHsWKcDxf5McMqhBmAcEhFQQrNtUEYUH1rRCPkUBY6RhLOgR7+eW/pH1RtevV2m2t3KjncRTBETgGFWCDS9AAN6AJWgCDR/AMXsGb8WS8GO/Gx7y1YOQzh+AXjM8fypadcw==</latexit><latexit sha1_base64="8X3r5xMy8IvCqHr9ZnqDRNLhxTU=">AAAB/3icbVDLSgMxFL1TX7W+RgU3boJFqCBlppTqsuDGZQX7kHZaMmmmDc08SDJCGbvwV9y4UMStv+HOvzHTzkKrBwKHc+7lnhw34kwqy/oyciura+sb+c3C1vbO7p65f9CSYSwIbZKQh6LjYkk5C2hTMcVpJxIU+y6nbXdylfrteyokC4NbNY2o4+NRwDxGsNLSwDy66yfR+WSGSj0fq7HrJo1Zv3I2MItW2ZoD/SV2RoqQoTEwP3vDkMQ+DRThWMqubUXKSbBQjHA6K/RiSSNMJnhEu5oG2KfSSeb5Z+hUK0PkhUK/QKG5+nMjwb6UU9/Vk2lIueyl4n9eN1bepZOwIIoVDcjikBdzpEKUloGGTFCi+FQTTATTWREZY4GJ0pUVdAn28pf/klalbNfK1ZtqsV7L6sjDMZxACWy4gDpcQwOaQOABnuAFXo1H49l4M94Xozkj2zmEXzA+vgHRlpVN</latexit><latexit sha1_base64="+l4kjPZdY29dslCE3Rky9U7CtRk=">AAACBHicbVC7TsMwFHXKq5RXgLGLRYXEVCVVVRgrsTAWiT6kJqoc12msOo6xHaQqysDCr7AwgBArH8HG3+C2GaDlSFf36Jx7Zd8TCEaVdpxvq7SxubW9U96t7O0fHB7Zxyc9laQSky5OWCIHAVKEUU66mmpGBkISFAeM9IPp9dzvPxCpaMLv9EwQP0YTTkOKkTbSyK56oUQ4i0SeNXLoMXIPp8sWiZFdc+rOAnCduAWpgQKdkf3ljROcxoRrzJBSQ9cR2s+Q1BQzkle8VBGB8BRNyNBQjmKi/GxxRA7PjTKGYSJNcQ0X6u+NDMVKzeLATMZIR2rVm4v/ecNUh1d+RrlINeF4+VCYMqgTOE8EjqkkWLOZIQhLav4KcYRMKtrkVjEhuKsnr5Neo+626s3bZq3dKuIogyo4AxfABZegDW5AB3QBBo/gGbyCN+vJerHerY/laMkqdk7BH1ifPwOEl6c=</latexit><latexit sha1_base64="KAwUcNOy/gLQxC4CqparOMpHcNk=">AAAB7HicbVBNSwMxEJ2tX7V+VT16CRahXspuKdWLUPDisYJbC+1Ssmm2DU2yS5IVytLf4MWDIl79Qd78N6btHrT1wcDjvRlm5oUJZ9q47rdT2Njc2t4p7pb29g8Oj8rHJx0dp4pQn8Q8Vt0Qa8qZpL5hhtNuoigWIaeP4eR27j8+UaVZLB/MNKGBwCPJIkawsZJfHd/ULwfliltzF0DrxMtJBXK0B+Wv/jAmqaDSEI617nluYoIMK8MIp7NSP9U0wWSCR7RnqcSC6iBbHDtDF1YZoihWtqRBC/X3RIaF1lMR2k6BzVivenPxP6+Xmug6yJhMUkMlWS6KUo5MjOafoyFTlBg+tQQTxeytiIyxwsTYfEo2BG/15XXSqde8Zq1x36i0mnkcRTiDc6iCB1fQgjtogw8EGDzDK7w50nlx3p2PZWvByWdO4Q+czx+GiY3R</latexit>(h = 2)hp
 k  hp
2<latexit sha1_base64="CUdzfZ10Bfpqvxi7qBJqfPU09Io=">AAAB7HicbVBNSwMxEJ31s9avqkcvwSLUS9nVUr0IBS8eK7htoV1KNs22odlkSbJCWfobvHhQxKs/yJv/xrTdg7Y+GHi8N8PMvDDhTBvX/XbW1jc2t7YLO8Xdvf2Dw9LRcUvLVBHqE8ml6oRYU84E9Q0znHYSRXEcctoOx3czv/1ElWZSPJpJQoMYDwWLGMHGSn5ldHt10S+V3ao7B1olXk7KkKPZL331BpKkMRWGcKx113MTE2RYGUY4nRZ7qaYJJmM8pF1LBY6pDrL5sVN0bpUBiqSyJQyaq78nMhxrPYlD2xljM9LL3kz8z+umJroJMiaS1FBBFouilCMj0exzNGCKEsMnlmCimL0VkRFWmBibT9GG4C2/vEpal1WvXq091MqNeh5HAU7hDCrgwTU04B6a4AMBBs/wCm+OcF6cd+dj0brm5DMn8AfO5w+IDo3S</latexit>(h = 3)26

SL(3, Z) redundancy in toric diagram
Toric diagram encodes the intersection data of 2-, 4- and 6-cycles.
The freedom in basis choice results in SL(3,Z) redundancy in its presentation.
<latexit sha1_base64="5Q2jj/lXIyl02r3moGfYMlEJuyI=">AAACDHicbVDLSsNAFJ3UV62vqks3g0UQbGtSSnUjFNy4rGAf0IQwmUzaoZNJnJkIJfQD3Pgrblwo4tYPcOffOG2z0NYDA4dzzuXOPV7MqFSm+W3kVlbX1jfym4Wt7Z3dveL+QUdGicCkjSMWiZ6HJGGUk7aiipFeLAgKPUa63uh66ncfiJA04ndqHBMnRANOA4qR0pJbLHH3DF7BGFbg6LwG7XIZ2vcJ8iF3K1ofQS3qlFk1Z4DLxMpICWRoucUv249wEhKuMENS9i0zVk6KhKKYkUnBTiSJER6hAelrylFIpJPOjpnAE634MIiEflzBmfp7IkWhlOPQ08kQqaFc9Kbif14/UcGlk1IeJ4pwPF8UJAyqCE6bgT4VBCs21gRhQfVfIR4igbDS/RV0CdbiycukU6tajWr9tl5qNrI68uAIHINTYIEL0AQ3oAXaAINH8AxewZvxZLwY78bHPJozsplD8AfG5w9p5pbC</latexit>

n+ = pk/2 ,n = k/2<latexit sha1_base64="AlC0qVt1g13N8gVKTlkZxZTbwFY=">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</latexit>C ( 1, 0, k)<latexit sha1_base64="pvsTkgXfLF/nsilMr3/NrDtH7j4=">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</latexit>

P (0, 0, p)<latexit sha1_base64="pnpuwSU4mMdqWiTFZQqp9L73Ca0=">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</latexit>

D (0, 1, k)<latexit sha1_base64="vZMh+vOg5a6h3isX4KZDuhQk3Xo=">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</latexit>

B (0, 1, 0)<latexit sha1_base64="kvaPtBU/rTmhjoNMPnzF9Op2YEc=">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</latexit>

A (1, 0, 0)& translation in z-direction

NB

Y p,k(P1 x P1) with even k is achieved by orbifold reduction of F0.
27

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OCAMI Reports Vol. 4 (2022)

Y 2,k (P1 x P1) with even k
Periodic quiver of Y p,k (P1 x P1) with even k is obtained by accumulating p building
blocks of (F0 x C) periodic quiver.

Y 2,0 (n+ , n ) = (2, 0)

Y 2,2 (n+ , n ) = (1, 1)<latexit sha1_base64="mnwyjtx36rT5GRzNZqDLrkqVPzQ=">AAAB/3icbVDLSgNBEOyNrxhfUcGLl8EgJBjDbgjRixDw4jGCeUCyLLOTSTJkdnaZmRXCmoO/4sWDIl79DW/+jZPHQaMFDUVVN91dfsSZ0rb9ZaVWVtfWN9Kbma3tnd297P5BU4WxJLRBQh7Kto8V5UzQhmaa03YkKQ58Tlv+6Hrqt+6pVCwUd3ocUTfAA8H6jGBtJC97lBfeGeoWi0h45wV0hfLlol1AXjZnl+wZ0F/iLEgOFqh72c9uLyRxQIUmHCvVcexIuwmWmhFOJ5lurGiEyQgPaMdQgQOq3GR2/wSdGqWH+qE0JTSaqT8nEhwoNQ580xlgPVTL3lT8z+vEun/pJkxEsaaCzBf1Y450iKZhoB6TlGg+NgQTycytiAyxxESbyDImBGf55b+kWS451VLltpKrVRdxpOEYTiAPDlxADW6gDg0g8ABP8AKv1qP1bL1Z7/PWlLWYOYRfsD6+AQdpkjc=</latexit><latexit sha1_base64="M8wTpsn1dkqnOkoP+RDKuKG79Eg=">AAAB/3icbVDLSgNBEOyNrxhfq4IXL4NBSDCGXQnRixDw4jGCeUCyLLOTSTJkdnaZmRVCzMFf8eJBEa/+hjf/xsnjoIkFDUVVN91dQcyZ0o7zbaVWVtfWN9Kbma3tnd09e/+grqJEElojEY9kM8CKciZoTTPNaTOWFIcBp41gcDPxGw9UKhaJez2MqRfinmBdRrA2km8f5YR/htqFAhL+eR5do5xbcPPIt7NO0ZkCLRN3TrIwR9W3v9qdiCQhFZpwrFTLdWLtjbDUjHA6zrQTRWNMBrhHW4YKHFLljab3j9GpUTqoG0lTQqOp+ntihEOlhmFgOkOs+2rRm4j/ea1Ed6+8ERNxoqkgs0XdhCMdoUkYqMMkJZoPDcFEMnMrIn0sMdEmsowJwV18eZnUL4puuVi6K2Ur5XkcaTiGE8iBC5dQgVuoQg0IPMIzvMKb9WS9WO/Wx6w1Zc1nDuEPrM8fB2eSNw==</latexit>

28

Y p,k (P1 x P1) with odd k
Post-processing after orbifold reduction is required in obtaining the desired toric
diagram of Y p,k when k is odd.
<latexit sha1_base64="AlC0qVt1g13N8gVKTlkZxZTbwFY=">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</latexit>C ( 1, 0, k)<latexit sha1_base64="pvsTkgXfLF/nsilMr3/NrDtH7j4=">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</latexit>[1] 3d printing

P (0, 0, p)<latexit sha1_base64="pnpuwSU4mMdqWiTFZQqp9L73Ca0=">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</latexit><latexit sha1_base64="C9tkvjtdq+azRH6T6DjcaDMmjBU=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiCbUlKqd0IBTcuK9gHNKFMppN26GQSZyZCCfkVN/6KGxeKdCf+jNM2C9t6YOBwzrncuceLGJXKsr6Njc2t7Z3d3F5+/+Dw6Ng8OW3LMBaYtHDIQtH1kCSMctJSVDHSjQRBgcdIxxvfzfzOMxGShvxRTSLiBmjIqU8xUlrqm3WowfvX8BZGsAQdXyCcjEt2mlRSp1iEzlOMBjpQ0oEls28WrLI1B1wndkYKIEOzb06dQYjjgHCFGZKyZ1uRchMkFMWMpHknliRCeIyGpKcpRwGRbjK/MIWXWhlAPxT6cQXn6t+JBAVSTgJPJwOkRnLVm4n/eb1Y+XU3oTyKFeF4sciPGVQhnNUFB1QQrNhEE4QF1X+FeIR0D0qXmtcl2Ksnr5N2pWzXytWHaqFRy+rIgXNwAa6ADW5AA9yDJmgBDF7AG/gAn8ar8W58GdNFdMPIZs7AEoyfXyjYn+I=</latexit>

n+ = p

k

1
2

,

n =

k

1

D (0, 1, k)<latexit sha1_base64="vZMh+vOg5a6h3isX4KZDuhQk3Xo=">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</latexit>

B (0, 1, 0)

2<latexit sha1_base64="kvaPtBU/rTmhjoNMPnzF9Op2YEc=">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</latexit>

A (1, 0, 0)[2] Higgsing

29

Geometry, Representation Theory and Quantum Fields

109

Thank you

30

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OCAMI Reports Vol. 4 (2022)

Cheshire cat resurgence in quantum field theories
Philip Glass

Abstract. The resurgence program aims to deepen our understanding ofquantum systems such as string theories and quantum field theories by writing observables in these systems as ambiguity-free Borel-Ecalle resummationsof transseries. Many interesting features of resurgence theory, such as Stokes
phenomena, rely on the existence of a divergent, asymptotic perturbative expansion of an observable. I will discuss how these features manifest in certain
examples where the perturbative series is in fact convergent.

(P. Glass) Keio University

Geometry, Representation Theory and Quantum Fields

Fermionic characters, knots, and BPS quivers
Sergei Gukov(S. Gukov) California Institute of Technology

111

112

OCAMI Reports Vol. 4 (2022)• Q-state Potts model (= Ising w/ spins: Q values)
Percolation (c=0)• O(n) model (another generalization of Ising)
Ising

self-avoiding
walk• QHE plateau phase transitions

John Cardy(density of lops with area greater than A = C / A)

Geometry, Representation Theory and Quantum Fields

113

Lattice models with unusual scaling (anothergeneralization of Ising w/ spins that take N values)chirality
parameters

D.HuseS.Ostlund
D.Huse, M.Fisher
D.Haldande, P.Bak, T.Bohr
D.Huse, A.Szpilka, M.Fisher:

J.Jose, L.Kadanoff, S.Kirkpatrick, D.Nelson
F.Alcaraz, R.Koberle
V.Dotsenko
S.Elitzur, R.Pearson, J.Shigemitsu
A.Ukawa, P.Windey, A.Guth
R.Savit:
G.Ortiz, E.Cobanera, Z.Nussinov

Non-chiral:

114

OCAMI Reports Vol. 4 (2022)

G. von Gehlen, V.Rittenberg
D.Haldane, P.Bak, T.Bohr
H.Schulz
P.Bak:

S.Howes
D.Huse, M.Fisher
W.Selke, J.Yeomans
P.Duxbury, J.Yeomans, P.Beale
M.Martins, C.Tsallis:

H.Everts, H.Roder (1989)

S.Nyckees, J.Colbois, F.Mila (2020)

Geometry, Representation Theory and Quantum Fields

Integrable lines:

N=7

N=4

H.Au-Yang, J.Perk (‘94)

Y.Zhuang, H.Changlani, N.Tubman, T.Hughes

R.Samajdar, S.Choi, H.Pichler, M.Lukin, S.Sachdev

115

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OCAMI Reports Vol. 4 (2022)

Geometry, Representation Theory and Quantum Fields

117

118

OCAMI Reports Vol. 4 (2022)

Definition: A conformal field theory
is a table of integrals.
- Brian Greene

A conformal field theory is alist of operators (states) andtheir correlation functions(many determined by scaling
dimensions).

• Spectrum• Correlation functions• Scaling dimensions

Geometry, Representation Theory and Quantum Fields

119“Exotic” Conformal Field Theories (log-CFT)• Spectrum and scaling dimensions ??

Hamiltonian appears in a Jordan block with other operators.

• Correlation functions contain logarithms“Non-local QFT”
(e.g. Ising with loops)

see also: Hee-Joong Chung (2018, `19, `20)
arXiv: 2201.08351t
3d T[M 3 ]1

120

OCAMI Reports Vol. 4 (2022)t

BPS spectrum

3d N = 21
3d-3d correspondence

Many newlogarithmic CFTs
• Spectrum• Scaling dimensions

Counting
BPS states

Space of BPS states with q-degree bounded by nasymptotically grows as:
1d QM
2d CFT

Geometry, Representation Theory and Quantum Fieldst

BPS spectrum

3d N = 2
1

Many newlogarithmic CFTs
• Spectrum• Scaling dimensionst

BPS spectrum

3d N = 2
1

Many newlogarithmic CFTs
• Spectrum• Scaling dimensions

BPS quiver

121

122

OCAMI Reports Vol. 4 (2022)t

BPS spectrum

3d N = 2
1

Many newlogarithmic CFTs
• Spectrum• Scaling dimensions

BPS quiverquasiparticles

Geometry, Representation Theory and Quantum Fields

Quantum groups

Kazhdan-Lusztig

123

Vertex Algebras
2d CFT

Integrable latticemodelsH.Bethe (1931):
A.Zamolodchikov, Al.Zamolodchikov (1979)
A.Zamolodchikov (1989)
Al.Zamolodchikov (1990)
F.Smirnov (1990)
N.Reshetikhin, F.Smirnov (1990)

Yangian symmetry,
Bethe ansatz equation, …

S.Kerov, A.Kirillov, N.Reshetikhin (1986)
A.Kirillov, N.Reshetikhin (1988)

Rogers-Ramanujan

R.Kedem, B.McCoy (1993)
R.Kedem,
T.Klassen,
B.McCoy,
E.Melzer (1993):
S.Dasmahapatra, R.Kedem, T.Klassen, B.McCoy, E.Melzer (1993)
R.Kedem, B.McCoy, E.Melzer (1993)
A.Berkovich, B.McCoy, A.Schilling, S.Warnaar (1997):
E. Frenkel, A. Szenes (1993)
W.Nahm, A.Recknagel, M.Terhoeven (1993)

124

OCAMI Reports Vol. 4 (2022)

Bethe
ansatzBPS spectrumt
3d T[M 3 ]1share infinite
symmetry

Many newlogarithmic CFTs
• Spectrum• Scaling dimensions

Geometry, Representation Theory and Quantum Fields

125

3-manifold

3d N = 2 theory

Topologicalinvariants

Various BPS
observables andpartition functions

WRT
RokhlinSeiberg-Witten& Turaev torsion

S.G., P.Putrov, C.Vafa

126

OCAMI Reports Vol. 4 (2022)

WRT
RokhlinNon-perturbative
SL(2,C) Chern-Simons
theorySeiberg-Witten& Turaev torsiont
3d T[M 3 ]1

S.G., P.Putrov, C.Vafa

WRT
RokhlinNon-perturbative
SL(2,C) Chern-Simons
theory:- decorated TQFT

Seiberg-Witten& Turaev torsiont
3d T[M 3 ]1

S.G., P.Putrov, C.Vafa
S.G., S.Park, P.Putrov
J.ChaeF.Costantino, S.G., P.Putrov

Geometry, Representation Theory and Quantum Fields

127

Non-perturbative SL(2,C) Chern-Simons theory via
at generic

Theorem [Lickorish, Wallace]:
Every connected oriented closed 3-manifold arises byperforming an integral Dehn surgery along a link in .

S.G., C.Manolescu

Surgery formula: as a starting point, one first needs toconstruct invariants for knot (or link) complements:

128

OCAMI Reports Vol. 4 (2022)

as of
2019:

[GPV’16][GPPV’17][GM’19][GPPV’17]

?

?

?

?

[GPPV’17]

?

?

?

?

?

?

For knot and link complements, a very efficient
diagrammatic approach based on the R-matrix for
Verma modules and quantum groups at generic wasproposed by S. Park (2020, 2021).
For example, using this approach and the GM surgery
formula one finds:Geometry, Representation Theory and Quantum Fields

129

Non-perturbative SL(2,C) Chern-Simons theory via
at genericx

x -1
S.G. (2003)

Spectral curve:A(x,y) = 0 in^
A = quantum Hamiltonian

Quantization of Algebraic Curves

Lagrangiansubmanifold

A(x,y) = 0“wave-function”^
A = quantum Hamiltonian

130

OCAMI Reports Vol. 4 (2022)

Quantization of Algebraic Curves• For any closed cycle:• Has a nice interpretation in termsof algebraic K-theory and the Blochgroup of
S.G., P.Sulkowski

A(x,y) = 0

quantization“wave-function”

Geometry, Representation Theory and Quantum Fields

P.Kucharski
T.Ekholm, A.Gruen, S.G., P.Kucharski, S.Park, M.Stosic, P.Sulkowski

BPS spectrumt
3d T[M 3 ]1
BPS quiver

share infinite
symmetry

Many newlogarithmic CFTs
• Spectrum• Scaling dimensions

131

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OCAMI Reports Vol. 4 (2022)

Questions: prove or find counterexamples to
with

1.
2.

3. express in terms
of known topologicalinvariants

log-VOA
characters
Quivers

3d SUSY
QFT

3-manifolds
BPS
statesCobordism
invariantsGeometry, Representation Theory and Quantum Fields

Problems and Prospects
Unification offor different
branches?
Quivers for different branches?
The role of log-VOAs and log-MTCs?
“correction terms”?
Relation to the affine Grassmannian?
Categorification of?
Quantum groups at q=0 and crystal
bases?
R-matrices for different branches?
:

133

134

OCAMI Reports Vol. 4 (2022)

Hasse diagrams and Quiver Gauge Theories
Amihay Hanany

Abstract. The moduli space of Supersymmetric gauge theories with 8 supercharges has a rich structure of symplectic singularities associated with degenerations in which additional massless states arise. These are neatly arranged
into phase diagrams that encode diﬀerent the sets of massless states, with information on the moduli that are needed to be tuned in order to move from one
phase to another. We will present results from studies of families of quivers,
including those on the aﬃne grassmanian of finite dimensional Lie algebras.

(A. Hanany) Imperial College London

Geometry, Representation Theory and Quantum Fields

135

Before folding
Orthosymplectic quiver

After folding
Orthosymplectic quiver

Coulomb branch

C

2

e8
Omin
2

2

4

4

6

6

8

6

6

4

4

2

2

8

6

6

4

4

2

2

2

1

e7

Omin

2

2

2

4

4

6
1

4

4

2

2

2

4

4

4

2

2

–4–

Before folding
Orthosymplectic quiver

Symplectic Singularities
4

4

6

6

8

6

6

4

4

Coulomb Branch — Higgs Branch
Hasse Diagram — Magnetic Quiver
2

2

4

4

6
1

4

4

2

2

2

e8
2

4

2

2

8

6

6

4

4

Amihay Hanany March 2022

4

4

2

2

4

2

2

6

4

4

2

2

2

1
1
Omin

1

4

4

2

2

2

so(10)Omin2

so(10)

4

so(8)

1

4

2

2

Omin

1

1

2

2

Omin

1

sl(5)

2

sl(4)

Omin

1

2

2

Table 2: The orthosymplectic quivers on the left have Coulomb branches that are closures of exceptional algebras En for n = 4, . . . , 8.
Folding these quivers along the identical legs gives the non-simply laced orthosymplectic quivers on the right. The Coulomb branchesof these theories are given as well. In all the quivers here, there is an overall Z2 which is ungauged, see [14] for more details.

Symplectic Singularities
Mathematics and Physics• Symplectic Singularities are of great interest in both physics and mathematics• In Physics

Slice
Slice

Framed quiver
1

Framed quiver

Unframed quiver
1

Unframed quiver

1

1

1

f4

an

• Higgs branches in 3, 4, 5, 6 dimensions• Coulomb branch in 3 dimensions• Closures of nilpotent orbits• Slices in A ne Grassmanians

1

1
1

···

1

1

1

1
1

1

···

1

2

3
1

1

2

1

1
1

2

1

1
1

···

2

1

1
1

2

···

1
1

1

1

1
1

···

2

1

2

···

2

1

1

1

···

1

2

···

1

1
1

1

1

2

3

2

1

1

1

1

1

1

2

3

1

2

2

···

1

1

1

2

3

4

1

3

1

2

2

3

1

1

1
1

1

1
1

1

1

3

1
1

2

1
4

5

6

4

2

2

3

4

5

6

1

···

1

1

4

2

An

1

1

1

···

1

···

1

1
n+1

1

k

k

hn,k

3

e8
3

1

1

hn,k

1

4

1

3

···

1

1

1

1

2

e7
1

2

cg2

2
2

1

2

ag2

1

2

e6

3
1

acn

dn
1

2

g2

cn

2

ffi

1

2

bn

–4–

• In Mathematics

4

e7

2

2

Omin
2

1

2

2

2

sl(5)

1

1

2

2

2

–4–
4

4

1

e6

2

4

Coulomb branch

Omin

1

Omin

2

After folding
Orthosymplectic quiver

2

Omin
2

6

2

2

Table 2: The orthosymplectic quivers on the left have Coulomb branches that are closures of exceptional algebras En for n =Folding these quivers along the identical legs gives the non-simply laced orthosymplectic quivers on the right. The Coulomb bof these theories are given as well. In
e7 all the quivers here, there is an overall Z2 which is ungauged,e6 see [14] for more details.
Omin
Omin

1

2

so(10)Omin

2 Coulomb2
4 branch
2
1

2

2

2

e6

Omin
1

2

1

2

1
1

1
1
k

k
1

1

1

k
1

···

n+1

1
k

1

1

1
1

Table 1. Most up-to-date, but incomplete list of unitary quivers without loops for elementary slices usable in the quiver subtraction algorithm.
In each case we provide two quivers, a framed version and an equivalent unframed version, where a U(1) should be ungauged on the long node.
For an , bn , cn , dn , acn , hn,k and h̄n,k there are n gauge nodes in the framed quiver and n + 1 gauge nodes in the unframed quiver. Notice thathn,1 = Hn , hn,2 = cn , h2,3 = cg2 , hn,1 = an , hn,2 = acn , and h2,3 = ag2 .

136

OCAMI Reports Vol. 4 (2022)

Symplectic Singularities
alternative names7• HyperKähler cones6• HyperKähler singularities

53• holomorphic symplectic varieties

A5
4

2

A4

• Gorenstein singularities are to CY manifolds

3

2• like Symplectic singularities are to HK manifolds
↵ =↵
1

A3
2

_
1

1

A2
1

$1 = $1_

A1

Component 0 2 Z2Gauge theories

3

A6

Component 1 2 Z2Figure 2. On the left: weight and coweight lattice of A1 . On the right: Bottom of the Hasse
diagram for PSL(2, C). See detailed caption on page 17.

symplectic singularities• These moduli spaces are relevant to gauge theories as they show as Higgs
branches• 3d Coulomb branches• hence physically relevant– 18 –

su

5Focsg.ltiGeometry, Representation Theory and Quantum Fields

137

Brane systems
symplectic singularities• These singularities naturally arise on brane systems• most notably through the A ne GrassmanianSU(2)-[SO(8)] brane system with O5

ffi• hence physically relevant

3d/4d, 5d (inf), 6d (inf) magnetic quiver:
1
1

2

1

1

1

1

1
2

2

1

–2–

Symplectic singularities
3d/4d, 5d (inf), 6d (inf) brane system, Higgs ph
Construction
symplectic singularities

HK

3d MS

HK - hyper-K
CB

CB - Coulom
3d MS - 3d Mirr

Figure 1: One has to make a distinction between the variou
with in the realm 3d N = 4 gauge theories. Examples of sy
given through both the hyper-Kähler quotient and the Coulomb
space of symplectic singularites which can be constructed as a C
hyper-Kähler quotient, is the realm of the famous 3d mirror sy
are Coulomb branches for which no hyper-Kähler quotient const
versa. Furthermore there are hyper-Kähler quotients and Coulom
symplectic singularities. If the hyper-Kähler quotient is a union othe individual cones may be described as the Coulomb branches o
The hyper-Kähler quotient need not accurately describe the Hig

17
18
19
+
5173382t+
9731196t+
17732334t+ 31384129t
r
r
Y
Y
23
24
x = + 89958111t
ziai = z0 22 +uai i145882550t.
+ 230128561t(3.10)+ 35309
i=0

i=1

27
28
29
+
767161840t+
1087152304t+
1501274126t+ 20
138
OCAMI Reports Vol. 4 (2022)
we have used a0 = 1 and the32identification (3.8).
(3.10) can be34
33
+ 3400290035t+ 4250584996t
5186895160t
terms of x andu. The constraint(3.7) from the+removalof the + 6
8168673774t37 + 9067212695t38 + 9832235886t39 + 1
determines z +
1 as

7
ons

42
43
+
10910252456t+
10784743772t+ palindromekz space= x of. instantons(3.11) up
Moduli1
Coulomb branch

ADHM quiver is a HK quotient for a classical gauge groupk F•4Theinstantons• Problem — no such quotient for exceptional gauge groups• The a ne Dynkin diagram is a Coulomb branch construction for any gauge
Thetheorywhose
Coulomb
branchmoduli on
space
group
oulomb branch
is the moduli
spaceof k isG2theinstantons
C2 ofis k Fby the quiver diagram
ver described
diagram • exceptional• non simply laced

•
k

2k

V

k

1

•
k

2k

3k

)

2k

k

(4.1)

ffi

1

denotestheeachranknumberof each unitarygroupofandan overall
U (1)groupwhere
denotes gauge
the ranka unitary
gauge
out.
symmetry
is factored out.of Symplectic Singularities
Characterization
rmula forG2 instantons
cank befromusingusing
by easeof computation
Thek ordered
Hilbertseriesof
F4extracted
instantons
canthisbe quivercomputedection3: by• (3.4).
given
For k
2, (3.4) is more easily calculated using
Dimension (quaternionic)
symmetry• Global
discussedin[26].
Indeed X
quiverkk2k
2k X
k (7.1) can be constructed from thX
XX
Hasse (Phase) diagram|mi | + •|mi nj | +|3nj s` |Representation
contentofthechiralring(Hilbert Series)•
T(k,k,kj=1(SU(3k)) : (1) (k) (2k) [3k1,1)`=1=1i=1 j=1!
• Highest weight generating function (HWG)
X
X
X
T(k,k,k)[3k] (2k) (k) ,relations(SU (3k)) :• Chiral ring - generators and|mi mi0 | +|nj nj 0 | +|s` s`0 | ,
0 2k
0 k
1i<i
once0 kthe edge [3k] 1j<j(2k)in the second1`<`building
block is converdoubling the value of the background magnetic charges (4.2)
in the Co(3k)).
twoblocksmk ),seriesn = of(n1T, (k,k,k)..., n2k(SU) ands = The(s1 , ...,sk ).building
Note thefactorareof glued
3 in byU (3k)/U(1).
iplyflavor
lacedsymmetry
bifundamental
contribution.
Here we have gauge fixedThe
expressionof thethe over-extended
Hilbert seriesnodein questionis givenby setting
thefinal
monopole
flux of(indicated
X
2

(m)

2

Geometry, Representation Theory and Quantum Fields

Higgs Branchchanges at special points with extra massless states• stays xed along the Coulomb branch• stays xed along continuous variation of gauge coupling• until we get extra massless states• At such points they become very di erent from the classical result

ff

fi

fi

• So how do evaluate the new Higgs branch?

3

2

8

13

A2
1

5

7

13

A4

6

6

12

A1

A5
1

4

6

11

5

10

4

5

8

A1

A1

1

2

4

7

1

2

6

10

A1

1

1

3

5

2

1

8
3

1

7

11

12

3

5

8

6

4

9

Magnetic Quivers

2

A1

ag2

6

1

cg2

1

2

A2

A2

9

A2

4

4

A2

A1

6

A3

3

A3

2

2

12

A2

3

3

7

A1

A4
1

4

3

A3

5

A1

2

2

3

2

g2

Figure 13. Bottom of the Hasse diagram for G2 . See detailed caption on page 17.

– 29 –

139

140

OCAMI Reports Vol. 4 (2022)

Lagrangian theories
Coulomb branch• Lagrangians are very important for construction of 3d Coulomb branches• Many quantum corrections• But under control• as space of dressed monopole operators• The monopole formula (CHZ 2013)

Magnetic Quiver
Brane system• Computes the Higgs branch of some theory in 3,4,5,6 dimensions• especially when the Higgs branch is not a HK quotient• Extends the notion of 3d mirror symmetry• Gives an important window to physics away from the weakly coupled point• This is a major change in our understanding of such theories

Geometry, Representation Theory and Quantum Fields

Brane system
Magnetic Quivers• Given a theory with some interesting physics at strong coupling• Embed using brane systems• Compute the magnetic quiver• Extract the new physics from the quiver• Using the monopole formula or the phase diagram

Magnetic Quivers
physics at strong coupling• 6d — physics of tensionless strings• 5d — physics of massless gauge instantons• 4d — physics of Argyres Douglas points• 3d and above — physics of moduli spaces with multiple cones• moduli space of instantons

141

142

OCAMI Reports Vol. 4 (2022)

6d — small instanton transition
SU(2) with 10 avors• The Classical Higgs branch — minimal nilpotent orbit of SO(20)
SU(2)-[SO(20)]finite andinfinite6d:
space
of 1coupling
SO(20)in instanton
on ℂ2• The moduli
5
4

5

6

7

8

4

e8
1

4

1
2

2

2

2

2

2

2

1

1

d10

2

3

4

5

6

7

8

5

2

d10

fl

1

–4–

6d — small instanton transitiona family of Sp(k-1) theories with 2k+6 avors

Dimension and
Hasse diagram
Hasse diagram
ymmetry for k > •1 The Classical Higgs branch — a nilpotent orbit of SO(4k+12)
finite coupling
infinite couplingat infinite
TheorycouplingSp • The
Magnetic
quiverMagnetic
quiverOSp
Higgs branch
at inU nite coupling with magnetic
quiverand Hasse
diagram. The HWG is PE of the polynomial below

E8

··· .
.

3

4

k+1

k
P

..
.

2

D1

µ2i t2i

d8

k
2
P

d2k+6

+
2

2
✓k 1
P

E4 k + 1

i=1

✓

µ2i

t2i

e7

1

µ2i t2i

a1

d10

+

.

d10✓ Pk

a11µ22k 1 µ22k t4k

..

◆

d2k+4
✓
kP1
i=1

✓k 1
P

+ t2 + (µ2k+2 q + µ2k+3 q 1 )tk+1

C1

Ck+1 Dk+2 Ck+1
◆

C1

i=1

✓

D1

+ t2 + (qµ2k + q 1 µ2k+1 )tk+1

2k t2k+2 )1+µU
2k µ2k+1 (t◆· · ·µ2it2i + t2 + (q + 1 )µ2k tk+1·+· µ·22k t2kd 7 C1

D1

U1◆µ2i t2i

C1⌫ 2 µ22k+4 t2k+6· · i=1· µ2it2i + t2 + (q + q 1)µ2k+2· ·tk+1
·

e6

D1

µ2i

i=1

t2i

···

+⌫ t +t

i=1

..
.

a1

1

+

..

k

i=1

d7

µ2i t

e8

D..1 a1 C1◆
d2k+4
2i + µ µ2kµ2i t12k 2k+1 t
◆

◆
k+1
C✓k+2
P D2i
k+3 C2 k+2
2
4

✓ k
P

d8

d5kP1 µ2i t2i + (µ122k 1 + µ22k )t2k

2k

1 E3

···

HWG at infinite coupling◆C1µ2i t2i + t4 + µ2k+6 (tk+1 + tk+3 )+⌫µ2k+4 (tk+1 + tk+3 ) + µ22k+4 t2k+4

i=1

d2k+4

i=1

···

C1

i=1

k

..
.

+

E5

2k

···

d6

k
P

✓k+2
P

d10

i=1

E6

2k 2 + 5k + 4Sp(k)so(4k + 6) u(1)

+

+

D2k+4

D2k+3

k

2k

Sp(k)

µ2i t2i

i=1

d2k+5

E7

2k 2 + 7k + 8 1
so(4k + 8) su(2)
Sp(k)

k
P

d9

.

