Phases Of Adjoint QCD$_3$ And Dualities

We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in the adjoint representation. We apply general considerations of symmetries, anomalies, and renormalization group flows to determine the possible phases of the theory as a function of the gauge group, the Chern-Simons level $k$, and the fermion mass. We propose an inherently quantum mechanical phase of adjoint QCD with small enough $k$, where the infrared is described by a certain Topological Quantum Field Theory (TQFT). For a special choice of the mass, the theory has ${\cal N}=1$ supersymmetry. There this TQFT is accompanied by a massless Majorana fermion - a Goldstino signaling spontaneous supersymmetry breaking. Our analysis leads us to conjecture a number of new infrared fermion-fermion dualities involving $SU$, $SO$, and $Sp$ gauge theories. It also leads us to suggest a phase diagram of $SO/Sp$ gauge theories with a fermion in the traceless symmetric/antisymmetric tensor representation of the gauge group.


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 gauge group 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 structure of phase diagrams that suggest new fermion-fermion dualities and include phases that are inherently 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][2][3][4][5]. 1 The Lagrangian of adjoint QCD contains a Chern-Simons term with coefficient k bare which must be properly quantized. Since the theory has 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 with h the dual Coxeter number of G, e.g. 2 If h/2 is a half-integer so is k and if h/2 is an integer so is k. In other words, we must take k = h 2 mod 1, which in our cases implies The virtue of labeling the theory by k is that time-reversal (or parity) acts by k → −k Adjoint QCD has two semiclassically accessible phases for all G and k. When the mass of the fermion is much larger than the scale set by the Yang-Mills coupling (i.e. |M λ | ≫ g 2 ) the fermion can be integrated out at one-loop. This shifts the coefficient of the Chern-Simons level [10,11,12] k → k + sgn(M λ ) h 2 . (1.2) At low energies the Yang-Mills term becomes irrelevant and the infrared description is pure Chern-Simons TQFT with gauge group G and level k + h/2 for large positive mass and 2 Note that the dual Coxeter number of SO(3) is one, while that of SU (2) is two. Correspondingly, our notation is SO(3) k = SU (2) 2k /Z 2 . More generally, we label the TQFT by the corresponding Chern-Simons gauge group and its level. A quotient as in this expression is interpreted from the 2d RCFT as an extension of the chiral algebra [6] and from the 3d Chern-Simons theory as a quotient of the gauge group [7]. More abstractly, it can be interpreted as gauging a one-form global symmetry of the TQFT [8,9]. This quotient is referred to in the condensed matter literature as "anyon condensation." level k − h/2 for large negative mass. We denote these Chern-Simons theories by G k+h/2 and G k−h/2 respectively. Our goal is to fill in the rest of the phase diagram.
For a special value of the bare mass In the quantum theory these masses are corrected and the theory may end up being massless even if the bare Lagrangian has a mass term.
Let us consider the infrared dynamics of the supersymmetric theory; i.e. set m λ = 0.
For large k all the propagating degrees of freedom have mass of order kg 2 ≫ g 2 and the semiclassical analysis is again reliable. First, we integrate out the gauginos thus shifting the Chern-Simons coefficient [13,14] k IR = k − h 2 . (1.5) Second, the gauge fields are heavy and can be integrated out (except for some global modes) leading at low energies to a topological G k− h 2 theory. Witten argued that the large k infrared description G k− h 2 remains valid in the entire domain of k where supersymmetry is unbroken [14], that is for (1.6) Thus, the low energy description at the supersymmetric point is given by Chern-Simons theory. This infrared theory coincides with the asymptotic phase at large negative mass. 4 3 N = 1 supersymmetry in 2+1 dimensions entails 2 real supercharges. 4 Note that for k = h 2 , k IR = 0 and hence the infrared theory is rather simple. For simply connected gauge groups it is completely trivial. But for non-simply connected gauge groups, like SO(N ) 0 , it has a zero-form magnetic symmetry, which can be spontaneously broken, leading to several vacua in the infinite volume system. This suggests a natural scenario for the phase diagram of adjoint QCD for k ≥ h 2 depicted in fig. 1. The system has two asymptotic phases with topological order G k± h 2 .
