One-day workshop for QFT and string theory after Nambu Symposium

Plane Partition Realization of (Web of) W-algebra
Minimal Models
Yutaka Matsuo
The University of Tokyo

December 14, 2018

based on a work arXiv:1810.08512 with K. Harada
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Introduction

VOA Phase I: 1980s
Virasoro algebra and the generalization (vertex operator algebra=VOA) is a fundamental symmetry in string theory and in exactly
solvable statistical models.
[Ln , Lm ] = (n − m)Ln+m +c 3(n − n)δn+m,012

Due to its importance, many variations of the symmetry was studied
in the 80s. Famous examples are N = 2 superconformal algebra which
describes Calabi-Yau compactifications and W-algebras which
describes higher spin symmetry.
The subject flourished in 80-90s when the perturbative study of string
theory was essential, but became less popular since the mid-90s when
the revolution of non-perturbative studies (such as Seiberg-Witten
duality) began.
Yutaka Matsuo (The University of Tokyo) Plane Partition Realization of (Web of) W-algebra Minimal
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Introduction

VOA Phase II: 2010s
The situation began to change when Nekrasov and his collaborators
evaluated the instanton partition function of N = 2 super Yang-Millsin the omega background. Localization technique implied that it is asum of the contribution from the fixed point labeled by Young tables.
X
Z (q) =
q |λ| Zλ (m, a)λAnother dramatic change occurs when Alday, Gaiotto and Tachikawa(AGT) conjectured that these Nekrasov factors may be identified withthe conformal block function of 2D conformal field theory. It became
possible to use VOAs to describe the nonperturbative physics.
In order to understand the dual role of VOA, one needs to notice that
the VOA has apparently different descriptions.
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Introduction

Dual faces of VOA
In Phase I, VOA
by chiral fields (say, stress-energy
Pwas described−n−2
tensor T (z) = n Ln z
) where z is the coordinate of Riemannsurface. They are quantum fields on the world-sheet. They areappropriate to represent world sheet dynamics.
On the other hand, their action on the Nekrasov factor Zλ iscomplicated. Better set of operators in VOA is affine Yangian whichis generated by Drinfeld currents e(ζ), f (ζ), ψ(ζ) where ζ1 , ζ2 arespectral parameters. Their action of Nekrasov functions is simpler, forinstance, e (f ) is the operator which add (remove) a box to λ. On theother hand the algebra looks very different from VOA. As the namesuggests, affine Yangian is deeply related with the integrable models.
4D/2D duality implies that these two descriptions are equivalent.

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Introduction

c1
W∞ and affine Yangian of gl
The correspondence between the first and the second picture is best
described by W∞ and affine Yangian of gl(1). These two algebras areequivalent after the translation of parameters.
W∞ is a VOA which
by infinite currents W (n) (z). If we
P is described(n)−n−mwrite W (z) = m Wn,m z, the generators have two indicesn, m ∈ Z. The power of z corresponds to the second index m.
On the other hand, theP
Drinfeld currents e(ζ) (resp. f (ζ)) is(roughly) identified as n≥0 Wn,±1 ζ −n , expansion with respect to thefirst index which represents the spin of the operator.
In this sense, the duality in VOA is realized by the rotation of indicesS : (n, m) → (m, −n)Sometimes VOA Phase I (resp. II) is referred to as horizontal (resp.
vertical) representation.
The other VOA is obtained by a truncation of W∞ .
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Introduction

Gaiotto-Rapcak VOA and WoW
Recently, Gaiotto and Rapcak introduced a family of VOAs (GRVOA) defined geometrically by intersecting branes.
It may be either described by DS reduction of WZW model (horizontal
picture) or a truncation of affine Yangian (vertical picture).
We note that the general representation of affine Yangian is labeledby plane partition (3D Young diagram) which may have three
asymptotic Young diagram.
By connecting two plane partitions by identifying the asymptotic
Young diagrams, one may define a new algebra (tensor product of two
Yangians with extra generators which modify the shared Youngdiagram). Each diagram represents a new VOA. It is called as Web-of
W (WoW).
Thus GR VOA is an infinite family of algebras which has dual
descriptions.
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Introduction

Plane partition – from Wikipedia

When it is sliced in one direction, it is decomposed into a set of partitions
λ1  · · ·  λn . Plane partition may represent random surface or ”melting
crystal”.

