What’s New With
Q?
Clay Córdova
School of Natural Sciences, IAS
June 26th 2018

The Road Ahead
Disclaimer:

Supersymmetry is an enormous subject. I make no claim at
being complete. I apologize in advance for the variety offascinating papers that are not discussed below

Themes:•Classification of superconformal field theories•

Exact calculations in SUSY QFT•

Connections between SUSY and Non-SUSY dynamics

The Classiﬁcation Problem
Goal:

In each spacetime dimension d>2, classify
possible superconformal field theories.

Why is it interesting? :•SCFTs are highly symmetric universality classes. Understanding them is a
starting point to classifying the possible dynamics of QFTs•

SCFTs exhibit extremely rich mathematical structure. New SCFTs might lead tonew mathematics•

The special case d=3 is related holographically to AdS vacua crucial to thestring landscapeThe Nahm Classification

Foundational algebraic results due to Nahm mean that the range of spacetime
dimensions where SCFTs are possible is limited
If we demand that our theories admit an energy-momentum tensor, then #Poincare supercharges, N Q, is also bounded[Cordova-Dumitrescu-Intriligator]
d=6:
N Q=8, 16. Highly constrained by consistency (e.g. anomaly
cancellation). Plausible conjectured classification for N Q=16. Constructive
progress using string theory for N Q=8d=5:
N Q= 8. New examples and proposed classifications for simple theories(low rank) using string theory techniques
d=4:
N Q=4, 8, 12, 16. Plausible conjectured classification for N Q=16. Recent
results for simple theories (low rank) with smaller N Q
d=3:

N Q=2, 4, 6, 8, 10, 12, 16. Classification wide open/wild (even for big N Q)

The Status of Maximal SUSY•

d=6: N Q= 16 theories admit a conjectural ADE classification [Witten]. The A
series is constructed by M5-branes.

Can motivate the ADE classification by reducing on a circle to 5d SYM. On the
Coulomb branch, the monopole strings and W-boson particles must uplift to thesame 6d string which requires simply laced group

The Status of Maximal SUSY•

d=6: N Q= 16 theories admit a conjectural ADE classification [Witten]. The A
series is constructed by M5-branes.

Can motivate the ADE classification by reducing on a circle to 5d SYM. On the
Coulomb branch, the monopole strings and W-boson particles must uplift to thesame 6d string which requires simply laced group•d=4: N Q= 16 conjectural classification in terms of gauge theories, (i.e. N=4
SYM) Can it be made into a theorem?

Every N Q= 16 theory has an exactly marginal complex coupling ! with fixed&' ($Zamolodchikov metric [Papadodimas]: "# ~)* ' (
Try to prove that there is a weak-coupling limit (tricky … depends on the
topology of the coupling space i.e. the duality group)Maximal SUSY Zoo in 3d
Bagger-Lambert-Gustavsson:&' 2 $ × &' 2 *$!"#$ =
+,

G1

G2

+k

-k

!"#$- = &' 2 $ × &' 2 *$Maximal susy for any k. Small k can be thought of as the worldvolume theory of
two M2 branes. But for general k there is no known string theory embedding!

Maximal SUSY Zoo in 3d
Bagger-Lambert-Gustavsson:&' 2 $ × &' 2 *$!"#$ =
+,

G1

G2

+k

-k

!"#$- = &' 2 $ × &' 2 *$Maximal susy for any k. Small k can be thought of as the worldvolume theory of
two M2 branes. But for general k there is no known string theory embedding!
Aharony-Bergman-Jafferis-Maldecena:.!/01,3 = ' 4 3 ×' 4 *3.!/01,, = ' 4 , ×' 4 *,.!/153,1,, = ' 4 + 1 , ×' 4 *,Can be thought of as the low energy limit of SYM with gauge group U(N),
SO(2N), and SO(2N+1)

Maximal SUSY Zoo in 3d
Bagger-Lambert-Gustavsson:&' 2 $ × &' 2 *$!"#$ =
+,

G1

G2

+k

-k

!"#$- = &' 2 $ × &' 2 *$Maximal susy for any k. Small k can be thought of as the worldvolume theory of
two M2 branes. But for general k there is no known string theory embedding!
Aharony-Bergman-Jafferis-Maldecena:.!/01,3 = ' 4 3 ×' 4 *3.!/01,, = ' 4 , ×' 4 *,.!/153,1,, = ' 4 + 1 , ×' 4 *,Can be thought of as the low energy limit of SYM with gauge group U(N),
SO(2N), and SO(2N+1)
List both incomplete (exceptional groups!) and redundant (dualities!)[Bashkirov-Kapustin, Agmon-Chester-Pufu]:!"#8 + 9:;; <ℎ;>:? = .!/08,3

