Phases Of Adjoint QCD 3 And Dualities
aa r X i v : . [ h e p - t h ] M a r Phases Of Adjoint QCD And Dualities
Jaume Gomis, Zohar Komargodski, , and Nathan Seiberg Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Israel Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in theadjoint representation. We apply general considerations of symmetries, anomalies, andrenormalization group flows to determine the possible phases of the theory as a function ofthe gauge group, the Chern-Simons level k , and the fermion mass. We propose an inher-ently quantum mechanical phase of adjoint QCD with small enough k , where the infraredis described by a certain Topological Quantum Field Theory (TQFT). For a special choiceof the mass, the theory has N = 1 supersymmetry. There this TQFT is accompanied by amassless Majorana fermion – a Goldstino signaling spontaneous supersymmetry breaking.Our analysis leads us to conjecture a number of new infrared fermion-fermion dualitiesinvolving SU , SO , and Sp gauge theories. It also leads us to suggest a phase diagramof SO/Sp gauge theories with a fermion in the traceless symmetric/antisymmetric tensorrepresentation of the gauge group.October 2017 ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. SU ( N ) Adjoint QCD Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . 122.1. Phase Diagram for k ≥ N/ k < N/ SO ( N ) and Sp ( N ) . . . . . . . . . . . . . . . . 223.1. Phase Diagram for k ≥ h/ k < h/ T -reversal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4. Consistency Checks for Low-Rank Theories . . . . . . . . . . . . . . . . . . 304. Phase Diagrams and Dualities for SO ( N ) with Fermions in the Symmetric Tensor and Sp ( N )with Fermions in the Antisymmetric Tensor . . . . . . . . . . . . . . . . . . . . . 31
1. Introduction
Consider adjoint QCD in 2+1 dimensions, that is Yang-Mills theory with a Chern-Simons term at level k and a Majorana fermion λ in the adjoint representation of the gaugegroup G . We study the low energy dynamics of this theory as a function of the gauge group G , the Chern-Simons level k and the mass M λ of the fermion. We uncover a rich structureof phase diagrams that suggest new fermion-fermion dualities and include phases that areinherently quantum mechanical (in the sense that they are invisible semiclassically).The infrared behaviour of the theory crucially depends on the level k of the Chern-Simons term. We denote by G k a Chern-Simons term with gauge group G and level k .We will be mostly interested in the classical gauge groups G = SU ( N ) , SO ( N ) , Sp ( N )and follow the conventions of [1-5]. The Lagrangian of adjoint QCD contains a Chern-Simons term with coefficient k bare which must be properly quantized. Since the theoryhas a fermion, we henceforth consider the theory on spin manifolds, which requires that k bare ∈ Z . The level k is related to k bare by k = k bare − h . Our notation is Sp (1) = SU (2). h the dual Coxeter number of G , e.g. G SU ( N ) SO ( N ) Sp ( N ) h N N − N + 1 Table 1 : Dual Coxeter number for classical groups. If h/ k and if h/ k . In other words, we must take k = h mod 1, which in our cases implies G SU ( N ) SO ( N ) Sp ( N ) k N mod 1 N mod 1 N +12 mod 1 Table 2 : Quantization of Chern-Simons levels in adjoint QCD.
The virtue of labeling the theory by k is that time-reversal (or parity) acts by k → − k (alongside with reversing the sign of the fermion mass M λ → − M λ ). Without loss ofgenerality we henceforth consider k ≥
0. Note that adjoint QCD for k = 0 and a vanishingmass ( M λ = 0) for the fermion is time-reversal invariant. This time-reversal invarianttheory therefore exists in SU ( N ) or SO ( N ) only for N even and in Sp ( N ) only for N odd.Adjoint QCD has two semiclassically accessible phases for all G and k . When themass of the fermion is much larger than the scale set by the Yang-Mills coupling (i.e. | M λ | ≫ g ) the fermion can be integrated out at one-loop. This shifts the coefficient ofthe Chern-Simons level [10,11,12] k → k + sgn( M λ ) h . (1 . G and level k + h/ Note that the dual Coxeter number of SO (3) is one, while that of SU (2) is two. Corre-spondingly, our notation is SO (3) k = SU (2) k / Z . More generally, we label the TQFT by thecorresponding Chern-Simons gauge group and its level. A quotient as in this expression is inter-preted from the 2 d RCFT as an extension of the chiral algebra [6] and from the 3 d Chern-Simonstheory as a quotient of the gauge group [7]. More abstractly, it can be interpreted as gauging aone-form global symmetry of the TQFT [8,9]. This quotient is referred to in the condensed matterliterature as “anyon condensation.” k − h/ G k + h/ and G k − h/ respectively. Our goal is to fill in the rest of the phase diagram.For a special value of the bare mass M λ = m SUSY ∼ − kg , (1 . N = 1 supersymmetric. At this point the fermion λ is the gaugino inthe N = 1 vector multiplet. Moving away from this supersymmetric point in the phasediagram can be interpreted as turning on a soft supersymmetry-breaking mass m λ for thegaugino M λ = m SUSY + m λ . (1 . m λ = 0.For large k all the propagating degrees of freedom have mass of order kg ≫ g and thesemiclassical analysis is again reliable. First, we integrate out the gauginos thus shiftingthe Chern-Simons coefficient [13,14] k IR = k − h . (1 . G k − h theory. Witten argued that the large k infrared description G k − h remains valid in the entire domain of k where supersymmetryis unbroken [14], that is for k ≥ h . (1 . G k − h Chern-Simons theory. This infrared theory coincides with the asymptotic phase at large negativemass. N = 1 supersymmetry in 2+1 dimensions entails 2 real supercharges. Note that for k = h , k IR = 0 and hence the infrared theory is rather simple. For simplyconnected gauge groups it is completely trivial. But for non-simply connected gauge groups, like SO ( N ) , it has a zero-form magnetic symmetry, which can be spontaneously broken, leading toseveral vacua in the infinite volume system. k ≥ h depicted in fig. 1. The system has two asymptotic phases with topological order G k ± h .The supersymmetric point m λ = 0 is in the G k − h phase. The two phases are separatedby a transition at some positive value of m λ . At that point we can think of the fermion asbeing effectively massless. We do not actually know whether this transition is first orderor second order. However, for very large k we can think about the model perturbatively,starting from the CFT of dim( G ) free fermions. The anomalous dimensions of the G -invariant local operators are only weakly corrected compared to the free model and thetransition is second order. Therefore, it is natural to expect that for (1.6) the transitionbetween the two topological theories G k + h/ and G k − h/ is second order. Fig. 1:
