Dualities and Dynamics in 2+1 Dimensions
Francesco BeniniSISSA(Trieste)

Strings 201825–29 JuneProgress on IR dynamics of certain gauge theories in 2+1 dimensions
with/without Chern-Simons interactions
with scalar and/or fermionic matter, mostly in fundamental rep.
Relativistic QFT in the continuum∗ Utilized in many condensed matter problems:
quantum phase transitions of spin liquids and quantum magnets
high-Tc superconductors
edge modes of topological insulators
half-filled Landau level∗ Engineer QED with CS interactions and fermions on graphene films[Lee, Wang, Zalatel, Vishwanath, He 18]

Progress driven by (conjectural) IR dualities (in HE sense)
Theory A

Theory B

IR physics
Particularly powerful for IR fixed points∗ Important role played by (’t Hooft) anomalies for global symmetries

Particle / vortex duality [Peskin 78; Dasgupta, Halperin 81]
O(2) vector model

gauged O(2) vector model

L = |∂φ|2 + |φ|4 + m2 |φ|2e 2 + |φ|e 4 − m2 |φ|e2
L = |Fµν |2 + |Da φ|

S1

gapped m2

U (1)0 with φeSimple, but with lots of beautiful physics:
gapped & broken phases↔

Higgsed & free-photon phases

perturbative excitations (m2 > 0)↔

(finite energy) vortices

fundamental field φ↔

monopole operator M

U (1) symmetry↔

U (1) magnetic symmetryO(2) Wilson-Fisher fixed point↔

IR CFT

Lattice Monte Carlo: [Nguyen, Sudbø 99; Kajantie, Laine, Neuhaus, Rajantie, Rummukainen 04]

Monopole operators
Local disorder operators: defined by boundary conditions in path-integral[’t Hooft 78; Borokhov, Kapustin, Wu 02]

CFT: radial quantization[Chester, Dyer, Iliesiu, Mezei, Pufu, Radicevic, Sachdev, . . . ]M(x)\π1 (Ggauge )• Charged under magnetic symmetry:
E.g.:U (1)gauge⇒

Magnetic (topological) symmetry U (1)MJµ = µνρ F νρSO(N ≥ 3)⇒Z2• Semiclassical monopoles can have electric charges
Dressing by matter field zero-modes⇒

spin and global quantum numbers

Flux attachment

Attaching one unit of magnetic flux to particles, their statistics changes[Wilczek 82; Polyakov 88; Zhang, Hansson, Kivelson 88; Jain 89; Shaji, Shankar, Sivakumar 90; Paul, Shankar,Sivakumar 91; Fradkin, Lopez 91; Chen, Fisher, Wu 93; Fradkin, Schaposnik 94; . . . ]Realized throughChern-Simons interactions
L=12ada + Da φ
4π

Semiclassical “bare” monopole Mbare is not gauge invariant
Gauge-invariant monopole

M = “Mbare φ”

has spin 12

Particle/vortex dualities with fermions

U (1)1 with φfree Dirac ψ1
L = 4πada + |Da φ|2 + |φ|4U (1) 12 with ψ
1/ aψ
L = 4πada + ψ̄ D←→/L = ψ̄ ∂ψO(2) vector model←→L = |∂φ|2 + |φ|4Also fermion/fermion duality: relevant for half-filled Landau level[Barkeshli, McGreevy 12; Son 15; Wang, Senthil 15; Metlitski, Vishwanath 15; Mross, Alicea, Motrunich 15;
Mulligan, Raghu, Fisher 16; Karch, Tong 16; Seiberg, Senthil, Wang, Witten 16; Murugan, Nastase 16]

Vector models at large N, kk, O(k)−Nλ̃ = N

Singlet sector ofSinglet sector of

critical O(N ) modelk free fermionsO(N )k , λ = Nk• High-spin symmetry (large N ): conserved currents J (s) with s = 2, 4, 6, . . .
J (s) = ϕi ∂ s ϕi + . . .

J (s) = ψ a γ∂ s−1 ψ a + . . .

All primary operators are products of J (s) ’s and J (0) = ϕi ϕi or ψ a ψ a• Parity-breaking deformation: couple to Chern-Simons gauge theory• Duality: identify the two families with λ̃ = 1/λ[Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin 11; Aharony, Gur-Ari, Yacoby 12]

Spectrum of primaries is independent of λ (at large N )[Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin 11; Aharony, Gur-Ari, Yacoby 12]

Correlation functions (of single-trace ops): 3 conformal structureshJ (s1 ) J (s2 ) J (s3 ) i = αs1 s2 s3 Tbos + βs1 s2 s3 Tfer + γs1 s2 s3 Todd
α, β, γ fixed by high-spin symmetry, in terms of two parameters[Maldacena, Zhiboedov 11; 12]

c1 , c2 (N, k)
Fix by direct computation.

