Dynamics of QCD{}_{3} with Rank-Two Quarks And Duality
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Changha Choi,111changha.choi@stonybrook.edu Diego Delmastro,222ddelmastro@perimeterinstitute.ca Jaume Gomis,333jgomis@perimeterinstitute.ca Zohar Komargodski,444zkomargo@gmail.com Physics and Astronomy Department, Stony Brook University, Stony Brook, NY 11794, USA Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada Department of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA Weizmann Institute of Science, Rehovot, Israel

1 Introduction

Strongly coupled Quantum Field Theories (QFTs) can develop interesting infrared “quantum” (nonperturbative) phases, distinct from those that can be inferred by semiclassical considerations. Recently the study of the infrared dynamics of dimensional gauge theories has resulted in the discovery of novel nonperturbative quantum phases [Komargodski:2017keh, Gomis:2017ixy, Cordova:2017vab, Bashmakov:2018wts, Benini:2018umh, Choi:2018ohn]. In addition, some related aspects have been already studied on the lattice [Karthik:2016bmf, Karthik:2018nzf]. The subject involves several conceptual differences from the (perhaps) more familiar setting of dimensional gauge theories coupled to matter:

  • Since the gauge coupling has positive mass dimension, dimensional gauge theories are always asymptotically free. In particular, these theories are interesting even when the gauge group is .

  • dimensional gauge theories are labeled by a gauge group, matter fields, and the Chern-Simons couplings, which are quantized.

  • Since in dimensions there is no notion of spinor chirality, one can typically add a mass term for the matter fields preserving all the continuous symmetries of the massless theory. In some theories, however, the massless point is distinguished by the presence of (a discrete) antilinear time-reversal symmetry. The phases of these theories can be studied as a function of the continuous mass deformations for the matter fields.

  • There are no continuous ’t Hooft anomalies for the symmetries of dimensional theories. However, there are many discrete anomalies and they have to be consistent with the infrared phases of these theories. Even though these anomalies are discrete they nevertheless provide highly nontrivial constraints on the infrared dynamics. In addition, such quantum phases are constrained by the matching of some counterterms. This matching of counterterms physically corresponds to consistency conditions on conductivity coefficients. We will encounter some examples in this paper.

The subject of dimensional gauge theories connects in an obvious way to condensed matter physics (where the gauge symmetry is typically emergent) and in somewhat less obvious ways to particle physics. Many dimensional gauge theories have degenerate trivial vacua. As a result, a domain wall connecting two vacua supports at low energies a dimensional theory, and one can often make this connection very natural (as we will see in this paper). In addition, one can study dimensional gauge theories compactified on a circle (e.g. in the context of finite temperature physics), a problem that similarly reduces to the study of dimensional systems.

In spite of the fact that dimensional gauge theories are asymptotically free, these theories admit regimes in parameter space where they are weakly coupled and the infrared dynamics can be inferred by a careful semiclassical analysis:

  • When all the matter fields have a large mass (in units of the Yang-Mills coupling constant) they can be integrated out at energy scales above the strong coupling scale. This only leads to a shift in the infrared Chern-Simons level (this shift is one-loop exact for fermions and trivial for bosons) and hence the deep infrared theory is given by the infrared dynamics of the pure Yang-Mills-Chern-Simons gauge theory, which is quite well understood. This typically leads in the infrared to a Chern-Simons Topological Quantum Field Theory (TQFT).111An exception to this is when the Chern-Simons level vanishes upon integrating out massive matter fields. In that case, for simply connected groups, the TQFT is trivial. For non-simply connected groups the situation is more complicated, but we will not need it here except in the case of , where the low energy theory is the gapless theory of a compact scalar. This fact will play an important role in this paper. Since the shift of the Chern-Simons level depends on the sign of the mass of the fermion [Niemi:1983rq, Redlich:1983kn, Redlich:1983dv], the large mass limit defines an asymptotically large-positive mass phase and an asymptotically large-negative mass phase, described by two distinct TQFTs.