The phase diagram of G k with an adjoint fermion for k ≥ h . Note thatthe physics at the supersymmetric point is smooth – there is no transition there.For k = h the negative mass phase is trivial and there is a transition at somepositive value of m λ . Here, and in all our later figures, the vertical line denotesrenormalization group flow from the UV at the top of the diagram to the IR at thebottom of the diagram. The solid diagonal line represents a flow to an IR fixedpoint. The physics across this line is not smooth. The theory is supersymmetricalong the dotted vertical line. Unlike the solid line, here the physics is smooth asthis line is crossed. Let us now turn to the interesting region k < h . We know from our discussion abovethat there are two asymptotic regions of large mass described by the TQFTs G k ± h/ . Theinfrared dynamics for small mass m λ is strongly coupled and requires a more sophisticatedanalysis. We will see that the physics at small m λ can lead to novelties in comparison tothe phase diagram for k ≥ h/ m λ = 0 supersymmetry is expected to be spontaneouslybroken for k < h/ m λ = 0 there is a massless Majorana4ermion (i.e. the Goldstino particle). An interesting question is whether the infrared theorycontains a TQFT in addition to the Majorana Goldstino particle. We will see that generalconsiderations including symmetries and ’t Hooft anomaly matching imply that this isindeed the case. We will also identify the TQFT accompanying the Goldstino.A first guess for the phase diagram of adjoint QCD for k < h/ m λ = 0 there is also a massless Goldstinoreflecting the fact that supersymmetry is spontaneously broken. In fact, we argue belowthat for G = SU ( N ), SO ( N ), and Sp ( N ) this is the correct scenario, but only for a specialvalue of k = h − This is depicted in fig. 2. Note that the low energy theory aroundthe point of the massless Goldstino includes a decoupled TQFT G − . There is a phasetransition to the right of the supersymmetric point m λ = 0, which we denote by a bulletpoint. We will not be able to determine whether this transition is first or second order. Wefind that this phase transition admits an alternative description in terms of another gaugetheory coupled to a fermion in a representation of the gauge group that depends on thechoice of G . This suggests new fermion-fermion dualities in nonsupersymmetric theories(see below). For other recent nonsupersymmetric dualities see [15-18,2,3,19-23,4,24,25].We are going to argue that the scenario in fig. 2 cannot be right for all lower valuesof k , i.e. all k < h −
1. To see that, consider the special case of k = 0. We knowthat for large | m λ | we have the two asymptotic topological phases G ± h exchanged by theaction of time-reversal. (Time-reversal changes the sign of M λ and indeed exchanges thetwo asymptotic phases.) Also, since supersymmetry is spontaneously broken, we expect amassless Goldstino at the supersymmetric point M λ = m λ = 0. If the system has only thetwo topological phases G ± h/ , the supersymmetric point must be at the transition point.So one might be tempted to consider a phase diagram with only two phases, one for m λ positive and the other for m λ negative. At the supersymmetric point the infrared theorymust be time-reversal invariant. This would be a consistent scenario if the two topologicalphases are level/rank dual to each other, G h ←→ G − h ; that is if the topological phase G h With the exception of adjoint QCD with gauge group SO ( N ) and a Chern-Simons term atlevel k = h/ − In our discussion below we will mostly exclude the case SO (4) = ( SU (2) × SU (2) ) / Z ,where supersymmetry is broken with two massless Goldstinos, one for each SU (2) factor. We have not analyzed whether this picture is valid also for other gauge groups. We use the symbol A ←→ B to denote that theories A and B are dual. ig. 2: The proposed phase diagram for G = SU ( N ), SO ( N ), and Sp ( N ) with k = h −
1. For low values of N the situations can be different. For example, for SU (2) the transition point occurs at the supersymmetric point m λ = 0. Andother than the massless Goldstino there, the TQFT does not change because ofthe duality SU (2) − ↔ SU (2) (as spin-TQFT). Below we will discuss anotherdual theory that flows to the same transition. It is denoted in the figure as “DualTheory.” The general structure of the figure is as explained in the caption of fig. 1,except that here supersymmetry is broken for m λ = 0 and we also suggest a dualtheory of the transition point. is time-reversal invariant. We could then have a massless Goldstino at m λ = 0 togetherwith the decoupled TQFT G h . However, this condition that G h is dual to G − h is notobeyed for generic G and thus invalidates this simplistic scenario. Another possibilityis that the theory at that supersymmetric point spontaneously breaks its time-reversalsymmetry such that the transition at m λ = 0 is first order. This possibility can be ruledout using the argument of [26]. Alternatively, the transition at that point could be secondorder and that would mean that the low energy theory includes more degrees of freedom.Such additional degrees of freedom would also need to match various anomalies of theultraviolet adjoint QCD theory. We cannot exclude this possibility. But we will argue foranother, more likely, and simpler option.Let us now present our scenario for k < h −
1: a new intermediate quantum mechan-ical phase opens up between the two asymptotic phases. The details of the scenario aresummarized in fig. 3. This picture is motivated by the analysis of QCD in [5], where itwas suggested that for small fundamental quark masses the system can have a new purely This is true for G = SU (2) where SU (2) ←→ SU (2) − , but this case has k = h − k < h −
1. See a comment in fig. 2. ig. 3: The proposed phase diagram for k < h −
1. The system has three phases.Two of them at large | m λ | are visible semiclassically and the middle phase at small | m λ | is purely quantum. The supersymmetric point m λ = 0 is in the interior ofthis middle phase. At that point there is a massless Goldstino. Even when themiddle phase is gapped it includes some TQFT, which we will determine. We willalso present two conjectured dual gauge theories, denoted by “Dual Theory A”and “Dual Theory B” that describe the two transition points. Again, the generalstructure of this diagram is as explained in the caption of fig. 1, except that herewe have two transition points and two dual theories. quantum phase, which cannot be understood semiclassically. It is also motivated by study-ing the effective field theory on domain walls and interfaces along the lines of [27-30], aswell as by the holographic realization [31,32] of adjoint QCD for m λ = 0 from which theintermediate infrared TQFT can be extracted by analyzing the effective low energy theoryon branes wrapping a noncontractible cycle threaded by flux. A detailed analysis of thesedomain walls and interfaces will appear in [30].General considerations imply that the TQFT in the intermediate, new phase cannotbe trivial. Adjoint QCD has a one-form global symmetry (see table 3) acting on its lineoperators [8,9]. This symmetry cannot act trivially in the infrared (i.e. it cannot be that allthe lines would be confined) because it has an ’t Hooft anomaly for generic k . Therefore,this symmetry must also be realized in the infrared. But since a single Goldstino clearlydoes not match the one-form global symmetry of adjoint QCD, this rules out the scenariowith just a Goldstino. This scenario can also be ruled out in the T -reversal-invariant cases7 = 0. In these cases adjoint QCD has a T -reversal anomaly, characterized by an integermodulo 16, which is often denoted by ν [33-41]. (For early work on mixed T -gravitationalanomalies, see e.g. [42].) The massless Goldstino does not saturate that anomaly. Belowwe will discuss how the proposed TQFT in the infrared does saturate that anomaly. SU ( N ) k Sp ( N ) k SO (2 N ) k SO (2 N ) k +1 SO (2 N + 1) k Z N Z Z Table 3 : One-form global symmetry of adjoint QCD.