[Aharony, Gur-Ari, Yacoby 12; Gur-Ari, Yacoby 12]

Thermal partition functions (also away from fixed point)[Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin 11; Aharony, Giombi, Gur-Ari, Maldacena, Yacoby 12;
Jain, Minwalla, Sharma, Takimi, Wadia, Yokoyama 13; Choudhury, Dey, Halder, Jain, Janagal,Minwalla, Prabhakar 18]

Large N : blind to many details

3D dualities among vector models
IR dualities between Chern-Simons gauge theories with matter in fundamental rep[Aharony 15; Hsin, Seiberg 16; Aharony, FB, Hsin, Seiberg 16]

scalars φ with |φ|4 interactions
SU (N )k with Nf φfermions←→

ψwith Nf ψU (k)−N + Nf
2U (N )k with Nf φ←→

SU (k)−N + Nfwith Nf ψ

2U (N )k,k±N with Nf φ←→U (k)−N + Nf ,−N ∓k+ Nf with Nf ψU Sp(2N )k with Nf φ←→U Sp(2k)−N + Nfwith Nf ψSO(k)−N + Nfwith Nf ψR

222SO(N )k with Nf φR←→valid for Nf less than a bound ( . N )2

Conjecturally, these theories have only one phase transition
SU (N )k with Nf φmψ

Nf < NTQFT

TQFT

SU (N − Nf )kSU (N )k

CFTU (k)−N +NfU (k)−NU (k)−N + Nf with Nf ψ
2Gapped phases: match via level-rank duality
Conjecture: 2nd order phase transitionm2φBosonization of free fermions
U (N )1 with Nf φSO(N ≥ 3)1 with Nf φR

Nf free Dirac ψNf free Majorana ψR(Nf ≤ N )(Nf ≤ N − 2)

[Aharony 15; Seiberg, Senthil, Wang, Witten 16;[Metlitski, Vishwanath, Xu 16;

Karch, Tong 16; Hsin, Seiberg 16]

Aharony, FB, Hsin, Seiberg 16]

Fermionization of Wilson-Fisher scalars
SO(k ≥ 3)− 21 with ψR←→O(1) (Ising) vector model

U (k)− 12 with ψ←→O(2) vector modelIR quantum-enhanced symmetries
Symmetries can enhance at IR fixed points∗ Sometimes dualities make this manifest
TATB

CFT
TB can have larger symmetry than TA
The symmetry groups of TA and TB might not commute[GA , GB ] 6= 0⇒

GIR ⊃ GA , GB

Caveat: assume IR fixed point with no symmetry breaking∗ E.g.: QED with one fermion∗ E.g.: QED U (1)0 with 2 ψU (1) 32 with 1 ψSU (2)−1 with 1 ψ

UV: G = O(2)UV: G = SO(3)self-duality

P in(2)−∪P in(2)− × SU (2)UV:Z2

CFT with SO(3) symmetry
IR:

Basic monopole operators M± : spin 1
IR conserved currents O(2) → SO(3)[Aharony, FB, Hsin, Seiberg 17]O(4)M±2 become extra IR currents[Xu, You 15; Karch, Tong 16; Hsin, Seiberg 16;
FB, Hsin, Seiberg 17; Cordova, Hsin, Seiberg 18]∗ Examples with emergent time-reversal symmetry
E.g.: U (N )1 with 1 φ←→free Dirac ψA few generalizations• More gauge groups from gauging global symmetries (choice of counter-terms)
E.g.: gauge groups O, Spin, P in± , . . .

[Cordova, Hsin, Seiberg 17]

Produce intricate nets, testing the conjectured dualities• Quiver gauge theories (inspired by 3D SUSY mirror symmetry)[Karch, Robinson, Tong 16; Jensen, Karch 17]



N −1



E.g.:U (1)− N with N ψ2←→U (1)N −1 with N “bifundamental” φCS matrix κab = SU (N ) Cartan matrix(hard to precisely identify scalar potential)• Vector models with scalars and fermions[Jain, Minwalla, Yokoyama 13; Gur-Ari, Yacoby 15; FB 17; Jensen 17]

SU (N )k− Nf with Ns φ, Nf ψ←→U (k)−N + Ns with Nf φ, Ns ψ22..
.
mψvalid for Ns , Nf less than a bound.