  • When the number of matter fields is very large222The relevant group theory factor is the index of the (possibly reducible) representation of the matter fields. [Appelquist:1988sr, Appelquist:1989tc] (i.e. many species or large representations) one can demonstrate that there exists a weakly coupled conformal field theory (CFT) which interpolates between the TQFTs describing the two asymptotically large mass phases. Such a theory does not develop interesting new quantum phases.

  • Likewise, when the Chern-Simons level is very large [Avdeev:1991za] one can show that there exists a weakly coupled CFT interpolating between the TQFTs describing the two asymptotically large mass phases. For large the theory does not develop interesting new quantum phases.

    While no new quantum phases emerge for “large representations” or large Chern-Simons level, it is a wide-open nonperturbative problem to determine for which representations and which levels new quantum phases develop. For some recent work on such questions in the context of quiver gauge theories see [Jensen:2017dso, Aitken:2018cvh] and references therein.

It follows from our discussion above that the dynamics of dimensional gauge theories is especially interesting when neither the Chern-Simons level, the dimension of the representation, nor the mass of the matter fields are too large. In this regime there is no semiclassical approximation to the dynamics of the theory. This is when we may expect quantum effects to dominate the dynamics and new interesting phenomena may emerge, including new nonperturbative phases.

In this paper we study a class of dimensional gauge theories for which we provide a large body of evidence that they indeed develop new nonperturbative phases along with new phase transitions, for which we propose novel dual descriptions. (We do not know, in general, if these phase transitions are 1st or 2nd order.) The theories we analyze are Yang-Mills gauge theories coupled to a fermion in the rank-two symmetric or antisymmetric representation of and a Chern-Simons term at level . (The fermion is a Dirac fermion, i.e. a complex fermion with two components.)

For generic , these models have a global baryon number symmetry, , acting on the fermion as . In addition, there is charge-conjugation symmetry. Both of these symmetries are unbroken by the mass term . For (which is only allowed for even ) the model also admits a time-reversal symmetry. The mass term breaks the time-reversal symmetry. Finally, since there are no dynamical degrees of freedom in the fundamental representation, these theories have a one-form symmetry when is even. (In the context of condensed matter physics, the one-form symmetry is expected to be accidental.)

Let us now summarize the main results:

  1. These theories have a critical value of the level below which a new intermediate quantum phase appears between the semiclassically accessible asymptotic large-positive and large-negative mass phases. The critical value is333The quantized level must be integer for even and half-integer for odd. See section 2.

    (1.1)
  2. For there is an intermediate quantum phase described by the following “emergent” TQFTs444We recall that with . The quotient by gauges an anomaly-free one-form symmetry [Kapustin:2014gua, Gaiotto:2014kfa].

    (1.2)
  3. For there is an intermediate quantum phase that includes a Nambu-Goldstone boson (NGB) corresponding to the spontaneous breaking of , along with a TQFT:

    (1.3)

    The notation stands for the linear sigma model of a compact real scalar field , which is dual to pure gauge theory in dimensions.555For , the factor in (1.2) should not be interpreted as a TQFT, but rather as a gapless gauge theory, which can be dualized to the compact scalar. The scalar couples to the TQFT by gauging a diagonal, anomaly-free one-form symmetry.

    The microscopic (ultraviolet) theory for is time-reversal invariant. It may then seem odd that we are proposing that this model flows to a TQFT coupled to a Nambu-Goldstone boson, as TQFTs are typically non-time-reversal invariant. It is encouraging to observe that the Chern-Simons theory is in fact also a time-reversal invariant (spin) TQFT [Aharony:2016jvv]. This is a nontrivial consistency check of our proposal.

  4. For these theories undergo two phase transitions as a function of the mass of the fermion. The phase transitions connect the intermediate quantum phase with the asymptotic large-positive mass phase and with the asymptotic large-negative mass phase, respectively. We propose that these transitions have a dual description in terms of another dimensional gauge theory. This leads us to propose new (fermion-fermion) dualities in dimensions.

