Theta, time reversal and temperature

Journal of High Energy Physics, May 2017

SU(N ) gauge theory is time reversal invariant at θ = 0 and θ = π. We show that at θ = π there is a discrete ’t Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. It may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies.

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Theta, time reversal and temperature

Received: March Theta, time reversal and temperature Davide Gaiotto 0 1 3 5 6 Anton Kapustin 0 1 3 6 Zohar Komargodski 0 1 3 4 6 Nathan Seiberg 0 1 2 3 6 Princeton 0 1 3 6 NJ 0 1 3 6 U.S.A. 0 1 3 6 0 Walter Burke Institute for Theoretical Physics, California Institute of Technology 1 Waterloo , Ontario, N2L 2Y5 , Canada 2 School of Natural Sciences, Institute for Advanced Study 3 Rehovot 76100 , Israel 4 Department of Particle Physics and Astrophysics, Weizmann Institute of Science 5 Perimeter Institute for Theoretical Physics 6 Open Access , c The Authors SU(N ) gauge theory is time reversal invariant at θ = 0 and θ = π. We show that at θ = π there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at θ = π the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at θ = 0 is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at θ = π, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for SU(2) gauge theory. The underlying symmetry at θ = π is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two O(2)-symmetric fixed points. It may also be that the four-dimensional theory around θ = π is gapless, e.g. a Coulomb phase could match the underlying anomalies. Anomalies in Field and String Theories; Confinement; Spontaneous Symme- 1 Introduction An analogous 2d system Outline 2 SU(N ) Yang-Mills theory in four dimensions 2.1 Symmetries 2.2 A CP anomaly for even N 2.3 A CP anomaly for odd N A continuum description of the CP anomaly Softly broken N = 1 SYM 2.6 Boundaries, interfaces, and domain walls 3 SU(2) Yang-Mills theory on Y × S The CP anomaly reduced to three dimensions The high temperature phases The domain wall at high temperatures 3.4 The confinement/deconfinement transition and the 3d Ising model 4 The confinement-deconfinement transition for nonzero θ A mixed gauge theory with dihedral symmetry The vacuum structure of the mixed theory The vacuum structure of SO(3)± 4.4 The CP domain wall 5.1 Domain walls 5 An inequality for the multicritical region 6 A Higgs phase at intermediate temperatures 6.1 O(2) phase diagrams 6.2 A trivial phase with unbroken D8 7 Scenarios with deconfinement at zero temprerature A A modification of the Z2 gauge theory B Another derivation of D8 symmetry C The (absence of ) anomalies for D8 symmetry C.1 An alternative proof D ’t Hooft anomalies in quantum mechanics D.1 A particle on a circle One of the central tools for analyzing strongly coupled systems is ’t Hooft’s anomaly Given a theory with symmetry G, we may try to couple G to classical background gauge fields. It is sometimes impossible to do that in spite of the fact that G is a true symmetry of the theory. ’t Hooft argued that the obstruction to coupling G to classical background gauge fields is preserved under the Renormalization Group flow. The obstruction is usually referred to as an anomaly. This idea has had powerful applications, especially when the associated obstruction and symmetry G are continuous and the theory has a weak coupling limit. This is due to the fact that the anomaly could be computed explicitly (for a review see [2]). Rather successful methods have been developed for computing continuous anomalies even for strongly coupled theories (especially for supersymmetric theories). Recently, there has been a lot of interest in discrete anomalies and their implications. There is indeed a rich landscape of theories that have discrete anomalies. Perhaps the simplest example is that of a free fermion in 2+1 dimensions, which has a parity anomaly [3, 4]. For more sophisticated examples see, for example, [5, 6] and references therein. In addition, discrete anomalies have been recently used extensively in the context of topological insulators and Symmetry Protected Topological phases, in which the anomalies are canceled by coupling to a higher-dimensional theory (this is known as anomaly inflow [7]). For a review of and references to applications in condensed matter systems see [8]. It has been shown recently that discrete anomalies can arise even in the absence of fermionic degrees of freedom [9, 10]. related discussion in two and four dimensions, see e.g. [11–18].) First, we review the fact on nontrivial background fields for its center. This discussion will lead us to a new time involving time reversal and the global center 1-form symmetry. (We can use time reversal and CP interchangeably.) The center symmetry is a 1-form symmetry [19, 20], because the associated conserved charge is associated to a co-dimension 2 surface (which can wrap a loop). Standard 0-form symmetries are of course associated to co-dimension 1 surfaces (which wrap points). We will review the results of [19, 20] that are needed to follow this paper. While time-reversal is a symmetry of the theory, if we couple the center 1-form Z symmetry to classical background gauge fields, time reversal is broken by a c-number. In other words, we have a mixed time-reversal/center symmetry ’t Hooft anomaly. As always with ’t Hooft anomalies, they do not invalidate the symmetry, but they lead to powerful constraints on the phases of the theory. In our case, assuming that the pure phase diagram of the theory at finite temperature. One can ask at which temperature the CP symmetry is restored. The anomaly implies that it has to be restored at a temperature that is not lower than the deconfinement transition. Otherwise, some Wilson loops would not admit an area law. If the restoration of CP is at a