The Exotic World of 4d N=3• In late 2015 [Gacía -Etxebarria-Regalado] constructed the first known
examples of 4d SCFTs with N Q =12• These theories are intrinsically strongly-coupled (no marginal parameters are
possible)[Aharony-Evtikhiev, Cordova-Dumitrescu-Intriligator]

The Exotic World of 4d N=3• In late 2015 [Gacía -Etxebarria-Regalado] constructed the first known
examples of 4d SCFTs with N Q =12• These theories are intrinsically strongly-coupled (no marginal parameters are
possible)[Aharony-Evtikhiev, Cordova-Dumitrescu-Intriligator]• Construction involves F-theory on (" # ×% & )/)* (can be thought of as an Ftheory uplift of ABJM geometry). For k=1,2 the + parameter of the torus canbe general. For k=3,4,6 this construction is possible if + is frozen.

•+=1+ = 2 &34/#

k= 4

k= 3

+ = 2 &34/5
k= 6

Probing with D3 branes leads to interacting 4d SCFTs with N Q =12. For large
numbers of D3 branes these theories have an ,-./ dual where the transverse
geometry is a quotient of . / and the quotient also acts as a duality inspacetime [Aharony-Tachikawa]

The Exotic World of 4d N=3
Sparse results on the dynamics of these theories• Chiral Algebras: [Nishinaka-Tachikawa] In rank one this leads to exact
computations of protected OPE coefficients and a formula for the central
charge.
! = #~2& − 1

The Exotic World of 4d N=3
Sparse results on the dynamics of these theories• Chiral Algebras: [Nishinaka-Tachikawa] In rank one this leads to exact
computations of protected OPE coefficients and a formula for the central
charge.
! = #~2& − 1• Moduli Space: [Nishinaka-Tachikawa, Argyres-Martone] In rank one ) * /,- ,with a chiral operator of dimension k• Also results on superconformal indices ([Imamura-Yokoyama, Burton-PiniPomoni]) and superconformal bootstrap ([Lemos-Liendo-Meneghelli-Mitev])

The Exotic World of 4d N=3
Sparse results on the dynamics of these theories• Chiral Algebras: [Nishinaka-Tachikawa] In rank one this leads to exact
computations of protected OPE coefficients and a formula for the central
charge.
! = #~2& − 1• Moduli Space: [Nishinaka-Tachikawa, Argyres-Martone] In rank one ) * /,- ,with a chiral operator of dimension k• Also results on superconformal indices ([Imamura-Yokoyama, Burton-PiniPomoni]) and superconformal bootstrap ([Lemos-Liendo-Meneghelli-Mitev])
4d N=3 theories seem like a natural candidate for a classification program. The
moduli space of vacua is necessarily flat
Assuming no relations among the chiral operators (not clear if this is correct!)
we are led to a quotient of a vector space by a complex reflection group.
Weyl groups give N=4 examples, other groups may give genuinely N=3 theories

Classiﬁcation
Requires a Tastefor the Exotic“Don’t encourage him, Sylvia.”

The NQ = 8 Universe: 6dOver the past few years there has been dramatic progress on 6d (1,0)
theories (on moduli space there are 2-form gauge fields ! and "! =∗ "!)F-Theory Constructions: classification of possible geometries that can
support a 6d SCFT [Heckman-Morisson-Rudelius-Vafa]
Dual Holographic Solutions: All %"&' solutions of type II string theory are
known [Appruzi-Fazzi-Rosa-Tomasiello]
QFT Analysis: strategy is to deform to the tensor branch and try tounderstand consistency conditions [Seiberg, Bhardwaj]. This led to progresson anomalies [Ohmori-Shimizu-Tachikawa-Yonekura, Intriligator, CordovaDumitrescu-Intriligator]