1

e8

d7 HWG at finite couplingfl2k 2 + 11k + 16
so(4k + 12)

Familyk + 2

fi

D2k+5

q
Dk+1 Ck+1
Dk+1
2 2k+2
◆

µ2k t

C1

D1

Geometry, Representation Theory and Quantum Fields

143

Non simply laced quivers
No known Lagrangian or path integral• A small modi cation of the monopole formula

Rank 1 SCFT

• A whole new set of moduli spaces

C5• A window to exotic moduli spaces

C 3 ⇥ A1

Magnetic quiver
1

2

3

1

4

2

3

5

2

1

2
1

• like rank 1 4d theories

C2 ⇥ U1

1

A3

2

1

A1 ⇥ U 1

2

3

1

A2

2

2

4

1

3

fi

1

1

Table 2. The magnetic quivers of 4d N = 2 rank 1 theories with enhanced Coulomb bra(labeled by their global symmetry).

quivers, i.e. the freedom to change between framed and unframed simply laced quive
Phase (Hasse) diagrams

the case of simply laced quivers there is no need to denote an ungauged node.
The magnetic quivers of the 4d N = 2 theories in question can be obtained
magnetic quivers of 5d N = 1 theories compactified with a Zk twist. In this case one
identifying
settheof5dmasslesseldsof k identical simply-laced• We characterize di erent phaseswith thebymagnetic
quiverthe
Q0 oftheory, whichcontains
the theory
and obtains the magnetic quiver Q of the 4d N = 2 theory by folding the k legs of Q

massless eldsare
• Theories with massive elds that have the same massless content
Fk : Q0 7!
Qconsidered to be equivalent.

where Fk is the action of folding k identical legs. Remarkably, most of the 5d theor

Hilbert series and chiral ring. The Coulomb branch Hilbert series and refined ple
tic logarithm (PL) of Q are given in Table 3. The refined PL encodes information ogenerators of the chiral ring and their relations [41]. The first few positive terms are
resentations of the generators whereas the first few negative terms are the representaof the relations. Higher order terms are often higher syzygies. The terms at ordtransform in the adjoint representations of the global symmetry group [42].
The C5 , C3 ⇥ A1 , and C2 ⇥ U1 magnetic quivers can be derived from 5d N = 1 S
magnetic quivers Q0 through folding [12], as detailed in Section 4 below. Therefore
can expect that the highest order relations exist at order t4 B where B = 3/2 i
conformal dimension of the baryonic/instanton generators, see for example [12, Sec.
As a result, there should be no relations beyond t6 . This is consistent with the fact
there are no negative terms in the PL at order t7 and t8 . At higher orders, negative
reemerge in the form of higher syzygies, i.e. relations between relations [41]. As a r
for these three families, all the generators and relations can be seen in Table 3.
fi

fi

ff

fi

• They are in the same phase.
question are all well known, simple theories.

144

OCAMI Reports Vol. 4 (2022)

Massive fields• Masses are functions of moduli in the theory• As we move along the moduli space, masses of massive states vary.
• At some critical points some states become massless• In such cases we say that the phase of the theory changes• It contains more massless states.

Natural questions• As we move from phase A to phase B with more massless elds:• Characterise each phase — give some names• How many moduli are tuned to get from A to B?
• What is the geometry of these moduli?

fi• These are called transition moduli, as they move from phase A to phase B.

Geometry, Representation Theory and Quantum Fields

145

Transition moduli• Necessarily conical• As we scale these moduli massive states remain massive• massless states remain massless• At the origin new massless states show up

Example: Free scalar field• Consider a scalar eld with mass m• There are two phases• m ≠ 0 one dimensional phase with 0 massless states• m = 0 zero dimensional phase with 1 massless statefi• The transition modulus m parametrizes ℝ+ which is conical, one dimensional

146

OCAMI Reports Vol. 4 (2022)

Minimal transitions• Given a phase, a minimal transition is a minimal set of tuned moduli formoving to a new phase• The Hasse (phase) diagram for such moduli consists of 2 points connected by
one edge.

• Two phases• The origin• Anything elsefi

• An important problem — nd such minimal cases

Hasse Diagram
Phase diagram• We form a diagram with two objects• nodes and edges• A node represents a phase (symplectic leaf)• An edge represents a minimal transition (transverse slice) between a node A
with some massless states to a node B with additional massless states• A minimal transverse slice has a Hasse diagram with 2 points. The origin andanything else.

Geometry, Representation Theory and Quantum Fields

147

1
a)

b)

1

1

c)

1

Hasse diagrams
2
2
1
1

2

...

...
1
n

1
Quiversubtraction
n−3

1

2

3

2

1

the work of Kraft• Recalle) and Procesi who classi3 ed degenerations in closures2

d)of nilpotent orbits

1

2

3•

4
3 degenerations
2
1
1 of two2
3
4
Minimal
are
types

5

6

4

2
1

by capital letters• Klein singularity (ADE)a)— denoted
b) 1
1

c)
1
Figure 2: Affine ADE Dynkin quivers. a) Ân b) D̂n c) Ê6 d) Ê7 e) Ê8 . Their Coulomb...
branches are theminimalnilpotent
orbit
closureofthecorresponding
algebra,writtena,of some algebra
lower case• closure of a minimal nilpotent. orbit
2 —n denoted
2..
1
1
dn , and en respectively. Their Higgs branches are the1 Kleinian 1singularities corresponding1n−3This isand generalized
withthe Coulomb
branchto the algebra, •written
Anreproduced, Dn , and En respectively.
Itbe clearfrom context,
n should
when a capital letter refers to the Kleinian singularity rather than a Dynkin diagram or an
algebra. The Hasse diagrams for both the Coulomb
and Higgs branches
are given in (1.5).e)d)2

1. Black dotswith a number n next to it: denote1 a leaf
2 of3 quaternionic4
3 dimension
2
1 n.

1

2

3

4

2

2

3

2

1

6

4

2

3

5

fi

2. A line � with a label next to it, between two black dots: denotes the elementary slice
Figure 2: Affine ADE Dynkin quivers. a) Ân b) D̂n c) Ê6 d) Ê7 e) Ê8 . Their Coulomb
between two neighbouring leaves.
branches are the minimal nilpotent orbit closure of the corresponding algebra, written an ,Now consider the affine Dynkin quivers of ADE(Figure2). Their Their
Coulomb
dn , anden respectively.
Higgsbranches
branches are the Kleinian singularities corresponding
are minimal nilpotent orbit closures, while their toHiggs
are Kleinian
the branches
algebra, written
An , Dnsingularities.
, and En respectively. It should be clear from context,
capital letterrefersto the Kleinian
singularity rather than a Dynkin diagram or an
Both their Higgs and Coulomb branches consist ofwhen
two asymplectic
leavesandtransverse
2 symplectic leaves, minimal
slicesalgebra.
The areHasse
diagrams for both the Coulomb1 and Higgs branches are given in (1.5).
slice is the the branch itself. The respective Hasse
diagrams

Quivers — ADE
x

a)

b)

1
...

1

n

HC =

c)

1

2

2

2

1

1
2 black
3 dots:2
1,nD
or Ennext to it, between−n
2. A lineA� N
with
a3 labeltwodenotes
the elementary slice
between two neighbouring leaves.

an , 1dn or en 10

1

1

1. Black dots . . . with a number n next to it: denote a leaf of quaternionic dimension n.

(1.5)of ADEe) affine .Dynkin quivers
2HH = Now consider the
3 (Figure 2). Their Coulomb branches0d)

are minimal nilpotent orbit closures, while their Higgs branches are Kleinian singularities.

Where closures of minimal nilpotent orbits are Bothdenotedwith aandlower
case and
Kleinian
Coulomb
consist12
3
4
3 their2 Higgs
1
1
2branches3
4
5of two
6 symplectic
42 leaves and the transverse
singularities with an upper case letter, note thatslice
A1 =isathe,and1 the branch itself. The respective Hasse diagrams are�n ADEforDynkin
an quivers. a) Ân b)x D̂n c) Ê6 d) Ê7 e) Ê8 . Their Coulomb�
Figure 2: Affine1���branches are�theminimalnilpotent
orbit closure of the corresponding algebra, written an ,
�
2n
−
3
for
dn
�
�
�
dn , andxen= respectively.
singularities
an , dn oren(1.6) correspondingAN , Dn or En� 11
forTheir
e6 Higgs branches are the Kleinian�to the algebra,
written An , Dn , and En respectively. It should be clear from context,
�
�
�
17
for
e
7
�
�
when a capitalletter refers to
the Kleinian singularity
�
0 rather than a DynkinHdiagram
0or an�
HCCoulomb=.
�
H = in (1.5).
29 diagramsfor e8 for both the� Hassealgebra. The
and Higgs branches are given

(1.5)

26
Where closures of minimal
nilpotent orbits are denoted with a lower case and Kleinian

1. Black dots

withsingularities
a number n next
denotecase
a leafof quaternionicn.
withtoanit:upper
letter,note thatdimension
A = a , and
1

1

2. A line � with a label next to it, between two black dots: denotesslice� nthe elementary
for an���
between two
leaves.
�– 6neighbouring–��
2n − 3 for dn���

Now consider the affine Dynkin quivers of ADE (Figure 2).
x =Their� 11 Coulomb
for e6branches�are minimal nilpotent orbit closures, while their Higgs branches are�
Kleiniansingularities.
�
�
17
for
e7
�
�
�
Both their Higgs and Coulomb branches consist of two symplectic leaves
and thetransverse
�
�
29
for
e8
�
slice is the the branch itself. The respective Hasse diagrams arex

1

AN , Dn or En

an , dn or en
HC =

0

HH =

–6–
0

.

(1.5)

(1.6)

Matrix
Partition dim
!
0300
3
3
0210
With the matrix formalism we can write a computer algorithm that is able to calculate all
A2!
matrices (i.e. all brane configurations and all quivers) and KP transitions for all nilpotent
1110
2,1
2
orbits of any slN algebra, starting from the matrix of the maximal nilpotent
orbit.
0110
a2
!
148
2001
1,1,1
0
0000

7

Results

7.1

sl3

OCAMI Reports Vol. 4 (2022)

Tables with Results from the Matrix Formalism

Table 6: Results obtained applying the matrix formalism to sl3 .

In this section we present all the results that have been produced with this algorithm. For
each value of N we include a table that contains all matrices for all models of the form
T t (SU (N )). The corresponding partition and quaternionic dimension can be read fromt
the matrix and are also included. The quivers for both T t (SU (N )) and T (SU (N )) can
easily be recovered from the matrices, as was shown in the example in the next section.
The algorithm can also provide the nature of the KP transition that is required in eachstep. These have been added to the matrix data in the form of Hasse diagrams9 .

Hasse diagrams for nilpotent orbits
Matrices
dim

Matrices

sl7 !

Partition
dim
MatricestakenMatrix
from Matrix
KP
Matrices
Partitiondim
sl6

07000000
06543210

06000006dim
0 5 4 3 2A1dim
0

dim

!

7

21

15 !

6

15100000
05543210

6,1
!
!
! A5
!
sl
1 4 1 0 0A0 0!
0 34 0 0 sl5
0 4 0 00 05 0 0 0 05,1 2 3 2 0 0 014
00
0
5,2
3 4
2
1 3
5 60 4 4 3 2 1 1004543210
0210!
0 3 2 10 04 3 A23 1 0a!
2220000
24010000
5,1,1
A
A1
! A3
A4
4,2 0 4 4 4 3 213
!
A2
!
!
10
0343210!
101
1 2 1 01 03 1a1 0 0 0
1 1 1,1
1031300000
0 2,1
4,152 3 0 1 A0 0 09!
4,3
3,1
2
03543210
000
4,1,1
12 !
0 2 2 10 03 3 2 1 A010110
0 3 3 3 2 1 0A
3
2
1
1
0
0
0
0
A2
!
4,2,1
a2
A1
!
!
!
03443210
3 0 3 0 0 0 0A!
2 0 2 02 01 A22 0 0 0
2001
3,240 2 4 3 a2 1 08 3,3 4 0 2 1 0 0120 0 3,3,1
Table 5: Results obtained applying the matrix
to2,2
sl2 .
1,1,1formalism
002443210
0 1 2 10 02 3 2 1 A02
0000
!
!
a1
3111000
33001000!
a1
4,1,1,1
A 3,2,1!
1103333210
0233210
2
2
0
1
0
0
!
21010
a2
3,1,1
7!
41020000
2,1,1
3
3,2,2
A
0 02 2a 2 formalism
10
3 2sl
0 30.1 0 002343210
Table 6: Results obtained applying
to
0 1the1 1matrix
3,1,1,19 !
92Note that there is an equivalence a1 = A1
0 2 2 2 2 1 0A
A
!
4
1
1
0
1
0
0
0
1
a3
3,2,1,1!
!
02333210
3011004 0 0 2 0 0 06a
30001
2,2,1
a1
2,2,2 5 0 0 1 1 0 90 0 !
1,1,1,1
0
0
1
2
3
2
1
0
a
2,2,2,1012210012332100 0 0 0 0 A1
!
!
a

sl2

sl

0 23 0
010

!

20

4

19

1

18

2

18

3

2

17

1

16

2

15

1

15

2

1

14

2

12

3

!

2

4010100
42000100
a2 2,2,1,1
802222210
3100100122210– 51 –
A1!
2,1,1,1
4
a3!
50100100
0
1
1
1
1
0
Table 7: Results obtained applying the matrix formalism
4 1 0 0 0 1 0 to sl4 . 0 1 2 2 2 2 1 0
2,1,1,1,1
5
a4
a
4
!
!
011111051000010!
4 0 0 a05 0 101111110
1,1,1,1,1
5 0 0 0 0 a06 10!
1,1,1,1,1,10000006 0 0 0 0 00
010000000
00000000

3,1,1,1,1

11

2,2,1,1,1

10

2,1,1,1,1,1

6

1,1,1,1,1,1,10Matrices
dim
Table 10:
Results
obtained
applying to
the slmatrixformalism to sl .
Table 9: Results obtained applyingthematrixformalism6.
! obtained
Table 8: Results
applying the matrix formalism to sl5 .
sl4
04000
4
6
– 55 –
03210
A3!
– 52 –
12100
3,1
5
02210
Slice
Framed quiver
A1
!
Slice
Framed quiver
Unframed quiver1– 54 –
20200
1
1
1
f4
2,2
4
an
01210
1 2 3 2
···
···
a1
!
1 1
1 1
1 1
1 1
1
1
1
A list of minimal 2transitions1010
g2
2,1,1bn
3
01110
1 2
···
···
1 2
2 1
1 2
2 1
1
1
a3
!
1
1
ac
n
30001
···
cn
1,1,1,1
0
1 1
1 1
···
···
00000
1 1
1 1
7

Minimal transverse slices
Symplectic singularities

•

1

dn

1
1

1
1

– 53 –

1

1

–4–

···
Table 7: Results obtained applying the matrix formalism
to sl4 .
1 2
2 1
e6
1

2

1

2

1
···
1

2

2

3

2

1

1

2

3

1

2

1

1

2

3

4

3

2

1

2

3

3

3

2

2

3

4

5

6

4

2

2

3

4

5

1

2

1

···

1

2

1

1

1

1
1

1

1
1

1

1

1

1

1

1
1

1

···

1

1

···

1

6

1

1
1

1

···

1

1

···

4

2

An

1
n+1

1

k

k
1

hn,k

3

e8

– 52 –

3
1

1

hn,k

1

4

1

2

1

1

1

2

2

1

cg2

e7

1

1

ag2

2

1

Unframed quiver

1
1
k

k
1

n+1

k
1

1
k

1

1

1
1

Table 1. Most up-to-date, but incomplete list of unitary quivers without loops for elementary slices usable in the quiver subtraction algorithm.
In each case we provide two quivers, a framed version and an equivalent unframed version, where a U(1) should be ungauged on the long node.
For an , bn , cn , dn , acn , hn,k and h̄n,k there are n gauge nodes in the framed quiver and n + 1 gauge nodes in the unframed quiver. Notice thathn,1 = Hn , hn,2 = cn , h2,3 = cg2 , hn,1 = an , hn,2 = acn , and h2,3 = ag2 .

Geometry, Representation Theory and Quantum Fields

149

4d — multiple cones
SQCD SU(6) with avors• a union of the mesonic and baryonic branches
a1

d3

d4

d5

a2

a3

a4

a5

a6

a3

a4

a5

a6

a7

a8

a4

a5

a6

a7

a8

a9

a10

a5

a6

a7

a8

a9

a10

a11

a12

Nf = 6

Nf = 7

Nf = 8A1
a1
a1– 26 –

Nf = 1

a1

a2

a1

a2

a3

a4

Nf = 2

Nf = 3

Nf = 4

Nf = 5

a3
A5

a2
A4

A3

A2

Nf = 9 Nf = 10 Nf = 11 Nf = 12 Nf = 13flFigure 7: This figure displays the explicit Hasse diagrams as given in generic form in Table 7, in the particular case Nc = 6, and for1  Nf  13.
The case Nf = 10 could also be represented as a d2 a3 a5 a7 a9 diagram.
Note that the maximal height of the diagram is 5, which is the rank of the gauge group. It is reached only when Nf 2Nc 2 = 10.
One can see that for Nf < Nc = 6 there is only the mesonic branch. The baryonic branch appears for Nf = 6, and then grows indimension from 1 (for Nf = Nc = 6) to 25 (for Nf = 2Nc 2 = 10) where it equals the dimension of the mesonic branch. Then theA2baryonic branch takes over,and contains the mesonic branch as a sub-cone for Nf 2Nc 1 = 11.
1

5

7

13

12

A1

A5
1

4

6

11

10

4

5

8

12

6

A1

2

2

6

10

A2

A2

9

A1

2

1

2

4

7

5

4

8
3

1

7

11

12

3

ag2

A1

1

1

3

5

6

1

9

Affine Grasmanian G2
8

2

A1

A2

4

A3

A2

A1

4

6

1

cg2

1

1

13

3

A3

2

2

7

A2

3

3

3

A1

A4
1

4

8

A1

2

A3

5

5

2

A4

6

6

3

2

3

2

g2

Figure 13. Bottom of the Hasse diagram for G2 . See detailed caption on page 17.

– 29 –

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Inversion of Hasse diagram
given a quiver• For a large class the Hasse diagram of the Coulomb branch is the inversion of
the Hasse diagram of the Higgs branch• upside down, with lower case exchanged with upper case
• Known as Spaltenstein map for nilpotent cones• Related to symplectic duality between the Higgs branch and Coulomb branchof a given quiver

Summary
Changing the way we think• Coulomb branch — a new way of constructing symplectic singularities• A quiver — analog of toric diagram to toric singularities• Phase (Hasse) diagrams — changes the way we analyze symplectic
singularities• Brane systems — very instrumental in getting the recent progress• Quiver subtraction• Monopole formula — opened the window to all recent achievements

Geometry, Representation Theory and Quantum Fields

Thank you !

151

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Finite N superconformal index via the AdS/CFT
correspondence
Yosuke Imamura
Abstract. I will present a prescription to calculate the superconformal index of the N=4 U(N) supersymmetric Yang-Mills theory with finite N on the
AdS side. The finite N corrections are included as contributions of D3-branes
wrapped around three-cycles in S5, which are calculated as the index of thegauge theories realized on the wrapped branes. I will also comment on applications to other theories.

(Y. Imamura) Tokyo Institute of Technology

Geometry, Representation Theory and Quantum Fields

Finite N superconformal index
via the AdS/CFT correspondence@ “Geometry, Representation Theory and Quantum Fields”
2022/3/22

Y. Imamura
Tokyo Institute of Technology
Based on[Arai, YI, arXiv:1904.09776][Arai, Fujiwara, YI, and Mori, arXiv:1907.05660][Arai, Fujiwara, YI, and Mori, arXiv:1911.10794][Arai, Fujiwara, YI, Mori, and Yokoyama, arXiv:2007.05213][YI, arXiv:2108.12090][YI and Murayama, arXiv:2110.14897]

Introduction

153

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Superconformal index [Kinney, Maldacena, Minwalla, Raju, hep-th/0510251]
Cartan generators of PSU(2,2|4)

𝐻 : Hamiltonian (Dilatation)
𝐽 , 𝐽 : Angular momenta
𝑅 , 𝑅 , 𝑅 : R-charges

Fugacities are not all independent but constrained by

We can calculate this quantity on the gauge theory side for an arbitrary N
by using localization method.

Free field theories
Pexp
Single-particle (letter) index → Index of free QFT

Pexp : Plethystic exponential (often denoted also by PE): numerical coefficients,: monomial of variables

Geometry, Representation Theory and Quantum Fields

155

Example 1 : U(1) theory

unrefinement

Example 2 : U(∞) theory( )This result strongly suggests the existence of free-field description → AdS/CFT

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AdS/CFT correspondence[Maldacena, hep-th/9711200]

AdS5×

N=4 U(N) SYM
Gauge invariant operators

Various objects

Large N𝐿

: AdS radius𝑙 : Planck length

Large

↔

↔

Classical analysis is justified.

→ We can reproduce the index by using classical supergravity.

Supergravity KK modes in[Kinney, Maldacena, Minwalla, Raju, hep-th/0510251]

Geometry, Representation Theory and Quantum Fields

157

Finite N
Parameter relations: AdS radius

finite N → quantum gravity: Planck length

finite N → Expanded D3-branes: Planck length

Giant gravitons[McGreevy, Susskind, Toumbas, hep-th/0003075]

Interesting possibility
If a quantity is protected from quantum gravity correction, it may bepossible to reproduce the finite N correction to the quantity as D3-brane
contributions.
Superconformal index seems to be such a quantity.

Main claim [YI, arXiv:2108.12090]

+

+

+

+…

or, equivalently,

=

(1+

+

+

+…)

Named “Giant Graviton expansion” in [Gaiotto, Lee, arXiv:2109.02545]

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Plan• Introduction✔
• Construction of the formula• Other examples• Additional comments

Construction of the formulaGeometry, Representation Theory and Quantum Fields

159

Strategy
Dynamical GG

Index formula

reductioncompletion(guess works)

Rigid GG
toy model

Weyl character formulasolvable

BPS D3 in S5
There exist stable BPS configurations of expanded D3-branes in S5.
(Giant gravitons [McGreevy, Susskind, Toumbas, hep-th/0003075])

BPS →holomorphic cycle [Mikhailov, hep-th/0010206]

To calculate their contribution to the index we have to take account of D3 of different shapes.

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Rigid branes (a toy model)=+ excitations

Rigid brane = D3 wrapped on a largeinA rigid D3 = A point particle in CP2

A particle in CP2
The wave function = holomorphic section of a line-buldle over CP2
RR flux → 1st Chern class =Degenerate states in

,

of

,

𝑁=3
𝑁=1

𝑁=2

𝑁=0

[0,0]

[1,0]

[2,0]

[3,0]

Geometry, Representation Theory and Quantum Fields

161“Index” of a rigidD3
𝑞 : the energy of D3𝜒,(𝑢) : SU(3) character

The SU(3) characters are given by
,

,
,
,

Numerical check
For N=1 we want to get
The supergravity reproduces four terms.

Inclusion of the rigid D3 contribution,

It does not improve the result but worsens it.
We need some completion of the formula.

gives

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Localization
We consider a particle in CP2 as a toy model.
Index of SQCD in CP2×R has been studied in the literature.
[Kim and Lee, arXiv:1210.0853], [Kim, Kim, Kim, and Lee, arXiv:1307.7660]

3 fixed points of U(1)→CP2

𝐶𝑃

CP2→3 Ω-backgrounds( )Our toy model can be regarded as the simplified (quantum mechanical)
version of such analysis.

Weyl formula
The Weyl formula gives,

𝐶𝑃
Contribution from each fixed pt = 2d harmonic oscillator.

Geometry, Representation Theory and Quantum Fields

163

Three three-cycles
,

𝑧 =0
𝑧 =0
𝑧 =0

: classical energy and charges of a brane wrapped around a large circle: harmonic oscillators of two rigid motions

1st Completion
Rigid motions (only two d.o.f.)

Include all fluctuations

spi on D3 is obtained from the standard U(1) spi by a simple variable change[Arai, YI, arXiv:1904.09776]

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Single-wrapping contributionForthis gives

Numerical checkWe want to get

SUGRA gives

Inclusion of[Arai, YI, arXiv:1904.09776]

Still we have mismatch, but this is encouraging!

Geometry, Representation Theory and Quantum Fields

165

2nd completion
Let us include the contribution from multiple-wrapping contributions.

𝑈(𝑛 )D3 x 𝑛𝑈 𝑛

D3 x 𝑛𝑈 𝑛

D3 x 𝑛
𝑈 𝑛
𝑈(𝑛 )

𝑈(𝑛 )

quiver gauge theory is realized on the brane system

Our proposal
The index is given by the following formula.

[YI, arXiv:2108.12090]

,
,

,

,

,

,

: index of the quiver gauge theory on the D3-branes (N-indep)

Remarks• The two-step completion is based on guess works, and we have no proof at present.
• As far as we have checked numerically, this formula works well.

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Numerical check
0

=
1

0

+
0

0

0

1

2

0

+
2

3
1

+
0

0
2

+
2

1

1

+
0

2

1

1

Numerical check
Inclusion of multiple-wrapping contributions gives

By summing up contributions withcomplete agreement up tofor
Expected error

0

+
0

1
1

1
1

+
1

+
0

1
2

+
0

1

+
1
0

+

0

1

0
3

+
0

1

+
0

0

=

0

+

0

0

0

2

+
0

3

0

0

=
2

1

+

we find. [YI, arXiv:2108.12090]

20Geometry, Representation Theory and Quantum Fields

167

Other examples

Application to other systems
AdS

CFT

Large N

Single-wrapping

Multiple-wrapping

4d N=4 SYM[Kinney, Maldacena, Minwalla, Raju, 05]✔[Arai, YI, 19]✔[Arai, Fujiwara, YI, Mori,
20][YI, 21]

AdS × 𝑆 /𝑍

4d N=3[IY, Yokoyama, 16]✔[Arai, YI, 19]

AdS × 𝑆 /Γ

4d N=1 quiver[Nakayama, 05]✔[Arai, Fujiwara, YI, Mori, 19]

AdS × SE

4d N=1 quiver[Nakayama, 06][Eager, Schmude, Tachikawa,
12][Agarwal, Amariti, Mariotti 13]✔[Arai, Fujiwara, YI, Mori, 19]

AdS × 𝑆

4d N=2 AD & MN

[Fayyazuddin, Spalinski, 98][Aharony,
Fayyazuddin, Maldacena, 98]✔[YI, Murayama, 21]

AdS × 𝑆 /𝑍

3d ABJM[Bhattacharya, Bhattacharyya, Minwalla, Raju,
08][Kim, 09]✔[Arai, Fujiwara, YI, Mori,
Yokoyama, 20]

AdS × 𝑆

6d (2,0)[Bhattacharya, Bhattacharyya, Minwalla, Raju,
08]✔[Arai, Fujiwara, YI, Mori,
Yokoyama, 20]

AdS × 𝑆 /𝑍

6d (1,0)[Ahn, Oh, Tatar, 98]✔[Fujiwara, YI, Mori, 21]

AdS × 𝑆（✔）[Fujiwara, in
preparation]（✔）[Arai, Fujiwara, YI,
Mori, Yokoyama]

As far as we have checked the formula reproduces finite N index correctly!

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Other examples : toric SE5
T1,1 caseS5 case𝑛generalization𝑛𝑛This is the toric diagramof C3 (=cone over S5).

𝑛

𝑛

𝑛

𝑛

The toric diagramof the conifold (=cone over T1,1).

We have numerically checked for many examples of toric SE5 that our prescription
reproduces correct indices (at least for single-wrapping sectors.)[Arai, Fujiwara, YI, Mori, 19] [Arai, Fujiwara, YI, Mori, 19]

Klebanov-Witten theory
A
N

N
B

fields𝑺𝑼 𝑵 𝟏𝑺𝑼 𝑵 𝟐

𝑺𝑼 𝟐 𝑨𝑺𝑼 𝟐 𝑩

𝐴

𝑁

𝑁

2

1

𝐵

𝑁

𝑁

1

2

Topological wrapping number= U(1)b charge

The red part is reproduced by

(

, , ,

, , ,

𝑼 𝟏 𝒃
1
+
𝑁
1
−
𝑁
,

𝑼 𝟏 𝑹
1
2
1
2

)

[Arai, Fujiwara, YI, Mori, 19]

Geometry, Representation Theory and Quantum Fields

Other examples : S-folds

169

(

)

We can check the consistency with the symmetry enhancement N→N[Aharony and Tachikawa, arXiv:1602.08638]
S-fold with rank 2 = N
The index of N

SU(3) SYM

SU(2) SYM is

The SUGRA contribution gives only the first trivial term.
Inclusion of GG finds a big improvement. [Arai, YI, arXiv:1904.09776]

The results are consistent with the anasysis in [Zafrir, arXiv:2007.14955]

Other examples : M-branes
Parameter relations
3d N=8M5 giants

6d N=(2,0)

M2 giants

These strongly suggest that we can get finite N corrections as GG contributions.
Single-wrapping sectors are analyzed in[Arai, Fujiwara, YI, Mori, Yokoyama, 20] [Fujiwara, YI, Mori, 21]
A1 type (2,0) theory
sugra

single GG

expected error

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More non-trivial examples : AD and MN
A class of N=2 theories (AD and MN) are realized as theories
on D3 in 7-brane backgrounds.
0

1

2

3

4

5

6

7

7-brane

o

o

o

o

o

o

o

o

D3

o

o

o

o

Holographic dual:

Finite N
corrections

8

9

Wrapped D3 (GG)

with a 7-branesugra mult.
in the bulk

vector mult.
on D3

Large Ncurrent algebra
at intersection

vector multon 7-brane

More non-trivial examples : AD and MN[YI and Murayama, arXiv:2110.14897]
7-brane𝐻
Argyres Douglas
theories

𝐻
𝐻
𝐷

𝑆𝑝 𝑁 SQCD

𝐸
Minahan Nemechanski
theories𝐸𝐸

index of the rank 1 theory
1 + 𝑞 − 2𝑞

+𝑞

+𝑞

+ 2𝑞

+⋯

1 + 𝑞 − 2𝑞

+ 3𝑞 + 𝑞 + 𝑞 + 2𝑞

− 4𝑞 − 2𝑞+⋯

1 + 𝑞 + 6𝑞 + 𝑞 − 6𝑞 + 12𝑞 + ⋯1 + 29𝑞 − 2𝑞 − 28𝑞 + 60𝑞 + 298𝑞 − 60𝑞 + ⋯1 + 78𝑞 − 78𝑞 + 156𝑞 + 2432𝑞 − 158𝑞 + ⋯1 + 133𝑞 − 134𝑞 + 268𝑞 + 7373𝑞 − 272𝑞 + ⋯1 + 248𝑞 − 249𝑞 + 498𝑞 + 27001𝑞 − 500𝑞 + ⋯sugra contribution

improvement by single GG

expected error

Geometry, Representation Theory and Quantum Fields

171

Additional comments

Triple expansion vs Simple expansion
Our formula gives the index as a triple expansion,,,

Gaiotto and Lee found a similar expansion in [Gaiotto, Lee, arXiv:2109.02545] by the
analysis on the gauge theory side.

This is a simple expansion. How can these be the same?
One possibility - due to the different expansion variables.

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Two expansions of q-Pochhammer -1

q-expansion (Expansion around q = 0)

q-1-expansion (Expansion around q = ∞)If this happens for,to a simple expansion.

with

,

, the triple expansion reduces

q-expansion,,

x-expansion,, ,Gaiotto and Lee gives a simple toy model whose partition function has this property[Gaiotto, Lee, arXiv:2109.02545].
I have not yet confirmed this because I do not have analytic expression of

,

,

.

Geometry, Representation Theory and Quantum Fields

173

Summary• We proposed a new method to calculate the index of N=4 SYM on the AdS side.
• Although we have no proof at present, numerical analysis showed it reproduces thecorrect answer.
• The method has been applied to many examples of AdS/CFT correspondence, andworks well.
• Similar expansion was proposed by Gaiotto and Lee based on the analysis on thegauge theory side.

Discussions• Localization?
The three brane configurations () inarefixed points inunder a generic U(1) action. It may be possible to proveby using localization that the GG contribution is given by the formula.
• Quantum gravity?
Although we simply neglected the quantum gravity effect, we could obtain
the index, and it seems possible to extract the AdS black hole entropy from it.
[S. Choi, J. Kim, S. Kim and J. Nahmgoong, [arXiv:1810.12067 [hep-th]]]How should we interpret it?
Can we regard GG effects as a dual description of QG?

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Thank you

Geometry, Representation Theory and Quantum Fields

ODE/IM correspondence and WKB periods
Katsushi Ito
Abstract. In this talk I will discuss the ODE/IM correspondence for the
linear problem associated with the aﬃne Toda field equations for aﬃne Liealgebras. In particular, we study the (r+1)-th order ODE with polynomial potential. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic
Bethe ansatz (TBA) equation. We show the equivalence between the logarithm of the Y-functions and the WKB periods, which is confirmed by solving
the TBA equation numerically. We also discuss the wall-crossing of the TBA
equations from the minimal chamber to the maximal chamber.

(K. Ito) Tokyo Institute of Technology

175

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OCAMI Reports Vol. 4 (2022)

ODE/IM correspondence and WKB periods
Geometry, Representation Theory and Quantum Fields@Osaka City Univ.