temperature strictly higher than the deconfinement transition, then there is a new phase of the theory with 2N vacua (but the space-like Wilson loop is confined). We will analyze in detail the case of SU(2) gauge theory and make some proposals regarding the phase diagram. In this particular case the deconfinement transition is second order and the possible phase diagrams are quite rich. In addition, there is no convincing We investigate these different scenarios in light of the constraints from anomaly matching. We suggest two possibilities that are particular to SU(2) gauge theory. One involves a new gapped phase with one ground state (with all the 0-form symmetries preserved) and a perimeter law for space-like Wilson loops. The phase diagram would contain two critical points with O(2) symmetry. This symmetry is “accidental” or “hidden”, i.e. arises only at long distances. This is like a finite temperature Higgs phase. The second scenario is a natural proposal involves a four-dimensional Coulomb phase. (There could also be an An analogous 2d system The anomaly that we described above and its consequences are reminiscent of the 2d Our discussion of this model will follow the lines of [6]. Naively, the global symmetry gauge invariant operators. Therefore, it is useful to couple the system to background 1This is widely believed to be the case, based on the results of lattice simulations, the analysis of supersymmetric Yang-Mills with softly broken supersymmetry, and holographic models (at large N ). This case of SU(2) gauge theory. 2We thank E. Witten for an extremely useful discussion about this analogy. (U(1) × SU(n))/Zn fields. Here the U(1) field a is dynamical and the PSU(n) fields A are lifted to an SU(n) bundle, the expression (1.2) is more subtle. In this case the dynamical TrFA = where w2(E) is the second Stiefel-Whitney class of the PSU(n) bundle E. (On a closed oriented 2-manifold R w2(E) is an integer modulo n, and on an open manifold its integral is well-defined only if one specifies a trivialization of E on the boundary.) In other words, fields (1.3) is added to the Lagrangian. and moved to the boundary to screen the background electric field. This is the physical is not precise. After the pair creation, the boundary of the system carries a nontrivial SU(n) representation. The whole system is still in a PSU(n) representation. But locally the boundary is not in a representation of the true global symmetry group PSU(n). The nontrivial term (1.3) is a reflection of that phenomenon. And its more subtle definition when the manifold has a boundary is associated with the zi particle there. to the nontrivial SU(n) representation at the boundary. This is similar to the phenomenon occurring in the famous Haldane chain. is naively time-reversal invariant.3 The argument for that uses the fact that under time reversal invariant, but it is not invariant for nontrivial A. This mixed anomaly between time reversal symmetry and the global PSU(n) symmetry leads to ’t Hooft anomaly conditions constraining the long distance behavior — the IR system should reflect the same ’t Hooft anomaly. Indeed, for n > 2 time reversal is spontaAnd for n = 2 the system is gapless. Our discussion of the 4d SU(N ) theory will be quite similar to that. Instead of the global PSU(n) symmetry, we will have a one-form global Z N symmetry. The periodicity of The outline of the paper is as follows. In section 2 we study some general aspects of SU(N ) Yang-Mills theory and derive the time reversal anomaly. We exhibit the consequences of this anomaly explicitly in the softly broken supersymmetric SU(N ) theory. In section 3 we begin our study of SU(2) Yang-Mills theory on a circle, or equivalently at finite temperature. We discuss in detail the symmetries, the consequences of the new anomaly, the high temperature phase, the various domain walls, and the confinement-deconfinement transition. In section 4 we define a mixed gauge theory, which is essentially an orbifold of the original gauge theory on a circle. It has the advantage that the consequences of the anomaly are clearer in the orbifold theory, while no information about the original theory is lost. The mixed gauge theory has a D8 symmetry, and we discuss its various phases. We also describe phases of SO(3) gauge theory at nonzero temperature. In sections 5, 6 we discuss two possible scenarios for the phase diagram of the SU(2) theory, assuming that the theory remains gapped at zero temperature for all theta. In section 5 we describe a scenario where there is a phase with 8 vacua and a completely broken D8 symmetry. We argue that this scenario, appropriately modified, is in fact very natural for SU(N ) Yang-Mills theory at sufficiently large N . In this scenario, a general inequality is derived relating the temperatures at which CP and the center symmetry are restored. In section 6 we describe a situation with a region of unbroken D8 symmetry. We explain the consequences of these two phases for the original SU(2) theory and make some detailed predictions about the phase diagrams. In section 7 we discuss possible phase diagrams assuming a phase transition at zero temperature but logical Z2 gauge theory in two dimensions, provide another derivation of the D8 symmetry of the mixed gauge theory, and verify that this symmetry is free from ’t Hooft anomaly. This is crucial for the consistency of the phase diagram with unbroken D8 symmetry. In the fourth appendix we present pedagogical quantum mechanical examples that exhibit anomalies. These theories will appear on some of the domain walls that we encounter. Let us comment on our notation. When discussing discrete 1-form symmetries and their gauging, it is convenient to adopt a lattice regularization of the gauge theory. We will use a slight modification of the Wilsonian lattice