The NQ = 8 Universe: 6dOver the past few years there has been dramatic progress on 6d (1,0)
theories (on moduli space there are 2-form gauge fields ! and "! =∗ "!)F-Theory Constructions: classification of possible geometries that can
support a 6d SCFT [Heckman-Morisson-Rudelius-Vafa]
Dual Holographic Solutions: All %"&' solutions of type II string theory are
known [Appruzi-Fazzi-Rosa-Tomasiello]
QFT Analysis: strategy is to deform to the tensor branch and try tounderstand consistency conditions [Seiberg, Bhardwaj]. This led to progresson anomalies [Ohmori-Shimizu-Tachikawa-Yonekura, Intriligator, CordovaDumitrescu-Intriligator]
Open Problems: Known interacting examples always have tensor
branches…why? How many distinct interacting theories are there which, at
low-energy, look like a single tensor? (F-theory predicts exactly one! The
small () instanton)

Applications of 6d SCFTs
What’s at Stake: Compactification on a manifold M produces field theories inlower spacetime dimension. We can trade some problems involving the
dynamics of the lower dimensional theory for problems involving thegeometry of M.
We can use our knowledge of 6d, together with geometry, to organize ourthoughts about SCFTs in lower spacetime dimensions (Famous example, class
S construction of 4d N=2 theories. [Gaiotto])

Applications of 6d SCFTs
What’s at Stake: Compactification on a manifold M produces field theories inlower spacetime dimension. We can trade some problems involving the
dynamics of the lower dimensional theory for problems involving thegeometry of M.
We can use our knowledge of 6d, together with geometry, to organize ourthoughts about SCFTs in lower spacetime dimensions (Famous example, class
S construction of 4d N=2 theories. [Gaiotto])
Recent Extension: NS5-branes embedded in k D6 branes
Reduce on a surface to get 4d SCFT

SU(k)

SU(k)

[Gaiotto-Razamat, Franco-Hayashi-Uranga, Del Zotto-Vafa-Xie, HananyMaruyoshi, Coman-Pomoni-Taki-Yagi, Morrison-Vafa, Ito-Yoshida, RazamatVafa-Zafrir, Bah-Hanany-Maruyoshi-Razamat-Tachikawa-Zafrir, Mitev-Pomoni,
Bourton-Pomoni, Kim-Razamat-Vafa-Zafrir, Razamat-Sabag]
Results: moduli, indices, anomalies, symmetry enhancement, duality

Applications of 6d SCFTs
What’s at Stake: Compactification on a manifold M produces field theories inlower spacetime dimension. We can trade some problems involving the
dynamics of the lower dimensional theory for problems involving thegeometry of M.
We can use our knowledge of 6d, together with geometry, to organize ourthoughts about SCFTs in lower spacetime dimensions (Famous example, class
S construction of 4d N=2 theories. [Gaiotto])
Recent Extension: NS5-branes embedded in k D6 branes
Reduce on a surface to get 4d SCFT

SU(k)

SU(k)

[Gaiotto-Razamat, Franco-Hayashi-Uranga, Del Zotto-Vafa-Xie, HananyMaruyoshi, Coman-Pomoni-Taki-Yagi, Morrison-Vafa, Ito-Yoshida, RazamatVafa-Zafrir, Bah-Hanany-Maruyoshi-Razamat-Tachikawa-Zafrir, Mitev-Pomoni,
Bourton-Pomoni, Kim-Razamat-Vafa-Zafrir, Razamat-Sabag]
Results: moduli, indices, anomalies, symmetry enhancement, duality
Open Problem: Are 6d SCFTs parents for all lower-dimensional SCFTs?

The NQ = 8 Universe: 5d
Field Theory: After relevant deformation, one often finds a non-abeliangauge theory and hypermultiplets. [Seiberg, Intriligator-Morrison-Seiberg]
Geometry/String-Theory: M-theory on a local geometry where a complex
surface collapses [Intriligator-Morrison-Seiberg]. Or in Type IIB, webs of (p,q)
5-branes [Aharony-Hanany].
Holography: All !"#$ solutions of type II known [Appruzi-Fazzi-Passias-RosaTomasiello]

The NQ = 8 Universe: 5d
Field Theory: After relevant deformation, one often finds a non-abeliangauge theory and hypermultiplets. [Seiberg, Intriligator-Morrison-Seiberg]
Geometry/String-Theory: M-theory on a local geometry where a complex
surface collapses [Intriligator-Morrison-Seiberg]. Or in Type IIB, webs of (p,q)
5-branes [Aharony-Hanany].
Holography: All !"#$ solutions of type II known [Appruzi-Fazzi-Passias-RosaTomasiello]
Puzzle: What are the correct low-energy consistency conditions on the field
theory? On the moduli space of vacua in the gauge theory we find an
abelian theory. Should this make sense for all vacua, or just a subregion?
For the field theory to be consistent with known string constructions we must
use the weaker (subregion) constraint. Explored in [Jefferson-Kim-Vafa-Zafrir]