The supersymmetric point m λ = 0 is in the G k− h 2 phase. The two phases are separated by a transition at some positive value of m λ . At that point we can think of the fermion as being effectively massless. We do not actually know whether this transition is first order or 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 Ginvariant local operators are only weakly corrected compared to the free model and the transition is second order. Therefore, it is natural to expect that for (1.6) the transition between the two topological theories G k+h/2 and G k−h/2 is second order. The phase diagram of G k with an adjoint fermion for k ≥ h 2 . Note that the physics at the supersymmetric point is smooth -there is no transition there. For k = h 2 the negative mass phase is trivial and there is a transition at some positive value of m λ . Here, and in all our later figures, the vertical line denotes renormalization group flow from the UV at the top of the diagram to the IR at the bottom of the diagram. The solid diagonal line represents a flow to an IR fixed point. The physics across this line is not smooth. The theory is supersymmetric along the dotted vertical line. Unlike the solid line, here the physics is smooth as this line is crossed.
Let us now turn to the interesting region k < h 2 . We know from our discussion above that there are two asymptotic regions of large mass described by the TQFTs G k±h/2 . The infrared dynamics for small mass m λ is strongly coupled and requires a more sophisticated analysis. We will see that the physics at small m λ can lead to novelties in comparison to the phase diagram for k ≥ h/2.
At the supersymmetric point m λ = 0 supersymmetry is expected to be spontaneously broken for k < h/2 [14]. This implies that at m λ = 0 there is a massless Majorana fermion (i.e. the Goldstino particle). An interesting question is whether the infrared theory contains a TQFT in addition to the Majorana Goldstino particle. We will see that general considerations including symmetries and 't Hooft anomaly matching imply that this is indeed the case. 5 We will also identify the TQFT accompanying the Goldstino.
A first guess for the phase diagram of adjoint QCD for k < h/2 is that there are two phases, as in fig. 1, except that at the point m λ = 0 there is also a massless Goldstino reflecting the fact that supersymmetry is spontaneously broken. In fact, we argue below that for G = SU (N ), SO(N ), and Sp(N ) this is the correct scenario, but only for a special value of k = h 2 − 1. 6 This is depicted in fig. 2. 7 Note that the low energy theory around the point of the massless Goldstino includes a decoupled TQFT G −1 . There is a phase transition to the right of the supersymmetric point m λ = 0, which we denote by a bullet point. We will not be able to determine whether this transition is first or second order. We find that this phase transition admits an alternative description in terms of another gauge theory coupled to a fermion in a representation of the gauge group that depends on the choice of G. This suggests new fermion-fermion dualities in nonsupersymmetric theories (see below). For other recent nonsupersymmetric dualities see [15][16][17][18]2,3,[19][20][21][22][23]4,24,25].
We are going to argue that the scenario in fig. 2 cannot be right for all lower values of k, i.e. all k < h 2 − 1. To see that, consider the special case of k = 0. We know that for large |m λ | we have the two asymptotic topological phases G ± h 2 exchanged by the action of time-reversal. (Time-reversal changes the sign of M λ and indeed exchanges the two asymptotic phases.) Also, since supersymmetry is spontaneously broken, we expect a massless Goldstino at the supersymmetric point M λ = m λ = 0. If the system has only the two topological phases G ±h/2 , 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 theory must be time-reversal invariant. This would be a consistent scenario if the two topological phases are level/rank dual to each other, With the exception of adjoint QCD with gauge group SO(N ) and a Chern-Simons term at level k = h/2 − 1 where the infrared description is just a Goldstino. 6 In our discussion below we will mostly exclude the case SO( where supersymmetry is broken with two massless Goldstinos, one for each SU (2) factor. 7 We have not analyzed whether this picture is valid also for other gauge groups. And other than the massless Goldstino there, the TQFT does not change because of the duality SU (2) −1 ↔ SU (2) 1 (as spin-TQFT). Below we will discuss another dual theory that flows to the same transition. It is denoted in the figure as "Dual Theory." 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 dual theory of the transition point.
is time-reversal invariant. 9 We could then have a massless Goldstino at m λ = 0 together with the decoupled TQFT G h 2 . However, this condition that G h 2 is dual to G − h 2 is not obeyed for generic G and thus invalidates this simplistic scenario. Another possibility is that the theory at that supersymmetric point spontaneously breaks its time-reversal symmetry such that the transition at m λ = 0 is first order. This possibility can be ruled out using the argument of [26]. Alternatively, the transition at that point could be second order 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 the ultraviolet adjoint QCD theory. We cannot exclude this possibility. But we will argue for another, more likely, and simpler option.