gauge theory where a triangulation is used in place of a hypercubic lattice. The advantage is that discrete gauge fields can be thought of as simplicial cochains, and an action for them can be written in terms of standard operations such as the coboundary operator and the cup product. We will denote by Cp(X, ZN ) the Abelian group of p-cochains with values in ZN , and by Zp(X, ZN ) its subgroup consisting of closed p-cochains (i.e. mod N p-cocycles). The coboundary operator N gauge field is represented on the lattice by a Z N 1-cocycle a ∈ Z1(X, ZN ), a Wilson loop in the charge-1 representation of Z φ is a 2π-periodic scalar. The dictionary between a and A is, roughly, A = 2Nπ a. Thus Lagrange multiplier field b ∈ Cn−2(X, ZN ), n = dim X, so that the action is The corresponding action in the continuum is B ∧ dA, field. The two actions are related by a formal substitution B → 2Nπ b, A → 2Nπ a, ∧ → ∪ SU(N ) Yang-Mills theory in four dimensions We begin by considering SU(N ) Yang-Mills theory on R4. The action is S = −1 F ∧ ⋆F + F ∧ F symmetry, as we said, we use CP and time reversal interchangeably.) Below we will argue 4The similarity transformation is implemented by the unitary operator U = e 4iπ RΣ3 T r(A∧dA+ 32 A3), where spontaneously broken CP symmetry. In the rest of this subsection we review some useful facts about the Z N 1-form global symmetry of our system [19, 20], which we will refer to as the the center symmetry. Given N charge generator. All local operators are neutral under this symmetry. Hence, this γ. If we take the Wilson line in the fundamental representation WF = T rF P ei Rγ A and Let us discuss the other line operators in the theory. We use the notation of [24] and organize line operators into families labeled by pairs (a, b) ∈ Z N . The fundamental Wilson line belongs to the (1, 0) family. More generally, a Wilson line in a tensor represenunder the center symmetry is not well defined since there could be counter-terms when the N symmetry does not suffer from an ’t Hooft anomaly. That is, the theory can be coupled to a background Z partition function is invariant under Z N 2-form gauge field B in such a way that the N 1-form gauge transformation. This is so because the usual Wilsonian lattice regularization of SU(N ) Yang-Mills theory has a manifest 1 N symmetry which acts locally [19]. On the lattice, B is represented by a 2-cocycle 1-cochain with values in ZN . Coupling the theory to B can also be thought of as inserting to [B] ∈ H2(X, ZN ). The choice of Σ2 within its homology class is immaterial, precisely because the ’t Hooft anomaly is absent. an area law), the 1-form Z N center symmetry is unbroken [20]. It is believed that SU(N ) is always unbroken at zero temperature. In more general SU(N ) gauge theories it might happen that for some nonzero a, which is a divisor or N , the lines (ak, 0) with integer k have a perimeter law, while other lines have area law [25]. In that case only a Z subgroup is unbroken [20]. This is the subgroup that leaves the line (a, 0) invariant. a ⊂ Z A CP anomaly for even N As we have remarked above, the identification (2.2) involves a nontrivial action on the line An immediate consistency check is that lines that are attached to a topological surface a mod N is invariant under the transformation and hence also the charge under the center symmetry is invariant. But (2.6) has consequences for other line defects, which do not have a well-defined Z Consider for instance the loop (0, 1) which is attached to our Z N surface. It transforms to the loop (1, 1), which is also attached to a surface. Now consider coupling the theory to a N two-form background gauge field B. This is equivalent to creating a network of surface and thus has intersection number 1 with the topological surface attached to the line defect. But there is a certain ambiguity in the gauging procedure, due to the possibility of adding counter-terms depending only on B (one may think of this counter-term as a seagull term that we are free to add). Geometrically, this ambiguity arises from the freedom to assign weights to intersection points of surface operators . We can choose the counter-terms speaking, changes the charge under the 1-form symmetry. Indeed, (1, 1) operator can be thought of as a composite of a (0, 1) operator and a (1, 0) operator, and the latter gives an even N , this counter-term takes the form B ∪ B . The case of odd N is considered in the next subsection. This counter-term guarantees the consistency of the mapping (2.6). More generally, if the counter-term was there in the It is a somewhat subtle fact that the expression (2.7) is well-defined (for even N ). Indeed, if we regard B as integer-valued 2-cocycle, then replacing B → B + N b for some b ∪ b − (b ∪ B + B ∪ b). the cup product is not commutative on the cochain level. It turns out one can correct the The corrected expression uses the Pontryagin square instead of the usual cup square. has many important consequences, which we will now explain. are equivalent. However, if we like to make B a dynamical gauge field then we have to Indeed, if we turn B into a dynamical gauge field, we find the PSU(N ) gauge theory, where where CP is a symmetry in the SU(N ) theory. We see that a CP transformation has to be accompanied by adding the counter-term (2.7).5 This means that there is a mixed ’t Hooft anomaly involving CP and the 1-form Z N symmetry. The discussion in the previous paragraph shows one manifestation of this mixed anomaly: if we gauge the 1-form Z then the CP symmetry is explicitly broken.6 This is a standard phenomenon in situations with a mixed ’t Hooft anomaly — if we gauge one symmetry the other is explicitly broken. There is another striking implication that we can derive by anomaly matching: if we the CP symmetry is either spontaneously broken or the vacuum supports a nontrivial theory has a trivial gapped vacuum. Hence either the CP symmetry is spontaneously broken at The anomaly (2.7) is reminiscent of the more familiar parity anomaly in three dimensions [3, 4]. There, a time-reversal transformation induces a properly-quantized ChernSimons counter-term for the background gauge field. The anomaly can be interpreted as a mixed time-reversal/U(1) anomaly. Here we see that SU(N ) Yang-Mills theory has a similar mixed time-reversal/ZN center anomaly. Let us make this analogy a little more precise. Here we discuss SU(N ) for even N , and in the next subsection we extend the discussion to odd N . 