Katsushi Ito (Tokyo Institute of Technology)
March 22, 2022
arXiv:2004.09856, 2104.13680(in collaboration with Takayasu Kondo, Kohei Kuroda, Hongfei Shu)arXiv:2111.11047 (w/ T. Kondo, H. Shu)

1

Introduction
ODE/IM correspondence and Aﬃne Toda Field Equations
generalized ODE/IM correspondence
WKB analysis for higher order ODE
Y-functions and WKB periods
Outlook2

Geometry, Representation Theory and Quantum Fields

177

Introduction

ODE/IM correspondence[Dorey-Tateo 1998, Bazhanov-Lukyanov-Zamolodchikov 1998]
ODE/IM correspondence is a relation between spectral analysis
approach of ordinary diﬀerential equation (ODE), and the“functional relations” approach to 2d quantum integrable model(IM). It provides an eﬃcient method to solve• integrable models• quantum mechanics (exact WKB method)• 4d N=2 supersymmetric ﬁeld theories• AdS/CFT correspondence

3

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Functional relation in Integrable models
two-dimensional Integrable ﬁeld theories: eﬀective ﬁeld theory in
the IR• pseudo particles
with mass ma with pseudo energy(
ma cosh θ massivea (θ) ∼(θ: rapidity)
ma e θ
massless• S-matrix Sab (θ) (Yang-Baxter eq.)
physical quantities( free energy, energy spectrum etc.) aredetermined by Qa (θ), Ta (θ)( transfer matrix), Ya (θ) = e−ϵaQ
• Bethe ansatz equations Q = const.
• Baxter’s T-Q relations T Q = Q• T-system and Y-system T T = T , Y Y = Y
4

ODE as the quantum SW curve
N=2 supersymmetric gauge theories in the NS limit of theOmega-background (1 , 2 ) = (, 0).
• Low energy-eﬀective theory is described by the quantum SW
curve [Mironov-Morozov].
• In the scaling limit, we obtain N = 2 superconformal ﬁeld
theory (the Argyres-Douglas theory). The quantum SW curvetake the form of the Schrödinger equation (and its higher
order generalization).
• The AD theories classiﬁed by (G, G′ ) [Cecotti-Neitzke-Vafa,
Xie, Wang-Xie]• eﬀective 2d CFT on R2 : 2d/4d-correspondence [Beem et. al.]
1 4d
c2d = − 12
c
2d/4d
• cIM
[I-Shu]2d = c2d5

Geometry, Representation Theory and Quantum Fields

179

ODE/IM correspondence and AﬃneToda Field Equations

massive and massless ODE/IM
Linear problem for aﬃne Toda ﬁeld theory ĝ∨⇓ conformal limit
holomorphic part of the linear prob.(ODE)⇐⇒⇐⇒

massive IM ĝ
⇓massless IM (CFT)• massless ODE/IM(1)• A1 [Dorey-Tateo, Bazhanov-Lukyanov-Zamolodchikov](1)• An [Dorey-Dunning-Tateo, Suzuki]• ABCD [Dorey-Dunning-Masorero-Suzuki-Tateo, Sun]• ĝ [Masoero et al. , IKKS]
(1)
(1)
(1)
(1)
• Br , Cr , F4 , G2 [IKKS]• massive ODE/IM(1)• A1 [Lukyanov-Zamolodchikov](2)• A2 [Dorey-Faldella-Negro-Tateo](1)• An [I-Locke, Adamopolou-Dunning, I-Shu](1)• B2 [I-Shu]
6

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Lie algebra: notation
Lie algebra g: generators {H a , Eα } (a = 1, . . . , r, α ∈ ∆)[H a , H b ] = 0[H a , Eα ] = αa Eα ,
α+β ∈∆
 Nα,β Eα+β ,∨[Eα , Eβ ] =α ·H
α + β = 0 (α∨ = 2α
: coroot)
α20otherwise• simple roots αa (a = 1, . . . , r)
• fundamental roots ωa , ωa · αb∨ = δab• Cartan matrix Cab = αa · αb∨P• highest root θ = ra=1 na αa ,P
Coxeter number h = 1 + ra=1 na• extended root α0 = δ − θ.
7aﬃne Toda ﬁeld equation and linear problem
aﬃne Toda ﬁeld equation: ϕ(z, z̄): r-component scalar ﬁelds
m2 X∂a ∂z̄ ϕ(z, z̄) + 2
ni exp(βαi · ϕ(z, z̄)) = 0
β
r

i=0

⇐⇒Lax form [∂z + Az , ∂z̄ + Az̄ ] = 0Xβ∂ϕ · H + meλ
2
r

Az =

q
β
α ·φ
2 i
n∨
i Eα i e

i=0XβAz̄ = − ∂z̄ ϕ · H − me−λ
2
r

q
β
α ·φ
2 i
n∨
i E−αi e

i=0

λ: spectral parameter⇐⇒ compatibility condition of the linear problem(∂z + Az )ψ = 0, (∂z̄ + Az̄ )ψ = 0
8

Geometry, Representation Theory and Quantum Fields

181

modiﬁed aﬃne Toda ﬁeld equation

conformal transformation z → f (z), ϕ → ϕ̃ = ϕ − β1 ρ∨ log ∂z f ∂z̄ f¯modiﬁed aﬃne Toda ﬁeld equation
( r
)
m2 X
∂a ∂z̄ ϕ + 2
ni exp(βαi · ϕ) + p(z)p̄(z̄)n0 α0 eβα0 ·φ = 0βi=1linear problem (∂z + Az )ψ = 0, (∂z̄ + Az̄ )ψ = 0β
A = ∂φ · H + meλ
2

( r
)
Xp
p
β
β
α
ϕ
α
ϕ
∨
∨
i
0
n i Eα i e 2,+ p(z) n0 Eα0 e 2i=1β¯Ā = − ∂φ· H − me−λ
2

( r
Xp

n∨
i E−αi eβ2 αi ϕ+ p̄(z̄)p)n∨0 E−α0 eβ
2 α0 ϕ.

i=1• p(z) = (∂f )h : a polyomial in z p(z) = z hM − shM• boundary condition of ϕϕ ∼ M ρ∨ log |z|2 + · · · for large |z|
ϕ ∼ g log |z|2 + · · · for small |z|

9Linear diﬀerential equation for aﬃne Lie algebra
1hM

conformal limit: z, z̄ → 0, x = (meλ ) M +1 z,E = shM (meλ ) M +1 ﬁxed
linear problem LΨ = 0• L is the 1st order diﬀerential operator
r
q
X
p
d
1
L=
− g·H +
n∨
E
+
n∨
a αa
0 p(x, E)Eα0
dx x
a=1

• p(x, E) = xhM − E• Ψ takes values in a representation V of g.
• singularity at x = 0, ∞• invariant under Symanzik-Sibuya rotation (x, E) → (ω k x, Ωk E)
If Ψ(x, E, g) is a solution,∨

Ψ[k] (x, E, g) = ω −kρ ·H Ψ(ω k x, Ωk E, g)
is also a solution.
10

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Examples
(1)

• A1

"

l1
d
dx − x00l1ddx + x!
+01p(x, E) 0

!#

ψ1
ψ2

!
= 0.


d2
l1 (l1 − 1)2M
− 2+
+ x − E ψ1 = 0.
dx
x2

(1)

• Ar [Dorey-Dunning-Tateo](r + 1)-th order ODE
(−1)r+1 D(µr+1 ) . . . D(µ1 ) − p(x) ψ(x) = 0,

D(µ) =d µ−
dx xµa = (ωa − ωa−1 ) · g
11

Asymptotic solution around the singularities
x = ∞, gauge transformation Ψ′ (x) = U (x)Ψ(x), L′ = U LU −1
r
q
X
p
1
d
′
∨
h
L =
+ p(x) Λ+ + · · · , Λ+ =na Eα a + n∨0 ζEα0
dx
a=1

Ψ

W KB

(x) =

n=dimVX
Ci exp −νi

i=1

Z x


1
∨
p (x )dx − log p(x)ρ · H ν i
h
1
h

′

′

Subdominant solution


Z x
1
1
′
′
∨
Ψ = ∼ C exp −µ
p h (x )dx − log p(x)ρ · H νhΛ+ ν i = νi ν i , µ: eigenvalue with largest real partdx = 0, L = dx
− g·H
x + ···

Xi = x−hi ·g ei + · · ·ei : weight vector of V with weight hi(e1 : highest weight vect., en : lowest weight vect.)12Geometry, Representation Theory and Quantum Fields

183

Q-function{Ψ[k] }: a basis of the solutions around x = ∞{Xi }: a basis of the solutions around x = 0

Ψ(x, E, g) =dimVX

Qi (E, g)Xi (x, E, g)

i=1

Q(E, g) = Q1 (E, g): Q-function (associated with the highest
weight state)
For V = V (a) : a-th fundamental representation with weight ωaQ(a) (E, g): Q-function associated with ωa13

ψ-system and BAE
Ar : V (a) = V (1) ∧ · · · ∧ V (1)N
V (a) ∧ V (a) ,→ V (a−1) ⊗ V (a+1) = rb=1 (V (b) )2δab −Cab ψ-system[Dorey-Dunning-Masoero-Suzuki-Tateo, Sun,. . . ](a)(a)Ψ[− 1 ] ∧ Ψ[− 1 ] =
2
2

r
O

(Ψ(b) )2δab −Cabb=1

quantum Wronskian relations
ω





r [
]2δab −Cab
∏
(a)
(a)
1 λ(a) −λ(a)−1λ1 −λ2
(a)
(a)
(a)
(a)
(b)
2
2
Q1[−1/2] Q2[1/2] −ω 2 1
Q1[1/2] Q2[−1/2] =
Q1
.
b=1

Bethe Ansatz equation
r
Y

(b)

Ω

Cab γb /2

b=1

Q[Cab /2]

= −1,

(b)

Q[−Cab /2]

i = 0, 1, . . . ,

(a)

Ei

2
γa = hM
ωa (g − ρ∨ )
14

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dual representation and the Q-function

Ψ(x, E, g) =nX

Qi (E, g)Xi (x, E, g)

i=1• Wronskian type formula
Qi (E, g) =det(X1 , . . . , Ψ, . . . , Xn )det(X1 , . . . , Xn )• inner product formula [I-Kondo-Kuroda-Shu]
V ∗ : dual representation, Ldual Ψ̄ = 0, Ldual = ∂ + Ā⟨Ψ̄, Ψ⟩ is x-independentX̄i : the basis of the solutions of the dual problem aroundx = 0 such that ⟨X̄i , Xj ⟩ = δij .
Qi (E, g) = ⟨X̄i , Ψ⟩
15

NLIE and ODE
Q-function from ODE
ĝ and the representation V (a)• Q(a) (E, g) = ⟨X̄1 , Ψ(a) ⟩(a)• Solve the dual problem Xi around the origin (Cheng’s
algorithm)(a)• ΨW KB : most divergent component is the lowest weight state• Use truncated solution and evaluate the Q-function at
x = xf ixed ≫ 1•
Qi (E, l) ∼ χ̄i,n (xﬁxed , E, l) ψn (xﬁxed )for xﬁxed ≫ 1.

(a)the lowest weight component of (X1 )n (xf ixed , E) = 0 =⇒(a)solution to Q(a) (E, g) = 0, Ei
16Geometry, Representation Theory and Quantum Fields

185

NLIE [Zinn-Justin]counting functiona(a)(E) :=

r
Y

(b)

Ω

−Cab γb /2

Q[−Cab /2] (E)(b),

a = 1, . . . , r.

Q[Cab /2] (E)

b=1
(a)

BAE eq. a(a) (Ek ) = −1.
ln a(a) (θ) = iπ α̂a − ib0 Ma eθ +

r Z
X
b=1

C1



dθ′ ϕab (θ − θ′ ) ln 1 + a(b) (θ′ )1−dθ ϕab (θ − θ ) ln 1 + (b) ′ ,a (θ )
b=1 C2Z ∞sinh(µπk)
dk ikθ
−1
e
δab −
Cab
(k)),
ϕab (θ) =sinh((hµ − 1)πk/h) cosh(πk/h)
−∞ 2π


a = b,
2,
Cab (k) :=Cab, a ̸= b,cosh(πk/h)
17
r Z
X



′

′

Ei
19.98072
38.50821
66.15524
97.97408(4)Ei
29.04567
80.49209152.5322
242.8648(5)Ei
21.54848
52.00437
93.90413
145.7213(4)Ei
29.04519
80.49126
152.5313
242.8597(5)Ei
21.54807
52.00351
93.90137
145.7216

Numerical Test
(1)

Bethe roots for E6 P = x2 − E,
g = (5/12, 1/3, 0, −1/3, −5/12, 1/10).
ODE
i
0
1
2
3

(1)
Ei

(2)
Ei

(3)
Ei

26.16492
76.14709
146.8766
236.0037

19.04286
37.06396
64.52501
95.98720

16.95324
28.48668
44.61201
63.83908

i
0
1
2
3

Ei
26.16452
76.14715
146.8773
236.0021(1)

Ei
19.04232
37.06297
64.52390
95.98496(2)Ei
16.95299
28.48688
44.61186
63.83727

(6)

IM
(3)

Ei
19.98020
38.50782
66.15433
97.97372

(6)

18

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OCAMI Reports Vol. 4 (2022)

Eﬀective central charge and 2d/4d correspondence

eﬀective central charge
3 ∑ib0 Maπ2 ar

cef f = 2 ×

[∫
C1

(
) ∫
dθ eθ log 1 + a(a) (θ) −(dθ eθ log 1 +C2

1
a(a) (θ)

)]

12
=r−
(−g + ρ∨ )2(M + 1)h2

g = 0, p = x2 − E, (G, A1 )-type AD theory cef f = r − dimg
h+22n
• (A2n , A1 ) cef f = 2n+3

• (E6 , A1 ) cef f = 67
• (E8 , A1 ) cef f = 12

19generalized ODE/IM
correspondenceGeometry, Representation Theory and Quantum Fields

187

generalized ODE/IM correspondence [I-Marinõ-Shu ]ODE (g = 0 p(z): polynomial potential)!
rX−∂z2 + z r+1 +ba z r−a ψ(z, ba ) = 0

z∈C

a=1

• invariant under the rotation (Symanzik rotation)
2πi(z, ba ) → (ωz, ω a+1 ba ),ω = e r+3 .

• asymptotically decaying solution along the positive real axis


r+3
1
2
y(z, ba ) ∼ √ z nr exp −
z 2 ,
r+3
2i
Stokesnsectoro2kππSk = z ∈ C : arg(z) − r+3 < r+3 .
decaying solutions in Sk :yk (z, ba ) = ω 2 y(ω −k z, ω −(a+1)k ba ).
k20

generalized ODE/IM
Introduce scaled variables2

q = ζ r+3 z,

2(a+1)

ua = −ζ r+3 ba ,−ζ 2 ∂q2 + q r+1 −rX

a = 1, 2, · · · , r.
!

ua q

r−a

ψ̂(q, ua , ζ) = 0.

a=1

• regard ζ as a new spectral parameter (ζ = ℏ in QM)• Symanzik rotation ↔ rotation of ζ2(a+1)ŷ(q, ua , ζ) = y(z, ba ) = y(ζ − r+3 q, −ζ − r+3 ua ).

2(a+1)
2−k−(a+1)kiπk − r+3iπk − r+3
(ω z, ωba ) = (e ζ)q, −(e ζ)ua .
2• subdominant solution in the sector Skkŷk (q, ua , ζ) = ω 2 ŷ(q, ua , eiπk ζ).

21

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OCAMI Reports Vol. 4 (2022)

Y-systemWronskian2Ŵk1 ,k2 (ζ, ua ) = ζ r+3 ŷk1 ∂q ŷk2 − ŷk2 ∂q ŷk1 (q, ua , ζ) = Wk1 ,k2 (ba ).
Y functions
Y2j (ζ, ua ) =πi

Y2j+1 (e− 2 ζ, ua ) =Y-system:πi2

Ys (ζe , ua )Ys (ζe− πi2Ŵ−j,j Ŵ−j−1,j+1(ζ, ua ),Ŵ−j−1,−j Ŵj,j+1
Ŵ−j−1,j Ŵ−j−2,j+1(ζ, ua ).
Ŵ−j−2,−j−1 Ŵj,j+1

, ua ) = 1 + Ys+1 (ζ, ua )


1 + Ys−1 (ζ, ua ) ,• ua ﬁxed, single spectral parameter ζ• massless limit of Y-system in AdS3 minimal surface withlight-like boundary [Alday-Maldacena-Sever-Vieira, Hatsuda-KI-Sakai-Satoh]
22

Exact WKB method
the Schrödinger equation (m = 1/2)−ℏ2 ψ ′′ (q) + (V (q) − E)ψ(q) = 0.
the WKB solution:

 Z q

i
′
′
ψ(q) = exp
Q(q )dq .
ℏRicatti equation:
Q2 (q) − iℏ
Q(q) =∞XdQ(q)= p2 (q),dqp(q) = (E − V (q))1/2 ,Qk (q)ℏk = Qeven + Qodd = P (q) +k=0

P (q) = Qeven =X

pn (q)ℏ2n ,iℏ dlog P (q).
2 dq

Qodd = total derivativen≥0

p0 (q) = p(q) and pn (q) are determined recursively.

23

Geometry, Representation Theory and Quantum Fields

189

WKB periods
potential: polynomial in q
V (q) = q r+1 + u1 q r + · · · + ur q
the WKB curve: hyperelliptic Riemann surface of genus g = r+1 2

ΣWKB : y 2 = E − V (q).
WKB periods (quantum periods)IΠγ (ℏ) = P (q)dq,

γ ∈ H1 (ΣWKB ).

γΠγ (ℏ) =

X

I

2n
Π(n)
γ ℏ ,

Π(n)
γ =

pn (q)dq.
γ

n≥0
(n)

asymptotic expansion: Πγ ∼ (2n)!
Borel resummation is necessary.
24

Y-function and quantum periods

We consider the case that all the branch points are real anddistinct(minimal chamber).

E<latexit sha1_base64="iyKsJ7k+bPLi33qq8l5aoG6aFEk=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSERRPBRE8tmA/oA1ls520azebsLsRSugv8OJBEa/+JG/+G7dtDtr6wsLDOzPszBskgmvjut9OYW19Y3OruF3a2d3bPygfHrV0nCqGTRaLWHUCqlFwiU3DjcBOopBGgcB2ML6d1dtPqDSP5YOZJOhHdCh5yBk11mrc9csVt+rORVbBy6ECuer98ldvELM0QmmYoFp3PTcxfkaV4UzgtNRLNSaUjekQuxYljVD72XzRKTmzzoCEsbJPGjJ3f09kNNJ6EgW2M6JmpJdrM/O/Wjc14bWfcZmkBiVbfBSmgpiYzK4mA66QGTGxQJnidlfCRlRRZmw2JRuCt3zyKrQuqp7lxmWldpPHUYQTOIVz8OAKanAPdWgCA4RneIU359F5cd6dj0VrwclnjuGPnM8floeMwQ==</latexit><latexit···πY2i−1 (θ ± i ) = exp
2




i
s± (Πγ2i−1 ) ,ℏ
Y2i (θ) = expis(Πγ2i )ℏθ = − log ℏs(∗): Borel resummation, s± (∗): lateral Borel resummation
25

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OCAMI Reports Vol. 4 (2022)

TBA equation• Asymptotic behavior of the Y-function (ℏ → ∞) can be
represented by the classical period integral on the WKB curve
I
1
mr−2k
log Y2k+1 (ζ, ua ) ∼ −p(q)dq = −,ℏ γr−2kℏIimr+1−2k ,log Y2k (ζ, ua ) ∼ −p(q)dq = −,ℏ γr+1−2kℏ• Discontinuity formula for the WKB periods
Y-system⇐⇒ TBA equations
Z
Z
La−1 (θ′ ) dθ′
La+1 (θ′ ) dθ′θa (θ) = ma e −−,′′R cosh(θ − θ ) 2π
R cosh(θ − θ ) 2π
ℏ = e−θ ,Ya (ζ) = e−ϵa (θ) ,La (θ) = log 1 + Ya−1 (θ)

26

TBA in the UV limit• constant solution a (θ) → ⋆a (θ → −∞)




πa
sin r+3
sin π(a+2)
r+3


Ya⋆ =
.
π
sin2 r+3• eﬀective central charge [Kirillov,Kuniba-Nakanishi-Suzuki]
6 X
ceﬀ = 2
ma
π
r

a=1

Z
eθ La (θ)dθ =Rr(r + 1).
r+3

• generalized paramermion SU (r + 1)2 /U (1)r [HISS]• TBA system for Homogeneous sine-Gordon model
SU (r + 1)2 /U (1)r [Fernandez-Pousa-Gallas-Hollowood-Miramontes]
27

Geometry, Representation Theory and Quantum Fields

191

Wall-crossing and TBA
For a general polynomial potential, turning points are complex.
(V (q) = q 3 = E)complex periods (complex mass )ma = |ma |eiϕa ,a = 1, · · · , r.
e a (θ) = La (θ − iφa ).
Le
a (θ) = a (θ − iφa ),TBA system:e a−1 − Ka,a+1 ? Le a+1ea = |ma |eθ − Ka,a−1 ? L
1
1
2π cosh (θ + i(φs − φr ))Z(K ? f )(θ) =K(θ − θ′ )f (θ′ )dθ′Kr,s =

R

This TBA formula is valid for φa satisfying |φa − φa±1 | < π2

28

When

π
π
−→ φ2 − φ1 > ,2
2
the pole φ1 − φ2 − i π2 crosses the real axis. The kernel picks thepole.
φ2 − φ1 <TBA equations are modiﬁed toiπe 2 − L2 θ − iφ1 −e
1 (θ) = |m1 |e − K1,2 ? L+ iδ ,2
iπθ
e 1 − L1 θ̂ − iφ2 +e
2 (θ) = |m2 |e − K2,1 ? L− iδ .
2θWall crossing phnenomena of the TBA equations[Gaiotto-Moore-Neitzke, AMSV, Toledo unpublished]
29

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OCAMI Reports Vol. 4 (2022)

3-term TBA
We can transform this 2-term TBA into 3-term TBA equations.
(n)Y-functions ˜na (θ − iφa ) = − log Ya (θ) (a = 1, 2, 12)cycles γ1 , γ2 and γ12 = γ1 + γ2nY1 (θ) =Y1 (θ),
1 + Y2 θ − iπ2nY2 (θ) =Y2 (θ),
1 + Y1 θ + iπ
2nY12 (θ) =Y1 (θ)Y2 θ − iπ2.
1 + Y1 (θ) + Y2 θ − iπ2e2 − K + ? L
e 12 ,e1 (θ) = |m1 |eθ − K1,2 ? L
1,12e 1 − K2,12 ? L
e 12 ,e2 (θ) = |m2 |eθ − K2,1 ? L−
e 1 − K12,2 ? Le2 .
e12 (θ) = |m12 |eθ − K12,1?L

m12 = m1 − im2 = |m12 |eiϕ12 , f ± (θ) := f (θ ± i π2 )For the monomial potential, there appears Z3 -symmetry. The TBA
eqs. reduce to the single one. (Yang-Lee edge singularity M2,5 )
√

Z

′

′

′

Φ(θ − θ )L(θ )dθ ,θ(θ) = me +RΦ(θ) =

3 sinh 2θ
π sinh 3θ
30

WKB analysis for higher order ODE

Geometry, Representation Theory and Quantum Fields

193

WKB analysis
for higher
order
ODE
WKB
analysisfor higher
order
ODE
WKB analysis
forhigher
order
ODE

箔箔

[Berk-Nervins-Roberts, Aoki-Kawai-Takei, Honda-Kawai-Takei, ...]

[Berk-Nervins-Roberts,
Aoki-Kawai-Takei,
Honda-Kawai-Takei,...][Berk-Nervins-Roberts,
Honda-Kawai-Takei,...](simple Aoki-Kawai-Takei,
turning points)(simple turning points)(simple turning points)! r+1"! r+1 "
d r+1 rdr r
d+du1 (x) d r + · · · + ur+1 (x) ψ(x) = 0
dr+1+ u1 (x)+ ·u· r+1
· + u(x)ψ(x)==00+u(x)+ · dx·· +r+1 (x)ψ(x)dx1
dxr+1 dxr
dxr
dxr+1
higher order generalization of the Schrödinger equation

higher order generalization of the Schrödinger equation(non-simple turning
higher order generalizationof the points)
Schrödinger equation(non-simple turning points)(non-simple turning points)N+1
X

r r+1 r+1[(−1)rrr+1∂xr+1r+1+" p(x)]ψ(x)= 0,p(x) = =uN
[(−1)
∂x +
p(x)]ψ(x)0,+1−i xi
r+1[(−1) "∂x

+ p(x)]ψ(x) = 0, i=0

N
+1
#

N
+1
#

p(x) =uNxi .
i +1−ip(x) =
uN +1−ii=0 x .
i=031

3rd order ODE 3 3 ∂x + p(x) ψ(x) = 0
 Z x

1
′
′
ψ(x) = exp
P (x ; )dx ,


28

28
P (x; ) =∞Xn pn (x).

n=0Riccati equation:
2 ∂ 2 P + 3P ∂P + P 3 + p(x) = 01

p0 = (−p) 3 ,p1 = −∂ log p0 ,p2 = −(∂p0 )22 ∂ 2 p0
+
,...
p30
3 p20

• podd : total derivative• p6n+4 = p4 , p10 , . . . : total derivativeΠγ () =

∞
X
n=0

Z


n

Π(n)
γ ,

Π(n)
γ =

pn (x)dx
γ

32

194

OCAMI Reports Vol. 4 (2022)

WKB curve: superelliptic curve
y 3 = p(x) = u0 xN +1 + · · · + uN +1
a basis of meromorphic diﬀerential [Farkas-Kra]dx xdxxN −1 dx
,
,...,
,
y
y
y

dx xdxxN −1 dx
,
,
.
.
.
,
y2 y2
y2

ydx, y 2 dx (SW diﬀerentials)∂ ua y =1 xN +1−a2 xN +1−a
2
,
,
∂
y
=
u
a
3 y2
3
y

classical SW periodsZy a dx(Π̂a )γ =(a = 1, 2)γquantum corrections
Π(n)
γ =

2
X

Oa(n) (Π̂a )γ ,Oa(n) = −

a=1

N
−1
X
i=1

3
(n)
B3−a,N +2−i ∂uN +2−ia33

(r + 1)-th order ODE[(−1)r r+1 ∂xr+1 + p(x)]ψ(x) = 0
Riccati equation(−1)r (∂ + S)r S + p(x) = 0

1

S0 = (−p) h ,S1=−

h − 1 ∂S0
,
2 S0

S2 =

h2 − 112

∂ 2 S03 (∂S0 )2
−
2
S0
2 S03


,...

h = r + 1: the Coxeter number of Ar• Sodd : total derivative• S2hn+h+1 : total derivative
34

Geometry, Representation Theory and Quantum Fields

195

WKB curve: (r + 1)-fold covering of the complex plane with
(N + 1) branch points
y r+1 = p(x) = u0 xN +1 + · · · + uN +1
a basis of meromorphic diﬀerentials on the WKB curve
xi−1
dx,
ya

i = 1, . . . , N − 1, a = 1, . . . , r.

SW diﬀerentials y a dx (a = 1, . . . , r)quantum corrections
PF(0),lnΠ(n)({ui }),γ ({ui }) = On ({ui }, {∂ui })Π̂γ

classical SW periods

I

ln

n
Π̂(0),l({ui }) =γ(p(x, {ui })) r+1 dx.
γ
35classical SW periods
γa,k :1-cycles on the WKB curve y 3 = u0 (x − x0 ) . . . (x − xN )classical SW periods
γa,k :1-cycles on the WKB curve y 3 = u0 (x − x0 ) . . . (x − xN )xN xN −1

···

xk

xk+1

γa,k

xk−1 xk−2
xN xN −1

···

xk+1

· · · γ x1
xk

龜

x0

a,k

xk−1 xk−2· · · x1

x0

Figure 2.1: The choice of the branch cuts and the definition of the cycle γa,k , where thesolid and dotted lines are on the a-th and a + 1-th sheet respectively.

Figure 2.1: The choice of the branch cuts and the definition of the cycle γa,k , where theHere ∆ais+
the1-thdiscriminantof thecurvesolid and dotted lines are on the a-th andrespectively.
! sheet"#2 2 2

2πi 3 2

2πi
3 3

3

2 4

a

2 3

= u1 u2 u3 − 4u30 ul2 u3 − 4u13u3 (l+1)
0 u3 −
1 u2 u 4
3 4u(Π̂a )∆γl,k=4e− e + 18u0u21u2u3 − 27u
p(x)
dx
3
2 2
2

+ 16u0 u2 u4 + 18u1 u2 u3 u4 − 80u0 u1u2 u3 uγ4 − 6u0 u1 u3 u4 + 144u0 u2 u23 u4(2.36)$N I(2l+1)a3 2 2πi 3 3 2πi = −2ie
πa x %
2 2 2
3
2 4 u0 sin
2 3 a(x − xi ) dx,l(l+1)∆ = u1 u2 u3 − 4u0 u2 u3 −3 4u1 u3 + 18u
u34 x
0 u1 u2 u3 − 27u0 u3 − 4u1 u2
3
3(Π̂a )γl,k= 3e− e2.2 Classicalperiod 2 2 p(x) 2 dx2 i=04
2Here ∆ is the discriminant of the curvel,k− 27u41 u24 + 144u0 u21 u2 u24 − 128u20 u22 u24 − 192u20 u1 u3 u24 + 256u30 u34 .

k−1
a
πi
3 in a similar way.
Higher order corrections 3can be calculatedk+ 16u0 u2 u4 + 18u1 u2 u3 u4 − 80u0 u1u2 u3 u4 − 6u0 uγ1 u3 u4 + 144u0 u2 u3 u4a3(2.36)We have seen that the WKB periods
l,k are expressed as the linear combinations of theZ x

2 2 integrals
2
2 differentials3definedof the 2meromorphic(2.19), that are expressed− 27u41 u24 + 144u0 u21 u2 u24 − 128uperiod+ 256uu34 . in(2.22).
0u2 ufirst-order4 − 192u0 u1 uof3 u
4 classicalbythederivatives
theSW0 periods
The periods are also

πi

πa
3

(2l+1)aHigher order corrections can be calculated
in a similar way.
3

= −2ieNY

u0 sin2.2 Classical period(Π̂a )γ =

!

a
3

γ

u0

N
"

k=0

xk

a
3

(x − xk ) dx,(x − xi ) dx,

i=0
a = 1, 2.

(2.37)To specify the cycle γ, we have to choose the branch cut. We label the branch points
such that Re(x0 ) ≥ Re(x1 ) ≥ · · · ≥ Re(xN ) when all the points align on the real axis.
Then we choose the branch cut to be as in figure 2.1. We introduce the one-cycle γl,k(l = 1, 2, 3, k = 1, . . . , N ) on Σ, where γl,k encircles the branch points xk−1 anticlockwise
and xk clockwise, respectively, on l-th and (l + 1)-th sheets. Here we defined the sheet lon which y = yl :2πi1
yl := e 3 l p(x) 3 ,l = 1, 2, 3.
(2.38)We have seen that the WKB periods are expressed as the linear combinations of theperiod integrals of the meromorphic differentials defined in (2.19), that are expressedby the first-order derivatives of the classical SW periods (2.22). The periods are also
specified by one-cycles on the WKB curve, which are determined by the branch points
xk (k = 0, 1, . . . , N ), i.e. p(xk ) = 0. In this paper, we consider the WKB curve whose8
branch points are distinct. The classical SW periods (2.22) are thengiven by(Π̂a )γ =

!

a

γ

33

k−1
a
specified by one-cycles
on the WKB curve, which
are determined by the branch pointsa
xk (k = 0, 1, . .3
. , N ), i.e. p(xk ) = 0. In this paper, we consider the WKB curve whose3
branch points are distinct. The classical SW periods (2.22) are then given by

u03

N
"

k=0

a

(x − xk ) 3 dx,

a = 1, 2.

(2.37)To specify the cycle γ, we have to choose the branch cut. We label the branch points
such that Re(x0 ) ≥ Re(x1 ) ≥ · · · ≥ Re(xN ) when all the points align on the real axis.
Then we choose the branch cut to be as in figure 2.1. We introduce the one-cycle γl,k

36

196

OCAMI Reports Vol. 4 (2022)

Y-functions and WKB periods

Y-functions and WKB periods

ODE: (−1)r r+1 ∂xr+1 + u0 xN +1 + · · · + uN +1 ψ(x) = 0.
subdominant solution along the real positive axis (u0 > 0)


1
r − r
2h
h
r(N +1)
N +h+1 2 u01
u0 hφ(x, u0 , uN ; ) ∼ r √x h ,
x− 2h exp −N +h+1i2 hSymanzik rotation:
2πi

φk (x, u0 , uN +1 ; ) = φ(x, u0 , uN +1 ; ek h ).
{φk , . . . , φk+r } form a basis of solutions
T-function Ta,l
[−l−1]
W ϕ−r+1+ a , ϕ−r+2+ a , . . . , ϕ1− a , ϕl+2− a , ϕl+3− a , . . . , ϕl+1+ a
,
2
2
2
2
2
2

W ϕ

−r+1+ a−12

,ϕ−r+2+ a−12,...,ϕ

− a−1
2

,ϕ

l+1− a−12,ϕl+2− a−12,...,ϕl+1+ a−12

a: even,[−l],a: odd.

37

Geometry, Representation Theory and Quantum Fields

197

T-system (followed from the Plücker relations for determinants)
[+1]

[−1]

Ta,l Ta,l= Ta,l+1 Ta,l−1 + Ta+1,l Ta−1,l ,T0,l = Tr+1,l = 1.
Y-functions
Ta−1,l Ta+1,l,Ta,l+1 Ta,l−1

Ya,l =l ∈ Z.

a = 1, . . . , r,

(Ar , AN )-type Y-system
[+1]

[−1]

Ya,l Ya,l=(1 + Ya+1,l )(1 + Ya−1,l ).
−1−1(1 + Ya,l+1)(1 + Ya,l−1)

Y0,l = Yr+1,l = 0.

38

Asymptotics of Y-functions: (A2 , AN )


− δϵk



Rx

• φk ∼ exp
xk yak dx : xk initial point at inﬁnity• W [φk1 , φk2 , φk3 ]: trivalent graph[ϕ−1 ,ϕ1 ,ϕ2 ]W [ϕ−2 ,ϕ−1 ,ϕ0 ]• Y1,1 = W
abelianization tree
W [ϕ figure.
,ϕ−1,ϕ2 ]W[ϕ−1 ,ϕpoint,] :relabel
0 ,ϕ1wepoint in the −2
After thered dotted
the line as 3̄, which lives on thesame sheet three with opposite arrows. The solution φ is subdominant before crossing[Neitzke,Hollands-Neitzke]
−1

Π

∼

Π

,

,

log Y

∼

Π

log Y

∼

Π

,

,

 → 0.

(3,2)

(3,1)

(1

2

1

)
,2

,1
)

1
2

(2,3)

(1,3)

x2

x1

x0

2
3

(2,1)

(1,2)

3
2
)
,3

3

1

(1,3)

,3
)

(2

3

3

(1

log Y

∼

(2

log Y

the red dot, while after the crossing, on line 3̄, the solution φ0 becomes subdominant.
[+1]labeled by
The line−1
with label(∗, 3).(0)Similarly, for the(0) 3 cannot cross the Stokes curve−1
1,1 Wronskian W [φγ−21,1
γ3,21 and 1.
1,2, φ−1 , φ2 ], we added a point on
the line with sheet labels
are independent of the location of the junction,
we can move the[−1] Since Wronskians−1 (0)−1 (0)
to the appropriate 2,2points and find that the integration path
γ1,1 +γ2,1
γ3,2 +γ1,22,1 junctions of the Wronskians
can be identified with the cycle γ1,1 . In the same way, one can identify the cycles for other

3
(2,3)

39

(a)

(b)

Figure 3.1: The stokes curve of (A2 , A2 )-type ODE with the potential p(x) = x3 − x andthe abelianization tree for Y1,1 (a) and the cycle γ1,1 (b).
Y-functions. For Y1,2 , it is convenient to draw the diagram for Y1,2 rather than Y1,2 :
[+1]

[+1]

πi

Y1,2 ({ui }; #) = Y1,2 ({ui }; e− 3 #) =W [φ−1 , φ2 , φ3 ]W [φ−2 , φ−1 , φ0 ],W [φ−1 , φ0 , φ2 ]W [φ−2 , φ−1 , φ3 ](3.25)

Then the corresponding Stokes curve is rather than that of adjoint ODE. The Stokes
graph becomes the same as in figure 3.1 but with the asymptotic directions replaced byπi(2k−1)2πi(k+1)$k = e N +4 R+ and $k = e N +4 R+ .
[−1]

198

OCAMI Reports Vol. 4 (2022)

TBA equations: (A2 , AN )Y-system + Asymptotics =⇒ TBA equations (Y1,k = Y2,k )log Y1,k (θ − iφk ) = |m1,k |eθ +K?L1,k −Kk,k−1 ?L1,k−1 −Kk,k+1 ?L1,k+1 ,

πi

(0)

• mass parameters ma,k = e 3 (k−a) Πγ̂a,k• kernel√
1 4 3 cosh θK(θ) :=,
2π 1 + 2 cosh 2θ

Kk1 ,k2 (θ) = K(θ − i(φk1 − φk2 )),• L̄1,k = log 1 + Y1,k (θ − iφk )−1• TBA is valid in the minimal chamber |φk − φk+1 | < π3+1)3
• eﬀective central charge cef f = 2NN(N+4+1) , coset CFT SUU(N(1)N40

Numerical Test (A2 , A2 )
(n−1)

m1,k

πi

(n)

= e 3 (1−k)(n−1) Πγ̂1,k ,log Y1,k (θ) = m1,k eθ +n = 0, 1, . . . ,∞Xm1,k e−nθ .
(n)

n=1
(n)
1,1

n

(n−1)

m1,1

Πγ̂

2
6
8
12
14

0.2172157436i−1.519567945i−20.48661777i
20065.20970i
1160395.676i

0.2172157436i−1.519567945i−20.48661776i
20065.20605i
1160393.422i

Table 1: The quantum corrections for p(x) = −x3 + 7x + 6.