Let us now present our scenario for k < h 2 − 1: a new intermediate quantum mechanical phase opens up between the two asymptotic phases. The details of the scenario are summarized in fig. 3. This picture is motivated by the analysis of QCD 3 in [5], where it was suggested that for small fundamental quark masses the system can have a new purely 9 This is true for G = SU (2) where SU (2) 1 ←→ SU (2) −1 , but this case has k = h 2 − 1 = 0 and here we discuss k < h 2 − 1. See a comment in fig. 2. The proposed phase diagram for k < h 2 − 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 of this middle phase. At that point there is a massless Goldstino. Even when the middle phase is gapped it includes some TQFT, which we will determine. We will also present two conjectured dual gauge theories, denoted by "Dual Theory A" and "Dual Theory B" that describe the two transition points. Again, the general structure of this diagram is as explained in the caption of fig. 1, except that here we have two transition points and two dual theories. quantum phase, which cannot be understood semiclassically. It is also motivated by studying the effective field theory on domain walls and interfaces along the lines of [27][28][29][30], as well as by the holographic realization [31,32] of adjoint QCD for m λ = 0 from which the intermediate infrared TQFT can be extracted by analyzing the effective low energy theory on branes wrapping a noncontractible cycle threaded by flux. A detailed analysis of these domain walls and interfaces will appear in [30].
General considerations imply that the TQFT in the intermediate, new phase cannot be trivial. Adjoint QCD has a one-form global symmetry (see table 3) acting on its line operators [8,9]. This symmetry cannot act trivially in the infrared (i.e. it cannot be that all the 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 clearly does not match the one-form global symmetry of adjoint QCD, this rules out the scenario with just a Goldstino. This scenario can also be ruled out in the T -reversal-invariant cases k = 0. In these cases adjoint QCD has a T -reversal anomaly, characterized by an integer modulo 16, which is often denoted by ν [33][34][35][36][37][38][39][40][41]. (For early work on mixed T -gravitational anomalies, see e.g. [42].) The massless Goldstino does not saturate that anomaly. 10 Below we will discuss how the proposed TQFT in the infrared does saturate that anomaly. We suggest that adjoint QCD for k < h (1.9) Note that the intermediate TQFTs for SU (N ) adjoint QCD coincide with the domain wall theories of 3+1 dimensional N = 1 SU (N ) super-Yang-Mills in [27]. This will be important in [30]. 10 We thank E. Witten for raising this point. 11 We follow the standard notation (1.7) U (M ) P,Q is described by the Chern-Simons Lagrangian with 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−P M is odd (see e.g. [1,3]).
The phase diagram has two phase transitions connecting the intermediate phase to the asymptotic phases. We find that the phase transitions can be described by a different gauge theory with a fermion in a representation of the gauge group that depends on the choice of G. We denote these dual theories in fig. 3 by "Dual Theory A" and "Dual Theory B." Specifically, we propose the new fermion-fermion dualities: 12 SU (N ) and k < N 2 : (1.10) trivialize when k reaches the upper limit, i.e. when k = h/2 − 1. It should also be pointed out that the dualities in the second line of (1.10)(1.11)(1.12) can be viewed as "analytic 12 As in other familiar examples, including most recently [5], the notion of this duality is valid only near the transition point. We say that G k + adjoint λ is dual to Theory A around one transition and it is dual to Theory B around the other transition, but we do not say that Theory A is dual to Theory B.
continuation" to negative k combined with orientation reversal of the dualities in the first line.