6If the 2-form ZN gauge field is not dynamical but it has some nontrivial value, then the CP symmetry is still present but the vacuum transforms nontrivially under a CP transformation. This can be interpreted by saying that with a nontrivial background B we have introduced the ’t Hooft twisted boundary conditions defect [26], which carries charge under CP. Indeed, nontrivial values of the 2-form Z N gauge field are classified by same cohomology group as the ’t Hooft twisted boundary conditions. In PSU(N ) we have to sum over all such defects and hence the CP symmetry is destroyed. 7This nontrivial theory could be either a gapless theory or a topological field theory. counter-terms to be invariant as well. Let us start with the the most general counter-term, adds the term (2.7) . In other words, it acts on p as follows: B ∪ B p → −p − 1 . Since gauge invariance forces p to be quantized, we cannot choose the counter-terms to preserve CP. This is exactly reminiscent of the famous time reversal anomaly of a free Dirac fermion in three dimensions. There, a time reversal transformation shifts the action i R A ∧ dA. Since there is no properly quantized 3d counter-term that can cancel this effect, there is a mixed anomaly that involves time-reversal and the U(1) symmetry of the free fermion. The only difference between this 3d example and our 4d problem is that the ordinary U(1) symmetry of the 3d problem is replaced by a Z one-form global symmetry in the 4d problem. invariance by not including any counter-terms. This reflects the fact that there is no mixed A CP anomaly for odd N For odd N a properly quantized counter-term has the form N there exists a choice of the counter-term that preserves both the 1-form Z 8This is a bit schematic. As explained above, one should really use Pontryagin square instead of the cup B ∪ B , B ∪ B . p −→ −p + N − 1 . mation shifts the counter-term by the shift. Therefore the coefficient of the counter-term transforms as follows under CP: Note however that at θ = 0 CP transformation acts by p → −p. For odd N , this almost as good as saying that there is an anomaly. Indeed, it implies that there cannot infrared consequences of this slightly weaker statement for odd N are similar to the ’t cannot be both trivial with unbroken CP invariance. spontaneously, or be in a topologically ordered state with both a CP symmetry and a N symmetry. The former option is much more probable, certainly for sufficiently large N . In the case of a topologically ordered state with both a CP symmetry and a A continuum description of the CP anomaly It is useful to present the above anomaly (2.7) from the continuum point of view. In (2.7) we have used a cup product between Z N valued 2-cocycles, but here we will show that it can be also understood in the continuum by embedding the 1-form discrete symmetry above into a continuous symmetry. This discussion uses some results that appeared in [19, 20]. In the continuum a Z N 2-form gauge field is represented by a pair (B, C), where B is a U(1) 2-form gauge field, C is a U(1) 1-form gauge field, and they satisfy a relation9 general observable with the same property: −i N 1-form gauge symmetry transformation is replaced by a We extend the original SU(N ) gauge field a to a U(N ) gauge field a . dynamical 2-form U(1) gauge field u and a background pair (B, C) as above such that our Lagrangian now has the term 9It is tempting to try to solve for B and work with C alone. But one must resist this temptation, since B is not a 2-form, but a 2-form gauge field, and N B does not completely determine B. u ∧ (TrF ′ − dC) , dC ∧ TrF ′ + dC ∧ dC Tr(F ′ ∧ F ′) − N dC ∧ dC . Finally, we can also add to the Yang-Mills action a gauge-invariant counter-term for the background field (B, C). The most general counter-term is B ∧ B = dC ∧ dC . to a gauge transformation. Clearly, for trivial (B, C) we find the original SU(N ) gauge theory. Under the gauge symmetry (2.15) the U(N ) gauge field a transforms as follows: reduce a U(N ) gauge field to an SU(N ) gauge field. Next, we would like to replace an action for the SU(N ) field a by an action for the U(N ) For example, the instanton number density becomes Here p must be integer for even N and an even integer for odd N . In both cases we have an U(N ) gauge field a′, RX Tr(F ′ ∧ F ′)/8π2 can be half-integral, but the expression 1 Z Tr(F ′ ∧ F ′) − TrF ′ ∧ TrF ′ = Tr(F ′ ∧ F ′) − dC ∧ dC 1 Z p → p + N − 1. p → −p + N − 1. is always integral. This is because the expression (2.21) is minus the second Chern number i(N − 1) Z dC ∧ dC. Since c2 is integral, the first term does not affect exp(S) and can be dropped. The second term can be absorbed into to a shift of the coefficient of the counter-term Note that for N odd it preserves the requirement that p is even. This transformation has no integral fixed points for even N and has a unique even fixed point for odd N . Thus we recover the analysis of the previous section. the one-form gauge symmetry and then a′ is traceless (up to a gauge transformation). This leads to the original SU(N ) theory. Otherwise, the periods of B are nontrivial and we find PSU(N ) bundles that are not SU(N ) bundles. If we make the two-form gauge field B dynamical, then p is interpreted as a coupling constant in the Lagrangian, which determines how we sum over the various bundles. If B is a background field then p is interpreted as a counter-term. Note that in our analysis thus far p has been a counter-term in the ultraviolet. Then, it has to be properly quantized. But there is another possible application of the discussion above. If we have a theory with Z N 1-form symmetry with a nontrivial ground state (gapless, or topologically ordered), it may lead to an effective fractional pir. We can then think of pir as an intrinsic observable of the infrared theory, defined modulo the counterterm (2.20). It would be interesting to study this observable. If the ground state is trivial, then pir would have to be properly quantized. But the difference between the ultraviolet and infrared p’s is still a meaningful observable associated to the RG flow. This point of view is analogous to the Hall conductivity in three dimensions, which is an intrinsic infrared observable that is defined modulo an integer, see [27]. Softly broken N = 1 SYM The anomalies described above follow just from the existence of the center 1-form symmetry The supersymmetric theory with gauge group SU(N ) has N vacua [28]. We define10 the instanton parameter η = Λ3N = µ 3N e− g8(πµ2)2 +iθ, where µ is a real renormalization point and induced potential is just Using the chiral anomaly the theory depends only on real and positive. Let us assume that the physics changes smoothly as a function of m. Therefore, the qualitative behavior of the low m theory is the same as the behavior of the 10We use the notations and conventions of [29]. theory at large m. This allows us to make contact with the non-supersymmetric SU(N ) and odd N . This is precisely consistent with the anomaly prediction. Furthermore, the anomaly implies that this degeneracy is not lifted for any m. A similar analysis of the softly QCD at large N are discussed in [31]. Finally, as discussed before, it may be that as we increase |m| the two vacua approach each other and lead to a nontrivial gapless ground state. Indeed, the supersymmetric 2d Other interesting continuous deformations of Yang-Mills theory were considered in detail both at zero and nonzero temperature, see e.g. [32, 33]. In these models spontaneous Boundaries, interfaces, and domain walls stressed an anomaly in-flow mechanism and proposed a U(1)N topological field theory on the domain wall. (A similar TQFT had been discussed earlier in the context of dynamical domain walls in N = 1 supersymmetric theories in [35].) The anomaly inflow alone, of course, does not completely determine the dynamical theory on the interface. The U(1)N proposal is just one possible solution to the anomaly inflow constraint. Note in particular that U(1)N is a spin-TFT when N is odd. This was Simons theories. This identification, though, is precise only for fermionic theories: the quasi-particles of the two theories match only when dressed by transparent fermion spin structure. It is natural to conjecture that the domain wall in the non-supersymmetric SU(N ) gauge theory supports at low energy an SU(N )−1 TFT. An alternative way to produce such interfaces is to consider a non-dynamical modifiSU(N )k Chern-Simons action supported at the interface. We may wonder what low energy degrees of freedom will appear at the interface. As the bulk is gapped, the degrees of freedom can be expressed as a 3d theory, Tk, equipped with N 1-form symmetry, which matches the anomaly of the UV Chern-Simons an anomalous Z The obvious candidate for Tk is an SU(N )k Chern-Simons TFT. This is not the only over a distance larger than the strong coupling scale, we would likely obtain a low energy theory of the form (SU(N )1)k. UV interface thickness. This phase transition may be first or second order. We can make a crude model for this transition. Consider a 3d quiver [SU(N )1]k theory with bi-fundamental scalars between consecutive nodes. If the scalars are massive, we get an [SU(N )1]k theory in the IR. However, in the Higgs phase we get a SU(N )k TFT in the IR. Whether the transition is first or second order may depend on the couplings in the Lagrangian. SU(2) Yang-Mills theory on Y × S We now study the SU(2) theory at temperature T = β1 by putting the theory on Y × S1, where Y is a 3-manifold (compact or non-compact, as the case may be), and S1 has 3.4) is valid, after some appropriate adjustments, for SU(N ) gauge theory. An important element in the analysis is the Polyakov loop, which measures the holonomy around the S1 U = P ei HS1 A coordinates on Y . The first natural question to ask when studying the theory on Y × S1 concerns the theory on Y . The symmetries of the theory are inferred by dimensional reduction from 0 − form 1 − form space − time The four-dimensional 1-form symmetry splits upon reduction to three dimensions to a 1global symmetry from the point of view of the three-dimensional theory, since it can be time reversal must be a symmetry of the three-dimensional theory. In 4d we had classes of line operators labeled by (a, b) ∈ Z Z2. These line operators operator attached to them. From the viewpoint of the three-dimensional theory, a genuine loop wrapping the S1 becomes a local operator. A wrapped loop that is attached to a topological surface becomes an operator attached to a topological line. Loops that do not wrap the S1 remain loops from the viewpoint of the three-dimensional theory. We often refer to such loops as “space-like” loops. They are labeled, as before, by (a, b) with a, b defined mod 2. For example, the wrapped (0, 1) is attached to a topological line that generates the Z2 1-form symmetry in three dimensions. The space-like (0,1) loop is attached to a topological surface that generates the 0-form Z2 symmetry in three dimensions. The CP anomaly reduced to three dimensions adds to the action the counter-term B ∪ B, B ∈ Z2(X, Z2) . a ∪ A ∪ B , where a is extended to the auxiliary four-dimensional space M with boundary Y (which theory was defined) as a standard 1-form gauge field. This can be viewed as a variation of the Dijkgraaf-Witten [43] construction, which includes a 2-form gauge field. 