We suggest that adjoint QCD for k < h − G k ± h/ . The intermediate phase is another gappedphase except at the supersymmetric point m λ = 0 where there is a massless Goldstino. TheTQFT describing the gapped phase differs from the asymptotic TQFTs G k ± h/ . Below weidentify the TQFT that governs the intermediate phase for G = SU ( N ) , SO ( N ) , Sp ( N )gauge theory with a fermion in the adjoint representation U (cid:0) N − k (cid:1) N + k,N for SU ( N ) k with k < N − SO (cid:0) N − − k (cid:1) N − + k for SO ( N ) k with k < N − Sp (cid:0) N +12 − k (cid:1) N +12 + k for Sp ( N ) k with k < N − . (1 . SU ( N ) adjoint QCD coincide with the domainwall theories of 3+1 dimensional N = 1 SU ( N ) super-Yang-Mills in [27]. This will beimportant in [30]. We thank E. Witten for raising this point. We follow the standard notation U ( M ) P,Q = SU ( M ) P × U (1) MQ Z M U ( M ) P ≡ U ( M ) P,P . (1 . U ( M ) P,Q is described by the Chern-Simons Lagrangian P π Tr( a ∧ da + 23 a ∧ a ∧ a ) + Q − P πM (Tr a ) ∧ ( d Tr a ) , (1 . a a U ( M ) gauge field. This expression makes it clear that it is well defined for integer P, Q with P = Q mod M and that the theory is a spin TQFT when P + Q − PM is odd (see e.g. [1,3]). G . We denote these dual theories in fig. 3 by “Dual Theory A” and “Dual TheoryB.” Specifically, we propose the new fermion-fermion dualities: SU ( N ) and k < N : SU ( N ) k + adjoint λ ←→ U (cid:18) N k (cid:19) − N + k , − N + adjoint ˜ λSU ( N ) k + adjoint λ ←→ U (cid:18) N − k (cid:19) N + k ,N + adjoint ˆ λ . (1 . SO ( N ) and k < N − : SO ( N ) k + adjoint λ ←→ SO (cid:18) N −
22 + k (cid:19) − N + k + + symmetric ˜ SSO ( N ) k + adjoint λ ←→ SO (cid:18) N − − k (cid:19) N + k − + symmetric ˆ S . (1 . Sp ( N ) and k < N +12 Sp ( N ) k + adjoint λ ←→ Sp (cid:18) N + 12 + k (cid:19) − N + k − + antisymmetric ˜ ASp ( N ) k + adjoint λ ←→ Sp (cid:18) N + 12 − k (cid:19) N + k + + antisymmetric ˆ A . (1 . SO ( N ) and anti-symmetric representations of Sp ( N ) we mean the irreducible symmetric-traceless andantisymmetric-traceless representations. The dualities in the second line of (1.10)(1.11)(1.12)trivialize when k reaches the upper limit, i.e. when k = h/ −
1. It should also be pointedout that the dualities in the second line of (1.10)(1.11)(1.12) can be viewed as “analytic As in other familiar examples, including most recently [5], the notion of this duality is validonly near the transition point. We say that G k + adjoint λ is dual to Theory A around onetransition and it is dual to Theory B around the other transition, but we do not say that TheoryA is dual to Theory B. k combined with orientation reversal of the dualities in the firstline. A way to understand fig. 3 is as follows. We start with large and negative m λ andfind G k − h . Then we use level/rank duality to express this theory as G Ak A with some gaugegroup G A and some level k A . The Dual Theory A has gauge group G A with some level andsome matter fields. Specifically, this theory can be found in (1.10)(1.11)(1.12). Then wemove to the right in fig. 3 and cross a phase transition. Here the matter fields in theory Achange the sign of their mass and k A changes to ˆ k A . We end up in the intermediate phasewith the TQFT G A ˆ k A with some level ˆ k A . In our examples, these are the TQFTs in (1.9).We then repeat these steps for large and positive m λ , where again we use level/rank dualityand then a transition in Dual Theory B to find the TQFT in the intermediate phase G B ˆ k B .Our proposal for the intermediate phase and the dual theories (1.10)-(1.12) was motivatedby requiring that the two descriptions of the TQFT in the intermediate phase are dual toeach other, i.e. G A ˆ k A ←→ G B ˆ k B .Below we will discuss this process in a lot of detail in our three examples SU ( N ), SO ( N ), and Sp ( N ).As with most dualities, these are merely conjectures. In fact, as we emphasized above,our entire proposed phase diagram fig. 3 is conjectural. A common check of dualities isthe matching of their global symmetries and their ’t Hooft anomalies. A simple argumentshows that many of these tests are automatically satisfied in our proposal. We have justmentioned that our scenario was designed such that the two descriptions of the intermediatephase are dual to each other, G A ˆ k A ←→ G B ˆ k B . Therefore, throughout our phase diagramwe used either level/rank duality or a phase transition in a weakly coupled theory. Thisguarantees that all the symmetries that are preserved through this process and their ’tHooft anomalies must match. These include all the zero-form and the one-form globalsymmetries [8,9].A notable exception to this statement is a symmetry that is violated in our process ofchanging m λ . Consider the special theories with k = 0. For m λ = 0 they are time-reversalinvariant. This symmetry is not present as we vary the mass and get to the intermediatephase and therefore there is no guarantee that it is present there. As a nontrivial check,10or k = 0 the TQFT in the intermediate phase (1.9) is T -invariant U (cid:18) N (cid:19) N ,N ←→ U (cid:18) N (cid:19) − N , − N SO (cid:18) N − (cid:19) N − ←→ SO (cid:18) N − (cid:19) − N − Sp (cid:18) N + 12 (cid:19) N +12 ←→ Sp (cid:18) N + 12 (cid:19) − N +12 , (1 . T -reversal-invariant cases also lead to additional consistency checks. As wementioned above, it is known that T -reversal-invariant 2 + 1 dimensional spin theories aresubject to an anomaly, which is characterized by an integer ν modulo 16. This integermust match between adjoint QCD and our proposed infrared theory, which includes theGoldstino and the TQFTs (1.13). In order to do that we will need the value of ν for theseTQFTs [44-47] (special cases had been found earlier) ν ( U ( n ) n, n ) = ± ν ( SO ( n ) n ) = ± n mod 16 ν ( Sp ( n ) n ) = ± n mod 16 . (1 . k .It is natural to examine them also for larger k , which after redefining k , are written for SO ( N ) k with k < N +22 as SO ( N ) k + symmetric S ←→ SO (cid:18) N + 22 + k (cid:19) − N + k − + adjoint ˜ λSO ( N ) k + symmetric S ←→ SO (cid:18) N + 22 − k (cid:19) N + k + + adjoint ˆ λ . (1 . Sp ( N ) k with k < N − as Sp ( N ) k + antisymmetric A ←→ Sp (cid:18) N −
12 + k (cid:19) − N + k + + adjoint ˜ λSp ( N ) k + antisymmetric A ←→ Sp (cid:18) N − − k (cid:19) N + k − + adjoint ˆ λ . (1 . k = N/
2. As above, the dualities inthe second line in (1.15)(1.16) are obtained from the first line by “analytic continuation”to negative k combined with orientation reversal.Furthermore, these dualities motivate us to conjecture also the phase diagram of thesetwo theories. Therefore, we will also study the theories with gauge group SO ( Sp ) witha fermion in the symmetric (antisymmetric) representation. We will find through ourdualities above that the infrared description also contains an intermediate quantum phase,given by the Chern-Simons TQFTs SO (cid:0) N +22 − k (cid:1) N +22 + k for SO ( N ) k + S with k < N Sp (cid:0) N − − k (cid:1) N − + k for Sp ( N ) k + A with k < N − (1 . ν modulo 16of the k = 0 ultraviolet gauge theory beautifully matches the T -reversal anomaly of theintermediate TQFT (1.14) SO (cid:18) N + 22 (cid:19) N +22 ←→ SO (cid:18) N + 22 (cid:19) − N +22 Sp (cid:18) N − (cid:19) N − ←→ Sp (cid:18) N − (cid:19) − N − . (1 . SO ( N )gauge theories will be presented in [48]. This will test our phase diagram and will allowus to derive similar results for other gauge groups, e.g. Spin ( N ).Another test of our proposal arises from the gravitational Chern-Simons counter-term . Starting in the ultraviolet theory we derive the two phases at large positive andlarge negative