    Dualities for for :

    (1.4)

    Dualities for for :

    (1.5)

    We note that the fermion in the dual gauge theory transforms in the other rank-two representation compared to the fermion in the original gauge theory.

  5. For the phase diagram has just two phases: the asymptotic large-positive mass and large-negative mass semiclassical phases, separated by a phase transition. For very large the phase transition is controlled by a weakly coupled CFT. The asymptotic large mass phases are the TQFTs

    (1.6)

    where the upper/lower sign is for the large positive/negative mass asymptotic phase. These phases are present in these theories for any , but for they are separated by the intermediate quantum phases discussed above while for they are separated by a single transition (which for sufficiently large must be given by a CFT).

Outline of the Paper

In section 2 we present our conjectures for the phases of Yang-Mills gauge theory coupled to a fermion in the symmetric and antisymmetric representation and with a Chern-Simons term. In section 3 we present a list of nontrivial consistency checks, involving a comparison to known special cases, and the highly nontrivial matching of some contact terms. In section 4 we study some domain wall solutions in four-dimensional gauge theories and show that at least in one case one can explicitly demonstrate baryon symmetry breaking on the domain wall, in agreement with the predictions for the infrared behavior of the corresponding three-dimensional gauge theory. In section 5 we make a few forward-looking comments about the time-reversal anomaly and about baryons.

2 Phase Diagrams

Consider Yang-Mills theory with gauge group , a Chern-Simons term and a Dirac fermion in the rank-two symmetric or antisymmetric representation of :

(2.1)

In this section we make a proposal for the phase diagram of these theories as a function of the effective Chern-Simons level and of the mass of the fermion.666The shift of the Chern-Simons level in (2.1) by arises from the determinant of the massless fermion. It is convenient to use when writing Lagrangians since is always an integer. However, the infrared phases of the theory are more conveniently labeled by because time-reversal symmetry acts on by simply reversing it, along with reversing the mass of the fermion. is the Dynkin index of the representation under which the Dirac fermion transforms. Since under time-reversal (along with reversing the sign of the mass), we restrict our discussion to . Since , it follows from table 1 that for even and for odd.

Table 1: Index for rank-two and adjoint representations. (Since the adjoint representation is real, one could also take the corresponding fermion to be a Majorana fermion – that model was discussed in detail in [Gomis:2017ixy].)

We now discuss the global symmetries of these theories. There is a flavor symmetry and a charge-conjugation symmetry acting as777For the action of on the gauge field is a gauge transformation.

(2.2)

These transformations do not commute and generate the group . Since the center of the gauge group acts as , the global symmetry group is for even and for odd. The operators charged under this symmetry are baryons, which will be discussed briefly at the end of this paper.

Since the gauge group is simply-connected, the magnetic symmetry group is trivial. For even a subgroup of the center acts trivially on and the theory has a one-form global symmetry. For odd the one-form symmetry is trivial. Finally, for the theory is time-reversal invariant. Time-reversal symmetry acts on the fermion by

(2.3)

Therefore, in these theories, . It is easy to verify that the mass term is odd under time-reversal symmetry but it preserves all other symmetries.

We proceed now to analyzing the phase diagram. We start with the phases that can be established by a semiclassical analysis. When we can reliably integrate out the fermion before the interactions become strong. Integrating out a massive fermion shifts the Chern-Simons level to , and the resulting effective theory is pure Yang-Mills with an integer-quantized Chern-Simons term at level . This theory, which now has no matter fields, flows at low energies to the topological Chern-Simons theory at level , which we denote by . Therefore the infrared dynamics is captured by the TQFTs for large positive and negative mass respectively. These asymptotic large mass phases are present for all and all . In the above discussion of the asymptotic phases, is an exception since the infrared theory (after integrating the fermions out) is pure Yang-Mills theory (without a Chern-Simons term) with gauge group . In this case the infrared theory is trivial and gapped due to confinement and due to the fact that the gauge group is simply-connected.

Figure 1: Phase diagram of gauge theory with a symmetric fermion for . The solid circle represents a phase transition between the asymptotic phases. For sufficiently large we know for certain that the phase transition is associated with a CFT.