Two-dimensional phase diagram undermψ > 0

m2φ < 0m2φ > 0

symmetry-preserving deformationsm2φ

m2φ |φ|2 and mψ ψψ
m2φ < 0mψ < 0∗ Semiclassically: match gapped phases and gapless lines
Origin: multicritical fixed point? more intricate topology?

mψ < 0• Multi-critical point from extra tuning of vector models (at large N ):Legendre transform ofU (N )k with Nf φtune |φ|2 and |φ|4to zero←→U (k)−N with Nf ψw.r.t. ψψ(Gross-Neveu critical point)[Aharony, Jain, Minwalla to appear; see Minwalla’s talk last year]

How to test such conjectures?
Large N, k computations (already described)
Consistency under relevant deformations(reduce to level-rank duality of TQFTs or other dualities)
Deformation from SUSY dualities[Jain, Minwalla, Yokoyama 13; Gur-Ari, Yacoby 15][Kachru, Mulligan, Torroba, Wang 16]’t Hooft anomaly matching (more later)[FB, Hsin, Seiberg 17]

Embedding in String Theory[Jensen, Karch 17; Armoni, Niarchos 17; Argurio, Bertolini, Bigazzi, Cotrone, Niro 18]

Lattice Monte Carlo computations? (not many results for parity-breaking theories)[Hands, Kogut, Scorzato, Strouthos; Karthik, Narayanan]d = 4 −  expansion?

[Di Pietro, Komargodski, Shamir, Stamou 15]

Anomalies
Theory with internal symmetry G can be coupled to background G gauge fields→

observables

E.g.:

Z[Abackground ]Might be impossible (by local counterterms) to make Z
a well-defined or gauge-invariant function of G-bundles→’t Hooft anomaly• Extend G-bundles to 4-dim bulk:

B40

M3

Z[Abackground ] depends on extension

M3B4Quantify ’t Hooft anomaly by dependence on extension→

classical TQFT (characteristic class) in 4-dim∗ ’t Hooft anomalies are independent of RG flow

(anomaly inflow)• Example in 3D (no anomalous currents)
SU (2)k with Nf φ:[FB, Hsin, Seiberg 17]

U Sp(2Nf )/Z2 faithfully-acting symmetry
SU (2)k × U Sp(2Nf )LZ2

Gauge + Global Chern-Simons counter-terms:
Quantization:k∈Z,L∈Z,k + Nf L ∈ 2Z

k odd, Nf even: ’t Hooft anomaly
ZP(w2 )
,Sanom = π2eiSanom = ±1

• Anomalies can be computed and matched across dualities[FB, Hsin, Seiberg 17; Komargodski, Seiberg 17;Cordova, Hsin, Seiberg 17]

The dualities suggest
other interesting physical phenomena

Quantum phases: spontaneous symmetry breaking
Dualities suggest quantum phases with SSB in 3D QCD for:2k < Nf < Nf∗[Komargodski, Seiberg 17]

Nf > 2k

SU (N )k with Nf ψU
Nf2 − k N with Nf φUNf
2 + k −N with Nf φmψ

NLSM
TQFT
SU (N )TQFTNk− 2fU (Nf )
N

N

U 2f + k × U 2f − kSU (N )Nk+ 2f

Two phase transitions with different dual descriptions
Anomalies match. Reproduced by a WZ term in NLSM
For k = 0, compatible with [Vafa, Witten 84]
Cannot be true for arbitrarily large Nf[Appelquist, Nash 90; Grover 12; Sharon 18]
Some numerical evidence (in SU (2)0 with Nf ψ)[Karthik, Narayanan 18]

3D transitions from 4D domain walls
4D SU (N ) QCD with Nf flavors ( < Nf(conf.) )[Gaiotto, Kapustin, Komargodski, Seiberg 17][Gaiotto, Komargodski, Seiberg 17]
Nfmquarkseiθ

Yang-Mills
chiralLagrangian
CPNf −1

SU (N )1

4D bulk 1st order transition for θ = π (broken CP)
Large |mquark |: SU (N )1 TQFT
Small |mquark |:Nf −1

CP→(e.g. from [Acharya, Vafa 01] )NLSM

3D phase transition for some value m∗quark
described by 3D vector model:

3D domain wall

SU (N )1− Nf with Nf ψ2Spontaneous Supersymmetry Breaking
Dualities of gauge theories with adjoint fermions shed light on SUSY breaking[Witten 99; Gomis, Komargodski, Seiberg 17]

0 ≤ k < N2 − 1SU (N )k with adj λU N2 − k 3N + k ,N with λ4


U N2 + k − 3N + k ,−N with λ242Goldstino ηTQFT
SU (N )− N +k2U



TQFT


N+k2−N+k,−N2SUSY breaking point: massless Goldstino η + TQFT
Three phase transitions, each with a dual descriptionmψTQFTSU (N ) N +k2

Concluding remarksLessons from SUSY QFTs⇒

Predictions about non-SUSY QFTs

Explore a variety of directions
E.g.: boundary conditions[Gaiotto; Aitken, Baumgartner, Karch, Robinson]

more general defects
Interactions with other non-perturbative methods. E.g. conformal bootstrap
Experimentally testable?