mass at weak coupling. So the difference between the coefficients of this We thank F. Benini for raising this point. SU groups and in [4]for SO and Sp groups) and the change when we go through the phase transitions to checkthat we find the same answer in the middle phase regardless of whether we arrive to itfrom positive mass or negative mass. When we perform this check we should make surethat in the theories with a massless Goldstino this coefficient changes as we go throughthat point. Since this computation is straightforward and tedious and since similar com-putations were done in [49,50], we will not present it in detail here. We will simply statethat this consistency check is satisfied in all the cases we discuss here.The plan of the paper is as follows. In section 2 we consider in detail SU ( N ) Chern-Simons theory with an adjoint fermion and discuss some interesting special cases. Insection 3 we consider the analogous problem for SO ( N ) and Sp ( N ) adjoint QCD. Insection 4 we discuss the phase diagram of SO ( N ) theories with a fermion in the symmetric-traceless representation and Sp ( N ) theories with a fermion in the antisymmetric-tracelessrepresentation. SU ( N ) Adjoint QCD Phase Diagrams
In this section we study the phase diagram of Yang-Mills theory with gauge group SU ( N ), level k Chern-Simons term and a fermion λ in the adjoint representation. The twoasymptotic infrared phases that describe the domain of large fermion mass (for any valueof k ) are the TQFTs SU ( N ) k + N/ and SU ( N ) k − N/ . We discuss the infrared dynamicsfor k ≥ N/ k < N/ k ≥ N/ SU ( N ) k ± N/ thatcan be reliably studied at large mass with the expectations from the supersymmetric theoryat m λ = 0. At m λ = 0 the infrared theory is believed to consist simply of SU ( N ) k − N/ all the way down to k = N/ SU ( N ) adjoint QCD for k ≥ N/ SU ( N ) k ± N/ . The supersymmetric theory with m λ = 0 is in the phase SU ( N ) k − N/ . This is true also for k = N/
2, where the supersymmetric theory is in thetrivial phase SU ( N ) (and hence has a unique ground state). There is a phase transitionbetween these two phases at some nonzero (positive) value of the supersymmetry-breakingmass m λ . The phase diagram is depicted in fig. 4. We emphasize again that the phasetransition is not coincident with the supersymmetric point.13 ig. 4: The phase diagram of SU ( N ) k with an adjoint fermion for k ≥ N/ k = N/ An interesting special case is G = SU (2) k with odd k . Here we we can gauge the Z one-form symmetry and arrive at SO (3) k/ = SU (2) k / Z with a single real fermion inthe three-dimensional (adjoint) representation. (The supersymmetric point of this theoryplayed in an important role in [14].) This theory was argued in [23,4,43] to be dual to an SO (cid:0) k +12 (cid:1) − gauge theory coupled to a scalar in the vector representation at a Wilson-Fisher point. Following [43] (see footnote 3 there) we identify the point where the dualityis relevant with the transition point in fig. 4, which differs from the supersymmetric point.We can recover the SU (2) k theory by gauging the magnetic Z global symmetryin this duality. This is done in detail in [48] with the conclusion that the SU (2) k theorycoupled to a fermion in the adjoint is dual to O (cid:0) k +12 (cid:1) − , − coupled to a scalar in the vectorrepresentation. Here the superscript and the two subscripts label various topological termsin the gauge theory. The first subscript is the Chern-Simons level of the continuous group.The superscript denotes a coupling between the continuous group and the discrete Z (itwas introduced in [51] and denoted there by ± ). In this case it is 0, which means thatthis coupling is absent. And the second subscript is a topological term in the Z sector.On a spin manifold such terms have a Z classification [53,54] (see also [55,56]) and thesubscript − Z . See [48] This Z classification of the anomaly is closely related to the standard anomaly in perform-ing a chiral GSO projection in two-dimensional theories. In this form this Z is crucial in theconsistency of type II and heterotic string theories [52]. We thank the referee for pointing thisout to us. O (cid:0) k +12 (cid:1) − , − theory coupled to a scalar in the vector representationhas two weakly coupled gapped phases with TQFTs. The TQFT of the phase where thescalar condenses is level/rank dual to an SU (2) k − TQFT, while the TQFT in the phasewhere the scalar is massive is level/rank dual to an SU (2) k +1 TQFT [48].
Fig. 5:
The phase diagram of SU (2) with an adjoint fermion. Note that thephysics at the supersymmetric point is smooth in this example, which correspondsto k = N/
2. We also present the dual description. Since O (1) ∼ Z , the dualdescription of the transition is in terms of the gauged 2+1d Ising model with acertain topological term in the Z gauge theory. Fig. 6:
The phase diagram of SU (2) with an adjoint fermion. We also presenta bosonic dual description – a gauged version of the O (2) Wilson-Fisher point. Two special cases k = 1 and k = 3 are particularly simple. For k = 1 we have O (1) ∼ Z , so the dual theory is a gauged version of the 2 + 1 dimensional Ising model[48]. This is depicted in fig. 5. For k = 3 the dual theory is O (2) − , − coupled to a scalar inthe vector representation [48]. This is a gauged version of the O (2) Wilson-Fisher model,where again, the two subscripts denote the topological terms in the gauge field. (In thiscase the superscript + of the general case is superfluous.) This case is depicted in fig. 6.15 .2. Phase Diagram for k < N/ k . The TQFTs SU ( N ) k ± N/ still describe theasymptotic large mass phases of the diagram. Since at m λ = 0 the theory is believed tobreak supersymmetry spontaneously [14], this implies the presence of a Majorana Gold-stino particle at the supersymmetric point. Therefore the phase diagram of fig. 4 needsto be somewhat modified. We have already argued in the introduction that a possible,well-motivated, modification of the phase diagram consists of introducing another phasetransition so that the system has generically three distinct phases. Two phases are visi-ble semi-classically and are described by Chern-Simons TQFTs and the third phase is anew quantum phase. The new phase cannot consist just of a Goldstino as this would notmatch various symmetries and anomalies of adjoint QCD. These include the one-form Z N symmetry of adjoint QCD and its ’t Hooft anomaly and the anomaly in the T -reversalsymmetry of the k = 0 theory. We will discuss this latter anomaly below. We will suggestthat the infrared theory in the new phase contains a TQFT (which we identify below) inaddition to the Majorana Goldstino.Recall that at large positive and negative m λ the theory is described in the infraredby SU ( N ) k ± N Chern-Simons theory, respectively. Using level/rank duality it is useful torewrite these TQFTs as [3] SU ( N ) k ± N ←→ U (cid:18) N ± k (cid:19) ∓ N = SU (cid:0) N ± k (cid:1) ∓ N × U (1) ∓ N ( N ± k ) Z N ± k . (2 . SU ( N ).We suggest that at small m λ the infrared is described by a different Chern-SimonsTQFT (see fig. 7) U (cid:18) N k (cid:19) − N + k, − N , (2 . U (cid:18) N − k (cid:19) N + k,N . (2 . m λ = 0, the infrared theory consists of a Goldstinoand a TQFT, which we now have identified with (2.2).16 ig. 7: The phase diagram of SU ( N ) k with an adjoint fermion and k < N/ − N >
2. In this theory there are two transitions where the infrared TQFThas to change and in addition one point where the Goldstino becomes massless, inbetween the two transitions. A fermionic dual for both of the nontrivial transitionsis proposed.