As long as is sufficiently large, the above two topological phases are separated by a single transition. The question is below which value of additional phases appear. Our proposal is that as long as the above picture holds true, namely, the two semiclassically accessible phases are separated by a phase transition at some value of the mass. The phase diagrams for are summarized in figures 1 and 2. Note that the boundary of the above region, , is exactly where one of the phases becomes a trivial infrared theory, with no Chern-Simons TQFT.

Figure 2: Phase diagram of with an antisymmetric fermion for . The solid circle represents a phase transition between the asymptotic phases. For sufficiently large we know for certain that the phase transition is associated with a CFT.

For we propose that there is a new intermediate “quantum phase” in between the asymptotic large mass phases.888In the case of symmetric fermion the new phase appears at . See below. This phase is inherently quantum mechanical, and is not visible semiclassically. This new quantum phase connects to each of the asymptotic phases through a phase transition. The phase diagrams for are summarized in figures 3 and 4. The reason that is excluded from the figures is that it requires a separate discussion, as we shall see below.

Figure 3: Phase diagram of with a symmetric fermion for . The solid circles represent a phase transition between the asymptotic phases and the intermediate quantum phase. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by and .

Figure 4: Phase diagram of with an antisymmetric fermion for . The solid circles represent a phase transition between the asymptotic phases and the intermediate quantum phase. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by and .

The way we arrive at the phase diagrams in figures 3 and 4 is as follows. As mentioned above the asymptotic positive and negative mass phases are described by the TQFTs . These TQFTs admit a level/rank dual description [Nakanishi:1990hj, Hsin:2016blu]999Level/rank dualities are generically valid only as spin TQFTs, and therefore, whenever the theory on one side of the duality is not spin (i.e. it does not have a transparent spin line) we must tensor that theory with a trivial spin TQFT. is never spin and is spin for odd.

(2.4)

We start with the level/rank dual description of the asymptotic positive mass phase and search for a dual description that would allow us to understand the quantum phase semiclassically in the dual variables. Similarly, we consider the asymptotic negative mass phase and search for an ultraviolet gauge theory that could describe the phase transition between the semiclassical asymptotic negative mass phase and the quantum phase. These steps involve some guesswork. The fact that this can be done at all is already a highly nontrivial consistency check. Indeed, the two required dual descriptions are mutually non-local101010By “mutually non-local” we mean that there exists no local map between the fields in the two dual descriptions, yet, they have some region of overlap in the physics they describe. This is reminiscent of the Seiberg-Witten solution, which has two mutually non-local theories describing two different massless theories, with a region of overlap. but there is only one quantum phase, which both of them have to describe simultaneously (in our case this happens thanks to the new level/rank duality in [Hsin:2016blu]). In the present context, luckily, we were able to find consistent dual descriptions describing the same quantum phase. Furthermore, this guess satisfies very nontrivial additional consistency checks, as we shall see. One of the dual theories is based on the gauge group and the other on the gauge group with appropriate Chern-Simons levels and matter representations.

We are thus led to propose the following new fermion-fermion dualities for :

Dualities for for :

(2.5)

Dualities for for :

(2.6)

For in the theory of the fermion in the symmetric representation the intermediate phase coincides with the asymptotic large negative mass phase and the first phase transition is therefore unnecessary. Indeed, the associated duality in the second line of (2.5) trivializes since the antisymmetric representation of is trivial.

The case of is particularly interesting and requires a separate discussion. The quantum phase that has appeared in the figure 3 and 4 would seem to make sense also for . However, while for it is a pure TQFT, for it is not. Indeed, after integrating the fermion in the dual theory with gauge group , we are left with pure Yang-Mills-Chern-Simons theory, with in the symmetric/antisymmetric case. The crucial point is that the factor is not topological. This latter theory can be dualized to the theory of a compact, real scalar field

(2.7)

with and the “decay constant”.