For (A2 , AN )

1log Ya,k =
Πγ̂
 a,k
(a = 1, 2,

[a−k]

k = 1, . . . , N ),γ̂a,k := γ2−k,k + · · · + γa+1−k,k ,41

Geometry, Representation Theory and Quantum Fields

199

Wall-crossing of TBA (A2 , A2 )Wall-crossing of TBA (A2 , A2 )• potential p(x, t) = (x − x0 (t))(x − x1 (t))(x − x2 (t))(0 ≤ t ≤ 1) which interpolates p(x, 0) = x3 − 7x − 6 and• potential p(x, t) = (x − x0 (t))(x − x1 (t))(x − x2 (t))
x3 −(08.≤ t ≤ 1) which interpolates p(x, 0) = x3 − 7x − 6 andtx3 − 8.

x0 (t) = 3 − t,

x0 (t) = 3 − t,→√

x1 (t) = −1 + √3it,

x1 (t) = −1 +

t

3it,

x2 (t) = −2 + t −

√

√
3it

x2 (t) = −2 + t −3it• two wall-crossing points• There occur two wall-crossings.
π
φ 2 − φ1 = ,3π

t = 0.162117...,t = 0.162117...,

φ 2 − φ1 =,2π3
φ 2 − φ1 =,3

t = 0.397459...,t = 0.397459...,Im

φ 2 − φ1 =

!

(0)

Πγ3,2

Π

2π
,
3

!

(0)= 0,(0)Πγ1,1 γ3,2

Im
!

Im

"

= 0,
"
(0) (0)Πγ2,2Πγ1,1
=!
0.
(0)
Πγ1,1Π(0)
γ
2,2

Im

(0)
Πγ1,1

= 0.
40

42

First Wall-crossing of TBA (A2 , A2 )New Y-functions(1)

Y1,1 (θ) = Y1,1 (θ) 1 +(1)
Y12 (θ) =

1+ Y


1
,
πi
Y1,2 (θ − 3 )

1πi
1,2 (θ− 3 )+ Y1,11(θ)

1
Y1,1 (θ)Y1,2 (θ− πi
3 )

(1)

Y1,2 (θ) = Y1,2 (θ) 1 +


1
,
πi
Y1,1 (θ + 3 ),TBA equations:
(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

−
? L12 ,log Y1,1 (θ − iφ1 ) =|m1,1 |eθ + K ? L1,1 − K1,2 ? L1,2 + K1,12(1)−
log Y1,2 (θ − iφ2 ) =|m1,2 |eθ + K ? L1,2 − K2,1 ? L1,1 − K2,12? L12 ,++
? L1,1 − K12,2? L1,2 .
log Y12 (θ − iφ12 ) =|m12 |eθ + K ? L12 + K12,1

New Y-functions are identiﬁed with the WKB periods
(1)

(1)

πi

log Y1,1 (θ) = eθ Πγ̂1,1 , log Y1,2 (θ) = e 3 eθ Πγ̂1,2 (θ +

πi
),
3

(1)

log Y12 (θ) = eθ Πγ̂1,1 +γ̂1,2 (θ),
43

200

OCAMI Reports Vol. 4 (2022)

Second Wall-crossing of TBA (A2 , A2 )New Y-functions

(
(2)
(1)
Y1,1 (θ) = Y1,1 (θ) 1 +
1+
(2)
Y12
f (θ) =)

1
(1)
Y1,2 (θ − 2πi
)
3

1
(1)

Y1,2 (θ− 2πi
)
3

(
(2)
(1)
Y1,2 (θ) = Y1,2 (θ) 1 +

,

1
(1)
Y1,1 (θ + 2πi
)
3

)
,

1

+

(1)

Y1,1 (θ)

1

,

(2)

(1)

Y12 (θ) = Y12 (θ).

(1)
(1)
Y1,1 (θ)Y1,2 (θ− 2πi)
3TBA equations
(2)

(2)

θ

(2)

−

(2)

(2)

−

(2)

log Y1,1 (θ − iϕ1 ) =|m1,1 |e + K ⋆ L1,1 − K1,2 ⋆ L1,2 + K1,12 ⋆ L12+ (K1,12
f +K
(2)

(2)
−
) ⋆ Lf ,
f
1,12
12
(2)

θ

log Y1,2 (θ − iϕ2 ) =|m1,2 |e − K2,1 ⋆ L1,1 + K ⋆ L1,2 − K2,12 ⋆ L12
− (K
(2)

(2)
−
+ K2,12,f ) ⋆ L12
f
f
2,12
θ

(2)

+

+

(2)

(2)

log Y12 (θ − iϕ12 ) =|m12 |e + K12,1 ⋆ L1,1 − K12,2 ⋆ L1,2 + K ⋆ L12+ (K12,12
f +K
(2)
12

(2)
+
) ⋆ Lf ,
f
12,12
12
(2)
(2)
+
+
) ⋆ L1,1 − (K f + K12,2
f ) ⋆ L1,2
12,1
12,2

θ

log Y f (θ − iϕ12
+ Kff ) =|m12
f |e + (K12,1
f

(2)
(2)
−
) ⋆ L12 + 3K ⋆ L f ,
12,12
12

+ (K12,12
+ Kf
f

44

monomial potentail
At t = 1, p(x, 1) = x3 − 8 monomial potential

√
|mf
|
=
3|m1,1 |12ππ
φ12 − φ1 = , φf
− φ1 = .
12
3
6|m1,1 | = |m1,2 | = |m12 |,φ2 − φ1 = π,Z3 -symmetry
(2)

(2)

(2)

log Y1,1 (θ − iφ1 ) = log Y1,2 (θ − iφ1 − πi) = log Y12 (θ − iφ1 −

πi
).
3

reduced TBA system (D4 /Z3 )log Y1,1 (θ − iϕ1 ) =|m1,1 |eθ + 3K(θ − θ′ ) ⋆ L1,1
(2)

(2)

+ K(θ − θ′ +(2)

log Y12f (θ − iϕ1 −

πi
πi
(2)
(2)
′
) ⋆ L12) ⋆ L12
f + K(θ − θ −
f ,
6
6

√
πi
(2)
) = 3|m1,1 |eθ + 3K(θ − θ′ ) ⋆ L12
f
6
πi
πi
(2)
(2)
+ 3K(θ − θ′ + ) ⋆ L1,1 + 3K(θ − θ′ − ) ⋆ L1,1 .
6
645

Geometry, Representation Theory and Quantum Fields

201

Wall-crossing of TBA (arXiv:2111.11047)• (A2 , A2 )minimal chamber 2-TBA =⇒ maixmal chamber 4-term TBA
For the monomial potential, D4 -type TBA appears (cef f = 1:
sine-Gordon model at reﬂectionless point) [Al. Zamolodchikov,
Klassen-Meltzer]• (A2 , A3 )minimal chamber 3-TBA =⇒ maixmal chamber 12-term TBA
For the monomial potential E6 -type TBA appears (cef f = 67 : tri-critical
3-state Potts model) [Al. Zamolodchikov, Klassen-Meltzer]• Duality of AD theories (Ar , A1 ) ∼ (A1 , Ar ), (A2 , A2 ) ∼ (D4 , A1 ),(A2 , A3 ) ∼ (E6 , A1 ) [Cecotti-Neitzke-Vafa]

46

Outlook202

OCAMI Reports Vol. 4 (2022)

Outlookexact WKB ↙
quantum period
4d N=2 theory

aﬃne Toda ﬁeld equationODE↘ ODE/IM
Y-function⇐⇒
2d/4d correspondence

2d IM

SU (N +1)r+1(N +1)• (Ar , AN ) ODE, cef f = rN
N +r+1 , CFT
U (1)N• (A2 , A4 ) ∼ (E8 , A1 )• monodromy at the origin (centrifugal potential)hM• Integrable model with SUSY p(x) = (x K − E)K (K = 2)
supersymmetric aﬃne Toda ﬁeld equation (work in progress
w/ M. Zhu)
47

Geometry, Representation Theory and Quantum Fields

Topological recursion and BPS structures for
hypergeometric spectral curves
Omar Kidwai
Abstract. We describe joint work with K. Iwaki relating computations in
the theory of topological recursion (TR) to Gaiotto-Moore-Neitzke’s countingof BPS states in 4d N=2 class S theories, in the case where the latter structureis ”uncoupled”. In particular, we describe a simple formula expressing the TR
free energies as a sum over BPS states for the relevant quadratic diﬀerential.
We will explain how this picture ought to generalize, and if time permits, therelation to so-called ”quantum curves”.

(O. Kidwai) Graduate School of Mathematical Sciences, University of Tokyo

203

204

OCAMI Reports Vol. 4 (2022)

Topological recursion and BPS structures for
hypergeometric spectral curves
Omar Kidwai
Graduate School of Mathematical Sciences
University of Tokyo

Based on arXiv:2010.05596, joint w/ K. Iwaki(see also arXiv:2108.06995 w/ K. Iwaki + upcoming w/ K. Osuga)

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

1 / 21

Two theories

Topological recursion [Eynard-Orantin, Chekhov-Eynard-Orantin]:• Matrix models, random matrices, loop equations• Integrable systems, WKB analysis (quantum curves) . . .
• Enumerative geometry (Gromov-Witten, Hurwitz,
Weil-Petersson volumes, Masur-Veech volumes...)
BPS structures [Gaiotto-Moore-Neitzke, Bridgeland, Kontsevich-Soibelman]:• 4d N = 2 QFT, hyperkähler geometry, Hitchin systems• Stability conditions on triangulated CY3 categories• Generalized DT invariants, wall-crossing, Joyce structures . . .

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

2 / 21

Geometry, Representation Theory and Quantum Fields

205

Let X compact Riemann surface, usually P1(Physically, we obtain the 4d theory of class S by compactifying"6d (2, 0) theory" on X . In this talk, gauge algebra g = A1 )Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

3 / 21

Summary
φ = Q(x)dx 2
quadratic differential(Coulomb vacuum)Σ ⊂ T ∗X

SW curve(TR spectral curve)
spectral networks(diff. eqn, counting trajectories...)topological recursion(residues, combinatorics...){Γ, Z , Ω(γ) . . .

"Wg ,n , Dℏ ϕ = 0 . . .

I’ll focus on describing one such result.
Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

4 / 21

206

OCAMI Reports Vol. 4 (2022)

ResultTogether with K. Iwaki, we showed:

2g −2

X
B2g
2πi
,
Fg =
Ω(γ)
2g (2g − 2) γ∈Γ
Z (γ)

(1)

Z (γ)∈H• Bk Bernoulli numbers• TR: Fg ∈ C "topological recursion free energies"• BPS structure: Γ lattice, Z "central charge", Ω(γ) ∈ Z "BPS
indices"

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

5 / 21

What we did (arXiv: 2010.05596)• Prove (1) for "hypergeometric type" examples:
φHG =m∞ 2 x 2 − (m∞ 2 + m0 2 − m1 2 )x + m0 2 ⊗2dx
x 2 (x − 1)2+ 8 other examples φ• (x) arising from limits/confluence.
• In particular,• Extend GMN definition of BPS indices• Compute BPS spectra (existence, location, classification of
saddles)

Main simplifying property: all BPS structures are uncoupled, (BPS
particles mutually local)

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

6 / 21

Geometry, Representation Theory and Quantum Fields

207

BPS counting in class S theories

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

7 / 21

Setting
Coulomb vacuum φ: quadratic differential (mero. section of ωX⊗2 ).
Given φ, construct the Seiberg-Witten curve as (Σ, π, λ):Σ = λ ∈ T ∗ X | λ2 = π ∗ φ ⊂ T ∗ X
together withπ|Σ : Σ → X ,λ = θcan |Σ

inherited from T ∗ X .
(Σ, π, λ) is a branched double cover with a meromorphic one-form(SW differential)

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

8 / 21

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Setting
Bridgeland gave a definition capturing the structure of BPS
particles in theories of class S.
Definition. A BPS structure is a tuple (Γ, Z , Ω):• finite rank free abelian group Γ "charge lattice", equipped w/
antisymmetric pairing ⟨·, ·⟩ : Γ × Γ → Z• homomorphism of abelian groups Z : Γ → C "central charge"• map of sets Ω : Γ → Q (for us, Z) "BPS invariants/indices"such that• Ω(γ) = Ω(−γ)• "Support property"
Terminology: γ ∈ Γ with Ω(γ) ̸= 0 are called BPS states.

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

9 / 21

GMN construction

Gaiotto-Moore-Neitzke gave an algorithm for constructing BPS
structures – consider rank 2 case.
e denote Σ with simple poles of φ filled in.
Let ΣDefine:e Z) | ι∗ γ = −γ}• Γ := {γ ∈ H1 (Σ,H
H p• Z (γ) := γ λ = γ Q(x)dx
Harder part: Ω : Γ → Z. Count degenerate “trajectories” (“spectral
networks”) of the quadratic differential φ.

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

10 / 21

Geometry, Representation Theory and Quantum Fields

209

GMN construction
Trajectories of phase ϑ:
Im e
Simple example (Weber):

−iϑ

Z z
λ = const
1
QWeb (x) = x 2 − m∞4as ϑ is varied. Lifting the degenerate portion to the cover, we haveΩ(γBPS ) = 1, all other Ω vanish
Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

11 / 21

GMN construction
New example (Legendre):

QLeg (x) =m∞(x − 1)(x + 1)Ω(γa ) = 4, Ω(γb ) = −1
Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

12 / 21

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Topological recursion

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

13 / 21

Spectral curves
A (TR) spectral curve is a tuple (C, x, y , B):• C compact Riemann surface• x, y : C → P1 nonconstant meromorphic functions, dx and dy
do not vanish simultaneously• “Bergman bidifferential" meromorphic section
B(z1 , z2 ) ∈ p1∗ (T ∗ C) ⊗ p2∗ (T ∗ C)
with some properties (pi : C × C → C projection).
For us, C = P1 so there is a canonical B,
B(z1 , z2 ) :=Omar Kidwai

dz1 dz2(z1 − z2 )2Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

14 / 21

Geometry, Representation Theory and Quantum Fields

211

Spectral curves

Zeroes and high-order (> 2) poles of dx: ramification points,
denoted r ∈ R
Note: If we choose x, y satisfying(y (z)dx(z))2 = Q(x(z))dx(z)2
we get an isomorphism C ≃ Σ, and the 1-form λ = ydx, projection
to x gives us the SW curve (π, Σ, λ).
One may write down explicit such spectral curves for all
hypergeometric type Q(x), so the TR may be applied.

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

15 / 21Topological recursion
Start with W0,1 (z0 ) := y (z0 )dx(z0 ), W0,2 (z0 , z1 ) = B(z0 , z1 ). Then"correlators" Wg ,n ∈ p1∗ (T ∗ C) ⊗ . . . ⊗ pn∗ (T ∗ C), are defined by"
Wg ,n+1 (z0 , z1 , · · · , zn ) :=X

Resz=r Kr (z0 , z) Wg −1,n+2 (z, z, z1 , · · · , zn )

r ∈R

+

′
X

#
Wg1 ,|I1 |+1 (z, zI1 )Wg2 ,|I2 |+1 (z, zI2 )g1 +g2 =g
I1 ⊔I2 ={1,2,··· ,n}

Kr a certain kernel, z̄ “local conjugation" near ramification point r .

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

16 / 21

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Topological recursion
Definition. Let Φ(z) be any primitive of y (z)dx(z). The g th freeenergy (g ≥ 2) is
Fg =

1 X
Res [Φ(z)Wg ,1 (z)]2 − 2gr ∈R[Iwaki-Koike-Takei] showed (for example):
FgHG (m) =B2g
2g (2g − 2)



1
1
+
2g
−2
(m0 + m1 + m∞ )(m0 + m1 − m∞ )2g −211++(m0 − m1 + m∞ )2g −2(m0 − m1 − m∞ )2g −2

1
1
1
−
−
−
.
(2m0 )2g −2(2m1 )2g −2(2m∞ )2g −2+ formulas for the other 8 examples.
Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

17 / 21

ResultTheorem. [Iwaki-K] For the spectral curves of hypergeometrictype, m generic, we have


X
B2g
2πi 2g −2
Fg (m) =Ω(γ),
2g (2g − 2)
Z (γ)

g ≥2

γ∈Γ
Z (γ)∈H

where H is any half-plane whose boundary rays are not BPS.
Conjecture. [Iwaki-K] This holds in higher rank too, under theassumption the BPS structure is uncoupled. (some evidence
presented, more forthcoming)

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

18 / 21

Geometry, Representation Theory and Quantum Fields

213

What we have done

• TR can tell us something about spectral networks. Practicallyspeaking, TR can help us compute information about BPS
counts without ever having to draw a spectral network!
• On the other hand, spectral networks can teach us about thestructure of TR (sequel paper). Can we predict new examples?
• Our formula is one simple example of this. What can we learn
in more complicated or exotic cases?

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

19 / 21

More results (arXiv:2108.06995)

Joint w/ K. Iwaki, we upgrade previous results via “quantum curves”• Solve natural “BPS Riemann-Hilbert problem” associated to
the BPS structure using Voros symbols of quantum curves• Observe Voros symbols are generated by a potential• Natural TR intepretation of Bridgeland’s τ -function for thesolution.

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

20 / 21

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Further• Higher rank: conjecture + a few experiments (ongoing)• Relation to Joyce structures / Joyce function• Relation to Nekrasov partition function• Coupled case?
• β-deformed case (ongoing w/ K. Osuga)• q-deformed case (5d BPS states)• TBA equations• ...

Omar Kidwai

Mathematical Sciences, University of Tokyo

Topological recursion and BPS structures for hypergeometric spectral curves

21 / 21

Geometry, Representation Theory and Quantum Fields

Elliptic chiral homology and quantum masterequation
Si Li
Abstract. We present an eﬀective BV quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection
between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of thecurved beta-gamma/b-c system and establish a coupled equation relating tochiral homology groups of chiral diﬀerential operators. This can be viewed as
the vertex algebra analogue of the trace map in algebraic index theory. The
talk is based on the recent work arXiv:2112.14572 [math.QA].

(S. LI) YMSC, Tsinghua University

215

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Elliptic chiral homology and Chiral index
Si Li
YMSC, Tsinghua University

Geometry, Representation Theory and Quantum Fields
Osaka City University, March 22-25, 2022

Motivation

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217

Given a deformation quantization Aℏ (M) = (C∞ (M)[[ℏ]], ?) on a
symplectic manifold (X, ω), there exists a unique linear map
Tr : Aℏ (M) → C((ℏ))
satisfying a normalization condition and the trace property
Tr(f ? g) = Tr(g ? f).
ZTheneωℏ /ℏ Â(M).

Tr(1) =
M

This is the algebraic index theorem which was first formulated by
Fedosov and Nest-Tsygan as the algebraic analogue ofAtiyah-Singer index theorem.

3 / 39In [Grady-Li-L 2017] [L-Xu-Gui, 2020] A rigorous connection
between the effective BV quantization for topological quantum
mechanics and the algebraic index theorem.
Topological quantum mechanics(M, ω)

Algebraic index theory

Tr : HH• (Aℏ (M)) → C((ℏ))
R
BV

R
(0)
Conf m (S1 )

D
E
(1)
(1)
dt1 · · · dtm O0 (t0 )O1 (t1 ) · · · Om (tm )4 / 39

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Replace S1 by an elliptic curve E. (Witten: index of dirac
operators on loop space).

···

···

2d Chiral analogue of algebraic index?

5 / 39We will be mainly interested in σ-models about the mapping space
ϕ:Σ→X

BV quantization

gluing:localized effective theory

local model:
factorization algebra

worldsheet:Σ

Target: X

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Geometry, Representation Theory and Quantum Fields

219

Observables and Factorization algebras

7 / 39Rich structures in QFT
A QFT is usually described by a manifold X and the data of fields
Spacetime : X=⇒

Fields : E = Γ(X, E).

▶ E is the space (called fields) where we will do calculusZ⟨O⟩ :=

E

OeS/ℏ

R
E

”Path integral”▶ Topology of X leads to new structures in ∞-dim geometry
When X =point, E = Rn . We arrive at the usual calculus.
Calculus = 0-dim QFT
When dim X > 0, the geometry and topology of X come in!
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One algebraic structure associated to the topology of X isobservables=functions on fields
Given an open subset U ⊂ X, we can talk about
Obs(U) = observables supported in U
Example: δ-function.

9 / 39Observables form an algebraic structure as `follows: given disjoint
open subset Ui contained in an open V:i Ui ⊂ V

we have a factorization product for observablesO
Obs(Ui ) → Obs(V).
i▶ Physics: OPE (operator product expansion)▶ Mathematics: factorization algebra.

▶ Origin: Beilinson-Drinfeld in 2d CFT▶ Costello-Gwilliam: (perturbative renormalized) QFT.
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Example: dim X = 1 (topological quantum mechanics)
QFT in dim = 1 is quantum mechanics.

In the topological case, for any contractible open U, Obs(U) = A.
The factorization product doesn’t depend on the location and size:
A ⊗ A → A.
We find an (homotopy) associative algebra.

11 / 39

Example: dim X = 2 (chiral conformal field theory)
The factorization product of 2d chiral theory is holomorphic.
O1 (z)O2 (w) ∼O1

P

O1(n) O2 (w)n (z−w)n+1

O2

which is the 2d analogue of “associative product”. We find∞-many binary operations O1(n) · O2 !
In this case, observable algebra forms a vertex algebra.

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An important class of quantities are correlation functions ofobservables. They capture “global” information of the theory.
▶ Local correlation⟨O1 (x1 ) · · · Oi (xi ) · · · On (xn )⟩ ,xi ∈ X.

It is singular when points collide, hence a function on
Confn (X) := {x1 , · · · , xn ∈ X|xi ̸= xj fori ̸= j}.
▶ Many interesting non-local information is hidden inZ⟨O1 (x1 ) · · · Oi (xi ) · · · On (xn )⟩Z⊂Confn (X)which might be divergent and require further renormalization.

13 / 39

Batalin-Vilkovisky (BV) Quantization formalism

14 / 39

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223

Homological methods (such as BRST-BV) arises in physics as a
general method to quantize theories with gauge symmetries.

15 / 39

BV algebra
A Batalin-Vilkovisky (BV) algebra is a pair (A, ∆) where▶ A is a Z-graded commutative associative unital algebra.
▶ ∆ : A → A is a linear operator of degree 1 such that ∆2 = 0.
▶ The BV bracket {−, −} : A ⊗ A → A by{a, b} := ∆(ab) − (∆a)b − (−1)|a| a∆b, a, b ∈ A.
{−, −} satisfies a version of graded Leibnitz rule.

Example (Polyvector fields)
The space of smooth polyvector fields with a divergence operator(PV• (X) = Γ(X, ∧• TX ),∆ = divergence)is a BV algebra.
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Roughly speaking, BV quantization in QFT leads to▶ Factorization algebra Obs of observables.
▶ (C• (Obs), d): a chain complex via algebraic structures of Obs.
RR
▶ A BV algebra (A, ∆) with a BV- map BV : A → C.
▶ A C[[ℏ]]-linear map satisfying⟨−⟩ : C• (Obs) → A((ℏ))
satisfies quantum master equation (QME)(d + ℏ∆)⟨−⟩ = 0.

(QME)

This means it is a chain map intertwining d and ℏ∆.
R▶ Partition function:
Index = BV ⟨1⟩.

17 / 39

Example: Topological Quantum Mechanics (TQM)

18 / 39

Geometry, Representation Theory and Quantum Fields
W2n

225

W2n

▶ Local observables: Weyl algebra
Obs1d = W2n = C[[pi , qi ]][[ℏ]], ?

▶ (C• (Obs1d ), b) = the Hochschild chain complex.
b • (R2n ), LΠ ). Here Π = Poisson▶ BV algebra (A1d , ∆) = (Ωtensor. In physics, this describes the geometry of zero modes.

19 / 39▶ ⟨−⟩1d : C• (W2n ) → A1d ((ℏ)) where⟨O0 ⊗ O1 · · · ⊗ Om ⟩1d
Oi ∈ W2n
Z
D
E
(1)
(1)
=
O0 (ϕ(t0 ))O1 (ϕ(t1 )) · · · Om (ϕ(tm ))free

t0 =0<t1 <···<tm <1

O0 (t0 )O1 (t1 )Om (tm )R···

t0 = 0 < t1 < · · · < tm < 1

Ok (tk )It satisfies
QME(b + ℏ∆)⟨−⟩1d = 0

Here b is the Hochschild differential.
Ref: [Gui-L-Xu, 2020]
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X
These data glues [Fedosov] to give a Weyl bundle W(X) → X.
▶ ⟨−⟩1d leads to a trace map on deformation quantized algebra,
as explicitly described by [Feigin-Felder-Shoikhet].
▶ [Grady-Li-L, Gui-L-Xu]: BV quantization of TQM givesZ⟨1⟩ =eωℏ /ℏ Â(X).
X

This is the simplest version of algebraic index theorem which was
first formulated by Fedosov and Nest-Tsygan as the algebraic
analogue of Atiyah-Singer index theorem.

21 / 392d Chiral Conformal Field Theory

22 / 39

Geometry, Representation Theory and Quantum Fields

227

1d TQM

2d Chiral CFT

S1Associative algebraΣ
Vertex operator algebra
Operator product expansion

Associative product
O1 ? O2O1

O1 (z)O2 (w) ∼P O1(n) O2 (w)n(z−w)n+1

O2
O1

O2

23 / 39

A chiral σ-model
ϕ:Σ→Xwill produce a bundle V(X) of chiral vertex operator algebras
V(X)


X
This is the chiral analogue of Weyl bundle in TQM.

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Theorem (L)The BV quantization of the 2d chiral model is equivalent tosolving a flat connection on the vertex algebra bundle V(X)
I

1
D=d+
L, − , D2 = 0ℏH
where L ∈ Ω1 (X, V(X)) and L is the associated chiral vertex
operator fiberwise.
▶ This is the chiral analogue of Fedosov connection.
▶ BRST reduction of chiral models falls into this setup
I
L = BRST operator.
Ref: [L] Vertex algebras and quantum master equation. To appear
in JDG.
25 / 39

Elliptic chiral homology▶ In [Zhu, 1994], Zhu studied the space of genus 1 conformal
block (the 0-th elliptic chiral homology) and establish themodular invariance for certain class of VOA.
▶ Beilinson and Drinfeld define the chiral homology for general
algebraic curves using the Chevalley-Cousin complex.
▶ Recently, [Ekeren-Heluani,2018,2021]: an explicit complex
expressing the 0th and 1st elliptic chiral homology.

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229

Intuitively, the chiral differential in the chiral complex can beviewed as a 2d chiral analogue of the Hochschild differential b.
dch

=

P

=

b

P

b(a0 ⊗ · · · ⊗ ap )p

= (−1) ap a0 ⊗ · · · ⊗ ap−1 +p−1X(−1)i a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ap .

i=0
27 / 39

We briefly review the construction of Beilinson and Drinfeld.
▶ M(X): category of (right) D-modules on X = Σ▶ M(XS ): category of (right) D-modules on XS
M ∈ M(XS ) is rule that assigns to each finite index set I ∈ S
a right D − module MXI on XI .
(satisfying some compatibility conditions.)▶ There is an exact fully faithful embedding∆∗ : M(X) ,→ M(XS )
(S)

(S)

(I)

defined by (∆∗ M)XI := ∆∗ M, where ∆(I) : X ,→ XI .
The category M(XS ) carries a tensor structure ⊗ch and a chiralalgebra A is a Lie algebra object via ∆S∗ .
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We consider the Chevalley-Eilenberg complex(C(A), dCE ) = (⊕Sym•⊗ch (∆∗ A[1]), dCE ),(S)

which is a complex in M(XS ).
The chiral homology (complex) Cch (X, A) is defined by
RΓDR (XS , C(A)), where
Dolbeault Resolution

Chevalley-Eilenberg


RΓDR XS , C(A)Spencer Resolution

29 / 39

Example: βγ − bc system

The VOA V βγ−bc of βγ − bc system is the chiral analogue of
Weyl/Clifford algebra. It gives rise to a chiral algebra (in the senseof Beilinson and Drinfeld) Aβγ−bc on a Riemann surface X = Σ.
The factorization homology (complex)(C• (V βγ−bc (h)), dch )in the BV formalism

will be the chiral chain complex Cch (X, Aβγ−bc ).

30 / 39

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231

Theorem (Gui-L)
Let X be an elliptic curve Eτ . We can construct an explicit map⟨−⟩2d : Cch (X, Aβγ−bc ) → A2d ((ℏ))
satisfying
QME :(dch + ℏ∆)⟨−⟩2d = 0.

Roughly speaking, this map is defined byZ⟨O1 ⊗ · · · ⊗ On ⟩2d := − ⟨O1 (z1 ) · · · On (zn )⟩.
Σn▶ ⟨O1 (z1 ) · · · On (zn )⟩ is local correlation (via Feynman rules).
R▶ − is the regularized integral introduced by [L-Zhou]. This is ageometric renormalization method for 2d chiral QFT.
31 / 39

We need to understand the integral of local correlatorsZ?
⟨O1 (z1 ) · · · On (zn )⟩”=”
Σn

Unlike the situation in topological field theory, ⟨O1 (z1 ) · · · On (zn )⟩is very singular along diagonals and there is no way to extend it tocertain compactification of Conf n (Σ).

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Regularized integral (L-Zhou 2020)
Let us first consider the integral of a 2-form ω on Σ withmeromorphic poles of arbitrary orders along a finite subset D ⊂ Σ.
Locally we can write ω = zηn where η is smooth 2-form and n ∈ Z.
We can decompose ω into
ω = α + ∂β
where α is a 2-form with at most logarithmic pole along D, β is a∂(0, 1)-form with arbitrary order of poles along D, and ∂ = dz ∂z
isthe holomorphic de Rham. We define the regularized integral
ZZZ− ω :=
α+
Σ

Σ

β

∂Σ

This does not depend on the choice of the decomposition.
33 / 39R− is invariant under conformal transformations. The conformalΣR

geometry of Σ gives an intrinsic regularization of the integralΣ ω.

The regularized integral can be viewed as a “homologicalintegration” by the holomorphic de Rham ∂ZZ− ∂(−) =
(−).
Σ

∂Σ

¯
The ∂-operator
intertwines the residueZ¯− ∂(−)= Res(−).
Σ

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233

In general, we can defineZ
Z Z
Z− (−) := − − · · · − (−) .
Σn

Σ Σ

Σ

This gives a rigorous and intrinsic definition ofZ⟨O1 ⊗ · · · ⊗ On ⟩2d := − ⟨O1 (z1 ) · · · On (zn )⟩.
ΣnIt exhibits all the required properties:▶ Holomorphic Anomaly Equation. (L-Zhou, in preparation)▶ Contact equations. (Gui-L-Tang, in preparation)▶ ···35 / 39

Elliptic chiral index (after Douglas-Dijkgraaf)
The partition function of a chiral deformation by a chirallagrangian L is given by
D 1R Eeℏ Σ L.
2d

If we quantize the theory on elliptic curve Σ = Eτ ,
lim

τ̄ →∞

D 1R
E
e ℏ Eτ Lc2d1= TrH qL0 − 24 e ℏ

H

dzL

,

q = e2πiτ

where the operation limτ̄ →0 sends
almost holomorphic modular forms=⇒

quasi-modular forms.

This can be viewed as an chiral algebraic index on the loop space.
The regularized integral [L-Zhou] precisely explains τ̄ → ∞.
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Theorem (L-Zhou 2020)
Let Φ(z1 , · · · , zn ; τ ) be a meromorphic elliptic function on Cn × H
which is holomorphic away from diagonals. Let A1 , · · · , An be ndisjoint A-cycles on Eτ . Then the regularized integral
!
Z
n
Y
d2 zi
1
] and
−
Φ(z1 , · · · , zn ; τ ) lies in OH [
im τ
im τ
Enτ
i=1

)
∫ (∏
∫
∫
n
1 ∑
d2 zi
Φ(z1 , · · · , zn ; τ ) =dzσ(1) · · ·dzσ(n) Φ(z1 , · · · , zn ; τ ) .
lim −τ̄ →∞ En
im τ
n!
A1
An
τ
i=1
σ∈Sn

Ai

Ai+1

R
In particular, −En gives a geometric modular completion forτquasi-modular forms arising from A-cycle integrals.

37 / 39

Algebraic Index vs Elliptic Chiral Index

1d TQM
Associative algebra
Hochschild homology
QME:(ℏ∆ + b)⟨−⟩1d = 0⟨O1 ⊗ · · · ⊗ On ⟩1d =integrals
on the compactified
configuration spaces of S1
Algebraic Index theory

2d Chiral CFT
Vertex operator algebra
Chiral homology
QME:(ℏ∆ + dch )⟨−⟩2d = 0⟨O1 ⊗ · · · ⊗ On ⟩2d =regularized
integrals of singular forms on Σn
Elliptic Chiral Algebraic Index

Joint work with Zhengping Gui. arXiv:2112.14572 [math.QA]

38 / 39

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235

Thank you!

39 / 39

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Duality Cascades and Aﬃne Weyl Groups
Sanefumi Moriyama(S. Moriyama) Osaka City University

Geometry, Representation Theory and Quantum Fields

Poincare Series, 3d Gravity and Averages ofRational CFT
Sunil Mukhi
Abstract. I will survey the Poincare series approach to computing 3d gravity
partition functions dual to Rational CFT. For a single genus-1 boundary, theSU(2) WZW models provide unitary examples for which this series is a positive
linear combination of two modular-invariant partition functions. This supports
the interpretation that the bulk gravity theory - topological Chern-Simons - isdual to an average over CFT s sharing the same Kac-Moody algebra. I will
discuss the generalisation to SU(N) at level 1, where one finds averages overarbitrarily many boundary CFT’s.

(S. Mukhi) Indian Institute of Science Education and Research (IISER), Pune

237

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OCAMI Reports Vol. 4 (2022)

Poincaré Series, 3d Gravities and Averaged Holography
Sunil Mukhi
Geometry, Representation and Quantum Fields
Osaka City University, March 22-25, 2022

Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Geometry, Representation Theory and Quantum Fields

239

AdS3 /CFT2 and Poincaré Series• The AdS3 /CFT2 correspondence is expected to be more tractable thanhigher-dimensional versions, since the latter require embedding into
string/M-theory.
• Of course AdS2 /CFT1 is even more tractable – and essentially solved –using mathematical results on volumes of moduli spaces[Saad-Shenker-Stanford 2019].
• In D = 3 there are no propagating graviton modes in the bulk, thoughin AdS3 one has boundary degrees of freedom – boundary gravitons.
• This means that the quantum version of pure (Einstein) gravity might

be consistent in D = 3, unlike D > 3 where it is presumably
inconsistent due to non-renormalisability.