A way to understand fig. 3 is as follows. We start with large and negative m λ and . Then we use level/rank duality to express this theory as G A k A with some gauge group G A and some level k A . The Dual Theory A has gauge group G A with some level and some matter fields. Specifically, this theory can be found in (1.10)(1.11)(1.12). Then we move to the right in fig shows that many of these tests are automatically satisfied in our proposal. We have just mentioned that our scenario was designed such that the two descriptions of the intermediate . Therefore, throughout our phase diagram we used either level/rank duality or a phase transition in a weakly coupled theory. This guarantees that all the symmetries that are preserved through this process and their 't Hooft anomalies must match. These include all the zero-form and the one-form global symmetries [8,9].
A notable exception to this statement is a symmetry that is violated in our process of changing m λ . Consider the special theories with k = 0. For m λ = 0 they are time-reversal invariant. This symmetry is not present as we vary the mass and get to the intermediate phase and therefore there is no guarantee that it is present there. As a nontrivial check, where we used results in [3,4,43].
These T -reversal-invariant cases also lead to additional consistency checks. As we mentioned above, it is known that T -reversal-invariant 2 + 1 dimensional spin theories are subject to an anomaly, which is characterized by an integer ν modulo 16. This integer must match between adjoint QCD and our proposed infrared theory, which includes the Goldstino and the TQFTs (1.13). In order to do that we will need the value of ν for these TQFTs [44][45][46][47] (special cases had been found earlier) (1.14) So far we discussed the fermion-fermion dualities (1.11)(1.12) in the range of small k.
It is natural to examine them also for larger k, which after redefining k, are written for (1.16) The duality in the second line of (1.15) trivializes for k = N/2. As above, the dualities in the 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 these two theories. Therefore, we will also study the theories with gauge group SO (Sp) with a fermion in the symmetric (antisymmetric) representation. We will find through our dualities above that the infrared description also contains an intermediate quantum phase, given by the Chern-Simons TQFTs This corroborates our conjectures about the infrared phases of these theories and the dualities.
It is common to couple quantum field theories to various background fields. In our case we can couple them to background gauge fields for the various global symmetries and to the metric. Then, there can be tests of our picture involving the counterterms for these fields. The counterterms for the background gauge fields of the dualities involving SO(N ) gauge theories will be presented in [48]. This will test our phase diagram and will allow us to derive similar results for other gauge groups, e.g. Spin(N ).
Another test of our proposal arises from the gravitational Chern-Simons counterterm 13 . Starting in the ultraviolet theory we derive the two phases at large positive and large negative mass at weak coupling. So the difference between the coefficients of this 13 We thank F. Benini for raising this point.
term in these two phases is easily computable. Then, we can use the change in this coefficient when we use level/rank duality (using expressions in [3] for SU groups and in [4] for SO and Sp groups) and the change when we go through the phase transitions to check that we find the same answer in the middle phase regardless of whether we arrive to it from positive mass or negative mass. When we perform this check we should make sure that in the theories with a massless Goldstino this coefficient changes as we go through that point. Since this computation is straightforward and tedious and since similar computations were done in [49,50], we will not present it in detail here. We will simply state that 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. In

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 two asymptotic infrared phases that describe the domain of large fermion mass (for any value of k) are the TQFTs SU (N ) k+N/2 and SU (N ) k−N/2 . We discuss the infrared dynamics for k ≥ N/2 and k < N/2 in turn.

Phase Diagram for k ≥ N/2
The phase diagram can be understood by combining the TQFTs SU (N ) k±N/2 that can be reliably studied at large mass with the expectations from the supersymmetric theory at m λ = 0. At m λ = 0 the infrared theory is believed to consist simply of SU (N ) k−N/2 all the way down to k = N/2 [14].
A natural guess is that SU (N ) adjoint QCD for k ≥ N/2 has only two phases, described by SU (N ) k±N/2 . The supersymmetric theory with m λ = 0 is in the phase SU (N ) k−N/2 . This is true also for k = N/2, where the supersymmetric theory is in the trivial phase SU (N ) 0 (and hence has a unique ground state). There is a phase transition between these two phases at some nonzero (positive) value of the supersymmetry-breaking mass m λ . The phase diagram is depicted in fig. 4. We emphasize again that the phase transition is not coincident with the supersymmetric point. Note that the physics at the supersymmetric point is smooth. For k = N/2 the negative mass phase is trivial. In that case the transition is still to the right of the supersymmetric point.