This theory is combined with the Chern-Simons theory in the following way: the NGB theory (2.7) has a non-anomalous one-form symmetry, corresponding to the conserved two-form current . Likewise, Chern-Simons theory has a non-anomalous one-form symmetry (it is non-anomalous when one views as a spin TQFT). We gauge the diagonal symmetry, and denote this by

(2.8)

where is the diagonal one-form symmetry. The phase diagrams for are summarized in figures 5 and 6.

Figure 5: Phase diagram of gauge theory with a symmetric fermion for . The circle represents the corresponding sigma model. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by and .

An immediate consistency check of this scenario – which we have already discussed in the introduction – is that the ultraviolet theory is time-reversal symmetric, so it is reassuring to realize that Chern-Simons theory and the NGB theory are time-reversal invariant. The time-reversal invariance of the quotient can be shown from level/rank duality as in [Aharony:2016jvv].

Figure 6: Phase diagram of gauge theory with a antisymmetric fermion for . The circle represents the gapless sigma model with target space. Each phase transition has a dual gauge theory description, which appears with an arrow pointing to the phase transition. The mass deformations are related by and .

The main feature of the model is, of course, the nonperturbative spontaneous breaking of the baryon number symmetry. The symmetry breaking occurs due to the condensation of a baryon. We will discuss the baryon operators in this theory very briefly in the last section.

This spontaneous symmetry breaking is not in contradiction with the Vafa-Witten type theorems [Vafa:1984xg]. In essence, if a symmetry cannot be preserved by a time-reversal invariant mass term then there is no obstruction for the spontaneous breaking of that symmetry.

Indeed, in our class of theories, it is not possible to deform by a mass term while preserving both baryon number symmetry and time-reversal symmetry. This is obvious for since there is no mass term whatsoever that preserves time-reversal symmetry. However, the case of requires special attention. In the case of with an antisymmetric fermion the theory is always in the large two-phase regime, and there is no quantum phase and no spontaneous breaking occurs, of course, since the fermion is completely decoupled. For with a symmetric fermion, the situation is more interesting. A Dirac fermion in the rank-two symmetric representation is equivalent to two Majorana fermions in the adjoint representation. Let us denote the two Majorana fermions by , such that . The baryon symmetry simply rotates these two real fermions

(2.9)

Time-reversal symmetry can be taken to act as , and . Finally, we have a charge-conjugation symmetry that acts as while keeping intact. The minimal scalar baryon operator transforms with charge 2 under . The hermitian combinations , and are the components of this baryon operator.111111They are the real and imaginary parts of the baryon constructed with a Dirac spinor in the symmetric representation . The hermitian combination is instead invariant under baryon symmetry, but obviously breaks time-reversal symmetry when added to the Lagrangian.

Therefore, clearly, if we want to preserve a time-reversal symmetry, we must use the hermitian baryon operators above. Indeed, for example, adding to the Lagrangian would preserve . However, there is no way to add a time-reversal invariant mass term that also preserves the baryon symmetry. Therefore, there is no obstruction for baryon symmetry to be spontaneously broken. Interestingly, in the case of , the TQFT trivializes (see figure 5) because , which is a trivial spin TQFT. The fact that in the particular case the NGB is not accompanied by a TQFT will be crucial later, when we make contact with dimensional physics. Note also that in the case of it is quite clear that the operator which condenses and leads to the NGB is (without loss of generality) .

3 Additional Consistency Checks

3.1 Special Cases

Here we discuss special values of where we can compare our proposed dynamics with previously conjectured phase diagrams for other families of theories. We also embed with a rank-two symmetric fermion in a renormalization group flow of supersymmetric pure gauge theory, and contrast our proposed phase diagram with the expected infrared dynamics of the supersymmetric theory.

with antisymmetric fermion

A consistency check on our proposed dynamics follows from the isomorphism and the fact that the antisymmetric representation of is the six-dimensional, vector, real representation of . Since is a Dirac fermion we have the following equivalence of theories

(3.1)

where by we mean two Majorana fermions in the vector representation of .

For the phase diagram of has two asymptotic phases (see figure 2). The phase diagram of was derived in [Cordova:2017vab]. For and the phase diagrams agree trivially by virtue of the identity of the TQFTs .