Let us summarize our proposal. Consider SU ( N ) adjoint QCD with k < N/ − SU ( N ) k + N/ ←→ U (cid:0) N + k (cid:1) − N . As we decrease the mass we encounter a transition to the TQFT U (cid:0) N + k (cid:1) − N + k, − N ←→ U (cid:0) N − k (cid:1) N + k,N . As we proceed along the mass axis the Gold-stino becomes massless at the supersymmetric point and then massive again. Finally,we encounter the last transition to the SU ( N ) k − N/ ←→ U (cid:0) N − k (cid:1) N,N
Chern-SimonsTQFT.Let us consider more carefully the first transition, namely, the transition between SU ( N ) k + N ←→ U (cid:0) N + k (cid:1) − N, − N and U (cid:0) N + k (cid:1) − N + k, − N . This can be nicely repro-duced using a dual fermionic theory U (cid:18) N k (cid:19) − N + k , − N + adjoint ˜ λ , (2 . λ is the adjoint of SU (cid:0) N + k (cid:1) – there is no singlet. The fermion is denoted by ˜ λ rather than λ in order to distinguish it from the original fermion. Since there is no mattercharged under the U (1) factor, the phase diagram of this model is that of SU ( N + k ) − N + k λ . Since in the range k < N/ | N − k | > N + k , we see,according to our previous analysis (for k ≥ N/ SU ( N ) k − N/ ←→ U (cid:0) N − k (cid:1) N,N and U (cid:0) N − k (cid:1) N + k,N , described by adjoint QCD with afermion ˆ λ U (cid:18) N − k (cid:19) N + k ,N + adjoint ˆ λ . (2 . SU ( N ) k + adjoint λ ←→ U (cid:18) N k (cid:19) − N + k , − N + adjoint ˜ λ (2 . SU ( N ) k + adjoint λ ←→ U (cid:18) N − k (cid:19) N + k ,N + adjoint ˆ λ , (2 . k < N , and describe the two transitions discussed above. As we saidabove, the second duality follows from the first by “analytic continuation” to negative k combined with orientation reversal.We would like to clarify a possibly confusing point. SU ( N ) k adjoint QCD has an N =1 supersymmetric point, which we have denoted by m λ = 0. Our dual descriptions (2.6)and (2.7) are supposed to describe the phase transitions of SU ( N ) k adjoint QCD for k < N/ m λ = 0. Yet, the dual theories consistof U gauge groups with an adjoint fermion ˜ λ, ˆ λ , and if we added another massive singletfermion these theories would also have their own N = 1 supersymmetric point. Thereis no relation between these supersymmetric points and those of the original SU ( N ) k adjoint QCD theory. In our SO and Sp dualities in section 3 this point will become evenmore apparent, as the dual theories do not have an N = 1 supersymmetric point in thefirst place. There is no contradiction here because the dualities (2.6) and (2.7) describetransitions that are away from the supersymmetric points m λ = 0.An interesting special case occurs for k = N/ −
1. In that case the asymptotic theoriesat large positive mass and large negative mass are given, respectively, by U ( N − − N, − N and U (1) N (see fig. 8). The quantum phase is also given by (2.3), i.e. U (1) N Chern-Simons theory. Therefore, the Chern-Simons theory at small m λ is identical to one of theasymptotic TQFTs. The transition that is dual to (2.6) remains nontrivial, while the dualtheory (2.7) becomes U (1) N with no matter fields, and hence it is a trivial dual description.There is a massless Goldstino somewhere in the part of the phase diagram that is describedby a U (1) N TQFT. In other words, the N = 1 theory with m λ = 0 flows to a U (1) N TQFTaccompanied by a massless Goldstino. 18 ig. 8:
The phase diagram of SU ( N ) k with an adjoint fermion and k = N/ − N >
2. In this theory there is one transition where the infrared TQFT hasto change and in addition one point where the Goldstino becomes massless. Wepropose a fermionic dual for the nontrivial transition.
Fig. 9:
The phase diagram of SU (2) with an adjoint fermion. In this diagramthere are no necessary transitions other than the Goldstino becoming massless atone point. This is due to the fact that all the following four TQFTs are dual: SU (2) − ←→ SU (2) ←→ U (1) ←→ U (1) − . The phase diagram therefore is especially simple. It could be summarized by sayingthat there is a duality between SU (2) with an adjoint fermion and a free Majoranafermion accompanied by a pure U (1) TQFT. N = 2, the theory with k = N/ − m λ = 0 andthe picture is further simplified since the asymptotic phase at large positive m λ is U (1) − (see fig. 9). There is a massless Goldstino at m λ = 0 and the TQFT at m λ = 0 can bechosen to be U (1) or U (1) − , which are identical by level/rank duality. Therefore, we seethat in SU (2) with an adjoint fermion there is a single second-order transition at m λ = 0.At the second order transition point, the Goldstino becomes massless. Therefore, in thiscase one can summarize the situation by the statement that there is a duality SU (2) + adjoint λ ←→ neutral ψ + U (1) , (2 . ψ a neutral Majorana fermion. This duality is reminiscent of the supersymmetricduality [57].The SU (2) adjoint QCD that we have just discussed is also a special member of the T -invariant family of theories SU ( N ) adjoint QCD, which exhibit three phases for N > T -invariant theories below. Fig. 10:
This is the time-reversal invariant theory with an adjoint fermion. The N = 1 supersymmetric point coincides with the time-reversal invariant point. Thetheory flows to a massless Goldstino and a U ( N/ N/ ,N TQFT.
As with all dualities and proposed infrared behavior of strongly coupled theories, onehas to check that the symmetries of the dual theories and their anomalies match. As20e said in the introduction, most of this is guaranteed to work in our setup. We used asequence of steps: integration out of the fermion at large positive mass, level/rank duality,a transition described by the weakly coupled theory (2.6), level/rank duality, a transitiondescribed by the weakly coupled theory (2.7), and level/rank duality. This matched withthe integration of the fermion at large negative mass. Every step here either involves acomputation in a weakly coupled theory, or level/rank duality, which is rigorously proven.This does not prove that our proposed phase diagram is correct. But it does show that allthe phases we described have the same global symmetries and they have the same ’t Hooftanomalies.There is an exception to this reasoning. For k = 0 (which is possible only for N even)our system is T -reversal invariant for m λ = 0, but it is not invariant for nonzero m λ . Sinceour argument for the global symmetry and its anomaly matching between the ultraviolettheory and the infrared theory involved first making m λ nonzero, there is no guaranteethat our proposed infrared theory for k = m λ = 0 is T -reversal invariant. And even ifit is invariant, there is no guarantee that the ’t Hooft anomaly in this symmetry in theultraviolet matches that of the infrared.Regardless of our specific proposal, the infrared behavior of the k = m λ = 0 theoryeither has to be time-reversal preserving or the time-reversal symmetry must be sponta-neously broken. The latter is excluded by the Vafa-Witten theorem [26]. In our proposal(fig. 10) the infrared theory consists of a massless Majorana fermion ψ and the U (cid:0) N (cid:1) N ,N TQFT. The massless Majorana fermion is manifestly time-reversal invariant, while thefact that U (cid:0) N (cid:1) N ,N is a time-reversal invariant TQFT follows from level/rank dualityeven though this is not a manifest symmetry of the Lagrangian [4].The matching of the ’t Hooft anomaly in the time-reversal symmetry at k = m λ = 0 isnot obvious. This time-reversal anomaly is related to the eta invariant in 3+1 dimensionsand it is an integer ν modulo 16. This anomaly in adjoint QCD is easily calculated usingthe N − ν UV = ( N −
1) mod 16 = (cid:16) − − N/ (cid:17) mod 16 (2 . N has to be even for k = 0). In general the value of ν IR in our infraredtheory is not easy to compute, see for instance [39,59,44,45,60]. One contribution to it,due to the Goldstino, is 1. The contribution of the TQFT U (cid:0) N (cid:1) N ,N was worked outin [44,45,46] (see also references therein) and was found to be ±
2. We suggest that in thepresent context time-reversal in the infrared would act such that it is actually − − N/ .Therefore, we find ν IR = 1 − − N/ , as in the ultraviolet!