An interesting special case is G = SU (2) k with odd k. Here we we can gauge the Z 2 one-form symmetry and arrive at SO(3) k/2 = SU (2) k /Z 2 with a single real fermion in the three-dimensional (adjoint) representation. (The supersymmetric point of this theory played in an important role in [14].) This theory was argued in [23,4,43] to be dual to an We can recover the SU (2) k theory by gauging the magnetic Z 2 global symmetry in this duality. This is done in detail in [48] with the conclusion that the SU (2) k theory coupled to a fermion in the adjoint is dual to O k+1 2 0 −3,−3 coupled to a scalar in the vector representation. Here the superscript and the two subscripts label various topological terms in 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 2 (it was introduced in [51] and denoted there by ±). In this case it is 0, which means that this coupling is absent. And the second subscript is a topological term in the Z 2 sector.
On a spin manifold such terms have a Z 8 classification 14 [53,54] (see also [55,56]) and the subscript −3 means that our term is the third power of the generator of that Z 8 . See [48] for more details. The O k+1 2 0 −3,−3 theory coupled to a scalar in the vector representation has two weakly coupled gapped phases with TQFTs. The TQFT of the phase where the scalar condenses is level/rank dual to an SU (2) k−1 TQFT, while the TQFT in the phase where the scalar is massive is level/rank dual to an SU (2) k+1 TQFT [48].  Two special cases k = 1 and k = 3 are particularly simple. For k = 1 we have O(1) ∼ Z 2 , 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) −3,−3 coupled to a scalar in the 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 this case the superscript + of the general case is superfluous.) This case is depicted in fig. 6.

Phase Diagram for k < N/2
Here we turn to the theory with low k. The TQFTs SU (N ) k±N/2 still describe the asymptotic large mass phases of the diagram. Since at m λ = 0 the theory is believed to break supersymmetry spontaneously [14], this implies the presence of a Majorana Goldstino particle at the supersymmetric point. Therefore the phase diagram of fig. 4 needs to be somewhat modified. We have already argued in the introduction that a possible, well-motivated, modification of the phase diagram consists of introducing another phase transition so that the system has generically three distinct phases. Two phases are visible semi-classically and are described by Chern-Simons TQFTs and the third phase is a new quantum phase. The new phase cannot consist just of a Goldstino as this would not match 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 -reversal symmetry of the k = 0 theory. We will discuss this latter anomaly below. We will suggest that the infrared theory in the new phase contains a TQFT (which we identify below) in addition to the Majorana Goldstino.
Recall that at large positive and negative m λ the theory is described in the infrared by SU (N ) k± N 2 Chern-Simons theory, respectively. Using level/rank duality it is useful to rewrite these TQFTs as [3] The equivalence between the theories in (2.1) is only as spin-TQFTs [3], but we can use it since adjoint QCD requires a choice of spin structure due to the fermion in the adjoint representation of SU (N ).
We suggest that at small m λ the infrared is described by a different Chern-Simons TQFT (see fig. 7) 2) or, equivalently by level/rank duality [3] As we said, at the supersymmetric point m λ = 0, the infrared theory consists of a Goldstino and a TQFT, which we now have identified with (2.2). Let us summarize our proposal. Consider SU (N ) adjoint QCD with k < N/2 − 1.
At large positive mass the infrared theory is a Chern-Simons TQFT SU (N ) k+N/2 ←→ U N 2 + k −N . As we decrease the mass we encounter a transition to the TQFT U N 2 + k − N 2 +k,−N ←→ U N 2 − k N 2 +k,N . As we proceed along the mass axis the Goldstino becomes massless at the supersymmetric point and then massive again. Finally, we encounter the last transition to the SU (N ) k−N/2 ←→ U N 2 − k N,N Chern-Simons TQFT.