We now proceed to the nontrivial matching for where both theories have an intermediate phase, which we want to compare. Plugging in figure 4 we find that the intermediate phase of is described by

(3.2)

which, as we explained in detail, is simply a free compact scalar with periodicity . By contrast, the intermediate phase of is described by the following coset [Cordova:2017vab]

(3.3)

where is a Wess-Zumino term. The coset in (3.3) can be described more explicitly by the equivalence relation of matrices

(3.4)

where is a block-diagonal matrix with two blocks and

(3.5)

such that .

For the Wess-Zumino term vanishes (because ) and we are left with a sigma model on the space of matrices subject to the equivalence relation

(3.6)

with implies that . Therefore, after the quotient the space is still isomorphic to a circle (albeit with a radius smaller by a factor of 2). Therefore this precisely coincides with the result that we obtained for with an antisymmetric fermion.

In summary, our phase diagram for with a fermion in the antisymmetric representation for precisely matches the proposed phase diagram of with Majorana fermions in the vector representation for and . This supports the validity of both phase diagrams.

with an antisymmetric fermion

A somewhat more trivial consistency check can be made for with an antisymmetric fermion by noting that the rank-two antisymmetric representation of is the same as the complex conjugate of the fundamental, three dimensional representation of . Thus we have the equivalence of theories

(3.7)

where is a Dirac fermion. These theories, which always have two phases (i.e. there is no intermediate phase regime), can be seen to have the same phase diagram by comparing figure 4 for with the phase diagram of with fermions in the fundamental representation for and in [Komargodski:2017keh].

with an antisymmetric fermion

A very degenerate special case is with a fermion in the antisymmetric representation. In this case the fermion is decoupled from the gauge dynamics and there is no intermediate phase. The infrared is captured by Chern-Simons theory except at one point in the phase diagram, which coincides with the phase transition in figure 2. The phase transition simply corresponds to a neutral fermion becoming massless.

with a symmetric fermion

The rank-two symmetric representation of is the adjoint representation. Therefore, figures 1 and 3 for describe the infrared dynamics of QCD with adjoint Majorana fermions.121212We recall that our theory is based on a Dirac fermion and therefore Majorana adjoint fermions of . This extends the phase diagram of adjoint QCD with put forward in [Gomis:2017ixy].

The particular case with a symmetric fermion admits an embedding in the supersymmetric theory of one vector multiplet. In addition to the fields we have in our theory, this model also has a real scalar field in the adjoint representation, with couples to the fermions via Yukawa terms. What we called is naturally referred to as the -symmetry in the supersymmetric context. We can flow from the supersymmetric theory to our theory by simply adding a (supersymmetry-breaking) mass term for the real scalar field .131313A closely related preserving mass deformation was analyzed in [Bashmakov:2018wts]. Below we analyze what happens if that mass term is very small compared to the scale set by the gauge coupling.

The infrared of the supersymmetric model consists of [Affleck:1982as] a complex scalar field , whose imaginary part transforms inhomogeneously under the -symmetry, thus signaling spontaneous symmetry breaking of this symmetry. The kinetic term for is approximately canonical for large and the potential is the runaway potential . Adding a small (supersymmetry-breaking) mass term in the ultraviolet translates to adding a small (supersymmetry-breaking) mass term in the infrared. (The map between the deformations in the UV and IR is rather simple for large because the theory is weakly coupled there.) For small enough mass of the minimum of

(3.8)

is therefore at large and we can analyze the physics semiclassically. The fermions are all lifted due to the Yukawa couplings and is likewise massive at the minimum of the potential. The deep infrared theory therefore consists of just the (compact) Nambu-Goldstone boson without an additional TQFT, exactly as in the scenario we proposed above for with a Dirac fermion in the rank-two symmetric representation.

We shall return to this theory later when we discuss domain walls in dimensional Yang-Mills with adjoint Majorana fermions.