1 / 38• Semi-classically, the partition function of 3d gravity would be a sum

over saddle points of the Euclidean path integral with a thermal circle
at infinity:
X −S i1Z=
e E (det i )− 2iiwhere SEis the Euclidean action of the saddle point solution and1i −2(det  )is the one-loop contribution of boundary gravitons aroundeach saddle point.
• For 3d gravity this is thought to be exact with no higher-loop

corrections (because of the absence of bulk degrees of freedom).
• Thus one may hope to compute the exact partition function of 3d

gravity. This was attempted by [Maloney-Witten 2007, Keller-Maloney 2014,
Alday-Bae 2020, Benjamin-Collier-Maloney 2020].

2 / 38

240

OCAMI Reports Vol. 4 (2022)• At low temperatures the leading saddle point is thermal AdS3 , which isa solid torus with a boundary of modular parameter τ = τ1 + iτ2 . Theβinverse temperature β is given by τ2 = 2π.

• Defining q = e2πiτ , the corresponding action is:

SE = −`πτ2c=ln(q q̄)4GN
24

while the determinant factor is [Giombi-Maloney-Yin 2008]:
1(det )− 2 =∞Y

1|1 − q n |2
n=2

3 / 38• Thus the approximate partition function for low temperature is:c

ZAdS3 = (q q̄)− 24

∞
Y

1
|1
−
q n |2n=2• Away from low temperature, we need to include other saddle points.
• A conjecture of [Maloney-Witten 2007] says the relevant saddle-points are

Euclidean versions of (spinning) BTZ black holes, which include
thermal AdS3 as a special case.
• These black holes are labelled by a pair of co-prime integers (c, d)which determine the mass M and angular momentum J by:
1 c2 τ22 + (cτ1 + d)28GN|cτ + d|4` cτ2 (cτ1 + d)
J=
4GN |cτ + d|4

M=

in the Lorentzian case.
• They satisfy an extremality bound M > |J|.

4 / 38

Geometry, Representation Theory and Quantum Fields

241• Each of these Euclidean black holes corresponds to a “handlebody”(filled torus) whose boundary, like that of thermal AdS3 , is a 2-torus.
• The (c, d) black hole has a boundary torus with a modular parameterτγ where:aτ + bcτ + dHere c, d defines the black hole and a, b are integers such thatad − bc = 1. They are unique up to a shift:τγ ≡(a, b) → (a, b) + (c, d)• Thus the (c, d) black hole is a modular transform of AdS3 (wetransform the boundary τ → τγ and then continue into the bulk).
• Hence its partition function is ZAdS3 (τγ , τ̄γ ).

5 / 38• It follows that the full partition of AdS3 gravity, assuming BTZ black

holes are the only saddle points, is:X
Z(τ, τ̄ ) =ZAdS3 (τγ , τ̄γ )γ∈Z\SL(2,Z)=

X
γ∈Z\SL(2,Z)

2πiτγwhere qγ ≡ ec(qγ q̄γ )− 24∞
Y1|1 − qγn |2n=2.

• In this approach the contribution of each (c, d) black hole does notneed to be evaluated separately. One just has to perform the above
sum, called a Poincaré sum.
• The answer is manifestly modular-invariant, as one would expect since

this should also be the partition function of the boundary 2d-CFT dual
to AdS3 gravity.

6 / 38

242

OCAMI Reports Vol. 4 (2022)• Since an infinite sum is involved, the sum can be (and is) divergent.
• In [Maloney-Witten 2007] the sum was regularised using a zeta-function

method. At the end one has a finite, though very complicated, answer.
• Unfortunately this answer is unphysical in many ways. First, thespectrum is continuous rather than discrete. Second, the degeneraciesof states are negative in certain ranges of M, J.
• The problem was revisited in [Keller-Maloney 2015,
Benjamin-Ooguri-Shao-Wang 2019] among other works. They confirm thatwithout “manual intervention” these properties persist.
• Manual intervention was attempted in [Benjamin-Collier-Maloney 2019] byadding new saddle points corresponding to conical defects. Also
[Maxfield-Turiaci 2020] proposed to add new saddle points, inspired by JT
gravity in 2D.
• The suggestion was also made that, as happens in 2d JT gravity, the

AdS3 /CFT2 correspondence may involve an average over CFT’s[Cotler-Jensen 2020].

7 / 38Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Geometry, Representation Theory and Quantum Fields

243

The CFT Side• While the AdS3 side remains a puzzle, surprisingly the CFT2 side iseven more mysterious.
• As famously shown by [Brown-Henneaux 1986], the asymptotic

symmetries of AdS3 gravity are two copies of the Virasoro algebra,
with central charge:
3`
c=
2GN
where ` is the AdS3 radius and GN is Newton’s constant.
• We would like to match semi-classical gravity results with CFT

calculations. So we must identify a suitable CFT with a central chargec that can be taken to infinity. It should have no additional
symmetries beyond the Virasoro algebra.

8 / 38• Unfortunately no concrete CFT2 is known, or conjectured, that is

supposed to be dual to pure Einstein gravity on AdS3 . We can only
estimate some of its properties (e.g. bounds on operator dimensions),
assuming it exists.
• The only explicitly known CFT’s having only conformal symmetry are

the Virasoro minimal models, and they have c < 1 (so no semi-classicallimit).
• We know many more 2d CFT’s in great detail – but most of them are

Rational CFT’s, namely those whose partition function has the form:Z(τ, τ̄ ) =p−1Xχ̄i (τ̄ )χi (τ )i=0where the χi are generalised characters (not Virasoro characters) andthe sum is finite.

9 / 38

244

OCAMI Reports Vol. 4 (2022)• These theories typically have (non-Abelian) enhanced symmetries –Kac-Moody algebras. They do admit the limit c → ∞.
• They may fail other criteria for semi-classical AdS/CFT, such assparseness of the spectrum, needed for local bulk physics. Then again,
there is no bulk physics in pure AdS3 gravity!
• Also they are integrable, so would not have properties associated to thesemi-classical physics of black holes e.g. chaos.
• Still, it is worth exploring whether they provide some lessons for the

AdS/CFT correspondence – and with some assumptions, we will beable to explore this.

10 / 38• The simplest RCFT’s beyond Virasoro minimal models are WZWmodels based on simple Lie algebras. We will work with SU(N)k .
• The pioneering work of [Cappelli-Itzykson-Zuber 1987] followed by[deGiovanni 1990, Bauer-Itzykson 1990, Gannon 1992] showed that for a given
algebra and level, one can form multiple modular invariant partition
functions out of the characters.
• So there are finitely many distinct boundary partition functions. Next

we need to:
identify the bulk gravity theory (it cannot be pure Einstein gravity),
compute the partition function,
3 compare with the CFT.
1
2

• We will see that identifying the bulk theory is not too difficult given

the non-Abelian symmetry of the boundary theory, which is quite
constraining.

11 / 38

Geometry, Representation Theory and Quantum Fields

245

Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Poincaré Series and WZW Models• In [Meruliya-SM-Singh 2021, Meruliya-SM 2021] we considered an

SU(N)k ×SU(N)−k Chern-Simons theory in the bulk:eSbulk = SCS [A] − SCS [A]where:SCS [A] =

k
2π

Z
M


tr A ∧ dA + 23 A ∧ A ∧ A• This is a well-known topological gauge theory, but if we sum it overmanifolds then it can be thought of as an exotic gravity theory. Like
Einstein gravity it has no bulk degrees of freedom, but should haveboundary gluons [Porrati-Yu 2019].
• So the boundary CFT must have an SU(N)k ×SU(N)k current algebra.
By the Sugawara construction it also has a Virasoro algebra:
1: J a (z)J a (z) :2(k + N )From the bulk point of view, this should be dual to composite
boundary gravitons.
• This further supports the idea that the bulk is some kind of gravity
theory.
T (z) =

12 / 38

246

OCAMI Reports Vol. 4 (2022)• To compute the bulk partition function we use the Poincaré series

approach to perform the sum over manifolds.
• We have to pick a starting point for the sum, i.e. the “seed” (theanalogue of ZAdS3 ).
• This will be chosen to be one of the current-algebra characters χi (τ )where i = 0, 2, · · · , p − 1 (a similar approach using Virasoro characters
was adopted by [Castro et al 2011]). These are not elementary functions
like the thermal AdS3 partition function, because of null vectors.
• The characters have a q-expansion:cχi (τ ) = q − 24 +hi (ai0 + ai1 q + ai2 q 2 + · · · )• Given an element:


γ=

a
c


b
∈ SL(2, Z),dγ : τ → τγ ≡aτ + bcτ + dthe characters transform as:
χi (τγ ) =X(Mγ )ij χj (τ )j

13 / 38• Choosing the primary i as the “seed” primary, we write:

Zseed (τ, τ̄ ) = |χi (τ )|2(no sum)

where χi (τ ) is the Kac-Moody character for that primary.
• This has a double expansion in q, q̄:cc
Zseed (τ, τ̄ ) = q − 24 +hi q̄ − 24 +h̄i (ai0 )2 + ai0 ai1 (q + q̄) + (ai1 )2 q q̄ + · · ·• Next one performs the Poincaré sum to get:

Z Grav (τ, τ̄ ) =X

Zseed (τγ , τ̄γ )γ∈Γsub \SL(2,Z)• Here Γsub is a subgroup of SL(2,Z) that preserves the seed character. In

rational CFT this has finite index and hence the Poincaré sum is finite.
• The result of the above sum is modular-invariant and we interpret it as

the partition function of Chern-Simons gravity in the bulk.

14 / 38

Geometry, Representation Theory and Quantum Fields

247• To perform the Poincaré sum we need to know the subgroup Γsub ofSL(2,Z) that preserves the seed.
• For a CFT with central charge c and primary dimensions hi ,mathematicians define an integer N called the conductor of the CFTby:c
N ≡ LCM of denominators of −+ hi
24
For this N , the principal congruence subgroup Γ(N ) preserves all thecharacters.
• There is another integer N0 that we call the semi-conductor:

N0 ≡ LCM of denominators of hi
Now the corresponding congruence subgroup Γ(N0 ) preserves thecharacters up to a phase, which is good enough for us (the phase drops
out in Zseed ).
• Since N0 |N , we have Γ(N ) ⊂ Γ(N0 ). So it is more convenient to sum

over Γ(N0 )\SL(2, Z).
• Actually there is a bigger non-normal subgroup Γ1 (N0 ) that alsopreserves the characters, but the answer differs from the sum overΓ(N0 ) by a finite overall factor that can be removed by normalisation.
15 / 38• The result of the Poincaré sum has to be modular-invariant, byconstruction.
• Hence we can parametrise all possible answers in terms of all available

modular invariants for the given set of characters.
• Let us label all modular invariants for a given set of characters (i.e. agiven SU(N)k current algebra) as follows:X
ZJ (τ, τ̄ ) =χ̄i (τ̄ )(IJ )ij χj (τ )i,jwhere the IJ are certain constant matrices.
• Then, it has to be true that:

Z Grav (τ, τ̄ ) =X

cJ ZJ (τ, τ̄ )J

for some coefficients cJ (which we normalise asP

J cJ = 1).

• Note that not all modular invariants are physical. There are a few ways

in which a modular-invariant partition function can be unphysical:• The coefficients in the q-expansion may be fractional (after normalising

to have a non-degenerate vacuum state).
• The coefficients in the q-expansion may be negative.

16 / 38

248

OCAMI Reports Vol. 4 (2022)• Now several possibilities arise:• J takes just one value.
• J takes at least two values and all the cJ are non-negative.
• J takes at least two values but at least one of the cJ is negative.
• J takes at least two values and for some non-zero cJ , the invariant ZJ is

unphysical in the sense explained above.
• Only the first two possibilities are physically acceptable: the firstwould be an example of ordinary holography while the second would
be averaged holography.
• Averaged holography is familiar from the case of JT gravity, where thepartition function corresponds to a sum over an ensemble of randommatrices. But in the present context it can only be an average over afinite set.
• There has been much discussion about the physical reason for averaged

holography and what are the circumstances in which it arises (e.g.
[Schlenker-Witten 2022]).

17 / 38• Let us now describe the method to derive the coefficients cJ (this issimilar to the one used in [Castro et al 2011] for specific minimal models).
• Consider a general seed for the Poincaré series. This can be written as:

Zseed (X, τ, τ̄ ) = χ̄i (τ̄ ) (Xseed )ij χj (τ )where Xseed is a p × p matrix.
• Now under a modular transformation we have:Zseed (X, τγ , τ̄γ ) = χ̄(τ̄γ ) Xseed χ(τγ ) = χ̄(τ̄ ) (Mγ† Xseed Mγ ) χ(τ )• Then:Z Grav (X, τ, τ̄ ) =X

Zseed (X, τγ , τ̄γ ) = χ̄(τ̄ )γ• Thus, the matrix

XMγ† Xseed Mγ χ(τ )γ†γ Mγ Xseed Mγ determines what linear combinations

Pof invariants will appear in the final answer.

18 / 38

Geometry, Representation Theory and Quantum Fields

249• Thus we may write:X

Mγ† Xseed Mγ =

γ

D
X

cJ IJ

J=1

where IJ is the basis of matrices we defined earlier, that lead tomodular invariant partition functions ZJ (τ, τ̄ ).
• We define an inner product on the space of these matrices:dJK = tr(IJ IK )Using this in the above equation, we find:
XX
dKJ cJ =tr(IK Mγ† Xseed Mγ )
γ

J

=

X

tr(IK Xseed ) = |Γsub \Γ| tr(IK Xseed )γ• This matrix equation can be inverted to give:cJ = |Γsub \Γ|X

d−1
JK tr(IK Xseed )K

These are the desired coefficients which, after normalisation, should beinterpreted as the weights with which different CFT’s appear.
19 / 38• We first consider the case of SU(2)k .
• It was shown by [Cappelli-Itzykson-Zuber 1987] that there is a diagonal

invariant ZA for all values of k. For all even k we have an additional
physical invariant ZD , and for k = 10, 16, 28 there is a third physicalinvariant ZE .
• For k > 12 there also arise unphysical modular invariants in the sense

described above.
• Let us list some explicit results for the Poincaré sum. We will see thatsometimes only one invariant appears, in other cases all the invariants
appear. In some cases the coefficients are non-negative but in others,
there are negative coefficients.
• Also for k ≤ 12 only physical invariants appear, but beyond that theunphysical invariants (called Znew ) also arise.

20 / 38

250

OCAMI Reports Vol. 4 (2022)

21 / 38

22 / 38

Geometry, Representation Theory and Quantum Fields

251• Some observations about these results:• The answers depend on the choice of seed.
• Within a given model, several distinct seeds give the same linear

combination of invariants.
• The vacuum primary seed always leads to a linear combination with

non-negative integers, though unphysical invariants can arise.
• For k + 2 = 4ρ, (ρ ≥ 2), the ratio of coefficients cD = 1 for the vacuum
cA

seed.
• For k + 2 = 4ρ + 2, (ρ ≥ 1), the ratio of coefficients cD = − 1 for thecA
2seed χ2 .
• For certain values of k and for specific seeds, one finds only ZA or only
ZD as the answer.

23 / 38• We were able to predict the result for the entire family of SU(2)ktheories.
• If k + 2 = p or k + 2 = p2 for any prime p, there is a unique modular

invariant and the Poincaré sum reproduces it.
• Whenever k + 2 = 2p for a prime p, the gravity partition function is alinear combination over precisely two CFT’s.
• In these cases, for the identity seed we get:

p+1
p−2
ZA +
ZD
2p − 1
2p − 1
which again agrees with the explicit calculations shown before.
ZAdS (τ ) =• Thus these are infinite families of unitary examples of both normal andaveraged holography.
• Modular invariants for SU(3)k are also classified. We calculated the

Poincaré series for several examples, with similar results.

24 / 38

252

OCAMI Reports Vol. 4 (2022)• The SU(2)k and SU(3)k examples had two major limitations:• There are no more than 3 physical modular invariants.
• For large k the central charge asymptotes to 3 and 8 respectively, so

there is no c → ∞ limit.
• We can go beyond these restrictions by considering the SU(N )k case.
• Modular invariants for SU(N )1 were classified in [Itzykson 1988,
deGiovanni 1990]. Define:m = N, N odd
N= , N even2
• Then the number Ninv of independent modular invariants is given by:σ(m), σ(m) even2
σ(m) + 1=, σ(m) odd
2P
where σ(m) := d|m 1 is the Euler divisor function.
Ninv =

25 / 38• In the SU(N )1 case some of the explicit results (starting with theidentity seed) look like this:26 / 38

Geometry, Representation Theory and Quantum Fields

253• Again we were able to predict the answer for every N .
• If we decompose m into its prime factors:

m=

Y n
pi i
i

then
σ(m) =

Y(ni + 1)i

• For this discussion let us restrict to the simplest infinite sub-class:m = pnand take p to be a generic (odd) prime. Then m = N andσ(N ) = n + 1
We can take this as large as we like. In this way we generate arbitrarily
many modular invariants.
• Another welcome feature is that for SU(N )1 there are no unphysicalinvariants.

27 / 38• Recall that to evaluate the Poincaré sum one has to compute thematrix:dJK = tr(IJ IK )where IJ is a basis of matrices such that χ̄IJ χ is modular invariant.
• For N = pn we found that:dJK = pn−|J−K| ,J, K ∈ {0, 1, 2, · · · , n}

• This is a symmetric Toeplitz matrix (the J, K element depends only on|J − K|). One finds the inverse to be:2p + 1 − δJ0 − δJn ,
1
d−1
−p,
JK = n 2
p (p − 1) 

0,

J =K
|J − K| = 1
else• From this we can compute the coefficients cJ in the Poincaré sum:

cJ ∼

n
X

d−1
JK tr(IK Xseed )K=028 / 38

254

OCAMI Reports Vol. 4 (2022)• Up to an overall normalisation, the Poincaré sum for the vacuum seedis:pZ0 + (p − 1)Z1 + (p − 1)Z2 + · · · + (p − 1)Zn−1 + pZn• This agrees with explicit calculations for N = pn with odd p. Moreover

it is easily extended to the general case.
• A remarkable feature of these results is that the result of the Poincarésum is always an average over physical CFT’s with non-negative
weights. This is in contrast to the SU(2)k case.

29 / 38

Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Geometry, Representation Theory and Quantum Fields

255

Relation to Narain Lattices• Let us comment on the “more favourable” results for the SU(N )1 case,as compared to SU(2)k .
• This seems to be related to the fact that SU(N )1 CFT is equivalent toN − 1 free bosons compactified on a special lattice. In fact, these
theories are special cases of Narain-lattice compactifications.
• A duality between free scalar CFT on a D-dimensional lattice and abulk U(1)2D Chern-Simons theory was proposed in [Maloney-Witten 2020,
Afkhami-Jeddi et al 2020].
• Toroidal theories have a continuous (Narain) moduli space, as is wellknown from string theory. Thus their partition function can be written
Z CFT (m, τ, τ̄ ) where m labels the moduli.

30 / 38• The above authors proposed that the bulk dual for Narain latticetheories is U(1)2D Chern-Simons theory in 3d with action:
Z
1S=
ΛIJ AI ∧ dAJ
2π
where ΛIJ is an even integral unimodular form of signature (D, D).
• They perform a Poincaré sum over the vacuum character of this

Abelian action and compare it to an integral:Z
dm Z CFT (m, τ, τ̄ )where dm is the Siegel-Weil measure on moduli space.

31 / 38

256

OCAMI Reports Vol. 4 (2022)• As is well-known, for special lattices the symmetry of the compactified

bosons is enhanced from U(1)2D to SU(D+1) × SU(D+1).
• For such lattices we recover a modular-invariant partition function

with the Kac-Moody algebra of the SU(D+1)1 WZW model.
• When there are multiple lattices with the same enhanced symmetry,

we find multiple modular-invariant CFT’s. Thus, these cases are adiscrete subset of the continuous Narain lattice examples.
• This suggests that the weights we found (by computing an entirely

different Poincaré sum over non-Abelian characters) correspond to
special discrete values of the Siegel-Weil measure for Narain lattices.
But we have not yet been able to prove this.
• Since the latter is a positive measure, it would explain why our discretemeasure is found to be positive.

32 / 38

Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Geometry, Representation Theory and Quantum Fields

257

Multiple boundaries• So far we discussed Poincaré sums over manifolds with a single toroidal

boundary. This can be generalised to multiple toroidal boundaries, andalso single or multiple higher-genus boundaries.
• We have only studied the former case, and only for SU(2)k . The results

are interesting.
• Let us consider NB toroidal boundaries with independent modular

parameters τ 1 , τ 2 , · · · , τ NB .
• If the single-boundary Poincaré sum, which gave:

Z Grav (X, τ, τ̄ ) =X

cJ ZJ (X, τ, τ̄ )J

is to be interpreted as averaged holography, then the multi-boundary
sum should give:
X
Z(X, τ 1 , τ 2 , · · · , τ NB ) =cJ ZJ (τ 1 )ZJ (τ 2 ) · · · ZJ (τ NB )J
i(the τ̄ arguments are suppressed for ease of notation).

33 / 38• However the contribution from disconnected manifolds, each with asingle boundary, gives instead:
X
X
 X


Z disconn (X, τ i ) =cJ ZJ (τ 1 )cJ ZJ (τ 2 ) · · ·cJ ZJ (τ NB )JJJ• As is well-known in 2d JT gravity, the extra contribution must comefrom “wormhole” manifolds (connected manifolds having multiple
boundaries).
• Unfortunately the present examples are very far from a semi-classicallimit where we can think about manifolds at all.
• We have to find a “proxy” for wormholes in terms of Poincaré series,

just as the single-boundary sum was defined through these series.

34 / 38

258

OCAMI Reports Vol. 4 (2022)• Let us restrict to the case of two independent modular invariants

ZA , ZD (since otherwise unphysical invariants start to appear). Then:Z Grav,{1} (τ 1 ) = cA ZA + (1 − cA )ZD• Now requiring:
1 2
12
Z Grav,{2} = cA ZAZA + cD ZD
ZD

we work out that the connected contribution must be:1
1
2
2
Z Grav,conn,{2} = cA (1 − cA )(ZA
− ZD
)(ZA
− ZD
)

• Generalising, we find that the connected “wormhole” analogue with

NB boundaries must be:Z

Grav,conn,{NB }= f (cA , NB )

NB
Y

i
i
(ZA
− ZD
)

i=1

where f (cA , NB ) is a calculable function.

35 / 38• Now we work backwards and ask if there is any seed over which themultiple Poincaré sum yields the above expression.
• It is not obvious that any such expression should exist. However wefind that there are indeed many choices of seed that do this job.
• They are not unique and they do not correspond to the simplest

choice, namely the identity seed for each boundary.
• Still it is remarkable that seed functions exist that work for an

arbitrary number NB of boundaries. The discussion is too technical topresent in more detail here.

36 / 38

Geometry, Representation Theory and Quantum Fields

259

Outline

1 AdS3 /CFT2 and Poincaré Series

2 The CFT Side

3 Poincaré Series and WZW Models

4 Relation to Narain Lattices

5 Multiple boundaries

6 Conclusions

Conclusions• We have generalised AdS3 /CFT2 to the case of non-Abelian

Chern-Simons gravity at positive integer levels k. We found acalculable pattern of averaging over CFT’s and explicitly verified it fora number of examples.
• There is as yet no clear understanding of why we find a single CFT

dual in some cases and an average over CFT’s in others.
• Recent work of [Schlenker-Witten 2022] argues, in the conventional case,that there is no averaging for observables below the black holethreshold, and that averaging reflects the chaotic behaviour of black
holes.
• Our examples are integrable and are not expected to have any chaotic

behaviour, yet we find (discrete) averaging. This seems puzzling.

37 / 38

260

OCAMI Reports Vol. 4 (2022)• For SU(N)1 we find only physical averages (with positive weights).

These are precisely the cases that correspond to compactified freebosons, and our result is consistent with the existence of a positive
Siegel-Weil measure for such theories.
• Very recently [Hikida-Nishioka-Takayanagi-Taki 2021] it has been suggested

that the same bulk Chern-Simons theories that we consider, but on S 3and at level k = −2, describe deSitter gravity. This is very intriguing.
Perhaps one can apply some analogue of Poincaré sums to this(non-unitary) case.

38 / 38

Thank you

Geometry, Representation Theory and Quantum Fields

261

Confinement in Adjoint QCD from Seiberg
Witten Geometry
Emily Nardoni

Abstract. Standard lore suggests that four-dimensional SU(N) gauge theory
with 2 massless adjoint Weyl fermions (”adjoint QCD”) flows to a phase withconfinement and chiral symmetry breaking. In this talk, we will test andpresent new evidence for this lore. Our strategy involves realizing adjoint QCD
in the deep IR of a renormalization group flow descending from SU(N) SeibergWitten theory, deformed by a soft supersymmetry-breaking mass for its adjoint
scalars. A crucial role in the analysis is played by a dual Lagrangian thatoriginates from the multi-monopole points of Seiberg-Witten theory, and which
can be used to explore the phase diagram as a function of the supersymmetrybreaking mass.

(E. Nardoni) Kavli Institute for the Physics and Mathematics of the Universe (IPMU),
University of Tokyo

262

OCAMI Reports Vol. 4 (2022)

The Nekrasov partition function and Schur index
of gauged Argyres-Douglas theories
Takahiro Nishinaka
Abstract. We study the Nekrasov partition function (in the classical limit)
and Schur index of two 4D N=2 conformal SU(2) gauge theories coupled to
Argyres-Douglas (AD) theories, and show that these quantities are obtained,by a simple change of variables, from the same quantities of the SU(2) gauge
theory with four fundamental flavors. This change of variables maps ”q” to”q 2 ” or ”q 3 ”, where q” is the exponential of the UV gauge coupling for
the Nekrasov partition function while it is the superconformal fugacity for the
Schur index. Using this relation, we can particularly read oﬀ how the S-duality
acts on the UV gauge coupling of these gauged AD theories.

(T. Nishinaka) Osaka City University

Geometry, Representation Theory and Quantum Fields

263

Symmetries, universal defects, and holography
Natalie Paquette

Abstract. In this talk, we discuss and characterize certain line and surface defects in supersymmetric quantum field theory and string theory. In
particular, we study the constraints arising from gauge or BRST invarianceon admissible defects. This point of view has a mathematical translation interms of a subject called Koszul duality, and its generalizations, which we will
explain. We also discuss applications of our Koszul duality point of view forholography: both celestial holography and AdS/CFT. We will show that thispoint of view provides an expedient way to derive universal symmetry algebrasin (holographic) systems. This is based on various works in collaboration with
Kevin Costello, and with Brian Williams.

(N. Paquette) University of Washington

264

OCAMI Reports Vol. 4 (2022)

Symmetries, universal defects, and holography
Geometry, Representation Theory, & Quantum Fields
Natalie M. Paquette
University of Washington
collaborations w/ Kevin Costello & Brian Williams

What we will talk about today:
1.
The picture: bulk/defect couplings
2.
The setting: twisted theories
3.
The translation: Koszul duality
4. The application: AdS/CFT & celestial symmetries

Geometry, Representation Theory and Quantum Fields

265

Basic constraints from gauge invariance de ne universal defects in perturbative QFT.
As with anything universal in QFT, they capture information about symmetries.

The defects can be lines, surfaces, ….

fi

Formalizing this notion at the level of algebras of local operators leads to Koszul duality, and various
generalizations

Coupling an order-type defect to some Lagrangian theory

S = SA + SB + SAB

generalized Wilson
line

di erential gradedalgebras⊗ℬδ = δA + δB

What are the constraints on SAB (actually PExp) imposed by gauge (BRST) invariance?

SAB ↔ α ∈ MC(⊗ ℬ)

Koszul dual

α:

!

→ℬ

ff

𝒜

𝒜

𝒜

𝒜

δα + α ⋆ α = 0

Claim:!

is the algebra of operators on the “universal line defect” for theory A

266

OCAMI Reports Vol. 4 (2022)

The concrete setting: twisted theories
Begin w/ a supersymmetric eld theory on at space, Euclidean signature
General twisted theories: δ → δ + Q

Q2 = 0choice of nilpotent supercharge

Compute cohomology with respect to the twisted BRST di erential{Q, Qμ} = iPμcf.

ϕ : Spin(d) → GR

T ϕ = {Q, . . }A- and B-twist of 2d= (2,2) theories•• Half-twist of 2d = (0,2) theories (CDOs, cdR,..)= 2 theories (chiralHolomorphic twist of 4d•to obtain topological eld theories→ Ed-algebras[Lurie,…]algebra)

ff

fl

fi

fi

𝒩

𝒩

𝒩

= 2 theories𝒩• Holomorphic-topological twist of 3d⋮•

E1 /A∞-algebra: topological theory in 1d

Descent [Witten, Moore-Witten,…]

L∞, E2-algebra: topological theory in 2d

vertex/chiral algebra: holomorphic theory in 2d

local
operators,
not Q-closed

Q-closed
k-formoperators• higher brackets/products on algebras of local ops
generate couplings/deformations•{Q, Qμ} = iPμ:=1Qμ (…Qμk( ))dx μ1 ∧ … ∧ dx μk
k! 1

→

∫γ

(k)

Q(O (k)) = d (k−1)
𝒪
𝒪

𝒪

𝒪

𝒪

𝒪

[Beem-Ben-ZviBullimore-DimofteNeitzke](d → dA in gauge thys)

(k)

(k)

is Q-closed for γk a cyclekHol’c-top’l: [Yagi, Costello-Dimofte-Gaiotto]Geometry, Representation Theory and Quantum Fields

267

The Maurer-Cartan equation & bulk/defect couplingŝ{Q, Q} = iPt

PExp

( ∫ℝt

)

(1)

∈

translations along ℝt are cohomologically trivial∴(homotopy) associative algebra
Add in to a top’l QM, thy B with op algℬ (homotopy associative)⊗ℬ

SA + SB

local operator
coh. degree/ghost number 1

Claim: This ``generalized Wilson line”/bulk-defect coupling is BRST invariant to all orders in perturbation
theory i=0[Costello-Paquette, Paquette-Williams]

δ + μ2( , ) + μ3( 3) + … = 0[Gaiotto-Oh]

𝒪

𝒪

𝒪

𝒪

𝒪

⋆

ff

𝒜

𝒪
𝒪

𝒪

𝒜

𝒪

δ +

↔

Koszul Duality & the Universal Line Defectα ∈ MC(⊗ ℬ)
s.t.

(ϵ ⊗ 1ℬ)α = 0 ∈ ℬ

ϵ:

ϕα :
!

→ℂ

,

→ℬ

:= ℝHom (ℂϵ, ℂϵ)!

ℝt × Mϵ → ℝt

αuniv ∈

Require coupling preserves vacuum

mutually commuting symmetries of vacuum

⊗

ϕα

→ℬ

!

α = ϕ̃α(αuniv)𝔤

any other coupling can be obtained from theuniversal coupling by application of an algebra-preserving
map ϕ̃α : ! → ℬ
𝔤

𝒜

𝒜

𝒜

𝒜

𝒜

𝒜

𝔤
fi

𝒜

ϕ̃α

U( )

example commuting diagram

𝒜

𝒜[Loday-Vallette,
Lurie, …]

α ∈ g* ⊗ ℬ ⊂ C*( ) ⊗ ℬ[1]augmentation(choice of massive vacuum
s.t. compacti ed thy trivial)

𝒜

!

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OCAMI Reports Vol. 4 (2022)

ϕ∈V

Φ = ϕ + θ +ψ+ + θ −ψ− + …
Q-closed: ϕ, ψ̄−≃

Descendants along ℝt:

ϕ(1)[Paquette-Williams]= 2 free chiral multiplet

Example 1: Twisted 3d⨂n≥0∼ ψ−dt∨S*(V(n))⊗holomorphic-topological twist[Aganagic-Costello-McNamara-Vafa,
Costello-Dimofte-Gaiotto]

⨂
m>0

∂nzϕ

ψ̄ (1)− ∼ ∂z̄ϕdtℝt × ℂz∧* (V(m))
∂m−1
ψ̄−
z

1
1
J[k]∂kzψ̄ (1)+K[ j]∂ jzϕ (1)−∑ ∫ k!
∑ j!
k≥0 z=0
j≥0

!

≃

⨂
n<0

commuting algebrasbulk ops. annihilated by Q
MC eqn trivially satis ed∨
S*(V(n))⊗⨂m≤0∧* (V(m))
K[m]

fi

𝒜

𝒜

𝒩

J[n]

Example 2:∫− BF theory in hol’c/top’l twist
B = B(z, t)dz
A = Az̄(z, t)dz̄ + At(z, t)dt

B ∧ FA

0 = C*(

Q0 =

=

Take

[Paquette-Williams]

MC elements ofii
N: {ψ , χj} = ℏδj ,0⊗Ni, j = 1,…, N

∑

n+m≥0

[[z]] ⊕ *[[z]][−1], Q0)c m a n bfab∂z c ∂z cα=∑ne.g. suppose ρ = 0 for n ≥ 1δδb m a n+ fac
∂z c ∂z Bb n+m
c
δ∂n+m
c
δ∂
Bc
z
z

α[n]

α[n] =

Q0α + α ⋆ α = 0
α[0] :

𝔤

𝔤
𝒞
𝒞

𝒜

𝔤

𝔤

𝔅

𝒜

𝔤

The full computation says α represents[[z]]→ End(ℝN )1 jρ [n]∂nz Ata χjψ i∑ ∫ n! a,in≥0 ℝt1
ρ[n]ji,a∂nzc aψ i χjn!

Geometry, Representation Theory and Quantum Fields

269[Paquette-Williams]

Example 2, cont’d: Koszul & quantum corrections

1
1 a
J [n]∂nz Ata +K [n]∂nzBt,a∑ ∫ n! a∑ ∫ m!
n≥0 ℝtm≥0 ℝtc[Ja[k], Jb[l]] = fab
Jc[k + l]U( [[z]] ⊕ *[[z]][−1])b c[Ja[k], K b[l]] = fac
K [k + l](ℏ) :dJa[n] =∑b
facJb[r]K c[s]

∑

a b
fbc
K [r]K c[s]r+s=n−1dK a[n] =𝔤𝔤𝒪r+s=n−1

Koszul duality beyond associative algebras
Defects beyond lines→

e.g. twisted theory with holomorphic surface ℂz[Costello-Paquette]

vertex algebra!
OPEs from Koszul duality
holomorphic descent

Formalizing Koszul duality for vertex algebras is a work in progress!
[Li,…]

Order-by-order Feynman diagrams•• Impose constraints on universal hol’c defect OPEs by canceling BRST variation
holomorphic Chern-Simons theory on ℂ3z,w1,w2(worldvolume of B-branes)1∂k1 ∂k2 A aJ [k , k ]∑ ∫ k1 !k2 ! w1 w2 z̄ a 1 2

k1,k2≥0

see also [Gwilliam-Williams]

Jb[l1, l2](0)Jc[m1, m2](z) ∼Ja[1,0](0)Jb[0,1](z) ∼ℂz

1 af J [l + m1, l2 + m2]
z bc a 11 c
1c df Jc[1,1] + ℏ K fef ae
fbf Jc[0,0]Jd[0,0]
z ab
z

⋮

ℂ2w1,w2

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OCAMI Reports Vol. 4 (2022)

Koszul duality & Celestial Holography[Costello-Paquette]rich symmetry algebras at tree-level: those of at space holography!