3. Phase Diagrams and Dualities for SO ( N ) and Sp ( N )In this section we study the phase diagram of adjoint QCD with SO and Sp gaugegroups. We often denote the gauge group by G when it makes no difference which of thetwo cases we are discussing. We discuss the long distance behavior of these theories as afunction of the mass m λ of the fermion and of k . Our discussion will be along the lines ofthe general description in the introduction and will be quite similar to the analysis of SU theories above.For any value of k , the phase diagram has two semiclassically accessible phases wherethe fermion is very massive. These are described by Chern-Simons theories G k + h/ and G k − h/ respectively, where h = N − SO ( N ) and h = N + 1 for Sp ( N ). These TQFTs Given a Majorana fermion, its contribution to ν can be ± T on that fermion. Above we have assumed that T acts in the same way on all the N − SU ( N ) gauge transformation with the action of T and change theway T acts on some of the Majorana fermions. Indeed, consider an SU ( N ) gauge transformation g = diag( − , ..., − , , ..,
1) with 2 k entries of −
1. Then there are two groups of Majorana fermions:one of size 4 k + ( N − k ) − k ( N − k ). The orientation of the action of T onthe second group is the opposite of the orientation of the action on the first group. The anomalyis therefore ν = (cid:0) k + ( N − k ) − − k ( N − k ) (cid:1) mod 16 = ( N −
1) mod 16 , (2 . N is even. We see that our prescription for computing ν in theultraviolet is unambiguous. For a related discussion see [41,58]. The sign of the contribution of the Goldstino can be understood from the action of time-reversal on the supercurrent
T r ( F λ ), which interpolates to the Goldstino in the deep infrared as G α ∼ T r ( F λ ). This shows that the correct sign in the infrared is +1. The same argument holdsfor the other gauge groups we discuss later. k ≥ h/ k < h/ Fig. 11:
The phase diagram for SO ( N ) k gauge theory with an adjoint fermionfor k ≥ ( N − /
2. The physics at the supersymmetric point is smooth. For k = ( N − / Fig. 12:
The phase diagram for Sp ( N ) k gauge theory with an adjoint fermionfor k ≥ ( N + 1) /
2. The physics at the supersymmetric point is smooth. For k = ( N + 1) / k ≥ h/ m λ = 0 the infrared description can be reliably obtainedat large k by integrating out λ and yields the Chern-Simons theory G k IR based on the23auge group G with level k IR = k − h/
2. This infrared description has been argued in [14]to be applicable beyond the large k regime all the way down to k = h/
2. This leads to arather simple phase diagram. There are two asymptotic phases described by G k − h/ and G k + h/ separated by a phase transition. The supersymmetric point is inside the phasedescribed by G k − h/ . The physics at the supersymmetric point is completely regular andthe phase transition occurs to the right of the supersymmetric point in the phase diagram.For k = h/ G which meets the nontrivial phase G h ata phase transition. The phase diagrams for SO and Sp for k ≥ h/ k < h/ k = h/ − SU ( N ) N − in fig. 8. SO ( N ) N − − has a single phase transition separating SO ( N ) − (which is trivial) and SO ( N ) N − . And Sp ( N ) N +12 − has a single phase transition sep-arating Sp ( N ) − and Sp ( N ) N . Supersymmetry is spontaneously broken in this case [14]with a massless Goldstino point in the G − phase. This is depicted in fig. 13 and fig. 14.We will return to this special case below.As in our discussions above, this simple phase diagram needs to be modified for k < h − k < h/ − SO ( N ) gauge theory with Chern-Simons level k < h/ λ . The infrared Chern-Simons theory at large positive mass is SO ( N ) k + N − ←→ SO (cid:0) N − + k (cid:1) − N . As the mass is decreased we cross a phase tran-sition and encounter an intermediate phase described by a distinct Chern-Simons theory SO (cid:18) N −
22 + k (cid:19) − N − + k ←→ SO (cid:18) N − − k (cid:19) N − + k . (3 . SO ( N ) k − N − ←→ SO (cid:0) N − − k (cid:1) N . The phase diagram is summarized infig. 15. As we said above, our discussion will not apply to the special case with gauge group SO (4),where the low energy theory includes two Goldstinos, one from each of the SU (2) sectors of thetheory. ig. 13: The phase diagram of SO ( N ) k gauge theory with an adjoint fermion andChern-Simons level k = N −
2. There is one transition that connects a nontrivialTQFT and a trivial one. In addition at one point the Goldstino becomes massless.We propose a fermionic dual for the nontrivial transition.
Fig. 14:
The phase diagram of Sp ( N ) k gauge theory with an adjoint fermion andChern-Simons level k = N − . There is one transition that connects two nontrivialTQFTs. In addition at one point the Goldstino becomes massless. We propose afermionic dual for the nontrivial transition. ig. 15: The phase diagram for SO ( N ) k gauge theory with an adjoint fermionfor k < N/ −
2. There are two phase transitions between different infrared TQFTs.There is also a massless Goldstino at the supersymmetric point. We propose a dualfermionic description of the two transitions.
The transition between the left and intermediate phase can be reproduced by an SO (cid:0) N − − k (cid:1) gauge theory with a Chern-Simons term at level N + k − and a Majoranafermion ˆ S in the symmetric-traceless representation of the gauge group. Giving a massto ˆ S allows us to integrate it out and obtain the two infrared TQFTs neighbouring the26ransition. Likewise, the transition between the right and intermediate phase can be reproducedby an SO (cid:0) N − + k (cid:1) gauge theory with a Chern-Simons term at level − N + k + and afermion ˜ S in the symmetric-traceless representation of the gauge group.This suggests the fermion-fermion dualities: SO ( N ) k + adjoint λ ←→ SO (cid:18) N −
22 + k (cid:19) − N + k + + symmetric ˜ SSO ( N ) k + adjoint λ ←→ SO (cid:18) N − − k (cid:19) N + k − + symmetric ˆ S . (3 . S and ˆ S are symmetric-traceless representations. These dualities hold for k < N − .As above, these two dualities are related by “analytic continuation” to negative k combinedwith orientation reversal.Further consistency checks involving the counter-terms for the background gauge fieldsare discussed in [48,58]. This allows a determination of the low energy TQFT in othertheories, e.g. Spin ( N ) k with an adjoint fermion.A similar picture emerges for Sp ( N ) k with an adjoint fermion λ . There is a tran-sition between the large positive mass Sp ( N ) k + N +12 ←→ Sp (cid:0) N +12 + k (cid:1) − N phase and anintermediate phase described by Sp (cid:18) N + 12 + k (cid:19) − N +12 + k ←→ Sp (cid:18) N + 12 − k (cid:19) N +12 + k . (3 . Integrating out a massive Majorana fermion ψ in a representation R of a gauge group G shifts the level of the Chern-Simons term by k → k + sgn( M ψ ) T ( R )2 , (3 . T ( R ) is the index of G in the representation R (see table 4). SO ( N ) Sp ( N )antisymmetric N − N − N + 2 N + 1 Table 4 : T ( R ) for the symmetric and antisymmetric representations. Sp ( N ) k − N +12 ←→ Sp (cid:0) N +12 − k (cid:1) N . The phase diagram is summarized in fig. 16. massless Goldstinoino Fig. 16:
The phase diagram for Sp ( N ) k gauge theory with an adjoint fermionfor k < ( N − /
2. There are two phase transitions between different infraredTQFTs. There is also a massless Goldstino at the supersymmetric point. Wepropose a dual fermionic description of the two transitions.