3.2 Gravitational Counterterm Matching

Another nontrivial check of our proposed phase diagrams in figures 3 and 4 can be devised by coupling the theories to background gravity. A well-defined (scheme independent) observable is the difference in the gravitational counterterm between the asymptotic negative and asymptotic positive mass phases. This difference is closely related to the difference in the thermal conductance in the two phases. This is an interesting quantity to study because it can be easily computed in the original “electric variables” where it is one loop exact. But it can also be computed in the dual variables, followed by traversing the quantum phase, and using the dual variables again. Therefore, one can devise a concrete nontrivial consistency check. Such computations were done in the context of supersymmetric dualities in [Closset:2012vp, Closset:2012vg], where the connection to the physically observable thermal conductance (or the analogous charge conductance which we will study soon) is explained.

The jump in the gravitational counterterm in the electric variables is given by the number of fermions in the ultraviolet gauge theory (this is related to the parity anomaly [Niemi:1983rq, Redlich:1983kn, Redlich:1983dv]). Therefore, in our theories

(3.9)

where is either the rank-two symmetric or antisymmetric representation, respectively.

Our phase diagrams for in figures 3 and 4 provide us with another way to compute this difference. The two computations must agree for consistency. We start with the TQFT in the asymptotic negative mass phase and work our way towards the asymptotic positive mass phase . This requires tracking the jump of the gravitational counterterm across level/rank dualities, where a gravitational counterterm is generated, and across the phase transitions:

  • TQFT level/rank duality in the asymptotic negative phase [Hsin:2016blu]:

    (3.10)
  • Jump induced from positive to negative mass of the leftmost dual gauge theory:

    (3.11)
  • TQFT level/rank duality in the intermediate phase [Hsin:2016blu]:

    (3.12)
  • Jump induced from positive to negative mass of the rightmost dual gauge theory:

    (3.13)
  • TQFT level/rank duality in the asymptotic positive phase [Hsin:2016blu]:

    (3.14)

Here and denote the representation of the fermion in the leftmost and rightmost dual descriptions in figures 3 and 4.

Adding up the contributions to the jump following this path we find that it precisely matches that in (3.9)

(3.15)

both for the theory with a symmetric and antisymmetric fermion!

3.3 Baryon Counterterm Matching

An entirely analogous exercise to (3.15) is to match the difference in the baryon number conductivity coefficient. This coefficient is simply the difference between the two asymptotic phases in the Chern-Simons term for the baryon background gauge field , i.e., we are after the difference

(3.16)

Here we think about as an ordinary connection and the space-time is assumed to be a spin manifold.141414As all our theories are fermionic and the baryon number clearly satisfies a spin-charge relation, we can in principle also study our theories on spin manifolds. This leads to some additional nontrivial consistency checks which we do not present here.

We need to carefully normalize the baryon charge of the fermion. The most convenient choice is to imagine that the fermion is in the (anti-)symmetric representation of rather than and the diagonal of is the baryon number. This would lead to the fermion carrying charge . However, the corresponding baryon gauge field would then have possible fractional fluxes and in order to fix that we need to take the charge to be .

We can therefore compute straightforwardly in the electric variables as

(3.17)

As before, we can also compute using the dual description:

  • First, in the phase with large negative mass we need to perform level/rank duality between and , which leads to a jump151515Let us explain briefly how to derive this shift from [Hsin:2016blu]. Using the notation of [Hsin:2016blu], the Lagrangian for is (3.18) where us a gauge field. If we integrate out and remove the trace (with ), we get (3.19) The level/rank dual also has a term , so the relative shift by the contact term is only given by the term . [Hsin:2016blu]

    (3.20)
  • Next, there is a crucial difference with our computation of the thermal conductivity. Since the baryon symmetry maps to the magnetic symmetry in the dual variables, and since the dual fermions are not charged under the magnetic symmetry, we get that . In other words, the phase transitions do not lead to a jump in the conductivity.