• The universal defect of 6d holomorphic BF thy(twistor space uplift of SD Yang-Mills)•ℙ ≃ ℝ4 × ℂℙ1z

• The universal defect of 6d Poisson BF theory(twistor space uplift of SD gravity)•

Jb[r, s](0)Jc[k, l](z) ∼1 af Ja[r + k, s + l](z)z bc

K[r, s](0)K[k, l](z) ∼Jb[r, s](0)J*c [k, l](z) ∼

1 a
f J*
a [r + k, s + l](z)z bc

K[r, s](0)K*[k, l](z) ∼positive/negative helicity gluons & gravitons in conformal basis

1(rl − ks)K[r + k − 1,s + l − 1](z)z1(rl − ks)K*[r + k − 1,s + l − 1](z)z[Guevara-Himwich-Pate-Strominger, Strominger]
Koszul duality for a holomorphic
defect on ber of twistor space
governs universal at space
asymptotic symmetries

!

ℂz

fl

fl

fi

𝕋

𝒜𝒜

Correlators of currents reproduce some simple amplitudes,
esp. when including B^2 term

Koszul duality & Celestial Holography
quantum corrections order-by-order!
For SDYM + axion:[Costello-Paquette]

Geometry, Representation Theory and Quantum Fields

271[Costello-Paquette]

Curved Koszul duality & (Twisted) AdS/CFT

D-branes, even twisted, can backreact on the geometry.
In at least some special examples, one can fully incorporate this e ect.
e.g. B-model string theory:-holomorphic CS theory-Kodaira-Spencer “supergravity”!

S → S + SBRℂzδgaugeSBR ≠ 0 on ℂz
SBR =ϵ ijw̄idw̄j1μμBR μdzdw1dw2 ∼ Nμ∂ μdzdw1dw2∫ℂ3 (w1w̄1 + w2w̄2)2 z
2 ∫ℂ3• coupling to identity element of vertex algebra• total BRST variation, including this contribution, must cancel• curved Koszul duality (source term in MC equation)

In classical (N → ∞) limit, this was used to compute central extensions of boundary chiral algebrain an example of twisted holography (AdS3/CFT2)

see also [Costello, Costello-Li, Costello-Gaiotto,…]ff

𝒜𝒜

Loop corrections for these theories are in progress [Costello-Paquette-Williams]

Upshot:•Defects in QFT and holography are being increasingly studied. What are theconstraints imposed by gauge invariance on defects?
• Mathematical framework that answers this question is Koszul duality. The Koszul
dual algebra furnishes a universal defect. Any other defect operator algebras must
represent the Koszul dual.
• Practically, compute BRST variation of Feynman diagrams in pert. expansion.
Requiring their cancellation will produce commutators, OPEs,…
• Koszul duality for (curved) vertex algebras needs more input on the math side!
• Computing the Koszul dual has given us: line defects in various twisted SUSY
gauge theories, celestial algebras + loop corrections, some OPE data of protected
operators in AdS3/CFT2 (at loop level)!…
• New playground in mathematical physics, intersections with homological algebra

272

OCAMI Reports Vol. 4 (2022)

Thank you!

Geometry, Representation Theory and Quantum Fields

Quantum codes from conformal field theory
Rajath Radhakrishnan

Abstract. I will describe an explicit map relating rational conformal field
theories (RCFTs) and associated bulk Chern-Simons theories to quantum stabilizer codes. Using this map, I will show that point operators in an RCFT
correspond to the stabilizer group of the associated quantum code. 0-form
symmetries of an RCFT are implemented by topological line operators. I willshow that the point operators living at the end of these topological line operators correspond to general elements of the Pauli group. This correspondence
leads to a description of orbifolding at the level of the quantum code.

(R. Radhakrishnan) Queen Mary University of London

273

274

OCAMI Reports Vol. 4 (2022)

Quantum

Conformal

from

codes

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TheoryFieldsQuantum

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City University

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Raj ath Radhakrishnan
Queen
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Geometry, Representation Theory and Quantum Fields

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276

OCAMI Reports Vol. 4 (2022)

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278

OCAMI Reports Vol. 4 (2022)

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280

OCAMI Reports Vol. 4 (2022)

AbelianAn

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Geometry, Representation Theory and Quantum Fields

281

Non-invertible Symmetries in Higher Dimensions
Shu-Heng Shao

(S. Shao) Stony Brook University

282

OCAMI Reports Vol. 4 (2022)
Non invertible

higherfromsymmetries-

gauging*

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286

OCAMI Reports Vol. 4 (2022)2=* -0=1-12( boson )=L ( I +9A -1713 + RARB )

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288

OCAMI Reports Vol. 4 (2022)

Magnetic quivers and line defects
Marcus Sperling

Abstract. Supersymmetric Sp(k) quantum chromodynamics with 8 supercharges in space-time dimensions 3 to 6 can be realised by two diﬀerent Type II
brane configurations in the presence of orientifolds. Consequently, two types ofmagnetic quivers describe the Higgs branch of the Sp(k) SQCD theory. Thisis a salient example of a general phenomenon: a given hyper-Kahler Higgs
branch may admit several magnetic quiver constructions. It is then natural
to wonder if these diﬀerent magnetic quivers, which are described by 3d N
= 4 theories, are dual theories. In this talk, I discuss how the unitary andorthosymplectic magnetic quiver theories are subjected to a variety of tests,
providing evidence that they are IR dual to each other. For this, sphere partition function and supersymmetric indices are compared. Also, I will introduce
half BPS line defects and find interesting regularities from the viewpoints ofexact results, brane configurations, and 1-form symmetry.

(M. Sperling) Shing-Tung Yau Center, Southeast University

Geometry, Representation Theory and Quantum Fields

289

Magnetic quivers and line defects
Marcus Sperling
Shing-Tung Yau Center, Southeast University (Nanjing)“Geometry, Representation Theory and Quantum Fields”— Mar 23, 2022 —Based on: arXiv:2111.02831 with Satoshi Nawata, Hao Wang, and Zhenghao Zhong+ previous works arXiv:1904.12293, 1912.02773, 2004.04082, 2007.04667, etc.

Motivation
Focus on hyper-Kähler moduli spaces of theories with 8 supercharges
Higgs branches H in d=3,4,5,6

Coulomb branch C in d=3

Lagrangian theories: H = hK-quotientmany quantum corrections, but under control!

classical physics, good a weak coupling

space of dressed monopole operators

However, Higgs branches are not classically exact!
ÝÑ changes at special points with extra massless states
Examples: Higgs branch H changes at strong coupling§ 6d: physics of tensionless strings§ 5d: physics of massless gauge instantons§ 4d: physics of Argyres Douglas points

Question: How to evaluate these new Higgs branches?

at such points, H becomesvery different from theclassical result!

290

OCAMI Reports Vol. 4 (2022)

Magnetic Quivers
Rely on understand of 3d N “ 4 Coulomb branches of Lagrangian theories.
ÝÑ Hilbert series techniques [Benvenuti, Cremonesi, Felito, Hanany, Kalveks, Mekareeya, MS, Zaffaroni, ...]For a theory T in dimension d=3,4,5,6 with 8 supercharges and a given phase P ,derive a magnetic quiver QpT, P q such that
H ptheory T in phase P q “ C pmagnetic quiver QpT, P qq
holds as equality of moduli spaces.
Question: How to derive?
ÝÑ use brane systems in Type II string theory

Magnetic quivers expand understanding of novel Higgs branches[Akhond, Van Beest, Bourget, Cabrera,§ Dimension, global symmetry, etc.

Carta, Closset, Eckhard, Giacomelli,§ Representation content of chiral ring — Hilbert series

Grimminger, Hanany, Hayashi, Kalveks,§ Phase (Hasse) diagram

S-S. Kim, Mekareeya, Schäfer-Nameki,ÝÑ see Amihay Hanany’s talk for details

MS, Y. Wang, Yagi, Zafrir, Zhong, ...]

Emerging challenges: new classes of quiver theories inevitably arise
1

Exists any systematic to the new types of quivers? ÝÑ Recent [2111.00026] w/ Zhenghao

2

A given Higgs branch X may admit several magnetic quiver constructions
3

2

...
1

2

5

6

4

2

E8 Ñ
Ð
Omin

...
2

2

4

...
6

8

6

ÝÑ Q: Do these agree as honest 3d N “ 4 theories? ÝÑ Today’s focus!

4

2

2

Geometry, Representation Theory and Quantum Fields

Outline1 Two magnetic quivers for Sp SQCD

2 Exact partition functions

3 Line defects

4 Summary & Outlook

Outline1 Two magnetic quivers for Sp SQCD

2 Exact partition functions

3 Line defects

4 Summary & Outlook

291

292

OCAMI Reports Vol. 4 (2022)

3d Sp SQCD and its two mirror quivers
Well-known example: 3d N “ 4 Sppkq SQCD with Nf fundamental flavours
D two brane construction: with O3 or O5 orientifold ñ D two mirror quivers
for Nf ě 2k ` 2:
1...

O5 plane [Hannay & Zaffaroni ’99]
1

2

k

...
2k´1

2k

2k
2k
Nf ´ 2k ´ 1 nodesk

O3 plane [Feng & Hanany ’00]
1

1

...
2

2

...
2k

2k

...

2k`1

2k`1

2k

2k

2

2

pNf ´2k´1q ˆ SOp2k ` 1q nodes
pNf ´2kq ˆ Sppkq nodes

6d Sp SQCD and its two magnetic quivers
6d N “ p1, 0q Sppkq must have Nf “ 2k ` 8, but again two realisations exist
Magnetic quiver via O8 plane [Cabrera, Hanany, MS ’19]
magnetic gauge group

k`3

diag
GU “ GUwithnaive {Up1qśGUnaive “
I UpmI q...
12

2k`4 2k`5 2k`6

2

k`4

Magnetic quiver via O6 plane [Cabrera, Hanany, MS ’19] :magnetic gauge group
2

...
2

2

diag
GOSp “ GOSp
withnaive {Z2śGOSpnaive “I SOp2kI qˆSppnI q...
2k`6 2k`8 2k`6

2

2[2007.04667]

Geometry, Representation Theory and Quantum Fields

293

5d Sp SQCD and its two magnetic quivers
Same logic for 5d N “ 1 Sppkq with Nf ď 2k ` 5‚ 5-brane web + O7: unitary magnetic quiver‚ 5-brane web + O5: orthosymplectic magnetic quiver
Naturally grouped into exceptional En families [Bourget, Grimminger, Hanany, MS, Zhong ’20]
family# flavours

k “ 1 case

E8

Nf “ 2k ` 5

E7

Nf “ 2k ` 4

E6

Nf “ 2k ` 3

E5

Nf “ 2k ` 2

E8
Omin
E7
Omin
E6
Omin
E5
Omin

E4´2l

Nf “ 2k ´ 2l ` 1, k ě l ě 0

l “ 0 : Omin

E3´2l

Nf “ 2k ´ 2l, k ě l ě 0

l “ 0 : Omin

1 Two magnetic quivers for Sp SQCD

3 Line defects

4 Summary & Outlook

1
2

...
2

2

...
2

2k`2 2k`4 2k`2
k`1

D4

Outline2 Exact partition functions

Ex: E7 family

A2

...
1

2

2k`2

k`2

2

1

2

294

OCAMI Reports Vol. 4 (2022)

Exact partition functions — mirror quivers of 3d Sp SQCD

Mirror quivers are expected to be dual, sincekNfthey are mirrors of Sppkq SQCD, Nf flavours.

1

3

1

4

1

5

1

6

2

4

2

5

2

6

‚ Known subtleties: orthosymplectic quiver
has no apparent FI terms.
‚ Known data: CB and HB Hilbert series[Hanany, Kalveks, Cabrera, Zhong]Additional data [Nawata, MS, Wang, Zhong]‚ superconformal index for various examples
agree (in perturbative expansion)

superconformal index
ˆ
˙
ˆ
˙
ˆ
˙
?
15
84
300 173
1 ` q 2 ` t2 ` q ´17 ` 4 ` 3t4 ` q3{2´ 2 ´ 2t2 ` 3t6
t
t
t6
t
ˆ
˙
825 707` q2 138 ` 8 ´ 4 ´ 2t4 ` 5t8 ` ¨ ¨ ¨
t
t
ˆ
˙
˙
ˆ
?
28 q
300
1925 649
1 ` 2 ` q ´29 ` 4 ` 2t4 ` q3{2´ 2 ´ t2 ` t6
t
t
t6
t
ˆ
˙
8918 5643` q2 376 ` 8 ´ 4 ´ t4 ` 3t8 ` ¨ ¨ ¨
t
t
ˆ
˙
ˆ
˙
?
45 q
770
7644 1714
1 ` 2 ` q ´46 ` 4 ` t4 ` q3{2´ 2 ` t6
t
t
t6
t
ˆ
˙
52920 24574` q2 988 `´´ t4 ` 2t8 ` ¨ ¨ ¨
t8
t4
ˆ
˙
ˆ
˙
?
66 q
1638
5775 929
1 ` 2 ` q ´67 ` 4 ` t4 ` 4q3{2
´ 2
t
t
t6
t
ˆ
˙
222156 78299` q2´` 2143 ` 2t8 ` ¨ ¨ ¨
t8
t4
ˆ
˙
ˆ
˙
ˆ
˙
?
28
335
2492 1042
1 ` q 2 ` 3t ` q` 52 ` 6t4 ` q3{2` 2 ` 48t2 ` 13t6
t
t4
t6
t
ˆ
˙
13524 2524
` q2
´ 4 ´ 96 ` 40t4 ` 23t8 ` ¨ ¨ ¨
t8
t
ˆ
˙
ˆ
˙
?
45
980
1 ` q 2 ` t2 ` q ´2 ` 4 ` 3t4 ` ¨ ¨ ¨ .
t
t
?
66 q
1 ` 2 ` ¨¨¨tExact partition functions — magnetic quivers of 5d Sp SQCD
Magnetic quivers could potentially be dual as 3d N “ 4 theories.
Known indicators‚ CB and HB Hilbert series agree
Additional data [Nawata, MS, Wang, Zhong]‚ superconformal index for E4,5,6 quivers‚ Hasse diagrams agree
En

E4(Subtleties: SCI for unframed OSp quiver requires
E5

careful choice of magnetic lattice [Bourget, Grimminger,
E6

Hanany, Kalveks, MS, Zhong ’20] )superconformal index
ˆ
˙
ˆ
˙
ˆ
˙
5
3
?
24
200
1000 451
1 ` q 2 ` t2 ` q ´26 ` 4 ` t4 ` 2q 4 t5 ` q 2´ 2 ` t6
t
t
t6
t
ˆ
˙
˘
7 `
3675 2824` q 4 ´2t3 ` 2t7 ` q2 373 ` 8 ´ 4 ` t8 ` . . .
t
t
ˆ
˙
ˆ
˙
?
45 q
770
7644 1714
1 ` 2 ` q ´46 ` 4 ` t4 ` q3{2´ 2 ` t6
t
t
t6
t
ˆ
˙
52920 24574` q2 988 `´´ t4 ` 2t8 ` ¨ ¨ ¨
t8
t4
ˆ
˙
?
78 q
2430
1 ` 2 ` q ´79 ` 4
` ...
t
t

Rmk: Overall computational challenges remain due to sheer size of quivers.

Geometry, Representation Theory and Quantum Fields

295

Refining global symmetriesÑ Q: refine 0-form sym?
Unframed orthosymplectic quivers: gauge 1-form sym Zdiag
2

»

fi
x

—
—
—
—
–

1
2

2

2

z1

4
y1 P Z2

4

6
y2 P Z2

4

4
y3 P Z2

2

2

ffi
ffi
ffi
ffi
fl

z2

w7

+ one or two continuous Up1qs

Ø p Aq
1

2

3

4

3

2

w1

w2

w3

w4

w5

w6

´1 “z,w “w “q,w “w “1q|“HSpAq pw1 “w7 “w42
63
5
q 2 “1“HSpBq pu1 “u7 “z,u2 “u6 “q,u3,4,5 “p“1q| 2
q “1

1 Two magnetic quivers for Sp SQCD

2 Exact partition functions

3 Line defects

4 Summary & Outlook

fi

»
SUp2q

—
Ø pBq —–HSOSp pzi “z,yi “x“1,qq“

Outline1{ZdiagQq
2

Identified discrete Z2 in unitary counterparts´1 “z,v“HSpCq pv1 “v5 “v6
2,3,4,5 “w“1,r“qq

2

1

2

3

4

3

2

1

u1

u2

u3

u4

u5

u6

u7

»

ffi
ffi
fl
{Zdiag
Qp
2

fi
v6

2

—
Ø pCq –

ffi
fl
1

2

3

4

3

v1

v2

v3

v4

v5

SUp2q

1
w

{ZdiagQr
2

296

OCAMI Reports Vol. 4 (2022)

Line defects
As another probe, can compare 21 BPS line defect expectation values.
3d N “ 4 theories admit two types: Wilson lines and vortex linesÝÑ D3-D5-NS5 brane systems [Hanany & Witten] + defect branes [Gomis & Passerini; Yamaguchi]‚ additional D5’ + stretched F1 ÐÑ Wilson line
‚ additional NS5’ + stretched D1 ÐÑ vortex line
Mirror symmetry exchanges Wilson/vortex line [Assel & Gomis]
New challenges§ Brane configurations with orientifolds and 12 BPS line defects§ Partition functions/ indices for (unframed) OSp quivers:
may need regulator and/or careful choice of magnetic lattice§ carefully evaluate allowed Wilson lines for unframed orthosymplectic quivers

3d Sp(k) SQCD and its mirror quivers
Wilson lines in unitary mirror quiverVortex lines in Sppkq
2k

¨¨¨

¨¨¨

¨¨¨

¨¨¨

L

Nf ´ L

SQM
1
1d
3d

k

¨¨¨

¨¨¨

¨¨¨

2

2k

1

2k´1

L

2k
2k

k

Nf ´ L

L

1

Sppkq

...
1

2

W

...

L

...
2k ´ 12k

2k
2k
Nf ´ 2k ´ 1 nodeskkVerified mirror symmetry via branes and partition functions2§ A-twisted index on S ˆ S
for vortex line in Sp1ÐÑ

B-twisted index on S 2 ˆ S 1
for Wilson line in mirror§ 3-sphere partition function (need regulator for Sppkq side a la [Benvenuti & Pasquetti ’11])
Rmk: vortex lines in both mirrors are computationally challenging

Geometry, Representation Theory and Quantum Fields

297

Wilson line in Sp node
But Wilson lines in both mirrors can be evaluated via B-twisted index on S 2 ˆ S 1
Ñ Found an intriguing pattern for any fundamental Wilson line
Examplessym. axis

1
W

...
2

2

2r

...

2r

1

...
2k

2k

2k`1

...
2k

2k

2

2

2

2

1

...

ÐÑ
1

...

2

W

...

2r

k

...
2k´1

2k

2k
k

Wilson line in SO nodesym. axis

1

...
2

2

W
2r

...
2r

...
2k

2k

2k`1

...
1

2

W
2r´1

...
2r

k

...
2k´1

2k

2k
k

1
W

...

`
1

2

...
2r´1

2r

k

...
2k´1

2k

...
2k

1

ÐÑ

1

2k
k

2k

298

OCAMI Reports Vol. 4 (2022)

5d Sp(k) SQCD and its magnetic quivers
Now, only magnetic quivers available: aim to match line defect expectation values!
ÝÑ Successful for Wilson lines for all En families, but open challenge for vortex lines.
Example: E7 family (again, verified via B-twisted index)
W

1
2

W

...
2

2

2r

...

2r

...
2k`2 2k`4 2k`2

2

2

2

2

k`1

ÐÑ

W

...
1

2

2r

...

2r

2k`2

k`2

W

2

1

1
2

...
2

2

W
2r

...
2r

...
2k`2 2k`4 2k`2
k`1

ÐÑ

...
1

2

W
2r´1

...
2r

2k`2

k`2

2

1

2

1

k`1

` W
1

...
2

...
2r´1

2r

2k`2

k`2

Geometry, Representation Theory and Quantum Fields

W

299

1
2

...
2

W

...

2

W

ÐÑ

...
2

2k`2 2k`4 2k`2
k`1

...
1

2

2k`1 2k`2

k`2

2

1

2

1

2

1

k`1

`

W

...
1

2

2k`1 2k`2

k`2
k`1

` W
1

...
2

2k`1 2k`2

Outline1 Two magnetic quivers for Sp SQCD

2 Exact partition functions

3 Line defects

4 Summary & Outlook

k`2

2

300

OCAMI Reports Vol. 4 (2022)

Summary & Outlook‚ Further evidence for duality between magnetic quivers by exact partition fcts.
‚ Identified Zdiag(+ some Up1qs) in unframed unitary and orthosymplectic quivers
2ÝÑ implies unitary counterparts for OSp quivers with non-trivial 1-form Zdiag
2‚ Matching pattern of Wilson lines uncovered!
ÝÑ expected to extend to unframed orthosymplectic quivers withśGnaive “ I SOpkI q ˆ SppnI q and unitary counterparts
Future directions & open questions:‚ Improve the computability of partition functions§ for A/B-twisted index need to solve BAE systematically‚ Clarify origin of Wilson line matching pattern and extend to vortex lines‚ Study the algebra of line defects, not just single expectation values

Thank you!

Geometry, Representation Theory and Quantum Fields

301

Backup: Magnetic quiver appetiser

Example: 5d N “ 1 Spp2q Nf “ 9finite coupling

infinite coupling

d7
Spp1q
Nf “7
Spp2q
Nf “9

1

2

3

4

4

1

‚

2
1

‚
d9

4

¨¨¨

¨¨¨
1

2

2

2

2

2

‚
Higgs branchenlargment

e8 1

SOp18q 4t ` µ4t s

SOp20q 2

H8 “ PErµ2

7

8

1

5

2

1
¨¨¨

1

d10

‚

SOp18q 2

2‚
symmetryenhancement1

Hfin “ PErµ2

¨¨¨

2

2

2

2

quiversubtraction

Higgs

t1u ‚

2

quiversubtraction

2

1

SOp20q 4t ` µ4

SOp20qt ` t4 ` µ10

pt3 ` t5 qs

302

OCAMI Reports Vol. 4 (2022)

Twisted Elliptic Genera
Kaiwen Sun
Abstract. The elliptic genera of 2d (0,4) SCFTs associated to the BPS
strings in 6d (1,0) SCFTs have been extensively studied in the past decade.
Some special 6d SCFTs allow twisted circle compactification, for example
when the gauge algebra admits an outer automorphism. In such cases, theelliptic genera can be generalized to twisted elliptic genera which have many
extraordinary properties. We systematically study the twisted elliptic genera
including their localization, Higgsing, modular bootstrap, spectral flow symmetry and twisted elliptic blowup equations. Geometrically, they are related
to the topological string partition function on genus-one fibered Calabi-Yau
threefolds. This is based on a joint work with Kimyeong Lee and Xin Wang.

(K. Sun) Korea Institute for Advanced StudyGeometry, Representation Theory and Quantum Fields

Twisted Elliptic Genera
Kaiwen Sun (KIAS)
Geometry, Representation Theory and Quantum Fields, Osaka

March 24, 2022

Joint work with Kimyeong Lee and Xin Wang, to appear

An overview
For the last 10 years, there has been huge progress on the following
relation web in both classification and computation.
6d (1,0) SCFTs classified in (Heckman-Morrison-Rudelius-Vafa 15)
1

are engineered by F-theory compactified on local elliptic CY32contain BPS strings with worldsheet theory as 2d (0,4) SCFTs

3

compactified on S 1 give 5d Kaluza-Klein (KK) theories
6d (1,0) SCFT

E2d (0,4) SCFT = ZR4 ×T 2

ref. top.
KK= ZR5d4 ×S
1 = Zlocal elliptic CY3

A simple example
1

6d (1,0) E-string theory2

5d SU(2) + 8F KK theory3

local half-K3 Calabi-Yau threefold

4

a series of 2d (0,4) O(k) gauge theories

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

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An overview

When 6d (1,0) SCFT has a discrete global symmetry, we can do twisted
circle compactification to a 5d KK theory (Bhardwaj-Jefferson-Kim-TaraziVafa 19). There are two kinds
1gauge algebra allows outer automorphism: folding vector multiplet

2

quiver structure has discrete symmetry: folding tensor multiplet

We focus on the first kind of twisted compactification. In such cases, therelations are generalized as

2d (0,4) SCFT

Etwisted

6d (1,0) SCFT

ref. top.
KK= ZR4 ×S 1 ×S 1 ,twisted = ZR5d4 ×S
1 = Zlocal genus-one fibered CY3

We are interested the twisted elliptic genera arising here.

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Outline

6d (1,0) SCFTs and twisted compactification
Twisted elliptic genera
Twisted elliptic blowup equations
Modular bootstrape on Γ(N)Spectral flow symmetry
Summary

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields

6d SCFTs and elliptic non-compact Calabi-Yau

Six is the highest dimension for SCFT
6d (2, 0)
ADE classification
6d (1, 0)Tons
F-theory compactified on elliptic non-compact CY3
Elliptic fibration over non-compact base surface B in which all curveclasses can be simultaneously shrinkable to zero volume
Atomic classification (Heckman-Morrison-Rudelius-Vafa 15)
Generalized quiver from non-Higgsable clusters
Tensor branch dimension called “rank” i.e. H 1,1 (B, Z)

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Rank One 6d (1, 0) SCFTs

Rank one 6d (1, 0) SCFTs are the natural elliptic lift of 4d N = 2 and 5d
N = 1 gauge theories.
Associated to Calabi-Yau as elliptic fibration over O(−n) → P1
Kodaira singularity type gives the gauge algebra G supported on the−n curve
Global flavor symmetry F on a non-compact curve
matters in representation R at intersection points between curves(n, G , F , R) are highly constrained by Calabi-Yau condition.

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Full List of Rank One 6d (1, 0) SCFTse.g. n = 4, SO(r + 8) + r F theories
n
12
8
7
6
6
5
5
5
4
4
4
4
4
3
3
3
3
3
3
3
3
3
3
3

G
E8
E7
E7
E6
E7
F4
E6
E7
so(8)
so(N ≥ 9)
F4
E6
E7
su(3)
so(7)
so(8)
so(9)
so(10)
so(11)
so(12)
G2
F4
E6
E7

F
−
−
−
−
so(2)12−u(1)6so(3)12−sp(N − 8)1
sp(1)3su(2)6 × u(1)12
so(4)12−
sp(2)1
sp(1)a1 × sp(1)b1 × sp(1)c1
sp(2)a1 × sp(1)b2
sp(3)a1 × (su(1)4 × u(1)4 )b
sp(4)a1 × Isingb
sp(5)1
sp(1)1
sp(2)3su(3)6 × u(1)18
so(5)12Kaiwen Sun (KIAS)

2(RG , RF )−−(56, 1)−(56, 2)−27−1 ⊕ c.c.
(56, 3)−(N, 2(N − 8))(26, 2)(27, 2)−1 ⊕ c.c.
(56, 2 ⊕ 2)−
(8, 4)(8v ⊕ 8c ⊕ 8s , 2)
(9, 4a ) ⊕ (16, 2b )(10, 6a ) ⊕ [(16s )b1 ⊕ c.c.](11, 8a ) ⊕ (32, 1bs )(12, 10) ⊕ (32s , 1)
(7, 2)(26, 4)(27, 3)−1 ⊕ c.c.
(56, 5)

Twisted Elliptic Genera

Full List of Rank One 6d (1, 0) SCFTse.g. SU(r ) + 2r F theories
n
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2

G
su(1)
su(2)
su(N ≥ 3)
so(7)
so(8)
so(9)
so(10)
so(11)
so(12)aso(12)bso(13)
G2
F4
E6
E7

F
su(2)1
so(7)1 × Isingsu(2N)1
sp(1)a1 × sp(4)b1
sp(2)a1 × sp(2)b1 × sp(2)c1
sp(3)a1 × sp(2)b2
sp(4)a1 × (su(2)4 × u(1)8 )b
sp(5)a1 ×?bsp(6)a1 × so(2)8sp(6)a1 × Isingb × Isingc
sp(7)1
sp(4)1
sp(3)3su(4)6 × u(1)24
so(6)12Kaiwen Sun (KIAS)

2(RG , RF )−(2, 8s × 1s )(N, 2N) ⊕ c.c.
(7, 2a ) ⊕ (8, 8b )(8v , 4a ) ⊕ (8s , 4b ) ⊕ (8c , 4c )(9, 6a ) ⊕ (16, 4b )(10, 8a ) ⊕ [(16s , 2b )1 ⊕ c.c.](11, 10a ) ⊕ (32, 2b )(12, 12a ) ⊕ (32s , 2b )(12, 12a ) ⊕ (32s , 1bs ) ⊕ (32c , 1cs )(13, 14) ⊕ (64, 1)(7, 8)(26, 6)(27, 4)−1 ⊕ c.c.
(56, 6)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields

Full List of Rank One 6d (1, 0) SCFTse.g. SU(r ) + (r + 8)F + Λ2 theories, Sp(r ) + (2r + 8)F theories
n
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

G
sp(0)
sp(N ≥ 1)
su(3)
su(4)
su(N ≥ 5)
su(6)∗
so(7)
so(8)
so(9)
so(10)
so(11)
so(12)aso(12)b
G2
F4
E6
E7

F
(E8 )1so(4N + 16)1
su(12)1su(12)a1 × su(2)b1
su(N +8)1 ×u(1)2N(N−1)(N+8)
su(15)1
sp(2)a1 × sp(6)b1
sp(3)a1 × sp(3)b1 × sp(3)c1
sp(4)a1 × sp(3)b2
sp(5)a1 × (su(3)4 × u(1)12 )b
sp(6)a1 ×?bsp(7)a1 × so(3)b8
sp(7)a1 ×?b ×?csp(7)1
sp(4)3su(5)6 × u(1)30
so(7)12Kaiwen Sun (KIAS)

2(RG , RF )−(2N, 4N + 16)(3, 12)1 ⊕ c.c.
a[(4, 121 ) ⊕ c.c.] ⊕ (6, 2b )[(N, N + 8)−N+4 ⊕ (Λ2 , 1)N+8 ] ⊕ c.c.
[(6, 15) ⊕ c.c.] ⊕ (20, 1)(7, 4a ) ⊕ (8, 12b )(8v , 6a ) ⊕ (8s , 6b ) ⊕ (8c , 6c )(9, 8a ) ⊕ (16, 6b )(10, 10a ) ⊕ [(16s , 3b )1 ⊕ c.c.](11, 12a ) ⊕ (32, 3b )(12, 14a ) ⊕ (32s , 3b )(12, 14a ) ⊕ (32s , 2b ) ⊕ (32c , 1c )(7, 14)(26, 8)(27, 5)−1 ⊕ c.c.
(56, 7)

Twisted Elliptic Genera

Twisted 6d (1, 0) rank-one SCFTs
When the gauge algebra allows outer automorphism, i.e. folding the
Dynkin diagrams, we can construct twisted 6d theories
Twisted circle compactification to 5d KK theories(Bhardwaj-Jefferson-Kim-Tarazi-Vafa 19,. . . ).
In rank one, one can fold both 6d vector and hyper multiplets.
We denote the truncated gauge algebra, flavor algebra and matter
representation as G̊ , F̊ , R̊.
Upon twist, fractional KK charges naturally appear.

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

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Twisted 6d (1, 0) rank-one SCFTs
KK-momentum shifts of representations under twist
G

G̊

A2r

(2)

Cr

(2)

Cr

A2r −1

R
Adj

→
→

R̊
Adj0 ⊕ F1/4 ⊕ Λ21/2 ⊕ 11/2 ⊕ F3/4

F⊕F
Adj

→
→

F0 ⊕ 11/4 ⊕ F1/2 ⊕ 13/4
Adj0 ⊕ Λ21/2

F⊕FΛ ⊕ Λ2→→F0 ⊕ F1/2
Λ20 ⊕ Λ21/2 ⊕ 10 ⊕ 11/2

Adj
V
S⊕C
Adj
F⊕F
Adj
V⊕S⊕C

→
→
→
→
→
→
→

Adj0 ⊕ F1/2
V0 ⊕ 11/2
S0 ⊕ S1/2
Adj0 ⊕ F1/2
F0 ⊕ F1/2 ⊕ 10 ⊕ 11/2
Adj0 ⊕ F1/3 ⊕ F2/3
F0 ⊕ 10 ⊕ F1/3 ⊕ 11/3 ⊕ F2/3 ⊕ 12/3

2

(2)
Dr +1

Br

(2)

F4

(3)

G2

E6

D4

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Some typical twisted 6d (1, 0) rank one theories(2)

There exist four pure gauge twisted theories: n = 6, E6 ,
(3)
(2)
(2)
n = 4, D4 , D4 and n = 3, A2 .
n
6
4
4
4
4
3
3
3
2
2
2
2
2
2
2
1
1

G
(2)
E6
(3)
D4
(2)
D4
(2)
Dr +4
(2)
E6
(2)
A2
(2)
D4
(3)
D4
A2r
A2r −1
(2)
D4
(3)
D4
(2)
D5
(2)
D6
(2)
E6
(2)
A2
(2)
A3

G̊
F4
G2
B3
Br +3
F4
C1
B3
G2
Cr
Cr
B3
G2
B4
B5
F4
C1
C2

R̊
−
−
−
2r (V0 ⊕ 11/2 )F0 ⊕ F1/2 ⊕ 10 ⊕ 11/2−
V0 ⊕ 11/2 ⊕ S0 ⊕ S1/2
F0 ⊕ F1/3 ⊕ F2/3 ⊕ 10 ⊕ 11/3 ⊕ 12/3
(2r + 1)(F1/4 ⊕ F3/4 ⊕ 10 ⊕ 11/2 )2r (F0 ⊕ F1/2 )2(V0 ⊕ 11/2 ⊕ S0 ⊕ S1/2 )2(F0 ⊕ F1/3 ⊕ F2/3 ⊕ 10 ⊕ 11/3 ⊕ 12/3 )4(V0 ⊕ 11/2 ) ⊕ S0 ⊕ S1/2
6(V0 ⊕ 11/2 ) ⊕ 12 S0 ⊕ 12 S1/2
2(F0 ⊕ F1/2 ⊕ 10 ⊕ 11/2 )6(F0 ⊕ F1/2 ⊕ 11/4 ⊕ 13/4 )6(F0 ⊕ F1/2 ) ⊕ 10 ⊕ Λ21/2

F̊
−
−
−
sp(2r )
sp(1)−
sp(1) × sp(1)sp(1)
so(4r + 2)
so(4r )
sp(2) × sp(2)sp(2)sp(4) × sp(1)
sp(6)
sp(2)
so(12)
so(12) × sp(1)

1
1

D4
(3)
D4

(2)

B3
G2

3(V0 ⊕ 11/2 ⊕ S0 ⊕ S1/2 )3(F0 ⊕ F1/3 ⊕ F2/3 ⊕ 10 ⊕ 11/3 ⊕ 12/3 )sp(3) × sp(3)sp(3)

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields

Elliptic Genera of 6d SCFT
Consider a elliptic non-compact Calabi-Yau X
Compactify F-theory on C2ϵ1 ,ϵ2 × T 2 × X
The full partition function contains three parts
Z6d = Zpoly Z1−loop Zell ,2Zell = 1 + E1 Qell + E2 Qell
+ ...

