The Vacua of Some 2+1 Dimensional Gauge
Theories
Zohar Komargodski
Simons Center for Geometry and Physics

June 25, 2018

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We’ll explain some recent non-perturbative results about gauge
theories in 3 (2+1) dimensions. We will then see that there arealso interesting implications for gauge theories in 4 dimensions.
The presentation is mostly based on collaborations with Davide
Gaiotto, Jaume Gomis, Nathan Seiberg, as well as some work inprogress.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Different phases of Quantum Field Theory that we will encounter
today:
Trivial Gapped: No massless excitations, no topologicaltheory, trivial (product) wave function
Topological Field Theory (TFT): no massless excitations,but some long range entanglement and topological order(such as anyons). Nontrivial ground state wave function.
Massless phases: This could be due to a Conformal Field
Theory, or due to Nambu-Goldstone particles.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We will concentrate our attention on two models:SU(N) adjoint Majorana fermion λα coupled to SU(N) gauge
fields with a Chern-Simons term at level k. We refer to thistheory as “Adjoint QCD3 .”Nf SU(N) fundamental complex fermions λα coupled toSU(N) gauge fields with a Chern-Simons term at level k. We
refer to this theory simply as “QCD3 .”Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

These models have various weakly coupled limits where we canunderstand the dynamics in detail. There are also strongly coupled
regimes, where we have made some conjectures.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Typically, ’t Hooft anomaly matching conditions (along with manyother constraints) are an important ingredient in our study ofstrongly coupled theories. One may be worried that since we arestudying 3 dimensional theories, this important tool will not beavailable to us.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

However, recent developments uncovered a swath of new discreteanomalies.
If a [fermionic] QFT in 2+1 dimensions has a time reversal
symmetry with T 2 = (−1)F , there is a possible obstruction tostudying this theory on a non-orientable space. The
obstruction is valued in Z16 (see [Witten 1508.04715] for anexposition).
Mixed ’t Hooft anomalies between time reversal symmetry andglobal symmetries. ’t Hooft anomalies for global symmetries.
If there is a ZN one-form symmetry, there is an anomaly
valued in ZN (for fermionic theories).

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Matching these discrete anomalies is, for practical purposes, asconstraining and non-trivial as matching continuous ’t Hooftanomalies.
The challenge is often, however, in computing these anomalies.
Even in rather simple theories, the computation of these anomalies
may not be easy. We will see examples.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

The Lagrangian isL=−12 3k2
Tr
AdA+A+ Lmatter .
TrF+4g 24π3m/ + 4π
Adjoint QCD: Lmatter = i λ̄Dλλ̄λ
PNfm k/ k + 4π
QCD: Lmatter = k=1 i ψ̄ k Dψψ̄ ψkNote that m is a real parameter.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Consistency requires
Adjoint QCD :N+k ∈Z .
2Nf+k ∈Z .
2
Adjoint QCD has no ordinary global symmetries while QCD hasU(Nf ) symmetry. Note that m preserves the U(Nf ) symmetry!
QCD :In both cases, if k = 0 and m = 0, we have time reversal symmetry
and T 2 = (−1)F (cf. Seiberg’s talk).

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

While Adjoint QCD has no ordinary (zero-form) global symmetries,
it has a ZN one-form symmetry because the center does not act on
the matter fields. (We use the terminology of [Kapustin-Seiberg],[Gaiotto-Kapustin-Seiberg-Willett].)

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Decoupling Limits: |m|  g 2 or k  N.
In some corners of the parameter space the theory becomes weakly
coupled.
If |m|  g 2 the quarks decouple even before the interactions setin. One has to be careful integrating them out as there is a
non-decoupling effect [Redlich, Niemi-Semenoff] proportional tom/|m| = sgn(m).
This shifts k according toAdjoint QCD : k → k + sgn(m)QCD : k → k + sgn(m)Zohar Komargodski

N.
2Nf.
2

The Vacua of Some 2+1 Dimensional Gauge Theories

Another weak coupling limit isk N
The gauge field A now has a mass kg 2 and therefore it decouples
before the interactions set in:kg 2  g 2 N .
The remaining light fields are weakly interacting and there is aweakly coupled Conformal Field Theory if we tune m.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Therefore the physics of large m for any k is a (possibly trivial)
TFT. For large k there is a second order phase transition as wevary m, described by some (weakly coupled) Chern-Simons mattertheory. On the two sides of the transition we encounter different
TFTs so these transitions are typically non-Landau-Ginzburg.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We therefore suggest the following phase diagram for k  N(“large k”) in Adjoint QCD:

SU (N ) +↵,level k

SU (N )k N/2kN

SU (N )k+N/2

m

Conformal Field Theory

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

And an analogous phase diagram in QCD for k  1:SU (N ) + Nf ,SU (N )klevel k

k1

SU (N )k+ NfNf2m2

Conformal Field Theory

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

The interesting question is where does this picture break down.
This is analogous to asking where is the end of the conformalwindow. Surprisingly, in 3d there is a very concrete proposal forthe answer!
Adjoint QCD: k ≥ N/2
QCD: k ≥ Nf /2

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

These bounds are motivated by the consistency of the whole
picture we will shortly propose. Some preliminary observations:Adjoint QCD has N = 1 supersymmetry for m = −kg 2 . It is
known [Witten] to be unbroken for k ≥ N/2 and the Witten
index precisely agrees with the number of lines inSU(N)k−N/2 TFT.
QCD has a dual bosonic description in terms of U k + N2f
gauge theory coupled to Nf fundamental bosons [...Aharonyet al., Minwalla et al....]. This duality needs to be fixed fork < Nf /2 because the bosonic theory has massless Goldstone
modes in the Higgs phase.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

To describe what happens for k < N/2 and k < Nf /2 one has tomake a certain leap, as this is a strongly coupled regime. We ofcourse know what happens at large |m|, where we can justintegrate out the quarks and flow to a Chern-Simons TFT, asabove. The question is what happens when the matter fields arelight.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

It is useful to start from the most strongly coupled regime, wherek = 0. We have time reversal symmetry at m = 0 and an
associated anomaly.
Let us start with the conjecture for Adjoint QCD:
SU(N)+Adjoint, m = k = 0 −→−→ Ua massless Majorana fermion.

Zohar Komargodski

N2

N,N2⊗ Gα , where Gα is

The Vacua of Some 2+1 Dimensional Gauge Theories

The m = k = 0 theory has “accidental” N = 1 supersymmetry
and the Gα can be interpreted as the Goldstino particle. Adding
some small δm in the UV would lift this particle but the TFT
U N2 N ,N would remain. Therefore, this new quantum phase
2exists in a neighbourhood of m = 0 and there are certain phasetransitions to the semi-classical domain.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

The conjecture that the infrared theory contains U
miraculously saturates all the anomalies:

N2



N,N
2

It has ZN one-form symmetry and it can be checked that it
has exactly the right one-form symmetry anomaly.
Time Reversal Symmetry: From level-rank duality,



N
N
U−k'U+k,2
2
N− N +k,−N+k,N
22


N

we see that U 2 N ,N is time-reversal invariant if k = 0, as
2
required!! Furthermore it has a time reversal anomaly[...Tachikawa-Yonekura,Cheng...]ν = 2(−1)N/2+1 .

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

From the Goldstino Gα we get at ν(Goldstino) = 1. So the total
infrared time reversal anomaly at k = m = 0
3 mod 16 if N = 2 mod 4
N/2+1vIR = 1 + 2(−1)
mod 16 =−1 mod 16 if N = 0 mod 4
This is in agreement with the ultraviolet anomaly because it is thesame as N 2 − 1 mod 16.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

It may appear surprising that the massless k = 0 theory, which isessentially the 2+1 dimensional analog of 3+1 dimensional adjoint
QCD, is not confining. Indeed, the Wilson lines have a perimeterlaw and a fusion/braiding algebra that is dictated by the TFT inthe quantum phase.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We therefore propose the following phase diagram for0 ≤ k < N/2:

SU (N ) +↵,level kConformal Field Theory0  k < N/2

Conformal Field TheoryBroken
N = 1 SUSY

SU (N )k N/2U✓

N+k2U✓◆N2k◆

SU (N )k+N/2N
2 +k,NN
2 +k,N⌦G↵ @ m =Zohar Komargodski

mk

The Vacua of Some 2+1 Dimensional Gauge Theories

The transitions from the “quantum” phase to the semi-classicalphases have a new dual description, for example, the left transition
could be described withN− k + λ̂
U2with levels 34 N + k2 , N. λ̂ is a dual fermion in the adjoint
representation.
This is a new adjoint-adjoint Fermion-Fermion duality.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

The phase diagram for QCD for 0 ≤ k < Nf /2 is constructed using
an analogous procedure. For even (sufficiently small) Nf andm = k = 0 we make the following conjecture:
SU(N)+Nf ψ, m = k = 0 −→−→ M =U f )  and the U(NN
Nf×U 2f2sigma model is accompanied by a Wess-Zumino term, Γ, whose
coefficient is N.
In short, the symmetry is broken as  NfNfU(Nf ) → U×U
22Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

The UV theory has time reversal symmetry anomalyνUV = 2Nf N mod 16. It is not trivial to compute the time reversal
anomaly of the infrared sigma model. The Wess-Zumino term Γ iscrucial in order to get the right answer.
In the particular case of Nf = 2, this is the CP 1 model at θ = π
discussed in [Freed,ZK,Seiberg]. In work in progress[Hason,ZK,Thorngren] we show that the anomaly in fact matches.
In addition, we show that all the discrete flavour symmetry
anomalies match.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

In general, we propose the following symmetry breaking patternU(Nf ) → UNf+k2

×U

Nf−k2



(and the sigma model again needs to be accompanied by aWess-Zumino term).
This makes sense only for k < Nf /2 which is another way tounderstand the boundary between this phase and the “large k”
phase. (Note that it is consistent with the [Vafa-Witten] theorem.)