Let us consider more carefully the first transition, namely, the transition between This can be nicely reproduced using a dual fermionic theory whereλ is the adjoint of SU N 2 + k -there is no singlet. The fermion is denoted byλ rather than λ in order to distinguish it from the original fermion. Since there is no matter charged under the U (1) factor, the phase diagram of this model is that of SU ( N 2 +k) − 3 4 N+ k 2 with an adjoint fermion,λ. Since in the range k < N/2 we have 2| 3 4 N − k 2 | > N 2 +k, we see, according to our previous analysis (for k ≥ N/2 in the notations of the original theory), that there is one phase transition in this theory and it describes the transition between the two phases we need. Similarly, there is a dual fermionic theory for the transition between We therefore arrive at two new fermion-fermion dualities which are valid for k < N 2 , and describe the two transitions discussed above. As we said above, the second duality follows from the first by "analytic continuation" to negative k combined with orientation reversal.  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 the asymptotic TQFTs. The transition that is dual to (2.6) remains nontrivial, while the dual theory (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 described by a U (1) N TQFT. In other words, the N = 1 theory with m λ = 0 flows to a U (1) N TQFT accompanied by a massless Goldstino.   9: The phase diagram of SU (2) 0 with an adjoint fermion. In this diagram there are no necessary transitions other than the Goldstino becoming massless at one point. This is due to the fact that all the following four TQFTs are dual: The phase diagram therefore is especially simple. It could be summarized by saying that there is a duality between SU (2) 0 with an adjoint fermion and a free Majorana fermion accompanied by a pure U (1) 2 TQFT.
For N = 2, the theory with k = N/2 − 1 = 0 is time-reversal invariant at m λ = 0 and the picture is further simplified since the asymptotic phase at large positive m λ is U (1) −2 (see fig. 9). There is a massless Goldstino at m λ = 0 and the TQFT at m λ = 0 can be chosen to be U (1) 2 or U (1) −2 , which are identical by level/rank duality. Therefore, we see that in SU (2) 0 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 this case one can summarize the situation by the statement that there is a duality with ψ a neutral Majorana fermion. This duality is reminiscent of the supersymmetric duality [57].
The SU (2) 0 adjoint QCD that we have just discussed is also a special member of the T -invariant family of theories SU (N ) 0 adjoint QCD, which exhibit three phases for N > 2 (see fig. 10). We will discuss these T -invariant theories below.

Symmetries and 't Hooft Anomalies Matching
As with all dualities and proposed infrared behavior of strongly coupled theories, one has to check that the symmetries of the dual theories and their anomalies match. As we said in the introduction, most of this is guaranteed to work in our setup. We used a sequence 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 transition described by the weakly coupled theory (2.7), and level/rank duality. This matched with the integration of the fermion at large negative mass. Every step here either involves a computation 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 all the phases we described have the same global symmetries and they have the same 't Hooft anomalies.
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 . Since (recall that N has to be even for k = 0). 15 In general the value of ν IR in our infrared theory is not easy to compute, see for instance [39,59,44,45,60]. One contribution to it, due to the Goldstino, is 1. 16 The contribution of the TQFT U N 2 N 2 ,N was worked out in [44,45,46] (see also references therein) and was found to be ±2. We suggest that in the present context time-reversal in the infrared would act such that it is actually −2(−1) N/2 . Therefore, we find ν IR = 1 − 2(−1) N/2 , as in the ultraviolet!

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 gauge groups. We often denote the gauge group by G when it makes no difference which of the two cases we are discussing. We discuss the long distance behavior of these theories as a function of the mass m λ of the fermion and of k. Our discussion will be along the lines of the 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 where the fermion is very massive. These are described by Chern-Simons theories G k+h/2 and where we used the fact that N is even. We see that our prescription for computing ν in the ultraviolet is unambiguous. For a related discussion see [41,58]. 16 The sign of the contribution of the Goldstino can be understood from the action of timereversal 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 holds for the other gauge groups we discuss later.
describe the asymptotics of the phase diagram. We now proceed to complete the phase diagram for k ≥ h/2 and k < h/2. Fig. 11: The phase diagram for SO(N ) k gauge theory with an adjoint fermion for k ≥ (N − 2)/2. The physics at the supersymmetric point is smooth. For k = (N − 2)/2 the negative mass phase is trivial. In that case the transition is still to the right of the supersymmetric point. Fig. 12: The phase diagram for Sp(N ) k gauge theory with an adjoint fermion for k ≥ (N + 1)/2. The physics at the supersymmetric point is smooth. For k = (N + 1)/2 the negative mass phase is trivial. In that case the transition is still to the right of the supersymmetric point.