Zpoly comes from the perturbative partZ1−loop contains the contributions from tensor, vector and hyper
multiplets
One main goal to study 6d SCFT is to compute the d-string elliptic
genus Ed (τ, mG , mF , ϵ1 , ϵ2 ) of the associated 2d (0, 4) SCFTs on the
BPS strings
Ed is the natural elliptic lift the d-instanton Nekrasov partition
function, also certain Jacobi form with mG , mF , ϵ1 , ϵ2 as elliptic
parameters
Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Twisted elliptic genera

In the twisted cases, similarly one can define twisted elliptic genera
tw
tw
tw
tw
Z6d
= ZpolyZ1−loop
Zell
,

tw
tw 2
Zell
= 1 + Etw
1 Qell + E2 Qell + . . .

Now Etwd contains fractional qτ powers.
tw
KK
In twisted circle compactification, Z6d= Z5d.
Twisted elliptic genera for a few theories have been studied in(Kim3 -Lee 21, Kim3 21)
Now we would like go through all twistable theories
We want to keep everything elliptic and as 6d gauge theory, rather
than geometries or 5d KK theories.

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

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Approaches to twisted elliptic genera

In the following, I mainly talk about the last three approaches.
2d quiver gauge theories(2)

n = 4, D4 , (Kim3 21)

Twisting from Higgsing
many (Kim2 -Lee 19, Kim3 21, Hayashi-Kim-Ohmori 21)

Twisted elliptic blowup equations(2)

n = 3, A2 , (Kim3 -Lee 21)

Modular bootstrap
Spectral flow symmetry

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Background for blowup equations

Blowup equations are functional equations for Nekrasov partition
functions originally for 4d N = 2 SU(N) gauge theories(Nakajima-Yoshioka 03)
K-theoretic version – 5d N = 1 (Nakajima-Yoshioka 05, 09,Göttsche-Nakajima-Yoshioka 06, Keller-Song 12, Kim2 -Lee2 -Song 19)

further generalize to refined topological strings on local Calabi-Yau
threefolds (Huang-KS-Wang 17)
elliptic version for 6d (1, 0) SCFTs (Gu-Haghighat-Klemm-KS-Wang
18-20)
6d (2, 0) SCFTs, 5d N = 1⋆ (Duan-Lee-Nahmgoong-Wang 21)
6d twisted SCFTs and 5d KK theories (Kim3 -Lee 21)

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields

311

Twisted elliptic blowup equations
Let λ0 be a coweight of G invariant upon the twist. If (λ0 , λF ) is
admissible for the original 6d SCFT, and let λF̊ be the reduction of λF ,then the twisted elliptic genera Ed (τ, mG̊ , mF̊ , ε1,2 ) satisfy the following
Twisted elliptic blowup equations
2′′′
1
2 ||λG̊ || +d +d =d+δ

−nλG̊ · mG̊ + kF λF̊ · mF̊ +[a](−1)|λG̊ | θi nτ,(y − n2 ||λG̊ ||2 )(ϵ1 +ϵ2 )−nd ′ ϵ1 −nd ′′ ϵ2λG̊ ∈ϕλ0 (Q ∨ (G̊ ))X

R̊
× AG̊V (τ, mG̊ , λG̊ )AfracV (τ, mG̊ , λG̊ )AH (τ, mG̊ , mF̊ , λG̊ , λF̊ )× Etw
d ′ (τ, mG̊ + ϵ1 λG̊ , mF̊ + ϵ1 λF̊ , ϵ1 , ϵ2 − ϵ1 )× Etw
d ′′ (τ, mG̊ + ϵ2 λG̊ , mF̊ + ϵ2 λF̊ , ϵ1 − ϵ2 , ϵ2 )[a]= Λ(δ) θi (nτ, kF λF̊ · mF̊ + ny (ϵ1 + ϵ2 )) Etw
d (τ, mG̊ , mF̊ , ϵ1 , ϵ2 ).
For δ = 0, Λ(δ) = 1, we call it unity blowup equations. For δ > 0,Λ(δ) = 0, we call it vanishing blowup equations.
Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Universal formula for twisted elliptic genera E1
From the unity equations, we can solve twisted elliptic genera recursively.
In particular, the twisted one-string elliptic genera for the pure gauge
cases have the following universal formula:
∨

Etw
1 =

X

∨

(−1)|α |

α∈∆∨
l
∨

Y
Y
η4
η
η
Dα
.
[k]
D θ1 (mα )θ1 (mα −ε1,2 )θ1 (mα −2ε+ )θ1 (mβ )θ1 (mγ )β∈∆
α∨ ·β=1∨γ∈∆sα∨ ·γ=1∨ααHere D α = Det{1,0,0}
and D = Det{0,0,0}, and we define
[a]

[a]

θi,{d0 ,d1 ,d2 } =θi (nτ,−nmα∨ + (n − 2)(ϵ1 + ϵ2 )− n((d0 + d1 )ϵ1 + (d0 + d2 )ϵ2 )), [a1 ]

[a1 ]
[a1 ]
θi,{0,d,0}
θi,{0,0,d}θi,{d0 ,d1 ,d2 }∨ [a2 ]

[a2 ]
[a2 ]
α
Det{d
= det  θi,{0,d,0}
θi,{0,0,d}θi,{d.
0 ,d1 ,d2 }0 ,d1 ,d2 }[a ]3θi,{0,d,0}[a ]

3
θi,{0,0,d}

[a ]

3
θi,{d
0 ,d1 ,d2 }Besides, k are the fractional KK charges for the short roots.

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Twisted Elliptic Genera

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Structure of twisted elliptic genera
From the universal Etw
1 formula, we find the following universal expansion
for twisted pure gauge theories (v = e ϵ+ )∨hG̊ −1
Etw
1 =v

∞
X

1

χG̊kθ v 2n + qτN (1 + v 2 )∞XχG̊kθ+f v 2n + . . . .

n=0

n=0

qτ0 order gives 5d one G̊ -instanton Hilbert series(Benvenuti-Hanany-Mekareeya 10)
i/N

Fractional orders qτ , i = 1, . . . , N − 1 all have exact v expansion
formulas(3)

For example, for D4

2/3

theory the qτ

order of Etw
1 is

∞
X
2
(χG2f2+nθ (1 + v 2 + v 4 ) + χGf +nθ
(1 + v 2 ) + χGnθ2 )v 2n+3 .
n=0

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Modular bootstrape on Γ(N)We turn off all gauge and flavor fugacities to study the modular propertyof E1 (q, v ). In many cases, this can fully determine E1 (q, v ) with small
amount of initial data.
For untwisted theories, the modular ansatz was proposed in (Del
Zotto-Lockhart 18) on SL(2, Z)
For twisted cases, we propose the modular ansatz on congruencesubgroup Γ(N). Here N is the twist coefficient:

(2)
(2)
(2)

 2, A2r −1 , Dr , E6 ,(3)
N=3, D4 ,
(2)
4, A2r .
Inspired by the modular ansatz for genus-one fibered CY3(Cota-Klemm-Schimannek 19) (Knapp-Scheidegger-Schimannek 21)

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields

313

Modular bootstrape on Γ(N)For a theory on (−n)-curve with twist coefficient N = 2, 3, 4, we propose
modular ansatz for the reduced one-string elliptic genus:
Etw
1 (τ, ϵ+ ) =N (τ, ϵ+ )∨

η 12(n−2)−4+24δn,1 ∆2N ( Nτ ) N ϕ−2,1 (τ, 2ϵ+ )hG̊ −1s.

N
Here s = N−1(c − n−22 − δn,1 ). Numerator N (τ, ϵ+ ) is of weight6(n − 2) + 2Ns + 12δn,1 − 2hG̊∨ and index 4(hG̊∨ − 1) + n − hG∨ andN (Nτ, ϵ+ ) ∈ M⋆ (N)[ϕ−2,1 (Nτ, ϵ+ ), ϕ0,1 (Nτ, ϵ+ )].
∆2N are certain cusp forms on Γ(N):∆4 (τ ) =η(2τ )16,
η(τ )8∆6 (τ ) =η(3τ )18,
η(τ )6∆8 (τ ) =η(2τ )8 η(4τ )16,
η(τ )8and ϕ−2,1 , ϕ0,1 are Eichler-Zagier’s generators for weak Jacobi forms.
Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Modular bootstrape on Γ(2)1Consider the Z2 twist of E-string theory. 4 of the 8 masses are
shifted by π2 . The twisted one-string elliptic genus is
1
Etw
1 =
2

(2,3),(3,2)

X

4 Y
8
Y
θr (mi )θs (mj )η2(r ,s)=(1,4),(4,1) i=1 j=5.

Turing off the flavor fugacities, we find the modular ansatz
Etw
1 =
2

(2)

For pure D4

16∆4 ( τ2 ),
η8

∆4 (τ ) =η(2τ )16.
η(τ )8

theory on (−4) curve,
1Etw
1 (τ, ϵ+ ) =∆4 (τ /2) 2 N (τ, ϵ+ ).
η(τ )20 ϕ−2,1 (τ, 2ϵ+ )4Here N (τ, ϵ+ ) is of weight 0 and index 14. There are 64 coefficients
to fix N (2τ, ϵ+ ).
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Twisted Elliptic Genera

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Spectral flow symmetry

For twisted one-string elliptic genus, we find the spectral flow from R-R
sector to NS-R sector is induced by transformation:
KK
ER̊NS−Rv, q = ± q 1/4 n−hG∨v

KKER̊R−R q 1/2v

,q .

On the other hand, 2d (0, 4) analysis shows spectral flow shifts the KK
charges of hypermultiplets by 1/2:
R̊

KK+1/2KK
ER̊NS−R= ER−R.

These two properties together suggest interesting symmetry for thetwisted elliptic genera.

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Example: spectral flow symmetry for pure gauge theories

For 6d (1, 0) pure gauge G (k) theories on (−n)-curve, we find thefollowing spectral flow symmetry for the reduced twisted one-string
elliptic genera:
EG1

(k)

 q 1/2   q 1/2 n−3 (k)q,= − 2
EG1 (q, v ).
v
v
(2)

(2)

(3)

(2)

We explicitly checked this symmetry for A2 , D4 , D4 , E6 .
generalization of (Del Zotto-Lockhart 16, 18) for the untwisted cases
A2 , D4 , F4 , E6,7,8 .
relate twisted R-R elliptic genus to twisted NS-R

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

Geometry, Representation Theory and Quantum Fields(2)

Spectral flow symmetry for E6

Let us expand the twisted one-string elliptic genus as−3
Etw
1 (qτ , v , mF4 = 0) = v∞
Xbi,j q i v 2i+j .

i,j=0Then we find the following table for coefficients bij :
q\v
0
1
2
3
0
0
0
0
0
1/2
0
0
0
0
1
0
0
0
0
3/2
0
0
0
0
2
0
0
0
0
5/2 −1
0
0
0
0
−26
0
0
3
7/2 −520−378
04
0 −1079 0 −4056

Kaiwen Sun (KIAS)

4
0
0
0
0
0
1
0
0
0

5
1
0
0
0
−1
0
26
0
0

6
7
8
0
52
0
26
0
1079
0
378
0
0
0
4056
0
0
0
−26
0
0
0
−378
0
378
0
−4004
0
4004
0

Twisted Elliptic Genera

Summary

We find all twisted elliptic blowup equations for all twisted rank one
6d (1, 0) SCFTs
We compute the twisted elliptic genera for most of them
We determine the modular ansatz for most of the twisted elliptic
generaWe find study the spectral flow symmetry for the twisted elliptic
generaIn the cases with 2d quiver gauge description, we can directly prove
the spectral flow symmetry

Kaiwen Sun (KIAS)

Twisted Elliptic Genera

315

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On a rarely-mentioned class of 3d N=4
superconformal field theories
Yuji Tachikawa

Abstract. After giving an overview of known highly-supersymmetric conformal field theories in three dimensions, I would like to introduce a class of 3d
N=3 superconformal field theories which exhibit supersymmetry enhancement
to N=4 when certain strange numerical conditions on the Chern-Simons levels
are met. I also discuss if this enhancement can be understood in the context
of the 3d/3d correspondence. The talk is based on an unfinished project goingback at least five years, in collaboration with Jenny Wong, Benjamin Assel,
Alessandro Tomasiello, and Seyed Morteza Hosseini.

(Y. Tachikawa) Kavli Institute for the Physics and Mathematics of the Universe (IPMU),
University of Tokyo

Geometry, Representation Theory and Quantum Fields

317

On a rarely-mentioned class of
3d N =4 SCFTs

Yuji Tachikawa (Kavli IPMU)
Osaka City U., March 24, 2022
(Phrases in purple are hyperlinked if you download the slides.)1 / 49When Jaewon and Kazunobu invited me to this workshop in December,I accepted to give a talk, without thinking too much.
But later I realized that this is mostly a SUSY QFT workshop,
and that I don’t have any suitable recent work to talk about.
So I decided to revive in February an unpublished long-dormant projectbased on an idea I had long time ago (circa 2015?).
Since its inception I collaborated with many persons on this project:
Jenny Wong, Benjamin Assel,
Alessandro Tomasiello, Seyed Morteza Hosseini.
Luckily we made some progress since last month that I can report on.

2 / 49

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I have two introductions from distinct points of view:• One from highly-supersymmetric Chern-Simons-matter theories• Another from M5-branes on 3-manifolds
They somehow ﬂow to the same endpoint...

3 / 49Highly-supersymmetric
Chern-Simons theories

4 / 49

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319

Supersymmetric Chern-Simons terms are available up to N =3.
Realizing N ≥ 4 was thought impossible.
Breakthrough in Autumn 2007 – Early summer 2008:• N =8 [Gustavsson 0709.1260][Bagger-Lambert 0711.0955]• N =4 [Gaiotto-Witten 0804.2907][Hosomichi-Lee-Lee-Lee-Park 0805.3662]• N =6 [Aharony-Bergman-Jafferis-Maldacena 0806.1218]• N =5 [Hosomichi-Lee-Lee-Lee-Park 0806.4977]
They used various approaches,
which in the end can be explained uniformly as follows:

5 / 49Supersymmetric Chern-Simons terms are available up to N =3.
Realizing N ≥ 4 was thought impossible.
Breakthrough in Autumn 2007 – Early summer 2008:• N =8 [Gustavsson 0709.1260][Bagger-Lambert 0711.0955]

Contained ABJM as a specialcase. If only they realized

N =6 enhancement...

• N =4 [Gaiotto-Witten 0804.2907][Hosomichi-Lee-Lee-Lee-Park 0805.3662]• N =6 [Aharony-Bergman-Jafferis-Maldacena 0806.1218]• N =5 [Hosomichi-Lee-Lee-Lee-Park 0806.4977]
They used various approaches,
which in the end can be explained uniformly as follows:

5 / 49

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Mechanism of the enhancement
Let us start with an exercise in 4d.
We can achieve N =2 with general G and R.
When R is the adjoint rep., the superpotential in N =1 language is

W = tr Φ[A, B]
where Φ is from the vector multiplet and A, B form a hypermultiplet.
This has SU (3) acting on Φ, A, B .
This does not commute with SU (2)R of N =2 SUSY.

⇒ SUSY has to enhance, giving N =4 SYM.

6 / 49In 3d, we can achieve N =3 with general G and R.
The superpotential in N =2 language isW =X(tr Φi µi −

i

ki
2

tr Φ2i )where Φi is the adjoint scalar in the Chern-Simons supermultiplet
and µi is the moment map operator, constructed from the hypers.
As Φi ’s have no kinetic term, they can be integrated out, giving

W ∝

X 1
i

ki

tr µ2i .

When ki is chosen carefully, W can have
N =2 ﬂavor symmetry not commuting with N =3 R-symmetry.

→ SUSY enhancement!
7 / 49

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321

If the ﬁnal result has N =N SUSY,
we expect SO(N − 2) ﬂavor symmetry in the N =2 formalism:

For example, in the ABJM case,
we expect to see SO(6 − 2) = SO(4) emerging.

8 / 49

Indeed, U (N )k × U (N ′ )−k′ with two bifundamentals Ai , B i has

W ∝

1
k

tr(Ai B i )2 −1k′tr(B i Ai )2

which simpliﬁes, if k = k′ , to

W ∝

1
k

tr Ai B a Aj B b ϵij ϵab .

So we indeed see SU (2) × SU (2) = SO(6 − 2).

(I will be sloppy about the global structure of groups in this talk.)
9 / 49

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Enhancement to N =4 is in a sense simpler.

We couple N =3 Chern-Simons term to N =4 matter system.
In N =2 language, N =4 matter system has SO(2)F ﬂavor symmetry,
under which the moment map operator µ has charge +1, say.
This is broken by the superpartner of the Chern-Simons term:

W =X(tr Φi µi −

i

ki
2

tr Φ2i ).

10 / 49

After integrating out Φi , we have

W ∝

X 1
|i

tr µ2i .

ki{zcharge +2}

The SO(2)F symmetry assigning charge +1 to µi is broken generically,X 1
but it can happen that W ∝tr µ2i = 0.
kiiThen we see the enhancement back to N =4.

11 / 49

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323

The observation of Gaiotto and Witten was that,for G with a half-hyper in R,

W ∝

X 1
i

ki

tr µ2i = 0

is equivalent to G ⊕ R forming a super Lie algebra G .

• G is the bosonic part and R is the fermionic part of G .
• The ratio k1 : k2 : · · · is part of the structure constants of G .
• Rozansky-Witten twist of this N =4 theory is
the Chern-Simons theory of G . [Kapustin-Saulina 0904.1447]

12 / 49

For example,

U (N |M ) gives
U (N )k × U (M )−k with one bifundamental hyper.
(With two bifundamentals, it enhances further to N =6.)

OSp(N |M ) gives
SO(N )k × Sp(M )−2k with one bifundamental half-hyper.
(With two bifundamentals, it enhances further to N =5.)
In both cases, the structure constant of the Lie algebras is unique⇔ the ratio of Chern-Simons levels is ﬁxed

13 / 49

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The case OSp(4|2) is special.
The gauge group is SO(4) × SU (2) = SU (2)1 × SU (2)2 × SU (2)3 .
The half-hyper is in 4 ⊗ 2 = 21 ⊗ 22 ⊗ 23 .
Denote it by Qaiu as always. The condition for the SUSY enhancement is

W ∝

1
k1

tr(µ1 )2 +1k2

tr(µ2 )2 +1
k3

tr(µ3 )2 = 0

where µ1,2,3 are the moment map operators for SU (2)1,2,3 .

14 / 49

Three moment map operators areµ1 c b = ϵca µ1,ab ,µ1,ab = ϵij ϵuv Qaiu Qbjv ,µ2 k j = ϵki µ2,ij ,µ2,ij = ϵab ϵuv Qaiu Qbjv ,µ3 w v = ϵwv µ3,uv ,µ3,uv = ϵab ϵij Qaiu Qbjv

and it turns out

tr(µ1 )2 = tr(µ2 )2 = tr(µ3 )2 .
Therefore, the SUSY enhances to N =4 if and only if

W ∝

1
k1

tr(µ1 )2 +1k2

tr(µ2 )2 +1
k3

tr(µ3 )2 = 0i.e. if and only if

1
k1

+

1
k2

+

1
k3

= 0.

A weird condition!
15 / 49

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325

As I said, the ratio k1 : k2 : · · · is part of the structure constants of thesuper Lie algebra.
The condition

1

+

k1

1
k2

+

1
k3

=0

means that there is a one-parameter family of the ratio.
Therefore, the super Lie algebra OSp(4|2) comesin a one-parameter family.
In fact, it is the only simple super Lie algebra which comes in acontinuous family.
Often denoted as D(2, 1; α).

16 / 49

For us, k1,2,3 need to be integers. Are there integer solutions to

1
k1

Yes,

+

1
p(p + q)

1
k2

+

+

1
k3

=0 ?

1
q(p + q)

−

1
pq

= 0,

for example.

17 / 49

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We can generalize. If you like class S theories, you know thatthe trifundamental Qaiu of SU (2) is the TSU (2) theory.
More generally, there is a 3d N =4 theory known as
the TG theory with G3 symmetry.
It is obtained by putting 6d (2,0) theory of type G
on S 2 with three full punctures, further compactiﬁed on S 1 .
It has moment map operators µ1,2,3 for G3 symmetry.
They satisfy

tr(µ1 )n = tr(µ2 )n = tr(µ3 )nfor arbitrary n.

18 / 49

Consider, then, the 3d N =4 TG theory coupled with
N =3 Chern-Simons terms Gk1 × Gk2 × Gk3 .
Then, the condition for the enhancement to N =4 is

W ∝
i.e.

1
k1

tr(µ1 )2 +
1
k1

+

1
k2
1
k2

tr(µ2 )2 +

+

1
k3

1
k3

tr(µ3 )2 = 0= 0.

This is (the simplest type of) the rarely-mentioned classof 3d N =4 theory in the title.

19 / 49

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327

M5-branes on 3-manifolds

20 / 49

People love considering M5-branes on 3-manifolds,
starting from [Terashima-Yamazaki 1103.5748] and[Dimofte-Gaiotto-Gukov 1108.4389]
What kind of 3d theories do we get?
Before that, how many supersymmetries do we have?
That’s the question I’d like to raise today.

(N.B. In the ﬁrst day of the workshop, [Dongmin Gang] told us also about N =4
enhancement among these theories. There are some differences, though.
He talked about Tirred [M ] while I will talk about Tfull [M ], in his notation.)

21 / 49

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M5-branes on T 3 gives 3d N =8, obviously.
M5-branes on generic M3 can preserve 3d N =2.
Indeed, the holonomy of M3 is in SO(3)M .
The R-symmetry of the 6d theory is SO(5)R .
We turn on the R-symmetry background by embedding SO(3)M to

SO(3) × SO(2) ⊂ SO(5)R
Then SO(2) remains as a 3d R-symmetry, leading to 3d N =2.

22 / 49

As another big class, M5-branes on Σ2 × S 1 gives 3d N =4.
The holonomy of Σ2 is in SO(2)Σ .
The R-symmetry of the 6d theory is SO(5)R .
We turn on the R-symmetry background by embedding SO(2)Σ to

SO(2) × SO(3) ⊂ SO(5)R .
Then SO(2) × SO(3) remains as a 3d R-symmetry.
(Note that SO(2) commutes with itself!)
But in 3d N =N superconformal theories, the R-symmetry is SO(N )R .
By carefully studying how the R-symmetry acts on the supercharges,
we ﬁnd thatSO(2) × SO(3) ⊂ SO(3) × SO(3) ≃ SO(4)R ,i.e. we have 3d N =4.
Note that we only used the fact that the holonomy is in SO(2).
23 / 49

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329

How about the other cases?
To study these points, it is useful to recall
the classiﬁcation of 3-manifolds(N.B. [Dongmin Gang] also gave a nice review on the ﬁrst day.)24 / 49

Prime decomposition
First step is to decompose 3-manifolds along shared S 2 .

Such decomposition is known to be essentially unique:

M = M1 #M2 # · · · #Mn .
25 / 49

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Torus decomposition
Second step is to decompose prime 3-manifolds along shared T 2 .

In this step, as there are multiple ways to ﬁll T 2 in,we keep each piece to have (multiple) torus boundaries.
This decomposition is not quite unique
but all the non-uniqueness comes from well-understood examples.

26 / 49

A piece which cannot be further decomposedby cutting along T 2 is called atoroidal.

cut along S 2

Any 3-manifold −−−−−−→ prime 3-manifolds
cut along T 2

Prime 3-manifold −−−−−−→ atoroidal 3-manifolds(Essential uniqueness of these decompositions was proved by 1979.)
What are atoroidal 3-manifolds, then?

27 / 49

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331

Geometrization

Atoroidal manifolds are either• Seifert ﬁbrations, or• hyperbolic, i.e. H3 /Γ.
Originally conjectured by Thurston in 1982.
Proved by Perelman in 2006 using the Ricci ﬂow.

28 / 49

So, to understand M5-branes on 3-manifolds requires a few steps:• Understand M5-branes on Seifert ﬁbrations.
• Understand M5-branes on hyperbolic manifolds.
And then• Understand what happens when glued along T 2 .
• Understand what happens when glued along S 2 .

29 / 49

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What happens when glued along S 2

I haven’t see any paper about it.
As M5-branes on S 2 give a gapped theory, nothing happens, presumably.
Some TQFT effects? I don’t know.

30 / 49

What happens when glued along T 2

M5-branes on T 2 give N =4 SYM with gauge group G.
Then, a torus boundary means a G ﬂavor symmetry for a 3d theory.
Gluing the two = gauging them with an SL(2, Z) wall,
which can be decomposed to

T k1 ST k2 S · · · T kn−1 ST kn ,where T k : N =3 Chern-Simons term and S : N =4 T [G] theory.
31 / 49

Geometry, Representation Theory and Quantum Fields

333

M5-branes on hyperbolic manifolds
Tons of papers!
Goes back to:[Terashima-Yamazaki 1103.5748] for mapping tori
[Dimofte-Gaiotto-Gukov 1108.4389] for triangulations
Important reﬁnements e.g. in:[Chung-Dimofte-Gukov-Sułkowski 1405.3663][Gang-Yonekura 1803.04009]
This is not the place to review them.

32 / 49

M5-branes on Seifert ﬁbrations

Basically determined in a series of works, e.g.
[Gadde-Gukov-Putrov 1306.4320][Pei-Ye 1503.04809][Gukov-Putrov-Vafa 1602.05302][Gukov-Pei-Putrov-Vafa 1701.06567][Eckhard, Kim, Schäfer-Nameki, Willett 1910.14086][Cho-Gang-Kim 2007.01532]
But I have a bit more to say today.

33 / 49

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What is a Seifert ﬁbration, anyway?
It is a slight generalization of an S 1 ﬁbration over a surface Σ,
where you allow singular ﬁbers of the form

The 3d manifold is smooth, but the base Σ now has
singularities of the form C/Za .
(a can vary from a singular ﬁber to another singular ﬁber.)

34 / 49

Another way to phrase the construction is: start fromand perform

at a number of points. Seifert ﬁbrations of type
q1 q2
qn
(g;
,
,...,
).
p1 p2
pn
35 / 49

Geometry, Representation Theory and Quantum Fields

335

The corresponding 3d theory is therefore

So we can make it N =3, ﬁeld theoretically.
I don’t understand this enhancement from the geometric point of view,
since the holonomy is generically SO(3), which leads to N =2.
And the original paper [Gukov-Putrov-Vafa 1602.05302]
didn’t mention the enhancement.
36 / 49

More concretely, consider M5-branes on a Seifert of type.

This leads to.

37 / 49

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Now, let me point out something new.
The Seifert ﬁbration of type

(g;

q1 q2
qn
,
,...,
)
p1 p2
pn

is known to have SO(2) holonomy when

X qi
pi

= 0,

although Seifert ﬁbrations generically have SO(3) holonomy.
Why?

38 / 49

Well, given a Seifert ﬁbration M → Σ of type (g;we consider the following pull back:M′
genuine S 1 ﬁbration

Σ′

q1 q2
qn
,
,...,
),
p1 p2
pn

M
∏

Seifert ﬁbration

pi coverΣ

and the ﬁrst Chern class of M ′ → Σ′ isY
X qi
c1 (M ′ ) = ( pi ).
piSo, M is a Z∏ pi shift-orbifold of S 1 × Σ′ if

X qi
pi

= 0.

Then, it is has SO(2) holonomy.
39 / 49

Geometry, Representation Theory and Quantum Fields

337

Therefore, the theories

are N =4 when

X qi
pi

= 0.

40 / 49

Let us come back to M5-branes on the Seifert of type.

This led to.

41 / 49

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We just learned that the Seifert manifold

has SO(2) holonomy iff

X 1
ki

= 0,

leading to N =4.

42 / 49

And we already saw in the ﬁeld theory introduction thathas the superpotential

W ∝which vanishes iff

X 1kitr(µi )2 = (
X 1
ki

X 1
ki

) tr µ2= 0,leading to N =4. Neat, isn’t it?
43 / 49

Geometry, Representation Theory and Quantum Fields

339

The ﬁeld theory argument can be extended to all theories

with

X qi
pi

=0

by using the properties of the S-duality wall theory T [G].

click for details

44 / 49

Geometrically Unexplained
Enhancements

45 / 49

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For example, consider

The superpotential is

W =

X
ki
(tr µi Φi −tr Φi 2 + tr Φi µ′i ).
2iIntegrating out Φi , we get

W ∝(

X 1
i

ki

)(tr µ2 + tr µ′2 ) +

X 1
i

ki

tr µi µ′i .

46 / 49

When

X 1
i

ki

= 0, this theory ends up having W =

X 1
i

ki

tr µi µ′i .

We can now assign charge +1 to µi and charge −1 to µ′i ,leading to N =4 enhancement.
For two M5-branes, this reduces to a particular case of
N =5 theory of [Hosomichi-Lee-Lee-Lee-Park 0806.4977].

47 / 49

Geometry, Representation Theory and Quantum Fields

341

But geometrically, this is a mapping torus of a genus-2 surface over S 1 ,where we use Dehn twists around three necks to glue.
Nothing special happens when

X 1ki

= 0, at far as I can see.

48 / 49

Summary

I considered how many supersymmetries are realized
when we put M5-branes on 3-manifolds without hyperbolic parts.
From geometry, N =2 is expected generically,
with occasional enhancements to N =4.
From ﬁeld theory, N =3 is expected generically,
with occasional enhancements to N =4.
Sometimes they agree; sometimes they don’t.

49 / 49

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Back-up slides

50 / 49

More geometricallyunderstandable cases

51 / 49

Geometry, Representation Theory and Quantum Fields

343

The ﬁeld theory argument can be extended to all theories

with

X qi
pi

=0

by using the properties of the S-duality wall theory T [G].
(The notations T [G] and TG are confusing. Please blame Davide.)

52 / 49

The 3d N =4 theory T [G] implements the S-duality of 4d N =4 SYM:

As such, it has G × G symmetry, one acting on the Coulomb branch
and another acting on the Higgs branch.
Correspondingly, there are two moment map operators µC and µH .
We can perform N =4-preserving mass deformations

W = tr MH µC + tr MC µH .
This is known to affect the operators so thattr(MH )n = tr(µH )n ,tr(MC )n = tr(µC )n .
53 / 49

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This can be used to show the chiral ring relation

tr(µ1 )n = tr(µ2 )n = tr(µ3 )nof the 3d N =4 TG theory, because this theory has the descriptionof three T [G] theories coupled together, with the superpotentialXW =tr Φµi,H
i

Then we havetr(µi,C )n = tr Φnindependent of i.
54 / 49

Now, a Seifert ﬁber of type q/p has a 3d N =3 description

where ∥ is the T [G] theory and ki are connecting N =3 CS terms, andthe levels ki are determined by writing q/p as a continued fraction:

q
p

1

=

.

1

k1 −
k2 −

1
k3 − · · ·55 / 49

Geometry, Representation Theory and Quantum Fields

345This parthas the superpotential

W = tr Φ0 (µC )1 + tr(µH )1 Φ1 −+ tr Φ1 (µC )2 + tr(µH )2 Φ2 −

k1
2
k2
2

tr(Φ1 )2tr(Φ2 )2

+ ···+ tr Φn−1 (µC )n + tr(µH )n Φn −kn
2tr(Φn )2 .

We now integrate out Φi , and usetr((µC )i )2 = tr(Φi )2 = tr((µH )i+1 )2 .
56 / 49

At the end of the day, we ﬁnd

generates the superpotential

W ∝

q
p

tr(Φ0 )2 .

• I have done only a very crude ﬁeld theoretical analysis;
there are some gaps in the derivation.
• This is related to the fact that the (p, q) 5-brane is tilted by the factor p/q ,and the observation goes back to [Kitao-Ohta-Ohta hep-th/9808111].
They noticed that this implied a ‘fractional Chern-Simons level’ of p/q
and got confused.
57 / 49

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So, the theory

ends up generating

X qi
W ∝(
) tr(Φ0 )2 .
pi

Therefore the theory enhances to N =4 when
X qi
= 0.
pi

go back
58 / 49

Geometry, Representation Theory and Quantum Fields

Derived Geometry for Physicists
Philsang Yoo

Abstract. In this largely expository talk, I aim to explain basic ideas ofderived geometry and discuss how they are useful in uncovering a new relationship between geometric representation theory and quantum field theory.
The original perspective is based on a joint work with Chris Elliott.

(P. Yoo) Seoul National University

347

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OCAMI Reports Vol. 4 (2022)

Derived Geometry for Physicists
Philsang Yoo
Seoul National University

Geometry, Representation Theory and Quantum Fields
Media Center, Osaka City University
March 23rd, 2022

1/16

Introduction◮ Review of Algebraic Geometry◮ Introduction to Derived Algebraic Geometry◮ Classical Field Theory and BV Formalism◮ AKSZ Formalism◮ Applications to Geometric Representation Theory

2/16

Geometry, Representation Theory and Quantum Fields

349

Algebraic Geometry: Local Story≃

O : Schaﬀ "category
objectclosedpoint
closedsubscheme
fiber
productaﬃne
space! cRingop : Spec

Schaﬀ
X
Spec Ax ↩→ XSpec A/m ↩→ Spec A
Z ↩→ XSpec A/I ↩→ Spec A
X ×W Y
Spec(A ⊗C B)
AdC
Spec C[x1 , · · · , xd ]

cRing
O(X )
A
O(X ) ↠ C
A ↠ A/mO(X ) ↠ O(Z )A ↠ A/IO(X ×W Y )A ⊗C B
O(AdC )
C[x1 , · · · , xd ]◮ A point a = (a1 , · · · , ad ) ↩→ AdC corresponds to C[x1 , · · · , xd ] → C given
by xi $→ ai , or m(a1 ,··· ,ad ) ⊂ C[x1 , · · · , xd ].
◮ A subscheme Z ↩→ AdC corresponds to C[x1 , · · · , xd ] → C[x1 , · · · , xd ]/I ,where I = (f1 , · · · , fm ) for fi ∈ C[x1 , · · · , xd ]. Points of Z (I ) = Spec A/I
are {(a1 , · · · , ad ) | f1 (a1 , · · · , ad ) = · · · = fm (a1 , · · · , ad ) = 0}, asI ⊂ m(a1 ,··· ,ad ) ⇐⇒ f1 (a1 , · · · , ad ) = · · · = fm (a1 , · · · , ad ) = 0.

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Algebraic Geometry: Example
The equations x = 0 and x 2 = 0 define the same variety but diﬀerent schemes.
Schemes are more robust under a deformation.
curvesC ∩ CkO(C ∩ Ck )curvesC (a) ∩ Ck

O(C (a) ∩ Ck )C1 = {y = x}

C = {y = 0}

C2 = {y = x 2 }••C[x, y ]/(x, y ) ∼C[x, y ]/(y − x 2 , y ) ∼= C(0,0)= C[x]/(x 2 )C (a) = {y = a}, a ∕= 0
C1 = {y = x}
C2 = {y = x 2 }•C[x, y ]/(x − a, y − a) ∼= C(a,a)••C(√a,a) ⊕ C(−√a,a)O(C (a)∩C1 )=O(C (a)×A2 C1 )=O(C (a))⊗O(A2 ) O(C1 )=C[x,y ]/(y −a)⊗C[x,y ] C[x,y ]/(y −x)∼
= C[x,y ]/(y −x,y −a)2O(C (a)∩C2 )=O(C (a)×A2 C2 )=O(C (a))⊗O(A2 ) O(C2 )=C[x,y ]/(y −a)⊗C[x,y ] C[x,y ]/(y −x 2 )∼= C[x,y ]/(y −x ,y −a).