11This is true in the sense that the variation of the functional integral should be the same as that of the four-dimensional theory. Such anomaly matching across dimensions has appeared recently in the context of hydrodynamics (see e.g. [37–39]) and supersymmetry (see e.g. [40–42]). 12From the four-dimensional point of view, that means considering possibly non-orientable circle fibrations over the three-dimensional space-time. we have to split the background 2-form gauge field B in 4d into a 2-form gauge field and a 1-form gauge field in 3d. They are gauge fields for the 1-form Z2 center symmetry and the The anomalies of the four-dimensional theory should be matched by the threedimensional theory.11 Therefore, to reproduce (3.3) also the three-dimensional theory A ∪ B . Since from the point of view of the three-dimensional theory CP is just a global Z2 corresponding 3d 1-form background gauge field. Then, the anomaly (3.4) can be viewed as arising by anomaly inflow from The anomaly (3.5) forbids the three symmetries, i.e. CP, center 0-form and center 1form, to be simultaneously unbroken in a trivial gapped vacuum. This is therefore a mixed anomaly in three dimensions which involves two Z2 0-form symmetries and one Z2 1-form theory at finite temperature. The high temperature phases As we will see, this anomaly has profound implications for the phase diagram of the 0 −1 When the S1 is very small compared to the dynamically generated scale in Yang-Mills theory, we can study the three-dimensional theory simply by dimensional reduction. This is the standard high-temperature expansion in Yang-Mills theory [44, 45]. dimensional theory has an SU(2) gauge field with gauge coupling g3d ∼ gY M β−1 and 2 2 the trace of the holonomy In addition to the standard kinetic terms we have the interaction that is induced by tion then such a term in the Lagrangian is trivial. This is because we can integrate by parts. But in the context of this theory arising from a circle compactification we have the well-defined adjoint-valued scalar field. We can now identify the symmetries of the problem. The center symmetry acts as act on lines, not on local operators. Time reversal acts by combining a reflection in one of the coordinates in R3 with the gauge transformation Φ → −Φ. This is as expected, as the with two minima at U = ± low temperature phase These minima are distinct and indicate a spontaneous breaking of the 0-form center symmetry. As we decrease the temperature, the two vacua approach each other until they collide at βφ = π2 ∼ − 2 . Since θ only makes subleading contributions at very high temperatures, π Let us now consider the 1-form symmetry in three dimensions. The mass squared of than the dynamical scale g3d We can thus safely integrate it out and the low energy theory is a pure three-dimensional SU(2) Yang-Mills theory, which confines. Hence, the space-like Wilson loops in all representations on which the center Z N acts nontrivially enjoy an area law, and the 1-form Z symmetry is unbroken [20]. Finally, note that the CP symmetry is preserved in both vacua We can now summarize the vacuum structure in the high temperature phase and the 1 − form center 0 − form center 1 − form center 0 − form center low temperatures. Note also that we did not find a phase where the 1-form center symmetry is broken. A priori, such a phase would be consistent with the anomaly so it is noteworthy that it does not exist in the high temperature or low temperature regimes. In section 6 we will describe a scenario where such a phase exists at intermediate temperatures. We remind the reader again that we assume here that at zero temperature the system continues to The domain wall at high temperatures There is a two-dimensional domain wall that separates the two vacua (3.10) of the threedimensional theory. We would like to determine the low-energy theory on this domain wall. Furthermore, the domain wall should be consistent with the anomaly (3.5), and we would like to understand how this comes about. However, here the adjoint scalar needs to undergo a large field excursion of the order of the temperature in order to interpolate between the two vacua, and the theta-term (3.8) even at high temperatures. Far from the domain wall U = ± 1 and the microscopic SU(2) symmetry is unbroken. Strong dynamics confines it and creates a mass gap. But near the domain wall U must interpolate from +1 to − 1 and then the SU(2) symmetry is broken to U(1). Since this happens in a narrow region around the domain wall, we effectively have a U(1) theory in two dimensions, which does not have physical propagating degrees of freedom. Then, due domain wall world-volume as an Abelian 2d theta angle wall theory to be gapped (except for the translation zeromode, of course). 