  • Next we need to address , which arises from the level/rank duality in the quantum phase. We find again from [Hsin:2016blu] that

    (3.21)
  • Finally, the positive-mass level/rank duality leads to

    (3.22)

We see that if we add all the partial jumps we get precisely the same shift in the baryon counterterm (3.17) computed in the electric theory:

(3.23)

for both the symmetric and antisymmetric representation!

The matching of the gravitational contact term guarantees that the phase diagram remains consistent in curved space and the thermal conductivities are single-valued, as they should be in physical theories. The matching of the baryon contact term further guarantees that we can consistently gauge the ultraviolet baryon symmetry in all the phases. Therefore, one can derive from our phase diagrams and dualities also the phase diagrams and corresponding dualities for gauge theories coupled to two-index matter fields.

4 Domain Walls in Four Dimensional Gauge Theories

Let us consider the four-dimensional theory of a Dirac fermion in the symmetric/antisymmetric representation coupled to gauge fields. We can equivalently think about it as gauge theory coupled to a Weyl fermion in the symmetric/antisymmetric representation and another Weyl fermion in the conjugate representation. Let us begin with the massless theory. This theory has a () discrete chiral symmetry that acts by re-phasing the two Weyl fermions together,

(4.1)

In addition the theory enjoys baryon number symmetry which acts by re-phasing the two fermions in an opposite fashion

(4.2)

The special case of with a symmetric representation Dirac fermion is equivalent to gauge theory with two Weyl fermions in the adjoint representation. In this case the symmetry is in fact enhanced to flavor symmetry (and in addition, there is the discrete axial symmetry, where the order-two generator in is identified with the center of flavor symmetry. This order-two generator coincides with the fermion number symmetry, and it is hence unbreakable as long as the vacuum is Poincaré invariant.)

These theories admit a mass deformation, , and we can take to be non-negative without loss of generality, at the expense of having to keep track of the parameter of the gauge theory. The mass perturbation breaks the symmetry down to . However, the mass term preserves . For also the time-reversal symmetry is preserved.

In the special case of , the mass perturbation breaks symmetry, but it preserves baryon number, which can be identified with the Cartan subgroup of . The Vafa-Witten-like theorems would imply that the massless theory cannot break . We will assume that is broken to .

It is reasonable to assume that the massless theory breaks the chiral symmetry as161616This can be proven in the planar limit [Armoni:2003fb], where the theory in the meson sector is equivalent to Supersymmetric Yang Mills theory, which is known to develop a condensate. Therefore our statement about the symmetry breaking pattern certainly holds for large enough finite .

(4.3)

According to these assumptions, the vacuum structure of the theory is therefore:

  • : The massless theory has vacua, each of which is trivial and gapped. The order parameter distinguishing these vacua is the fermion bilinear , which is charged under .

  • : Here breaks spontaneously to but also the axial symmetry is spontaneously broken. The fermion bilinear is in the adjoint representation and it is the order parameter leading to this symmetry breaking pattern.171717Indeed, since the order parameter must be a scalar in space-time the Lorentz indices are contracted antisymmetrically, and since it must be gauge invariant, the gauge indices are contracted symmetrically and hence the flavor indices must be contracted symmetrically as well, leading to the symmetric product of the fundamental representation of with itself, namely, the adjoint representation. Let us parameterize it without loss of generality by . This leads to the breaking pattern , and the corresponding coset manifold is . Acting with the generator of we get a new vacuum, and therefore we have at least two copies of the coset . However, acting with the square of the generator of we get the matrix , but this is in fact on the same coset as the original condensate (more precisely, the Weyl group of relates these two configurations). Hence we have exactly two copies of , and the broken axial symmetry allows us to move from one copy to the other copy.181818This scenario of the gauge theory with two adjoint fermions flowing in the infrared to two copies of has been recently connected to the Seiberg-Witten solution of the vector multiplet theory [Cordova:2018acb]. Other possibilities for the infrared dynamics were recently discussed also in [Bi:2018xvr, Anber:2018tcj].

Let us now turn on a small positive mass . This corresponds to a small potential on the above space of vacua which is a function of . For , for any this lifts the degeneracy and picks up one of the vacua. At there is a first