The NQ = 8 Universe: 5d
Field Theory: After relevant deformation, one often finds a non-abeliangauge theory and hypermultiplets. [Seiberg, Intriligator-Morrison-Seiberg]
Geometry/String-Theory: M-theory on a local geometry where a complex
surface collapses [Intriligator-Morrison-Seiberg]. Or in Type IIB, webs of (p,q)
5-branes [Aharony-Hanany].
Holography: All !"#$ solutions of type II known [Appruzi-Fazzi-Passias-RosaTomasiello]
Puzzle: What are the correct low-energy consistency conditions on the field
theory? On the moduli space of vacua in the gauge theory we find an
abelian theory. Should this make sense for all vacua, or just a subregion?
For the field theory to be consistent with known string constructions we must
use the weaker (subregion) constraint. Explored in [Jefferson-Kim-Vafa-Zafrir]
The 6d to 5d connection has been explored [Hayashi-Kim-Lee-Yagi, Del
Zotto-Heckman-Morrison, Jefferson-Katz-Kim-Vafa]. Known 5d theories with
≤ 2 non-abelian gauge groups arise from compactification from 6d

Recent Results in 5d
New results about these theories have appeared in [Chang-Fluder-Lin-Wang]
Consider the [Seiberg] exceptional series. At low energies these are SP(2N)
gauge theory with N F < 8 fundamental hypermultiplets
In the UV the gauge coupling becomes strong and we reach a critical point
The topological current !"#$ ∼∗ '((* ∧ *) together with the IR flavor symmetry
acting on the hypers enhances in the UV to the exceptional group -./ 01

Recent Results in 5d
New results about these theories have appeared in [Chang-Fluder-Lin-Wang]
Consider the [Seiberg] exceptional series. At low energies these are SP(2N)
gauge theory with N F < 8 fundamental hypermultiplets
In the UV the gauge coupling becomes strong and we reach a critical point
The topological current !"#$ ∼∗ '((* ∧ *) together with the IR flavor symmetry
acting on the hypers enhances in the UV to the exceptional group -./ 01
Using both the superconformal bootstrap and exact results for sphere partition
functions they extract estimates for the two-point functions< '' >∼ 4" ,< !! >∼ 46

For instance in the rank one theory with -7 flavor symmetry one finds4" (-7 )84 (ℎ:;<() ≈ 153"

46 (-7 )84 (ℎ:;<() ≈ 243
6

The results also suggest that, among all theories with -7 flavor symmetry, thistheory minimizes 46

The NQ = 8 Universe: 4d
The problem of classifying 4d SCFTs has reached the stage of big data. For
instance [Chacaltana-Distler-Trimm-Zhu] recently constructed 49,836 isolated
SCFTs by compactifying the !" (2,0) theory on three-punctured spheres

The NQ = 8 Universe: 4d
The problem of classifying 4d SCFTs has reached the stage of big data. For
instance [Chacaltana-Distler-Trimm-Zhu] recently constructed 49,836 isolated
SCFTs by compactifying the !" (2,0) theory on three-punctured spheres
Field theory strategies for understanding the landscape of 4d N=2 SCFTs can bedivided into two approaches
Coulomb Branch: Directly constrain the [Seiberg-Witten] geometry. Works bestfor low-rank theories. Systematized by [Argyres, Lottito, Long, Lu, Martone,
Whittig] (Next talk!!)