Phase Diagram for k ≥ h/2
At the supersymmetric point m λ = 0 the infrared description can be reliably obtained at large k by integrating out λ and yields the Chern-Simons theory G k IR based on the gauge 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 a rather simple phase diagram. There are two asymptotic phases described by G k−h/2 and G k+h/2 separated by a phase transition. The supersymmetric point is inside the phase described by G k−h/2 . The physics at the supersymmetric point is completely regular and the phase transition occurs to the right of the supersymmetric point in the phase diagram.
For k = h/2 one phase is trivial, labeled by G 0 which meets the nontrivial phase G h at a phase transition. The phase diagrams for SO and Sp for k ≥ h/2 are summarized in fig. 11 and fig. 12.

Phase Diagram for k < h/2
The modification of this picture for k = h/2 − 1 is straightforward and is similar to SU (N ) N 2 −1 in fig. 8. SO(N ) N −2 2 −1 has a single phase transition separating SO(N ) −1 (which is trivial) and SO(N ) N−3 . 17 And Sp(N ) N +1 2 −1 has a single phase transition separating Sp(N ) −1 and Sp(N ) N . Supersymmetry is spontaneously broken in this case [14] with a massless Goldstino point in the G −1 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 2 − 1 and we propose that for k < h/2 − 1 there are two phase transitions connecting the two asymptotic phases that are visible semiclassically to an intermediate, inherently quantum mechanical, phase. The supersymmetric theory is in the intermediate phase.
Let us consider an SO(N ) gauge theory with Chern-Simons level k < h/2 and an adjoint fermion λ. The infrared Chern-Simons theory at large positive mass is As the mass is decreased we cross a phase transition and encounter an intermediate phase described by a distinct Chern-Simons theory The supersymmetric point, where there is also a massless Goldstino occurs in this phase.
As we decrease the mass further we encounter another phase transition to the asymp- The phase diagram is summarized in fig. 15. 17 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 the theory. There is one transition that connects a nontrivial TQFT and a trivial one. In addition at one point the Goldstino becomes massless. We propose a fermionic dual for the nontrivial transition. 2 . There is one transition that connects two nontrivial TQFTs. In addition at one point the Goldstino becomes massless. We propose a fermionic dual for the nontrivial transition. Likewise, the transition between the right and intermediate phase can be reproduced by an SO N−2 2 + k gauge theory with a Chern-Simons term at level − 3N 4 + k 2 + 1 2 and a fermionS in the symmetric-traceless representation of the gauge group.
This suggests the fermion-fermion dualities: HereS andŜ are symmetric-traceless representations. These dualities hold for k < N−2 2 . As above, these two dualities are related by "analytic continuation" to negative k combined with orientation reversal.
Further consistency checks involving the counter-terms for the background gauge fields are discussed in [48,58]. This allows a determination of the low energy TQFT in other theories, 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 transition between the large positive mass Sp(N ) k+ N +1 2 ←→ Sp N+1 2 + k −N phase and an intermediate phase described by (3.4) 18 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 where T (R) is the index of G in the representation R (see table 4). Decreasing the mass further leads to another phase transition to the asymptotic theory The phase diagram is summarized in fig. 16.

Consistency Checks for Low-Rank Theories
The isomorphism of SO and Sp Lie groups for low rank with other Lie groups can lead to 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 to match, since these can be obtained semiclassically. Thus, a further nontrivial check of the phase diagram can be made only when there is an intermediate phase, i.e. for k < h/2 − 1.
Our phase diagrams for Sp(N ) gauge theories are applicable for all N . Here we could look for tests based on Sp(1) ≃ SU (2) and Sp(2)/Z 2 ≃ SO (5). However, in these cases there 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, 2 and 4.