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Derived Algebraic Geometry: Example!

x =0
define the same
x =0
scheme but diﬀerent derived schemes. Schemes are not good enough:The equation x = 0 and the system of equations

space X

C = {x = 0}C ∩C

C[x, y ]/(x) ∼= C[y ]
C (a) = {x = a}C[x, y ]/(x) ∼= C[y ]C ∩ C (a), a ∕= 0

C[x, y ]/(x − a) ∼= C[y ]

C[x, y ]/(x, x − a) ∼=0

pictureO(X )space X

pictureO(X )O(C ∩C (a))=O(C ×A2 C (a))=O(C )⊗O(A2 ) O(C (a))=C[x,y ]/(x)⊗C[x,y ] C[x,y ]/(x−a)∼= C[x,y ]/(x,x−a).
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Derived Algebraic Geometry: Idea
How do we make a space more robust under deformation? The idea is toconsider a derived tensor product instead of an ordinary tensor product:O(C )⊗L

O(A2 )!
"x·
O(C (a))≃C[x,y ]/(x)⊗L
C[x,y ] C[x,y ]/(x−a)≃ C[x,y ][1]→C[x,y ] ⊗C[x,y ] C[x,y ]/(x−a)
!
" !
"
x·
a·
≃ C[x,y ]/(x−a)[1]−→C[x,y ]/(x−a) ≃ C[y ][1]−→C[y ]#$%C[y ][1]⊕C[y ]if a=0≃$&0
if a∕=0.

!
"x·where we use a resolution C[x, y ]/(x) ≃ C[x, y ][1] → C[x, y ] .
Note that◮ In degree 0, we obtain C[y ] for a = 0 and 0 for a ∕= 0.
◮ The Euler characteristic would always be invariant.

When a = 0, there is a hidden aﬃne line lurking in degree −1 in addition to
the aﬃne line in degree 0 from the picture. One should think that they wouldcancel out each other as a becomes non-zero to make the space invariant undera deformation.
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Classical Field Theory and BV Formalism
The input data for classical field theory is a spacetime manifold, space of fields,
and an action functional. For simplicity, we assume the space of fields is anaﬃne variety X , e.g., X = Ad , and S : X → C is a polynomial.
The output of classical field theory is the critical locus
Crit(S) = {dS = 0} = ΓdS ∩T ∗ X X = ΓdS ×T ∗ X X ,where ΓdS is the graph of dS. In terms of algebra, one hasO(Crit(S)) = O(ΓdS ) ⊗O(T ∗ X ) O(X ),where the algebra O(T ∗ X ) of polynomial functions on T ∗ X is by definition thesymmetric algebra SymO(X ) (TX ).
The classical BV formalism amounts to working with the derived critical locus
dCrit(S), defined byO(dCrit(S)) := O(ΓdS ) ⊗LO(T ∗ X ) O(X ).

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Classical BV Formalism and (−1)-shifted Symplectic SpaceTo proceed further, we find a resolution of O(X ) over O(T ∗ X ) given by∗' ):=(Sym
O(X )≃O(X
O(X ) (TX [1]⊕TX ),d )≃(SymO(X ) (TX [1])⊗O(X ) O(T X ),d )where the diﬀerential d is induced from IdTX : TX [1] → TX . In coordinates,O(X )C[x i ]

O(T ∗ X )C[x i , pi ]O(Γ$dS )%∂S
C[x , pi ]/ pi − ∂x
i
i

#)
O(X
C[x , pi , ξi ], dX' ξi = pii!
"' )=C[x i ,pi ]/ pi − ∂S ⊗ i
O(dCrit(S))=O(ΓdS )⊗O(T ∗ X ) O(X(C[x i ,pi ,ξi ],dX! )iC[x ,p ]∂xii= C[x ,ξi ],d , where dξi = ∂Si
∂x

(
)
!
"
or O(dCrit(S)) = SymO(X ) (TX [1]), d . As we define the (−1)-shifted

cotangent bundle T ∗ [−1]X of X by setting O(T ∗ [−1]X ) := SymO(X ) (TX [1]),we have
dCrit(S) = (T ∗ [−1]X , d)& ∂S ∂& iwhere d =. Note that it has a symplectic form ω =dx ∧ dξi of∂x i ∂ξi
cohomological degree −1.
Claim: Any classical field theory gives rise to a (−1)-shifted symplectic space.
Moreover, we define classical field theory to be a (−1)-shifted symplectic space.
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AKSZ Formalism: Statement
AKSZ formalism gives a way to produce a classical field theory.

Theorem (AKSZ–PTVV construction)
For X with m-orientation and Y with n-shifted symplectic structure,
Map(X , Y) is (n − m)-shifted symplectic.

Example (shifted symplectic structure)◮ BG is 2-shifted symplectic.
◮ T ∗ [n]Z is n-shifted symplectic.

Example (orientation)◮ If M is a closed oriented d-manifold, then MB has a d-orientation.
◮ If X is a smooth projective variety of dimension d, then•
XDol ≈ (X , (Ω•,•X , ∂)) and XdR ≈ (X , (ΩX , ddR )) have a 2d-orientation.
◮ If X is a smooth projective Calabi–Yau variety of dimension d, then X has
a d-orientation.
◮ If Xi has mi -orientation, then X1 × X2 has (m1 + m2 )-orientation.
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AKSZ Formalism: Examples
Theorem (AKSZ–PTVV construction)
For X with m-orientation and Y with n-shifted symplectic structure,
Map(X , Y ) is (n − m)-shifted symplectic.

Example◮ For a (0-shifted) symplectic space Y , 1d theory
EOMTQM (M 1 ) = Map(MB , Y ) is topological quantum mechanics.
◮ For a Calabi-Yau manifold Y , 2d theory EOMYB (Σ2 ) = Map(ΣB , T ∗ [1]Y )is B-model.
◮ EOMG3-CS (M 3 ) = Map(MB , BG ) is 3d Chern–Simons theory.
◮ EOMG4-CS (Σ2 × C ) = Map(ΣB × C , BG ), where C is a Calabi–Yau curve, is
4d Chern–Simons theory.
◮ EOMG5-CS (M 1 × X ) = Map(MB × X , BG ), where X is a Calabi–Yausurface, is 5d Chern–Simons theory.
◮ EOMGhCS (X ) = Map(X , BG ), where X is a Calabi–Yau 3-fold, isholomorphic Chern–Simons theory.
Note that a manifold of codimension k yields a (k − 1)-shifted symplectic space.
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From AKSZ to TQFT
A manifold of codimension k yields a (k − 1)-shifted symplectic space. Recall
that a manifold of codimension k should yield a linear (k − 1)-category in afunctorial TQFT.
Claim: There is a procedure, called shifted geometric quantization, promoting a(k − 1)-shifted symplectic space to a linear (k − 1)-category,
EOM(N d−k ) : (k − 1)-shifted symplectic space ⇝ Z (N) : (k − 1)-category.
If k = 1, this is the usual geometric quantization from the phase space to thespace of quantum states.

Example (B-model)
Recall the B-model described by EOMYB (Σ2 ) = Map(ΣB , T ∗ [1]Y ).
◮ EOMYB (S 1 ) = Map(SB1 , T ∗ [1]Y ) ∼= T ∗ (T [−1]Y ); its geometricquantization is O(T [−1]Y ) = Ω• (Y ), the space of de Rham forms on Y .
◮ EOMYB (pt) = T ∗ [1]Y ; its categorified geometric quantization is Coh(Y ),the derived category of coherent sheaves on Y .
Note that the result is what we would have gotten as ZB (S 1 ) and ZB (pt) for a2-dimensional B-model understood as a functorial TQFT ZB .
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4d N = 4 Gauge Theory
Theorem-Definition (Elliott–Y.)
For a smooth proper complex algebraic surface X , one has
EOMGHol (X ) = T ∗ [−1] Map(XDol , BG ) = Map(XDol , T ∗ [3]BG ). For
X = Σ × C with smooth proper curves Σ, C , one has
EOMGHT (X ) = Map(ΣdR × CDol , T ∗ [3]BG ) = Map(ΣdR , T ∗ [1] HiggsG (C ))EOMGB (X ) = Map(ΣdR × CdR , T ∗ [3]BG ) = Map(ΣdR , T ∗ [1] FlatG (C ))EOMGA (X ) = Map(ΣdR , T ∗ [1](BunG (C )dR )).

Remark◮ Our GL twists are diﬀerent from [Kapustin–Witten]’s twists.
◮ FlatG (C ) = Map(CdR , BG ) appears in a deformation family with
HiggsG (C ) as a special fiber; it is not LocG (C ) = Map(CB , BG )!
◮ The twists are not entirely topological: trivialization of infinitesimaltranslations is encoded by having de Rham stack.
◮ It leads to new structures in geometric Langlands theory.
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Categorified Geometric Quantization of 4d N = 4 Gauge Theory

We know that in d-dimensional TQFT, Z (S d−2 ) describe the category of line
operators. One can construct it even in an algebraic world by applying
categorified geometric quantization of EOM(“S 2 ”) = EOM(B), where
B = D ∐D× D is the formal bubble.
dual theories
EOM
cat. GQSYMGA [C ] = SYMǦB [C ]T [1](BunG (C )dR )QCoh(BunG (C )dR ) =: D-mod(BunG (C ))∗T ∗ [1](FlatǦ (C ))QCoh(FlatǦ (C ))Construction
One can identify the categories of line operators:
G◮ ZHT(S 2 ) = QCoh(HiggsG (B))◮ ZAG (S 2 ) = D-mod(BunG (B))
◮ ZBG (S 2 ) = QCoh(FlatG (B)).
Holomorphic-topological twist (aka Kapustin twist) is self-dual, and A-twist
and B-twist are dual to each other [Raghavendran–Y.].

Meta-Conjecture
QCoh(HiggsG (B)) ≃ QCoh(HiggsǦ (B))
D-mod(BunG (B)) ≃ QCoh(FlatǦ (B))

The latter is called the derived geometric Satake equivalence.

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Categorified Geometric Quantization of 4d N = 2 Gauge Theory

One way to construct 4d N = 2 theory is to use input data of a reductive
group G (or its compact form) and its representation V .

Theorem-Definition
4d N = 2 theory determined by (G , V ) has holomorphic-topological twist
EOM(Σ × C ) ≃ Map(ΣdR , T ∗ [1](Map(C , V /G ))).

Construction
The category of line operators of 4d N = 2 theory determined by (G , V ) is
G ,V
Z4d,HT(S 2 ) = QCoh(Map(B, V /G )).

In particular, the category of line operators of pure 4d N = 2 theory is
G ,V =0Z4d,HT(S 2 ) = QCoh(Map(B, BG )) = QCoh(BunG (B)).

Remark[Cautis–Williams] studied the structure of cluster algebras in 4d N = 2 theory
on the latter category (or rather, its heart of a perverse t-structure), called thecoherent Satake category.
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Categorified Geometric Quantization of 3d N = 4 Gauge Theory
Let us denote the 3d N = 4 theory determined by (G , V ) by T G ,V . There is aduality on 3d N = 4 theories. Sometimes T G ,V is dual to another 3d N = 4!
!
theory of the form T G ,V , in which case one can argue the following:dual
EOM
cat. GQ!

!

TAG ,V [S 1 ] = TBG ,V [S 1 ]!
!
T ∗ [1](Map(D× , V /G )dR ) T ∗ [1](Map(D×dR , V /G ))××!
D-mod(Map(D , V /G ))QCoh(Map(DdR , V /G ! ))Construction
The categories of line operators of T G ,V are
G ,V
Z3d,A
(S 1 ) = D-mod(Map(D× , V /G ))
G ,V
Z3d,B
(S 1 ) = QCoh(Map(D×dR , V /G ))Meta-Conjecture!

!

If T G ,V is dual to T G ,V , then!
!
D-mod(Map(D× , V /G )) ≃ QCoh(Map(D×dR , V /G )).

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Remarks

Meta-Conjecture!

!

If T G ,V is dual to T G ,V , then!
!
D-mod(Map(D× , V /G )) ≃ QCoh(Map(D×dR , V /G )).

◮ The conjecture was unpublished until it appeared for the first time in printin lecture notes of [Braverman–Finkelberg].
◮ [Dimofte–Garner–Geracie–Hilburn] studied these categories from thephysical perspective.
◮ [Hilburn–Raskin] proved the simplest case.
◮ [Hilburn–Y.] further develops the framework of categorified geometricquantization and studies its relationship to local geometric Langlands
correspondence. The relevant physics is based on [Gaiotto–Witten].
◮ [Ben-Zvi–Sakellaridis–Venkatesh] has an independent work relating theidea of Gaiotto–Witten to number theoretic objects.

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Exceptional moduli spaces for exceptional N=3
theories
Gabi Zafrir
Abstract. It is expected on general grounds that the moduli space of 4d
N = 3 theories is an orbifold of flat space by a special type of groups known
as crystallographic complex reflection groups (CCRG). For the case of 4d N
= 4 theories these further reduce to the Weyl groups of simple Lie algebras.
As in the case of Lie algebras, the space of CCRGs consists of several infinite
families, together with some exceptionals. To date, no 4d N = 3 theory withmoduli space labeled by an exceptional CCRG (excluding Weyl groups) has
been identified. In this talk, I shall discuss how one can identify 4d theories
realizing such moduli spaces and study some of their properties.

(G. Zafrir) SCGP

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357

Exceptional moduli spaces for
exceptional 𝓝=3 theories
Gabi Zafrir, Simons Center for Geometry and Physics
J. Kaidi, M. Martone, G.Z: 2203.04972

1

Motivation• The classification of CFTs with specific symmetries is of great interest in the
study of QFT.
• Expected to become easier with ample symmetry, particularly SUSY.
Indeed, believed settled for 4d 𝒩 = 4 SCFTs. However, still quite open for
4d 𝒩 = 3 SCFTs.
• We shall explore the existence and present a potential string theory
construction for a novel class of 4d 𝒩 = 3 SCFTs.
• Interest in the study of QFT: understanding endpoints of RG flows,
constraints on QFT.
• Interest in the study of string theory: can point to novel string theory
constructions.
2

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Outline1. Introduction
 𝒩 = 3 SCFTs The structure of moduli spaces

2. How to realize new 𝒩 = 3 SCFTs ?

3. 4d consistency checks
4. String theory realization

5. Conclusions3

𝒩 = 3 SCFTs• The 𝒩 = 3 vector multiplet = The 𝒩 = 4 vector multiplet. As such itis thought that any Lagrangian possessing 𝒩 = 3 SUSY has 𝒩 = 4
SUSY.
• Genuine purely 𝒩 = 3 SCFTs are then expected to be inherently
strongly coupled.
• Central charges obey a=c. Have a 𝑈(1) × 𝑆𝑈(3) R-symmetry, but noother global symmetry [Aharony, Evtikhiev, 2016].
• Can be engineered in string theory by D3-branes probing S-folds[Garcia-Etxebarria, Regalado, 2016, 2017][Aharony, Tachikawa, 2016].
4

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The moduli space of 𝒩 = 3 SCFTs• In 𝒩 = 3 SCFTs, the Higgs and Coulomb branch combine to a single
moduli space of vacua.
• The coordinates of this space are three complex scalars. In 𝒩 = 4
SCFTs, these are the six real scalars in the vector multiplets.
• For 𝒩 = 4 SCFTs, the moduli space has the form: ℂ3𝑁 /Γ, for Γ a Weylgroup of a simple Lie group.
• What about the moduli space of 𝒩 = 3 SCFTs?

5

Classification• 𝒩 = 4 SCFTs are believed to be classified by a choice of Lie algebra,
although there are some identifications due to S-duality (SO(odd)=USp).
• These identifications lead to: 𝒩 = 4 SCFTs are classified by a choice ofWeyl group.
• Weyl group = crystallographic real reflection groups.
• Conjecture: 𝒩 = 3 SCFTs are classified by crystallographic complex
reflection groups [Tachikawa, GZ, 2019].
• The moduli space will then have the form: ℂ3𝑁 /Γ, for Γ a crystallographic
complex reflection groups.
• This is indeed the moduli space of all known 𝒩 = 3 SCFTs (up to discrete
gauging).

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Complex reflection groups• Complex reflection group (CRG): a group whose action on ℂ𝑛 , withcoordinates 𝑧1 , 𝑧2 … 𝑧𝑛 , is generated by complex reflections 𝑧𝑖 →2𝜋𝑖𝑒 𝑘 𝑧𝑖 .
• Generators can also act as a reflection on a combination of
coordinates. Example: permutation 𝑧𝑖 − 𝑧𝑖+1 → −(𝑧𝑖 − 𝑧𝑖+1 ).
• Crystallographic: action preserves a lattice.
• Necessary for 𝒩 = 2: preserves EM charge lattice.

Classification of CRGs
• There is a classification of CRGs [Shephard, Todd, 1954].
• Similarly to the classification of simple Lie groups, CRGs can be broken
to one infinite family and 34 exceptionals.
• The infinite family is denoted as 𝐺(𝑘, 𝑝, 𝑁) (𝑝 a divisor of 𝑘), and isequivalent to Weyl groups of A, B=C, D for 𝑘 = 1,2. It iscrystallographic for 𝑘 = 1,2,3,4,6.
• The exceptionals are usually denoted by 𝐺𝑖 , for 𝑖 = 4,5, … , 37.
Include the Weyl groups of 𝐹4 , 𝐸6 , 𝐸7 , 𝐸8 (Weyl group of 𝐺2 is𝐺(6,6,2)). Out of the 34 exceptionals only 16 are crystallographic and
12 are genuine crystallographic CRGs (not real reflection groups).

Geometry, Representation Theory and Quantum Fields

Known cases• Only known 𝒩 = 3 SCFTs realize the groups: 𝐺(3,1, 𝑁), 𝐺(3,3, 𝑁),𝐺(4,1, 𝑁), 𝐺 4,4, 𝑁 , 𝐺(6,6, 𝑁).
• In particular, there are no known 𝒩 = 3 SCFTs realizing exceptional
CRGs.
• If the classification program is correct, we expect such theories toexist. Here we shall try to motivate for the existence of such for mostcrystallographic exceptional CRGs.
• We first consider how such theories can arise in string theory byunderstanding how the moduli space arises.
• Check that the moduli space is a consistent one for an 𝒩 = 3 SCFT.
• Identify a potential string theory realization for these theories.

Moduli spaces of D3-branes in flat space• Consider a collection of N D3-branes in flat space.
• The moduli space is given by the location of the branes in the six
transverse directions, ℂ3𝑁 . These can be spanned by the vector• However, a permutation of the D3-branes gives an equivalentconfiguration.
• As such we have that the moduli space is:

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Moduli spaces of orientifolds• Consider N D3-branes in the presence of an O3-plane.
• Before orientifolding: had 2N D3-branes in flat space with moduli
space, described by the location of the D3-branes, 𝜙𝑖 .
• After orientifolding: need to limit to configurations invariant under
the orientifold action 𝜙𝑖 → −𝜙𝑖 .
• This is achieved by restricting to the portion of the moduli space
where such an action is equivalent to a Weyl transformation.
• In other words, the portion of the moduli space that survives is theone on which the orbifold action can be realized as an element of the
Weyl group.

Moduli spaces of orientifolds• The moduli space is then spanned by the N coordinates 𝑣𝑖 = 𝜙2𝑖 − 𝜙2𝑖−1 .
This gives a space of the form.
• However we still need to identify configurations related by the original
Weyl action. These include: permutations of the N 𝑣𝑖 coordinates and theirreflections, 𝑣𝑖 → −𝑣𝑖 . These span the group.
• The moduli space is then.

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Moduli spaces of S-folds• Similarly, we can consider kN D3-branes in the presence of an S-fold.
• A 𝑘-type S-fold is a simultaneous orbifold by a ℤ𝑘 rotation in the ℂ 32𝜋𝑖

orthogonal to the D3-branes (𝑧𝛼 → 𝑒 𝑘 𝑧𝛼 ) and a ℤ𝑘 subgroup of the𝑆𝐿(2, ℤ) symmetry of type IIB [Garcia-Etxebarria, Regalado, 2015].
• Due to the latter: 𝑘 limited to 1,2,3,4,6. 𝑘 = 1 is flat space, 𝑘 = 2 isstandard O3-plane.
• Alternatively, can be described as F-theory on ℂ 3 × 𝑇 2 /ℤ𝑘 .
• This background preserves 𝒩 = 3 SUSY in 4d and so leads to an
𝒩 = 3 SCFT on the D3 branes (𝒩 = 4 for k=1,2 ).

Moduli spaces of S-folds• We can next analyze the moduli space.
2𝜋𝑖• We’ll now demand invariance under 𝜙𝑖 → 𝑒 𝑘 𝜙𝑖 .
• This is achieved by restricting to configurations where this action isequivalent to a Weyl transformation, leading to a moduli space of theform ℂ3𝑁 .
• Need to identify configurations related by the original Weyl action.
Again, these include: permutations of the N coordinates and theircomplex reflections, 𝑣𝑖 → 𝑒• These span the group

2𝜋𝑖
𝑘

𝑣𝑖 .
.

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CRG subgroups of Lie groups• The moduli space of 𝒩 = 3 SCFTs can be thought of as arising fromthe moduli space of 𝒩 = 4 𝑈(𝑘𝑁) SYM.
• The complex reflection group arises as a subgroup of the Weyl groupof 𝑈(𝑘𝑁).
• It is natural to expect to realize exceptional CRGs
as subgroups of exceptional Lie groups.
• The list of such CRGs as been classified in[Brewer, 1997].
• This includes almost all crystallographic
exceptional CRGs (omissions: 𝐺24 , 𝐺29 , 𝐺33 , 𝐺34 ).

S-folding exceptional groups• We then expect that performing an S-folding on the moduli space of
exceptional gauge theories can provide realizations of the mentioned
exceptional CRGs. We can check this as follows.
• The discussion so far can be carried out at the group theory level.
• As such we can also perform it for exceptional groups by abstractly
considering a vector 𝜙𝑖 spanning the moduli space: ℂ3𝑁 /Γ, for Γ a Weylgroup of an exceptional Lie group.
2𝜋𝑖• Then consider S-folding by demanding invariance under 𝜙𝑖 → 𝑒 𝑘 𝕆𝑖𝑗 𝜙𝑗 ,with 𝕆𝑖𝑗 taking into account the possibility of also combining other actionsin the orbifold.
• This indeed leads to exceptional CRGs.

Geometry, Representation Theory and Quantum Fields

Example: 𝐺4• Consider a theory with moduli space(like 𝒩 = 4 𝑆𝑝𝑖𝑛(8)SYM).
• We consider performing an S-folding by a ℤ3 action that is acombination of the ℤ3 S-fold and a triality outer automorphism.
• We can span the moduli space with a vector.
• The moduli space of the S-folded theory should be restricted to thesubspace of Φ where

Example: 𝐺4• We can solve this constraint for a chosen Weyl element to get:• So the moduli is locally ℂ6 spanned by 𝑣. Additionally, need to mod
out by Weyl transformations of 𝑆𝑝𝑖𝑛(8) that keep this form of Φ.
• These turn out to be generated by the elements:• These generate 𝐺4 . As such the full moduli space is ℂ6 /𝐺4 .

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General case• We can repeat the analysis also for othergroups.
• Indeed find the expected CRGs.
• Complications for non-simply lacedgroups as the S-duality group is slightly
different in that case.

4d consistency checks• We can perform consistency checks on the resulting theories.
• For this, need to know something about these theories:
 a and c central charges.
 Finer structure of the moduli space.

• a and c central charges: can calculate from [Shapere, Tachikawa,
2008] (used a=c for 𝒩 = 3 SCFT ):• Δ𝑖 are the dimensions of the operator spanning the Coulomb branchwhose complex dimension is 𝑟.

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Finer structure of the moduli space• We can go on the moduli space of the 𝒩 = 3 SCFT.
• At generic points: have only 𝑟 free vector fields.
• At special points in the moduli space: may have 𝒩 = 3 SCFTs ofsmaller rank.
• These arise at subspaces that are invariant under the action of someof the non-trivial elements of the group.
• Can be determined by group theory.

Example, USp(2𝑁) 𝒩 = 4 SCFT

USp(2𝑁) → 𝑈𝑆𝑝(2𝑁 − 2) → 𝑈𝑆𝑝(2𝑁 − 4) → …
SU 𝑁 → 𝑆𝑈 𝑁 − 1→ 𝑆𝑈 𝑁 − 2→…
2𝜋𝑖• 𝑈𝑆𝑝 2𝑛 : subspaces invariant under some reflections, 𝜙𝑎 → 𝑒 𝑘 𝜙𝑎 .
• 𝑆𝑈 𝑛 : subspaces invariant under permutations, but not reflections.

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Finer structure of the moduli space• Consider the collection of all non-trivial rank 1 theories that can ariseon the moduli space. Denote the set by .
• Described by a polynomial equation of dimension Δ𝑠𝑖𝑛𝑔 in the
Coulomb branch operators: 𝑃(𝑢𝑖 ) = 0.
• The a and c central charges of the original theory are related to thoseof the set of rank 1 theories [Martone, 2020] (specialized to 𝒩 = 3
SCFT):• Can compare the central charges derived from the two formulas.

Example: 𝐺4• Consider the example of 𝐺4 .
• Up to conjugation by elements of 𝐺4 , only one non-trivial rank 1
theory. Subspace can be described by the relation:• On this locus we get the moduli space• We can then flow from the 𝐺4 theory to a rank 1
theory with moduli space ℂ3 /ℤ3 .

Geometry, Representation Theory and Quantum Fields

Example: 𝐺4• We can next compare the central charges evaluated from the two
approaches.
• However, requires knowing the physical theory realizing this moduli
space. There are two known options for this theory.
• Option 1: ℤ3 gauging of free vectors. Central charge formula is thenconsistent with a discrete gauging of ℂ6 by 𝐺4 .
• Option 2: non-trivial 𝒩 = 3 SCFT with moduli space ℂ3 /ℤ3 . Central
charge formula is then consistent with the ones calculated from theShapere-Tachikawa formula.
• The structure of the moduli space and central charges are thenconsistent with a non-trivial 𝒩 = 3 SCFT.

String theory construction• We can next look for a string theory construction for the 𝒩 = 3 SCFTs
realizing these moduli spaces.
• Natural candidates: S-folding of E-type (2,0) theories.
• Indeed such a construction appeared in [Garcia-Etxebarria, Regalado,2016].
• Can be generalized to include outer automorphism twists.
• The resulting 𝒩 = 3 SCFTs are then naturally identified as ones basedon exceptional CRGs.

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String theory construction• Consider type IIB compactified on ℂ2 /Γ, where Γ is a discrete
subgroup of 𝑆𝑈(2). This leads to Γ-type (2,0) theories.
• Can compactify two directions in ℂ2 /Γ: a torus fibration over ℂ
defined by an 𝑆𝐿(2, ℤ) monodromy around the origin.
• We can further compactify on 𝑇 2 to get Γ-type 4d 𝒩 = 4 SYM.
• T-dualize to Type IIA and lift to M-theory on 𝑅3,1 × 𝑇 2 × 𝑇 2 × ℂ × 𝑆1 .
• Can then introduce an ‘S-fold’ by ℤ𝑘 orbifolding: rotations in 𝑇 2 × ℂ,
𝑆𝐿(2, ℤ) transformations of 𝑇 2 , and the non-geometric 𝑆𝐿(2, ℤ)
symmetry of M-theory on 𝑇 2 × 𝑆1 .
• This complicated background involving non-trivial fibration andorbifolding is expected to give a 4d 𝒩 = 3 SCFT on 𝑅3,1 .

String theory construction• From the preceding discussion: natural to identify the moduli space of
these theories with the ℤ𝑘 S-folding of E-type moduli spaces.
• Can be generalized to include outer automorphism twists.
• Specifically, can blow-up ℂ2 /Γ. We then expect a geometric symmetry
acting on the blow-up cycles as the outer automorphism action on
the Dynkin diagram.
• Can twist by this symmetry on 𝑇 2 → get non-simply laced groups,
give 𝐺12 .
• Can incorporate into the orbifold group → 𝐺4 .

Geometry, Representation Theory and Quantum Fields

Conclusions• Are there 𝒩 = 3 SCFTs associated with exceptional CRGs?
• Argued that 𝒩 = 3 SCFTs based on exceptional CRGs should be
realized by S-folding of exceptional (2,0) theories by studying
properties of the moduli space.
• Shown that the resulting 4d theories pass non-trivial consistency
checks.
• Identified some of these theories with a known string theory
construction, and shown how it might be generalized to get othercases.
• Overall, this suggests the existence of 8 new 𝒩 = 3 SCFTs based on
exceptional crystallographic CRGs.

Open questions• What about the remaining four exceptional crystallographic CRGs?
• Is it possible to learn more about the properties of these SCFTs(indices, generalized symmetries, mass deformations)?
• Study compactification to 3d that should lead to new 𝒩 = 6 SCFTs.
Should also be related to twisted compactification of 4d 𝒩 = 4 SYM.
• Can the string theory construction be generalized to lead to new
𝒩 = 2 SCFTs?
• Relation to the compactification of 6d (1,0) SCFTs?

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Thank you31

Geometry, Representation Theory and Quantum Fields

373

Symmetry Protected Topological Criticality
Yunqin Zheng

Abstract. Symmetry protected topological (SPT) phases have been extensively studied over the past decade. They are systems with a uniquely gapped
symmetric ground state but are not able to be connected to a trivially gappedsystems under symmetric local unitary transformation. In this talk, I will showthat the notion of SPT for gapped systems can be generalized to critical systems, which we denote as symmetry protected topological criticality (SPTC).
I present a general way of constructing topologically nontrivial SPTCs, study
their physical observables that characterize the nontrivial topological properties, and study the stability under perturbation. (This talk is based on workin collaboration with Linhao Li (ISSP) and Masaki Oshikawa (ISSP, IPMU).)(Y. Zheng) Kavli Institute for the Physics and Mathematics of the Universe (IPMU),
University of Tokyo

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Symmetry Protected Topological
Criticality
Yunqin Zheng
Kavli IPMU, ISSP @ U.Tokyo
Geometry, Representation Theory and Quantum Fields
Osaka City U.,
Mar 24th, 2022
arXiv:2203.XXXXX
with Linhao Li (ISSP), Masaki Oshikawa (ISSP,IPMU)

1 / 13Symmetry Protected Topological Criticality

1

What is it? Why do we study it?

2

How to construct it?

3

Physical properties? Stability?

2 / 13

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375

Gapped Quantum Matter
Gapped phases of quantum matter are relatively well-understood.

Trivially
GappedSymmetry
Protected
Topological
Phase

Symmetry
Enriched
Topological
Phase

Symmetry
BreakingPhase’t Hooft Anomaly (Hep-th)
Integer QH (Cond-mat)
Generalized cohomology (Math)
3 / 13Gapless Quantum Matter
Gapless quantum matter is much less well-understood.
Gapless quantum matter typically appear in two situations:1

Critical point/Critical region.

2

Goldstone boson.

We will mainly consider the first case - Quantum criticality.
What is the landscape of Quantum criticality?
We don’t know. But from many examples studied in the past, we wish
to organize them, analogous to the landscape of the gapped quantum
matter.

4 / 13

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Landscape of Quantum Criticality - A WishLandauTransition...

“Trivial
Criticality”

LandauTransition
⊕ SPT
...

Symmetry
Protected
Topological
Criticality(SPTC)

Neel-VBS
Transition,...

Symmetry
Breaking
at Criticality

Symmetry
Enriched
Topological
Criticality(SETC)
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Properties of Gapped SPT Phase

Systems belonging to a gapped SPT phase satisfy:● Global symmetry Γ: Non-anomalous.

● Single ground state under periodic boundary condition.

● Degenerate ground states under open boundary condition.

● Ground state under Γ twisted boundary condition carries Γcharge.

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Properties of SPTC
By minimally generalizing the properties of gapped SPT, we expect
SPTC should satisfy:● Global symmetry Γ: Non-anomalous.

● Single ground state under periodic boundary condition witha polynomial finite size gap.
● Degenerate ground states under open boundary condition witha exponential decaying finite size gap for 1+1d system.
● Ground state under Γ twisted boundary condition carries Γcharge.
How to construct models with these properties?

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Decorated Defect Construction of Gapped SPT
1→A→Γ→G →1

G SSB
Phase

Γ Gapped
SPT

1

G SSB phase, decorate the G -defects by A gapped SPT
consistently. ⇒ Γ is anomaly free and2

Fully fluctuate the decorated G -defects.
[Chen,Lu,Vishwanath,1303.4301][Wang,Ning,Cheng,2104.13233]

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Decorated Defect Construction of SPTC
1→A→Γ→G →1

WeakΓ SPTCG SSB
PhaseΓ GappedSPTG SSB phase, decorate the G -defects consistently. ⇒ Γ isanomaly free.
2 Fine-tune the fluctuation of the decorated G -defects to criticality.
1[Scaffidi,Parker,Vasseur,1705.01557][Li,Oshikawa,YZ,2203.?????]
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Decorated Defect Construction of
Weak and Strong SPTC
1→A→Γ→G →1
Γ Weak
SPTC
G SSB
Phase

1

2

Γ Strong
SPTC

Γ Gapped
SPT

G SSB
Phase

Non-SPTG SSB phase with certain anomaly, decorate the G -defects

hhhh(
consistently.
hh ⇒ Γ is anomaly free.
(((((Fine-tune the fluctuation of the decorated G -defects to criticality.
[Scaffidi,Parker,Vasseur,1705.01557][Thorngren,Vishwanath,Verresen,2008.06638][Li,Oshikawa,YZ,2203.?????]
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Example of Weak SPTC: Γ = ZA2 × ZG2

L

x
z
z z
z
HWeakSPTC ∶= − ∑ (σiz τi+σx τ z ) .
1 σi+1 + σi σi+1 + τ
i− 1 i i+ 1
2

i=1

2

2

1

It satisfies all the required physical properties:

2

Unstable under infinitesimal perturbation to Γ SPT.

3

Symmetry charge under TBC is a more rubust feature of
nontrivial SPTC, and generalizable to higher dimensions.

● Γ is anomaly free.
● Single ground state under PBC.
● Degenerate ground states under OBC.
● Under ZA2 /ZG2 TBC, ground state carries ZG2 /ZA2 charge.

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Example of Strong SPTC: 1 → ZA2 → ZΓ4 → ZG2 → 1
L

y
z
x y
z
x
x z
z
+ σi−1τi−
HStrongSPTC ∶= − ∑ (τi−+ τi−
1 σi τ
1 σi ) .
1 σi τ
i+ 1
i+ 1
i=1

2

2

2

2

2

1

It satisfies all the required physical properties:2

Stable under small perturbation to Γ SPT (Trivially gapped
phase).

3

Topological phase transition(s) at finite perturbation strength at
finite size, where the ground state symmetry charge jumps. ⇒Strong SPTC is more robust than weak SPTC.

● Γ is anomaly free.
● Single ground state under PBC.
● Degenerate ground states under OBC.
● Under ZA2 /ZΓ4 TBC, ground state carries ZΓ4 /ZA2 charge.

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Summary● There is a natural notion of SPT for quantum criticality –symmetry protected topological criticality.
● Decorated defect construction is a power tool to systematicallyconstruct SPTC.
● Symmetry charge of the ground state under TBC is a physicalobservable to probe the nontrivial SPTCs.
● Strong SPTC is more stable than weak SPTC against perturbing
to gapped phase with one ground state.
● Future directions: Classification? Fermionic system? Continuous
symmetry? Anomaly inflow picture? Entanglement feature?
SETC?
Thank you!
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