The NQ = 8 Universe: 4d
The problem of classifying 4d SCFTs has reached the stage of big data. For
instance [Chacaltana-Distler-Trimm-Zhu] recently constructed 49,836 isolated
SCFTs by compactifying the !" (2,0) theory on three-punctured spheres
Field theory strategies for understanding the landscape of 4d N=2 SCFTs can bedivided into two approaches
Coulomb Branch: Directly constrain the [Seiberg-Witten] geometry. Works bestfor low-rank theories. Systematized by [Argyres, Lottito, Long, Lu, Martone,
Whittig] (Next talk!!)
Higgs Branch: There is a tight interplay between the geometry of the Higgs
branch and the protected set of local operators encoded in the chiral algebraof [Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees]
The chiral algebra is a VOA constructed from (twisted) correlators of local
operators restricted to a 2d plane. The Higgs branch is an associated variety
constructed by truncating to commutative algebra [Beem-Rastelli]# $%&' AD Theory~( $ /*% Higgs Branch~ [Bershadsky-Polyakov] VOACalculations in SUSY QFT
Exact calculations are the bread and butter of supersymmetry• Superconformal index of the Argyres-Douglas CFT [Song’s Talk!]• Organizing principles for supersymmetric partition functions• Relations between moduli spaces of vacua and critical exponents of local
operators [Hellermans’s Talk!]• 3d dualities and boundary conditions [Paquette’s Talk!]• Exact calculations using deconstructed field theories• Relation between OPE coefficients and scattering amplitudes [Chester/Pufu’s
Talk!]• Non-planar correlation functions from integrability [Komatsu’s Talk]

Calculations in SUSY QFT
Exact calculations are the bread and butter of supersymmetry• Superconformal index of the Argyres-Douglas CFT [Song’s Talk!]• Organizing principles for supersymmetric partition functions• Relations between moduli spaces of vacua and critical exponents of local
operators [Hellermans’s Talk!]• 3d dualities and boundary conditions [Paquette’s Talk!]• Exact calculations using deconstructed field theories• Relation between OPE coefficients and scattering amplitudes [Chester/Pufu’s
Talk!]• Non-planar correlation functions from integrability [Komatsu’s Talk]

An Organizing Principle for
Localization Calculations
Over the past decade there has been significant interest in computing
supersymmetric partition functions and indices [many many papers]• 3d theories on M 3: S 3 (correlation functions, F-theorem); S 2 x S 1(superconformal index); ! g x S 1 (AdS 4 black hole entropy)….
• 4d theories on M 3 x S 1: S 3 x S 1 (superconformal index); ! g x T 2….

An Organizing Principle for
Localization Calculations
Over the past decade there has been significant interest in computing
supersymmetric partition functions and indices [many many papers]• 3d theories on M 3: S 3 (correlation functions, F-theorem); S 2 x S 1(superconformal index); ! g x S 1 (AdS 4 black hole entropy)….
• 4d theories on M 3 x S 1: S 3 x S 1 (superconformal index); ! g x T 2….
Q: How can we unify these calculations? Recent progress [Closset-Kim-Willett]An Organizing Principle for
Localization Calculations
Over the past decade there has been significant interest in computing
supersymmetric partition functions and indices [many many papers]• 3d theories on M 3: S 3 (correlation functions, F-theorem); S 2 x S 1(superconformal index); ! g x S 1 (AdS 4 black hole entropy)….
• 4d theories on M 3 x S 1: S 3 x S 1 (superconformal index); ! g x T 2….
Q: How can we unify these calculations? Recent progress [Closset-Kim-Willett]An important clue is that in the special case where M 3 is a product ! g x S 1 thenwe can interpret the partition function as an observable in a 2d topological Amodel on the surface ! g
Natural to interpret other partition functions of 3d or 4d theories as observables
in 2d theory. This organizes the different geometries and background fields into
correlation functions of 2d local operators

Geometry

Local Operators

In the 2d topological theory, changes in topology come from the handle gluing
local operator H [Vafa]

HGeometry

Local Operators

In the 2d topological theory, changes in topology come from the handle gluing
local operator H [Vafa]

H
In a partition function of a d-dimensional theory on ! g x T d-2 we may want tointroduce fluxes for background gauge fields (R or flavor symmetries)
By concentrating the flux on ! g , we can replace it by a 2d local flux operator

Geometry

Local Operators

In the 2d topological theory, changes in topology come from the handle gluing
local operator H [Vafa]

H
In a partition function of a d-dimensional theory on ! g x T d-2 we may want tointroduce fluxes for background gauge fields (R or flavor symmetries)
By concentrating the flux on ! g , we can replace it by a 2d local flux operator
One such flux is associated to the geometric fibration of the circles over ! g.
Using this idea we can change a partition function on a circle fibration (e.g. S 3)
to a partition function on the product (e.g. S 2 x S 1)
The price we pay is that we introduce a disturbance localized at a point in thesurface (e.g. the S 2)