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Introduction

Minimal models for GR VOA or WoW
In this talk, I will explain the description of minimal model for GR VOA or
WoW in the vertical frame.
Minimal models are important since
Modular property by finite number of fields
They describe solvable statistical models such as Ising.
building block of Calabi-Yau compactification (Gepner)

In the horizontal picture, we obtain the minimal models of VOA(W-algebra, for instance) by restricting the parameters to obtain thescreening charges provide infinite null states.
In the vertical frame, GR VOA or WoW is given by a truncation of affine
Yangian. We demonstrate that their minimal models by double truncation
which is significantly simpler than horizontal picture. We can provide agraphical representation of the Hilbert space.
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Gaiotto-Rapcak VOA

1Introduction

2

Gaiotto-Rapcak VOA

3

Diagrammatic construction of WoW

4

Minimal model by Double truncation

5

Some examples
WN algebra
N = 2 SCA

6

Conclusion and outlook

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Gaiotto-Rapcak VOA

Gaiotto-Rapcak VOA
Gaiotto and Rapcak introduced a VOAYLMN through an intersecting 5-branes.
Between two 5-branes, D3-branes withgiven number are placed. VOA is definedon the vertex of three 5-branes.

NS5
N
L

D5
M

On D3-branes in each domain, we have U(L) (U(M), U(N)) N = 4(twisted) SYM with coupling constant ψOn the boundary of each domain, we have 3D Chern-Simons withgauge groups. We obtain U(M|L) and U(N|L) CSs with ψ = k + h.
Finally on the vertex, we have the chiral algebra YLMN [ψ].
The algebra is isomorphic when we shuffle three numbers L, M, N but
we need perform SL(2, Z2 ) transformation for ψ.
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Gaiotto-Rapcak VOA

GR VOA as coset
The CFT appearing at the vertex becomes a coset CFT (for N > M),
DS N−M [U(N|L; Ψ)].
U(M|L; Ψ − 1)
DS N−M : Drinfeld-Sokolov reduction with principal su(2) embeddingin (N − M) × (N − M) part in U(N|L). The coset for N < M case or
N = M case is similar.
For M = L = 0 case, we obtain DS N [U(N; Ψ)] which is identical to
WN algebra (with decoupled U(1) boson) with central charge,c = (N − 1)(1 − Q 2 N(N + 1)) + 1,

Q = ψ 1/2 − ψ −1/2

This is analogous to N-reduction of KP hierarchy.
Thus, GR VOA is a generalization of WN algebra.
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Gaiotto-Rapcak VOA

Affine Yangian(Part of) defining relations of affine Yangian (AY):e(u)e(v ) ∼ ϕ(u − v )e(v )e(u),f (u)f (v ) ∼ ϕ(v − u)f (v )f (u),ψ(u)e(v ) ∼ ϕ(u − v )e(v )ψ(u),ψ(u)f (v ) ∼ ϕ(v − u)f (v )ψ(u),where
ϕ(u) =(u + h1 )(u + h2 )(u + h3 ).
(u − h1 )(u − h2 )(u − h3 )P
hi (with 3i=1 hi = 0) and ψ0 are the parameters of the algebra. They
have some redundancies.
Irreducible set of parameters λi = −ψ0 σ3 /hiP(σ3 := h1 h2 h3 , i λ−1=0). Relation to the parameters of W1+∞ :i
3Yc = 1 + (λi − 1),i=12

x = 144(c + 1)

3Y

(λi − 2)(λi − 3)−1 ,i=1

c: central charge of Virasoro algebra4 )2C 0 (C33
x 2 = 44
:OPE coefficients among primary fields.
0
(C )2
33