The transition between the left phase and the intermediate phase can be reproducedby an Sp (cid:0) N +12 − k (cid:1) gauge theory with a Chern-Simons term at level N + k + and afermion ˆ A in the antisymmetric-traceless representation of the gauge group. Likewise, thetransition between the right phase and the intermediate phase can be reproduced by an Sp (cid:0) N +12 + k (cid:1) gauge theory with a Chern-Simons term at level − N + k − and a fermion˜ A in the antisymmetric representation of the gauge group. Using the formula (3.2) for theshift of the Chern-Simons level induced by integrating out a massive fermion we reproducethe asymptotic infrared Chern-Simons theories (see table 4).28his suggests the fermion-fermion dualities: Sp ( N ) k + adjoint λ ←→ Sp (cid:18) N + 12 + k (cid:19) − N + k − + antisymmetric ˜ ASp ( N ) k + adjoint λ ←→ Sp (cid:18) N + 12 − k (cid:19) N + k + + antisymmetric ˆ A . (3 . A and ˆ A are antisymmetric-traceless representations. These dualities hold for k < N +12 . Again, these two dualities are related.We have already mentioned the case k = h/ − k . Herethe infrared TQFTs that describe the left and intermediate phase become identical. Thisimplies that for k = h/ − G h − and another TQFT that governs the restof the phase diagram. For SO ( N ) h/ − adjoint QCD both the left and the intermediateTQFTs become trivial spin-TQFTs (see fig. 13), since SO ( n ) is a trivial spin-TQFT.Therefore, for SO ( N ) h/ − adjoint QCD the infrared theory at the supersymmetric pointis just a massless Goldstino, without an extra topological sector. The transition admits adual description given in the first line of (3.3).For the Sp ( N ) gauge theory both the left and the intermediate TQFTs become Sp (1) N ←→ Sp ( N ) − (see fig. 14). Therefore, for Sp ( N ) h/ − adjoint QCD the infraredtheory at the supersymmetric point is a massless Goldstino with the Chern-Simons theory Sp ( N ) − . The transition admits a dual description given in the first line of (3.5). T -reversal symmetry The adjoint QCD theories with gauge group SO ( N ) k and Sp ( N ) k with an adjointfermion are time-reversal invariant when k = 0 and m λ = 0. This is the supersymmetricpoint for k = 0. This is possible for G = SO ( N ) only for even N and for G = Sp ( N )only for odd N (see table 2). In our scenario the infrared theory consists of a Gold-stino, which is time-reversal invariant, and a nontrivial time-reversal invariant TQFT: SO (cid:0) N − (cid:1) N − and Sp (cid:0) N +12 (cid:1) N +12 respectively. These TQFTs are time-reversal invariant byvirtue of level/rank duality among spin-TQFTs [4]: SO (cid:0) N − (cid:1) N − ←→ SO (cid:0) N − (cid:1) − N − and Sp (cid:0) N +12 (cid:1) N +12 ←→ Sp (cid:0) N +12 (cid:1) − N +12 . The fact that we find a time-reversal invarianttheory in the infrared is a nontrivial consistency check of our proposal.29e would like to analyze the T -reversal ’t Hooft anomaly in this theory. We startwith the Sp ( N ) adjoint QCD following the discussion of SU ( N ) in section 2. In theultraviolet the adjoint fermions contribute ν UV = N (2 N + 1) mod 16 = ( N + 2) mod 16,where we used the fact that N is odd. (As in footnote 14, it is possible to verify that thisprescription for computing ν is gauge invariant.) In the infrared the Goldstino contributes+1 (it is +1 rather than -1 for the same reason as in section 2). The contribution ofthe TQFT Sp (cid:0) N +12 (cid:1) N +12 to ν IR follows from ν ( Sp ( n ) n ) = ± n mod 16 [47], generalizing ν ( SU (2) ) = ± ν ( Sp (2) ) = ± ν IR = ( N + 2) mod 16, as in the ultraviolet. Thismatching is a highly nontrivial test of our phase diagram.Next, we move to the T -reversal ’t Hooft anomaly in the SO ( N ) adjoint QCD the-ory. It exists only for even N . Here we use ν ( SO ( n ) n ) = ± n mod 16 [47], generalizing ν ( SO (2) ) = ± ν ( SO (3) ) = ± N = 2 mod 4 we have ν UV = N ( N − mod 16 = (cid:0) − N + 2 (cid:1) mod 16, whichmatches the infrared contribution with the sign choice − N − from the TQFT and +1(as above) from the Goldstino.The situation for N = 0 mod 4 is more subtle. Here we claim that the naive time-reversal symmetry T of the ultraviolet theory is not mapped to the naive time-reversalsymmetry of the infrared theory. Instead, the relevant symmetry in the ultraviolet, whoseanomaly we match is CT [58]; i.e. the product of the naive time-reversal symmetry andcharge conjugation C . The latter acts on the fermions by reversing the sign of the fermions λ [1 ,i ] → − λ [1 ,i ] and not changing the sign of the other fermions. This leads to ν UV = (cid:16) ( N − N − − ( N − (cid:17) mod 16 = (cid:0) − N + 2 (cid:1) mod 16. This matches ν IR , which is thesum of +1 from the Goldstino and − (cid:0) N − (cid:1) from the TQFT. Again, this is a nontrivialtest of our proposal. (One can verify, as in footnote 14, that for N = 2 mod 4 the anomalyof T is gauge invariant and for N = 0 mod 4 the anomaly of CT is gauge invariant. For N = 2 mod 4 the anomaly of CT is not gauge invariant and for N = 0 mod 4 the anomalyof T is not gauge invariant. For a discussion of the implications of that see [41] and [58].) The isomorphism of SO and Sp Lie groups for low rank with other Lie groups can leadto further consistency checks of our proposal for the infrared dynamics of these theories.The asymptotic phases of theories related by a Lie group isomorphism are guaranteed to30atch, since these can be obtained semiclassically. Thus, a further nontrivial check of thephase diagram can be made only when there is an intermediate phase, i.e. for k < h/ − Sp ( N ) gauge theories are applicable for all N . Here we couldlook for tests based on Sp (1) ≃ SU (2) and Sp (2) / Z ≃ SO (5). However, in these casesthere is no intermediate phase and therefore there is no non-trivial test.Our phase diagrams for SO ( N ) gauge theories are applicable for all N = 1 , N = 1 there is no gauge group and for N = 2 the adjoint fermion is free. For N = 4the adjoint representation is reducible and up to a Z quotient the theory factorizes to twocopies of SU (2) with an adjoint.The group isomorphism SO (6) ≃ SU (4) / Z leads to a nontrivial consistency checkonly for k = 0, where it has an intermediate phase. We first note that even though the Z one-form global symmetry of SU (4) k has an ’t Hooft anomaly for k = 0 mod 4, its Z subgroup is anomaly free since the spin of that line is half-integer. This implies that thequotient Chern-Simons theory SU (4) k / Z can be defined for any k (on a spin manifold).The intermediate phase of the SU (4) gauge theory is described by the TQFT (see fig. 10) U (2) , = SU (2) × U (1) Z , (3 . SO (6) is SO (2) = U (1) (see fig. 15). The group isomorphism SO (6) ≃ SU (4) / Z requires gauginga Z subgroup of the Z one-form global symmetry of U (2) , . This leads to (cid:18) SU (2) × U (1) Z (cid:19) / Z ≃ SU (2) Z × U (1) Z ←→ U (1) , (3 . SO (3) is a trivial spin-Chern-Simons theoryand the second factor is U (1) . This matches the intermediate region of the SO (6) gaugetheory.