Geometry

Local Operators

There is also a 2d local operator that allows us to insert orbifold singularities in
the fibration, i.e. non-trivial Seifert fibers (to appear!)A 2d local operator uplifts to a defect operator in higher dimensions: in 3d aline, in 4d a surface

Geometry

Local Operators

There is also a 2d local operator that allows us to insert orbifold singularities in
the fibration, i.e. non-trivial Seifert fibers (to appear!)A 2d local operator uplifts to a defect operator in higher dimensions: in 3d aline, in 4d a surface

3d Statement: The partition function of a 3d N=2 theory on any Seifert 3manifold (roughly a circle bundle over a Riemann surface with orbifold
singularities) can be understood as a correlation function of line defects on thegeometry S 2 x S 1 where the lines wrap the circle
4d Statement: The partition function of a 4d N=1 theory on torus bundle over a
Riemann surface can be understood as a correlation function of surface
defects on the geometry S 2 x T 2 where the surface defects wrap the torus
Organizes a huge class of partition functions!

(2,0) Deconstruction
The 6d maximally symmetric (2,0) theory plays a starring role in the story of SCFTs
But what is this theory? How can we study it?
Various Ideas: light cone quantization, large N techniques, reduction to 5d,dimensional deconstruction [Arkani-Hamed-Cohen-Kaplan-Karch-Motl](2,0) Deconstruction
The 6d maximally symmetric (2,0) theory plays a starring role in the story of SCFTs
But what is this theory? How can we study it?
Various Ideas: light cone quantization, large N techniques, reduction to 5d,dimensional deconstruction [Arkani-Hamed-Cohen-Kaplan-Karch-Motl]
Setup: 4d quiver gauge theory with N Q=8
N nodes SU(k) gauge groups, bifundamental matter• Give hypermultiplets a vev v (gauge group now SU(k))• Consider a limit of large N, v, and gauge coupling g• Keep fixed: g/v = 2!R 1

andN/gv = 2!R 2(2,0) Deconstruction
The 6d maximally symmetric (2,0) theory plays a starring role in the story of SCFTs
But what is this theory? How can we study it?
Various Ideas: light cone quantization, large N techniques, reduction to 5d,dimensional deconstruction [Arkani-Hamed-Cohen-Kaplan-Karch-Motl]
Setup: 4d quiver gauge theory with N Q=8
N nodes SU(k) gauge groups, bifundamental matter• Give hypermultiplets a vev v (gauge group now SU(k))• Consider a limit of large N, v, and gauge coupling g• Keep fixed: g/v = 2!R 1

andN/gv = 2!R 2

At long distances the resulting theory is believed to describe the 6d (2,0) theoryof k M5 branes on a torus with sides R i (Note that the 4d theory is UV complete)
Nice idea, but how can we make it quantitative?

Quantitative Deconstruction[Hayling-Papageorgakis-Pomoni-Rodriguez-Gomez] provide quantitative
checks on the deconstruction idea using supersymmetric partition functions
Half BPS Operators: The partition function of half-bps primary operators in the(2,0) is [Bhattacharyya-Minwalla]:'! = #$%&

1
1 − *$(One can guess this by looking at coordinates on the moduli space of vacua )This is reproduced by the higgs branch primary operators in the quiverQuantitative Deconstruction[Hayling-Papageorgakis-Pomoni-Rodriguez-Gomez] provide quantitative
checks on the deconstruction idea using supersymmetric partition functions
Half BPS Operators: The partition function of half-bps primary operators in the(2,0) is [Bhattacharyya-Minwalla]:'! = #$%&

1
1 − *$(One can guess this by looking at coordinates on the moduli space of vacua )This is reproduced by the higgs branch primary operators in the quiver
These days we have to do better than half-BPS operators!
S 4 Partition Function: Using exact localization we can compute the S 4 partition
function in 4d and the S 4 x T 2 partition function in 6d. These match exactly in thedeconstruction limit (should be sensitive to the string spectrum in 6d)
Striking check! Motivation to work harder and learn more about the (2,0) theory

Connections Between SUSY
and Non-SUSY Dynamics
Lots of recent progress in 3d non-supersymmetric QFT (see e.g Benini’s talk)
These results have cousins in the supersymmetric world. For instance• 3d N=4 mirror symmetry: U(1) gauge theory with 2 hypers of charge 1 is selfdual and flows to a theory with enhanced SU(2)xSU(2) global symmetry[Intriligator-Seiberg]• Non Susy Analog: U(1) gauge theory with 2 electrons of charge 1 is self-dual
and flows to a theory with SU(2)xSU(2) global symmetry [Xu-You, Hsin-Seiberg]