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Gaiotto-Rapcak VOA

Plane partition realization of AY
Plane partition realization: |Λi the orthogonal basis labelled by a planepartition Λ,
ψ(u) |Λi = ψΛ (u) |Λi ,rX11e(u) |Λi =− resu→q+h ψΛ (u) |Λ + i ,
u
−
q
−
h
σ
3
∈Λ+
r
X
1
1
f (u) |Λi =− resu→q+h ψΛ− (u) |Λ − i ,
u−q−h
σ3
∈Λ−
Y
ψΛ (u) = ψ0 (u − q)ϕ(u − q − h )∈Λwith h = h1 x + h2 y + h3 z when the box is located at (x, y , z).
The recursion relation among the orthogonal basis |Λi is essentially thesame as Nekrasov factor (while they are reduced to Young diagrams).
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Gaiotto-Rapcak VOA

Affine Yangian and GR VOA
Vertical GR VOA YLMN appears as a truncation of the affine Yangianthrough constraints on its parameters. (Prochazka and Rapcak)
L
M
N
+
+
=1
λ1 λ2 λ3With this constraint, the plane partition contains a null state (pit) at(L + 1, M + 1, N + 1). Note: there is a shift symmetry(L, M, N) → (L + k, M + k, N + k) and one may set one of L, M, N
to be zero.
For instance, when (L, M, N) = (0, 0, N), the height of the planepartition is limited by N. Such diagram can be sliced horizontally todescribe N Young diagrams λ1  λ2  · · ·  λN , which can be
identified with the Hilbert space of WN algebra (plus U(1) boson).
This agrees with the horizontal picture.
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Diagrammatic construction of WoW

1Introduction

2

Gaiotto-Rapcak VOA

3

Diagrammatic construction of WoW

4

Minimal model by Double truncation

5

Some examples
WN algebra
N = 2 SCA

6

Conclusion and outlook

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Diagrammatic construction of WoW

Plane partition with infinite legs
Plane partition may have nonvanishing asymptotic Young diagrams inthree directions, say λ, µ, ρ. The partition function for original planepartition is MacMahon function,∞Y(1 − q n )−n = M(q)n=1With the infinite legs attached, the partition function is modified to,Cµνρ (q)M(q)where Cµνρ (q) is the (unrefined) topological vertex. The conformal weightand U(1) charge becomes also nonvanishing. They are computed from thecoefficients of ψ(u). This function remains finite since the contributions of
infinite boxes cancel with each other and only finite factor remains. This is
a way to generate nontrivial representations.
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Diagrammatic construction of WoW

Connecting two Y -algebras: Web of W (WoW)
One may connect two affine Yangian by sharing one asymptotic Youngdiagram with another one.

L

R

N
M

We have two AYs acting on two plane partitions. Since it adds (or remove)
only one box to each plane partition, they cannot change the shape ofshared leg. We need to introduce extra generators which changes theshape of the asymptotic Young diagram. Thus, WoW consists of tensor
product of two W∞ with extra generators (typically fermionic operator).
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Diagrammatic construction of WoWFree field example
This is the simplest example of Y-algebra
Y001 which represents free fermion.
The basis of the state may be written as,b−n1 · · · b−ng c−m1 · · · c−mg |0iwith n1 > · · · > ng ≥ 1
and m1 > · · · > mg ≥ 1which gives aYoung diagram.
Another example is Y110
which is described by bosonic ghost.
β−n1 · · · β−ng γ−m1 · · · γ−mg |0iwith n1 ≥ · · · ≥ ng ≥ 1
and m1 ≥ · · · ≥ mg ≥ 1
It is described by a plane partition
1
with a pit (2, 2, 1)

n1
n2
ng
m1m2

mg

3

n6
n5
n4
n3
n2
n1

m6
m5
m4
m3
m2
m1

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Diagrammatic construction of WoW