4. Phase Diagrams and Dualities for SO ( N ) with Fermions in the SymmetricTensor and Sp ( N ) with Fermions in the Antisymmetric Tensor In this section we analyze the phase diagram of SO ( N ) k with fermions in thesymmetric-traceless tensor representation and of Sp ( N ) k with fermions in the antisymmetric-traceless tensor representation. We denote a symmetric-traceless fermion of SO by S We recall that the adjoint of
SO/Sp is the rank-two antisymmetric/symmetric representation. A an antisymmetric-traceless fermion of Sp . These theories have already appearedin our proposed dual description of the transitions of SO/Sp adjoint QCD for k < h/ The analysis of this section provides further consistency checks on thepreviously discussed dualities and gives rise to new ones.The phase diagrams for these theories resemble those of adjoint QCD. For large posi-tive and negative mass and any value of k there are the two semiclassical phases describedby Chern-Simons theory G k + T ( R ) / and G k − T ( R ) / respectively.We propose that for k ≥ T ( R ) / G k + T ( R ) / and G k − T ( R ) / connected via a transition. Intuitively, it is at the point where the ultravioletfermion becomes massless. While this statement can be reliably established for k ≫ k = T ( R ) /
2, wherethe theory is strongly coupled (for k = T ( R ) / k < h/
2. Our dualities require thatthe dual theories in (1.11) and (1.12) have only two phases when k < h/
2. (Again, k hereis that of the gauge theory with adjoint fermions.) And indeed it is simple to verify thatthis is the case if SO ( N ) k with a fermion in a symmetric-traceless tensor S and Sp ( N ) k with a fermion in an antisymmetric-traceless tensor A have two phases for k ≥ T ( R ) / k < T ( R ) /
2. In SO ( N ) k with a fermion S with k < T ( R ) / − N/ This new phase is described by the nontrivial intermediate TQFT SO (cid:18) N + 22 + k (cid:19) − N +22 + k ←→ SO (cid:18) N + 22 − k (cid:19) N +22 + k . (4 . Note that k in this expression and in the previous sections is the level of the theory withadjoint fermions. In most of this section we label by k the level of the fermionic dual of thattheory. We hope that this change in notation will not cause confusion. For N = 2 the theory is U (1) with a fermion of charge two. Here the Z magneticsymmetry is enhanced to a global U (1) symmetry and the intermediate phase shrinks to a point– the infrared behavior of the massless theory includes a free Dirac fermion and a U (1) TQFT[58]. What follows holds for N = 2 if we add to the Lagrangian a charge-two monopole operator.This breaks the U (1) global symmetry down to Z , which coincides with the magnetic symmetryfor all other values of N . ig. 17: The phase diagram of SO ( N ) k gauge theory with a fermion in asymmetric-traceless tensor for k < N/
2. There are two phase transitions betweendifferent infrared TQFTs. We propose a dual fermionic description of the twotransitions.
The transitions from the semiclassical phases to this quantum phase admit dual fermionicdescriptions SO ( N ) k + symmetric S ←→ SO (cid:18) N + 22 + k (cid:19) − N + k − + adjoint ˜ λSO ( N ) k + symmetric S ←→ SO (cid:18) N + 22 − k (cid:19) N + k + + adjoint ˆ λ . (4 . k = T ( R ) / − N/ k the theories on the right hand side of (4.2) are in the regime where they have only onetransition according to the previous section. Hence, these dualities make sense.)In Sp ( N ) k with a fermion A and k < T ( R ) / N − / ig. 18: The phase diagram of SO ( N ) k gauge theory with a fermion in asymmetric-traceless tensor for k = N . There is one transition that connects anontrivial TQFT and a trivial one. We propose a fermionic dual for the nontrivialtransition. intermediate TQFT Sp (cid:18) N −
12 + k (cid:19) − N − + k ←→ Sp (cid:18) N − − k (cid:19) N − + k . (4 . Sp ( N ) k for k < ( N − / Sp ( N ) k + antisymmetric A ←→ Sp (cid:18) N −
12 + k (cid:19) − N + k + + adjoint ˜ λSp ( N ) k + antisymmetric A ←→ Sp (cid:18) N − − k (cid:19) N + k − + adjoint ˆ λ . (4 . k = 0 at the point where the fermion is masslessare time-reversal invariant, and the infrared TQFTs for k = 0 are SO (cid:0) N +22 (cid:1) N +22 and Sp (cid:0) N − (cid:1) N − , which are indeed time-reversal invariant.We would like to analyze now the time-reversal ’t Hooft anomaly in these theories.We start with Sp ( N ) with an antisymmetric-traceless fermion A . This theory exists only34 ig. 19: The phase diagram of Sp ( N ) k gauge theory with a fermion in anantisymmetric-traceless tensor for k < N − . There are two phase transitions be-tween different infrared TQFTs. We propose a dual fermionic description of thetwo transitions. for odd N . In the ultraviolet the fermions contribute ν UV = ( N (2 N − −
1) mod 16 =( − N + 1) mod 16. In the infrared we have the TQFT Sp (cid:0) N − (cid:1) N − . Using ν ( Sp ( n ) n ) = ± n mod 16 we find, with an appropriate choice of sign, that the anomaly of the infraredtheory is ν IR = ( − N + 1) mod 16, as in the ultraviolet.Next, we move to the time-reversal ’t Hooft anomaly in SO ( N ) with a symmetric-traceless fermion S , which exists only for even N . For N = 2 mod 4, the anomaly in theultraviolet is ν UV = (cid:16) N ( N +1)2 − (cid:17) mod 16 = (cid:0) N + 1 (cid:1) mod 16. In the infrared we havethe TQFT SO (cid:0) N +22 (cid:1) N +22 . Using ν ( SO ( n ) n ) = ± n mod 16 we find, with an appropriatechoice of sign, that the anomaly of the infrared theory is ν IR = (cid:0) N + 1 (cid:1) mod 16, as in theultraviolet.As already noted in discussing adjoint QCD, the symmetry, whose anomalies shouldbe matched for SO ( N ) with N = 0 mod 4 is CT rather than T [58]. (In particular, in allthe cases the symmetry that we discuss has an unambiguous, gauge invariant, anomaly.)For a symmetric-traceless fermion S the CT ’t Hooft anomaly in the ultraviolet is ν UV = (cid:16) N ( N − − ( N − (cid:17) mod 16 = (cid:0) N +22 (cid:1) mod 16. Using ν ( SO ( n ) n ) = ± n mod 16 and35hat the infrared TQFT is SO (cid:0) N +22 (cid:1) N +22 we find, with an appropriate choice of sign, thatthe anomaly of the infrared theory is ν IR = (cid:0) N +22 (cid:1) mod 16, as in the ultraviolet.It is noteworthy that unlike the theories with an adjoint fermion, here the anomaliesmatch without an added massless Majorana fermion. This is satisfying because the theoriesdiscussed in this section are not supersymmetric.We consider the matching of the T and CT ’t Hooft anomalies in these SO and Sp theories as nontrivial checks of our scenarios. Acknowledgments:
We would like to thank M. Barkeshli, F. Benini, M. Cheng, C. Cordova, D. Gaiotto,P.-S. Hsin, A. Kapustin, M. Metlitski, S. Todadri, C. Vafa, and E. Witten for usefuldiscussions. The work of NS was supported in part by DOE grant DE-SC0009988. J.G.would like to thank the Simons Center for warm hospitality. This research was supportedin part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute issupported by the Government of Canada through Industry Canada and by the Province ofOntario through the Ministry of Research and Innovation. J.G. also acknowledges furthersupport from an NSERC Discovery Grant and from an ERA grant by the Province ofOntario. Z.K. is supported in part by an Israel Science Foundation center for excellencegrant and by the I-CORE program of the Planning and Budgeting Committee and theIsrael Science Foundation (grant number 1937/12). 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