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge TheoriesSU (N ) + Nf ,level kConformal Field Theory

SU (N )kNf2M=Nf2

Conformal Field Theory

U (Nf )⌘⇣NNU 2f + k ⇥ U 2f⇣0k<+NZohar Komargodskik⌘SU (N )k+ Nfm2

The Vacua of Some 2+1 Dimensional Gauge Theories

As in the previous case, the transition has a new dual description.
For instance, for the left transition it is given by
Nf−k+ Nf φ .
U2NWhile the quantum phases are robust, the transitions in thesevarious cases do not actually need to be 2nd order.
See [...Jensen-Karch, Armoni-Niarchos,
Argurio-Bertolini-Bigazzi-Cotrone-Niro...] for stringy constructionsof these dualities and symmetry breaking phases.
See also [Karthik-Narayanan] for a recent lattice study, (strongly)
hinting that symmetry breaking indeed takes place!

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We will now mention (very) briefly a striking parallel between theproblem of 3d dynamics that we have been analyzing so far andthe problem of 4d dynamics and domain walls.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

4d Yang-Mills theory with an Adjoint fermion.
4d Yang-Mills theory with fundamental Dirac fermions (a.k.aQCD).
In both cases, under appropriate circumstances, there aredegenerate vacua, each of which is gapped and trivial. Therefore,
at low energies there are some 3d theories living on these domainwalls.

|V AC1i

Zohar Komargodski|V AC2i

The Vacua of Some 2+1 Dimensional Gauge Theories

These domain walls have the same symmetries and anomalies as
the pure 3d theories. In some cases their phases can be explicitly
computed using 4d techniques and one can discover the quantum
phases that have appeared above.
For instance, 4d QCD with Nf = 2 massive quarks and θ = π has
two ground states for any nonzero positive mass. For large m wehave SU(2)1 TFT on the wall and for small m we find the coset
U(2)1U(1)×U(1) ' CP on the wall (while the bulk is massive!).

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Conversely, our 3d techniques can shed light on some open
problems in 4d dynamics. We will now make a proposal for thedomain walls theories of N = 1 four-dimensional gauge theories
with simply connected gauge groups. We identify the domain walltheory with the quantum phase in the corresponding 3d adjoint
QCD model. This problem has been resistant to other approaches
for quite some time.

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

N = 1 Super Yang-Mills with gauge group SU(N) [Agreeswith Acharya-Vafa]:
U(i)N−i,N .
N = 1 Super Yang-Mills with gauge group Sp(N):Sp(i)N−i+1 .
N = 1 Super Yang-Mills with gauge group Spin(N)[Gomis-Seiberg-ZK, Cordova-Hsin-Seiberg]:
O (i)1N−i−2,N−i+1 .

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We have seen that discrete anomalies are a powerful constraint on
the dynamics of QFTs. For additional discussions, see, forinstance, the recent papers [Benini et al., Cordova et al.,Tanizaki, Sulejmanpasic, Kikuchi, Misumi, Sakai, Shimizu,
Anber-Poppitz, Aitken-Cherman-Ünsal, Yao-Hsieh-Oshikawa,Dunne, Hofman-Iqbal,Guo-Putrov-Wang,
Yamazaki-Yonekura, Kitano et al., Di Vecchia et al., ... ]It has been further argued that such discrete anomalies are, inspecial cases, the continuum avatar of Lieb-Schultz-Mattis
theorems in lattice theories [ZK-Sulejmanpasic-Ünsal,Metlitski-Thorngren, ...]

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

We have briefly seen that domain walls in four-dimensional(supersymmetric and non-supersymmetric) theories exhibit rich
dynamics. Some additional aspects are discussed in [...
Dierigl-Pritzel, Draper, Ritz-Shukla, Armoni-Niarchos,
Argurio-Bertolini-Bigazzi-Cotrone-Niro (embedding in theSakai-Sugimoto model – The bosonic dual appears naturally!),
Aitken-Baumgartner-Karch (interactions between domain walls
and the corresponding 3d quivers and dualities; to appear) ].

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories

Thank you for the attention !!

Zohar Komargodski

The Vacua of Some 2+1 Dimensional Gauge Theories