bcβγ system: the simplest WoW
One may connect these two to generate the simplest example of WoW
Y110 − Y001 . Originally the AF is described by bilinear combination of bc
and βγ separately. In the combined system new generators bγ and βc
appear which modifies tensor product of AY to WoW. In this case, thenumber of bc and βγ may not balance separately. It generates an infinite
leg with the same shape appearing in both sides. Supersymmetric
generators operate on them to modify the shape.
3
2
n6
n5
n4
n3

m̄1

m3
m2
m1

⊗
n̄
n̄2n̄3 4

m̄g3
n̄g4

n̄1

n2
n1

1

2

1

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Diagrammatic construction of WoW

bcβγ system: negative U(1) charge
The number #(c) − #(b) may be either positive or negative. Dependingon the sign, the direction of the leg changes. For the positive (resp.
negative) case, it goes to x 2 (resp. x 1 ) direction. In diagrammatic
representation, it implies that the direction of the connected legs depends
on the sign. Since this is inconvenient, we fix the direction of the leg andallows the Young diagrams to have negative rows.
3

n3
n2
n1

m̄1
m̄2
m̄3

m6
m5
m4

n̄1

⊗

2

m̄g3
n̄g4

m3
m2
m1

1

2

1

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Diagrammatic construction of WoW

WoW: generalization
By changing L, M, N, R, we obtain different types of VOA.

L

R

N

M

(L, M, N, R) = (0, 1, 1, 0) bcβγ system.
(L, M, N, R) = (0, 1, 2, 0) N = 2 SCA(L, M, N, R) = (0, 1, 3, 0) Polyakov-Bershadsky VOA
One may connect the third vertex (WoWoW) and so on, and it describes
infinitely many unknown algebras. The treatment of negative Youngdiagrams may be more complicated for the general models.
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Minimal model by Double truncation

1

Introduction

2

Gaiotto-Rapcak VOA

3

Diagrammatic construction of WoW

4

Minimal model by Double truncation

5

Some examples
WN algebra
N = 2 SCA

6

Conclusion and outlook

Yutaka Matsuo (The University of Tokyo) Plane Partition Realization of (Web of) W-algebra Minimal
DecemberModels14, 201822 / 34

Minimal model by Double truncation

Double truncation
In the following, we consider a special case where we put the second
constraint on the parameters.
L1
M1 N1
+
+
= 1,
λ1
λ2
λ3

L2
M2 N 2
+
+
= 1.
λ1
λ2
λ3

In this case, we seem to have two pits at (Li , Mi , Ni ) (i = 1, 2). This is,however, too naive. With such parametrization, we have periodicity in hfunction,hx+L1 ,y +M1 ,z+N1 = hx+L2 ,y +M2 ,z+N2
Since h governs the representation of the plane partition, above relation
implies that the plane partition becomes degenerate, namely the boxes inthe different locations should be identified. We will denote YL1 M1 N1 :L2 M2 N2
to describe such algebra. With two constraints, there is no free parameter.
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Minimal model by Double truncation

We illustrate the degeneration phenomena in the simplest case Y120:001 .
3

3

2

1

⇔

2

1

The algebra may be identified as Y001 described by one Young diagram orY120 by plane partition with a pit at (2, 3, 1). One may construct thelatter diagram by cutting Young diagram into pieces. One needs to imposethat both diagrams is consistent as a plane partition which imposes a tightconstraints on both.

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Some examples

Outline1

Introduction

2

Gaiotto-Rapcak VOA

3

Diagrammatic construction of WoW

4

Minimal model by Double truncation

5

Some examples
WN algebra
N = 2 SCA

6

Conclusion and outlook

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Some examples

WN algebra

WN -algebra
WN algebra is realized as GR VOA Y00N . In order to describe its minimalmodel, we restrict the parameters to have second pit : Y00N:(p−N)(q−N)0
where p, q > N.