1-form background connection. We thus conclude that the charge conjugation symmetry is spontaneously broken on vacuum degeneracy. Because of the counter-term shift, the kink between these two vacua carries charge 1 under the 1-form symmetry, much as the bulk Wilson line operator in the fundamental representation. of two vacua interchanged by charge conjugation, and a line which interpolates between them. This system reproduces the mixed charge conjugation/1-form anomaly that the domain wall needs to carry. This system is almost identical to the standard BF TQFT with Z2 symmetry, except that here orientation plays a role, suggesting that this is a nontrivial modification of the standard BF theory on unorientable spaces. This modification is described in detail in One can think about the domain wall theory intuitively as QED2 with massive even charges and theta angle (3.14). Since only even charges are present, we have a Z form symmetry. Since the charges are massive and the gauge field has no propagating degeneracy [12]. A probe unit charge particle, i.e. a Wilson line, would interpolate between these two vacua. The 1-form symmetry is spontaneously broken (this is because inside the Wilson loop of a unit probe charge we have the other vacuum and hence no confining string) and CP is spontaneously broken as well. We see that this domain wall is quite rich. We will have more to say about it in appendix D. The confinement/deconfinement transition and the 3d Ising model parameter for the confinement-deconfinement transition is a real field (i.e. the wrapped question marks we assume that the first order line continues to higher temperatures beyond the Let us make some further comments about the intersection of the first-order and second-order lines. We have at that point two vacua in SU(2) gauge theory and four vacua in the mixed theory. At each of these vacua, we have a three-dimensional Ising model. scale at this point. Therefore, there is no necessity that all the four Ising models would be simultaneously described by one three-dimensional effective field theory. The cutoff of the In the figure above the first order CP transition line ends with a second order transition in the 3d Ising universality class. While our description here of the phases of the theory (assuming that the space-like loops are confined) was for SU(2) Yang-Mills theory, in fact, a straightforward adaptation of this scenario makes perfect sense for SU(N ) Yang-Mills theory. The inequality (5.1) continues to hold for all N . are allowed to write any potential that is D8 invariant (D8 is generated by a rotation by 90 degrees and by a reflection). The general such potential is V = with ϕ the angle between T and D. If only the term cos(4ϕ) is present, then there are four vacua, but in general, there could be 4 or 8 vacua depending on the relative coefficients of the various terms. It takes fine tuning to have more than 8 vacua. Indeed, if there are four vacua, then D8 is broken to some Z2 subgroup. Depending on the situation, it could be either a Z2 reflecting around an axis or a Z2 reflecting around a diagonal. These two scenarios are realized at low and high temperatures, respectively. If there are 8 vacua, then D8 is broken completely. These are the only scenarios that do not require fine tuning. transition between 4 vacua with an unbroken Z2 in the conjugacy class of a reflection around one of the axes and 4 vacua with an unbroken Z2 in the conjugacy class of a reflection around a diagonal. The transition is through a phase with 8 vacua and a completely broken D8. finement transition are the same as described in subsections 3.3 and 4.4, respectively. The domain walls in the region above the deconfinement transition and below the CP restoration temperature could have some interesting dynamics. It is useful to look first at the mixed gauge theory. Then D8 is completely broken and we have eight massive vacua. We can pick any of the vacua, call it v1, and label all other vacua vg by the non-trivial D 8 group element g mapping v1 into them. The domain walls between these vacua will generically also have a single gapped vacuum. Notice that not all domain walls are guaranteed to exist: in principle one may interpolate between two vacua by a sequence of stable domain walls involving other vacua Although the vacua are locally equivalent, pairs of vacua are not all equivalent. The seven potential domain walls between v1 and the other vacua are all physically distinct and can potentially have distinct physical properties, such as tension. We can label the domain walls wall between vacua v1 and vg as dg. Because of the broken D8 symmetry, the domain wall between vacua vg′ and vg′g will have the same properties as dg. A 3d CPT transformation will both exchange the two walls have identical properties up to a space reflection. In the case at hand, the reflection relates the two domain walls labelled by the order 4 elements of D8. The other group elements are of order two and the reflection symmetry does not add more information. Upon gauging some Z2 subgroup of D8, vacua related by gauge transformations become equivalent. Lets denote the Z 2 generator as x. The domain wall dx which interpolated between v1 and vx will become a dynamical string in the gauged theory. Domain walls dg and dxg now interpolate between the same pair of vacua in the gauged theory. These two domain walls are related by a space reflection if g2 = x. A Higgs phase at intermediate temperatures Above we saw that assuming there are no tensionless color flux tubes leads to a prediction of an inequality, (5.1). But it is worthwhile to explore the possibility that a point with unbroken D8 symmetry exists. inverse temperature has to respect the D 8 symmetry. We know that at sufficiently low temperatures the symmetry is broken spontaneously to a Z2 subgroup while at sufficiently high temperatures it is broken to a different Z2 subgroup. In between, whether or not D8 is restored depends on the coefficients of various terms in the effective Lagrangian (5.2). It requires no fine tuning for the region in between low and high temperatures to have an unbroken D8 symmetry. But if this happens, it does require the existence of new critical points on the phase diagram. We will see one concrete realization below. There are two distinct scenarios: 1. There is some finite collection of points with unbroken D8. 