Connections Between SUSY
and Non-SUSY Dynamics
Lots of recent progress in 3d non-supersymmetric QFT (see e.g Benini’s talk)
These results have cousins in the supersymmetric world. For instance• 3d N=4 mirror symmetry: U(1) gauge theory with 2 hypers of charge 1 is selfdual and flows to a theory with enhanced SU(2)xSU(2) global symmetry[Intriligator-Seiberg]• Non Susy Analog: U(1) gauge theory with 2 electrons of charge 1 is self-dual
and flows to a theory with SU(2)xSU(2) global symmetry [Xu-You, Hsin-Seiberg]
It is natural to ask if such susy/non-susy dualities can be smoothly connected.
In 3d this has been shown for large N Chern-Simons matter dualities in [JainMinwalla-Yokoyama, Gur-Ari-Yacoby] and explored at small N in [KachruMulligan-Torroba-Wang] using soft supersymmetry breaking
Can we use this idea to guess the IR behavior of gauge theories in 4d?

Soft SUSY Breaking
We add a susy breaking mass term to a susy Lagrangian to remove the scalars! = ! #$#% + ' ( ) (Ideally, the mass operator should be positive definite and part of a shortmultiplet (so we can track it) and preserve all symmetries so that all ‘t Hooft
anomalies are maintainedThis problem of 4d soft susy breaking was explored in the heyday of duality[Aharony-Sonnenschein-Peskin-Yankielowicz, Alvarez-Gaume-Distler-Kounnas-MarinoZamora, Arkani-Hamed-Rattazzi, Luty-Rattazzi, Nelson-Strassler, Abel-Buican-Komargodski ]Soft SUSY Breaking
We add a susy breaking mass term to a susy Lagrangian to remove the scalars! = ! #$#% + ' ( ) (Ideally, the mass operator should be positive definite and part of a shortmultiplet (so we can track it) and preserve all symmetries so that all ‘t Hooft
anomalies are maintainedThis problem of 4d soft susy breaking was explored in the heyday of duality[Aharony-Sonnenschein-Peskin-Yankielowicz, Alvarez-Gaume-Distler-Kounnas-MarinoZamora, Arkani-Hamed-Rattazzi, Luty-Rattazzi, Nelson-Strassler, Abel-Buican-Komargodski ]In general (even if we can find such a mass) we face the following challenge:• The regime of parametric control is small M (using supersymmetry)• The regime of most interest is large M (the non-supersymmetric theory)

Soft SUSY Breaking
We add a susy breaking mass term to a susy Lagrangian to remove the scalars! = ! #$#% + ' ( ) (Ideally, the mass operator should be positive definite and part of a shortmultiplet (so we can track it) and preserve all symmetries so that all ‘t Hooft
anomalies are maintainedThis problem of 4d soft susy breaking was explored in the heyday of duality[Aharony-Sonnenschein-Peskin-Yankielowicz, Alvarez-Gaume-Distler-Kounnas-MarinoZamora, Arkani-Hamed-Rattazzi, Luty-Rattazzi, Nelson-Strassler, Abel-Buican-Komargodski ]In general (even if we can find such a mass) we face the following challenge:• The regime of parametric control is small M (using supersymmetry)• The regime of most interest is large M (the non-supersymmetric theory)
We will use the supersymmetric model to propose candidate phases for the
non-susy theory. We cannot be certain which, if any, holds dynamically. But by
construction any phases that we find this way will match all ‘t Hooft anomaliesAdjoint QCD
An ideal test case is 4d adjoint QCD with two Weyl flavors of adjoint quarks !" .
The index # is a doublet under a chiral SU(2) global symmetry
Lore is that this theory confines and undergoes chiral symmetry breaking via achiral condensate< %& !" ! ' > ≠ 0This leads to a non-linear sigma model of nambu-goldstone bosons on CP 1
Is this really correct? If so, what are the couplings in the sigma model?