0

N
0

p−N

⇔

q−N

0

The central charge becomesc = (N − 1) 1 − Q 2 N(N + 1) + 1 ,Q=pp
β − 1/ β,β = p/q

The identification of boxes is (x, y , z + N) ∼ (x + p − N, y + q − N, z). It
generates the correct partition function of WN minimal models.
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Some examples

WN algebra

Primary fields by asymptotic Young diagrams
In order to describe the primary fields of WN algebra, we need to havenontrivial asymptotic Young diagrams in x, y directions specified (~n andn~0 ) (z direction is not allowed since we have a pit at (0, 0, N)). We
decompose the plane partition in z direction which gives N Youngdiagrams.
2
0
ñN−1

1
YN

0
ñN−2

..

YN−1
YN−2

..

.
ñ10

.

Y2

ñN−1

Y1

ñN−2
..

.

ñ1
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Some examples

WN algebra

In order that they should be consistent as a plane partition and theperiodicity constraint
Yi,R − Yi+1,R+ni −1 ≥ −(ni0 − 1) ,with the periodicity YN+1 ∼ Y1 . This constraint is referred to as N-Burge
condition and it is known to characterize the Hilbert space of minimalmodels correctly.
The conformal dimensions of the primary fields can be derived
geometrically from the shape of the infinite legs as,P0 ω )2 − N(N 2 − 1)(p − q)212( N−1ii=1 (pni − qni )~∆(~n, n~ ) =24pq
0

which agrees with the known formula.

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Some examples

N = 2 SCA

N = 2 SCFT is a WoW (Y210 − Y001 ) described by following diagram
2

1

0

0

Y001 is described by a decoupled free fermion. It provides, however,the degree of freedom of intermediate channel. Supercharges aredefined to act on it.
We impose the parameter of Y210 such that it has the second pit at(1, 1, n + 1). Thus the algebra may be written as Y210:00n . The
central charge of the whole algebra becomes,
3n
c=
+1
n+2
which corresponds to the minimal models (plus decoupled U(1)factor).

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Some examples

N = 2 SCA

N = 2 SCA
There is a free leg in Y210;00n (between 2 and 1). Putting anonvanishing asymptotic Young diagram corretly describes every
primary field. By tuning asymptotic Young diagrams, one may
describe all the known primary fields
hl,m =

l(l + 2) − m2,
4(n + 2)

Jl,m =

mn+2

with 0 ≤ l ≤ n, −l ≤ m ≤ l, l − m ≡ 0 (mod2).
The number of states of the plane partition is identical to the known
character of N = 2 SCA (Dobrev, Kiritsis, Matsuo 1986)
χ(τ, z) =

X
r ∈Zn

(n)

cl,m+2r (τ ) Θ2m+2r (n+2),2n(n+2) (τ,zn+2) + Θ2m+2(r +n)(n+2),2n(n+2) (τ,zn+2

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Some examples

N = 2 SCA

Some technical issue
Here is a picture of the plane partition in Y210;00n
2

n
n
ν1 ν2

k

..
.
n
r

1

Two columns in x 1 direction describes a free leg. The heights ν1 , ν2
are related to l, m which parametrizes the primary fields.
The vacancy between two legs appears when we translate infinite legs.
These vacancy should be filled to make a consistent plane partition.
In k < 0 case, we need to have both legs in x 1 direction.
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Conclusion and outlook

Conclusion and outlook

Construction of minimal models in the conventional method issophiscated– DS reduction, screening currents, counting of null states
In the vertical frame, the reduction is straightforward and geometrical– adding the second pit in the plane partition.
It works for known VOA, WN algebra and N = 2 SCA. We hope that
it applies universally for other WoW as well. An interesting exercise is
to study the next simplest case Y310:00n − Y001 which produces
Bershadsky-Polyakov algebra (Bershadsky 1991, Arakawa 2010).
Use of plane partition may provide a new insight to understand therelation between the algebra and geometry in string theory.

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Conclusion and outlook

Thank you for your attention.

Bon Voyage, NITEP!
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