For N = 1 there is no gauge group and for N = 2 the adjoint fermion is free. For N = 4 the adjoint representation is reducible and up to a Z 2 quotient the theory factorizes to two copies of SU (2) with an adjoint.
The group isomorphism SO(6) ≃ SU (4)/Z 2 leads to a nontrivial consistency check only for k = 0, where it has an intermediate phase. We first note that even though the Z 4 one-form global symmetry of SU (4) k has an 't Hooft anomaly for k = 0 mod 4, its Z 2 subgroup is anomaly free since the spin of that line is half-integer. This implies that the quotient Chern-Simons theory SU (4) k /Z 2 can be defined for any k (on a spin manifold).
The intermediate phase of the SU (4) 0 gauge theory is described by the TQFT (see fig. 10) where we used the fact that the first factor SO (3)  In this section we analyze the phase diagram of SO(N ) k with fermions in the symmetric-traceless tensor representation and of Sp(N ) k with fermions in the antisymmetrictraceless tensor representation. 19 We denote a symmetric-traceless fermion of SO by S 19 We recall that the adjoint of SO/Sp is the rank-two antisymmetric/symmetric representation. and by A an antisymmetric-traceless fermion of Sp. These theories have already appeared in our proposed dual description of the transitions of SO/Sp adjoint QCD for k < h/2 (see (1.11)(1.12)). 20 The analysis of this section provides further consistency checks on the previously discussed dualities and gives rise to new ones.
The phase diagrams for these theories resemble those of adjoint QCD. For large positive and negative mass and any value of k there are the two semiclassical phases described by Chern-Simons theory G k+T (R)/2 and G k−T (R)/2 respectively.
We propose that for k ≥ T (R)/2 there are two asymptotic phases G k+T (R)/2 and We now consider the phase diagram for k < T (R)/2. In SO(N ) k with a fermion S with k < T (R)/2 − 1 = N/2 there is an intermediate phase, which is not visible semiclassically (see fig. 17). 21 This new phase is described by the nontrivial intermediate TQFT (4.1) 20 Note that k in this expression and in the previous sections is the level of the theory with adjoint fermions. In most of this section we label by k the level of the fermionic dual of that theory. We hope that this change in notation will not cause confusion. 21 For N = 2 the theory is U (1) 0 with a fermion of charge two. Here the Z 2 magnetic symmetry 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) 2 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 2 , which coincides with the magnetic symmetry for all other values of N .  Again, these two dualities are related.
In the theory with k = T (R)/2 − 1 = N/2 the negative mass phase and the intermediate phase are trivial and there is a single nontrivial transition (see fig. 18) with a dual description given by the first line of (4.2). (It is easy to verify that for this range of k the theories on the right hand side of (4.2) are in the regime where they have only one transition according to the previous section. Hence, these dualities make sense.) In Sp(N ) k with a fermion A and k < T (R)/2 = (N − 1)/2 there is an intermediate phase, which is not visible semiclassically. This new phase is described by the nontrivial Fig. 18: The phase diagram of SO(N ) k gauge theory with a fermion in a symmetric-traceless tensor for k = N 2 . There is one transition that connects a nontrivial TQFT and a trivial one. We propose a fermionic dual for the nontrivial transition.

intermediate TQFT
The phase diagram for Sp(N ) k for k < (N − 1)/2 has two transitions, which admit dual fermionic descriptions (see fig. 19). This suggests the fermion/fermion dualities , 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 ) 0 with an antisymmetric-traceless fermion A. This theory exists only . Using ν(SO(n) n ) = ±n mod 16 we find, with an appropriate choice of sign, that the anomaly of the infrared theory is ν IR = N 2 + 1 mod 16, as in the ultraviolet.
As already noted in discussing adjoint QCD, the symmetry, whose anomalies should be matched for SO(N ) 0 with N = 0 mod 4 is CT rather than T [58]. (In particular, in all the cases the symmetry that we discuss has an unambiguous, gauge invariant, anomaly.) It is noteworthy that unlike the theories with an adjoint fermion, here the anomalies match without an added massless Majorana fermion. This is satisfying because the theories discussed 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.