Adjoint QCD
An ideal test case is 4d adjoint QCD with two Weyl flavors of adjoint quarks !" .
The index # is a doublet under a chiral SU(2) global symmetry
Lore is that this theory confines and undergoes chiral symmetry breaking via achiral condensate< %& !" ! ' > ≠ 0This leads to a non-linear sigma model of nambu-goldstone bosons on CP 1
Is this really correct? If so, what are the couplings in the sigma model?
In [Cordova-Dumitrescu] (online now!) we construct candidate phases for thistheory (with gauge group SU(2)) by embedding it into pure N=2 SYM. This adds
a complex adjoint scalar +. The IR of the N=2 theory is solved ([Seiberg-Witten]).
There is a moduli space of vacua controlled by, = < %& + . >At generic , the IR is a U(1) vector multiplet. At , = ±Λ. there are additional
massless fields and the IR is N=2 SQEDA Deconﬁned Phase at Small M
The UV scalar mass operator ! " can be matched in the supersymmetric theory
and leads to a potential on the moduli space [Luty-Rattazzi]__! " ~$(& & ' − & & ' )At small soft mass * this traps the theory at the origin of moduli space + = 0
The remaining massless fields are an IR gaugino and a free photon

A Deconfined Phase at Small M

The UV scalar mass operator ! " can be matched in the supersymmetric theory
and leads to a potential on the moduli space [Luty-Ra:azzi]__! " ~$(& & ' − & & ' )At small soft mass * this traps the theory at the origin of moduli space + = 0
The remaining massless fields are an IR gaugino and a free photon
The physics of this phase is quite exotic:• The gaugino is a doublet under the global SU(2) so chiral symm is unbroken• The free photon means the theory is deconfined (and in a Coulomb phase!)• The gauge coupling of the photon is at the self-dual value . = $. An
unbroken discrete r-symmetry of the UV acts in the IR by S-duality
This candidate phase matches all ‘t Hooft anomalies (closely related to a
recent conjecture by [Anber-Poppitz, Senthil-Bi]).
Does it persist as the soft mass * is increased?

Monopole Condensation
At ! = ±Λ% the theory is SQED with a charge one hyper ℎ ' in a doublet of SU(2)The scalar potential for ℎ ' and the vector multiplet scalar ( is)~+ % ℎ , + ℎ % ( % + . % ( ( % /+ % − ℎ % + Λ 2+(())
If we naively extrapolate to large M (speculative! dangerous!) this suggests thatℎ ' will eventually condense.
If this happens we obtain the expected CP 1 sigma model with confinement
and chiral symmetry breaking. Remarkable that N=2 can do this!

Monopole Condensation
At ! = ±Λ% the theory is SQED with a charge one hyper ℎ ' in a doublet of SU(2)The scalar potential for ℎ ' and the vector multiplet scalar ( is)~+ % ℎ , + ℎ % ( % + . % ( ( % /+ % − ℎ % + Λ 2+(())
If we naively extrapolate to large M (speculative! dangerous!) this suggests thatℎ ' will eventually condense.
If this happens we obtain the expected CP 1 sigma model with confinement
and chiral symmetry breaking. Remarkable that N=2 can do this!
The embedding in SQED also allows us to determine subtle topologicalcouplings which have interesting implications for the physics.

Monopole Condensation
At ! = ±Λ% the theory is SQED with a charge one hyper ℎ ' in a doublet of SU(2)The scalar potential for ℎ ' and the vector multiplet scalar ( is)~+ % ℎ , + ℎ % ( % + . % ( ( % /+ % − ℎ % + Λ 2+(())
If we naively extrapolate to large M (speculative! dangerous!) this suggests thatℎ ' will eventually condense.
If this happens we obtain the expected CP 1 sigma model with confinement
and chiral symmetry breaking. Remarkable that N=2 can do this!
The embedding in SQED also allows us to determine subtle topologicalcouplings which have interesting implications for the physics. One highlight:
The 2d confining string of adjoint QCD is a 2d topological insulator
We can give the fermions a mass and flow to pure YM with zero theta angle.
The mass term leaves the bulk 4d physics gapped
Eventually the string undergoes a phase transition to a trivial phase

Conclusions• The universe of supersymmetric field theory is vast and growing. Each theory
is a potential playground for mathematical physics• New exact results in supersymmetry are teaching us about the guts ofquantum field theory• There is more to explore in the relationship between supersymmetric and nonsupersymmetric theories

Conclusions• The universe of supersymmetric field theory is vast and growing. Each theory
is a potential playground for mathematical physics• New exact results in supersymmetry are teaching us about the guts ofquantum field theory• There is more to explore in the relationship between supersymmetric and nonsupersymmetric theories

Thanks for Listening!