2. There is a whole region with unbroken D8. Note that straightforward ideas like joining the three lines of figure 2 and having and O(2) model at the intersection (with unbroken D8 symmetry embedded inside the O(2)) are inconsistent because the O(2) model admits an O(2) invariant deformation in which the O(2) symmetry is unbroken and the vacuum is unique and trivial. This is the usual disordered phase of the O(2) model. Such a phase does not exist in figure 2. A variation of this scenario is to postulate the existence of some 3d CFT which contains a D8 subgroup in its symmetry group, has two relevant deformations, but does not have a disordered phase. Such a 3d CFT would typically be expected to have a discrete D8 ’t Hooft anomaly (which would explain why a trivial disordered phase does not exist). But we show in appendix C that such an anomaly does not exist in the mixed theory. In view of these considerations, we regard the first scenario as unlikely and turn to the O(2) phase diagrams To prepare for the construction of the phase with unbroken D8, let us analyze the phases of the Ginzburg-Landau theory with two fields, M~ = (M1, M2) and potential diagram is two-dimensional, containing r, k. The phase diagram crucially depends on the which we essentially neglect here, are to curve the various lines in the figures below.) exchanges M1 and M2, which corresponds to the CP generator in (4.5). Clearly, for large positive r > |k| we always get one disordered vacuum. If r < |k| but r is not too small (in a sense that we will specify momentarily), then we get into an ordered phases. One finds The main difference between them is that the first-order line in figure 5 splits in figure 6 A trivial phase with unbroken D8 We have already noted above that the low temperature phase of the mixed gauge theory, figure 2, is very similar to the bottom half of figure 5. Similarly, the high temperature part of figure 2 is extremely similar to the lower half of figure 6. M 1 = M 2 = 0 r = k = 0 M 2 = 0 M 1 = 0 M 2 = 0 M 1 = M 2 = 0 r = k = 0 M 1 = 0 T = D = 0 D = 0 T = 0 Figure 7. A suggested phase diagram with an unbroken D8 phase in the shaded region. In terms of the SU(2) gauge theory, this is a Higgs phase. gauge theory is a line in this diagram starting at the lower left side and ending in the lower right side. The dashed line corresponds to the scenario in section 5 with figure 4, and the wiggly line corresponds to the scenario in section 6 with figure 7. It is therefore suggestive to glue these two descriptions together, as in figure 7. We obtain a phase diagram with a tricritical O(2) point and a tetracritical O(2) point. As we explained above, both of these are locally possible multicritical points for the O(2) CFT, depending on some dangerously irrelevant operators. The disordered phase is present in weakly-coupled or lattice-accessible regions of the theory is not a problem. The shaded region in figure 7 corresponds to a single, trivial (gapped), D8-preserving vacuum. This is possible only if there is no ’t Hooft anomaly for the D8 symmetry, a fact that we verify in appendix C. This phase can be interpreted as a Higgs phase of the SU(2) gauge theory. Indeed, an unbroken bonus Z2 symmetry of the mixed theory means that the 1-form Z2 symmetry of the SU(2) gauge theory is broken, and hence the spatial Wilson loops have a perimeter law. This is a hallmark of the Higgs phase. The 0-form center Z2 symmetry is unbroken, so the Polyakov loop still has zero expectation value, as do the 3d local operators T and D. Whether the present scenario or the scenario of the previous section is realized depends on whether the D8 symmetry is unbroken somewhere. None of these scenarios requires fine vacua and an unbroken Z2 symmetry in the conjugacy class of a reflection around one of but preserves Z′2 in the conjugacy class of a reflection around a diagonal. In between, for negative r, there is a phase with 8 vacua and completely broken D8 symmetry. For positive is irrelevant in the O(2) fixed point). There are also two lines with r < 0 which support the 3d Ising model. We see that the present scenario corresponds to the wiggly curve (the Broken Center D = 0 T = 0 transition at zero temperature. temperature increases from left to right) — it intersects the O(2) model lines twice and it includes a phase with unbroken D8 symmetry. The scenario of section 5 corresponds to the dashed line — it intersects the Ising lines twice and it does not include a phase with unbroken D8 symmetry. Scenarios with deconfinement at zero temprerature The scenarios outlined above assumed that SU(2) Yang-Mills theory is confining at zero supersymmetric Yang-Mills theory and (at large N ) the behavior of holographic models. However, none of these arguments are conclusive, and it is conceivable, especially for SU(2) Yang-Mills theory, that there exists a zero-temperature phase with a Coulomb or Higgs The simplest possibility, depicted in figure 9, is that the deconfineθ = π and in the second case it is not confining in the range θ ∈ (π − x, π + x).18 18For SU(N ) gauge theory with N > 2 the deconfinement line is a first order line, and therefore the zero-temperature phase transition can be either first order or second order. For SU(2) gauge theory, these transition points must be second order. both with broken 1-form Z2 symmetry. Let us consider the possibility that the physics in the interval θ ∈ (π − x, π + x) is in a Coulomb phase.19 has continuous magnetic and electric one-form symmetries. We imagine that the electric one-form symmetry is accidental and it is in fact entirely broken by some heavy charged particles. The magnetic one-form symmetry is broken to Z2, which matches the Z2 oneform electric symmetry of the original SU(2) degrees of freedom. This symmetry acts on ’t Hooft loops of the U(1) gauge theory and it is spontaneously broken in the Coulomb phase. Now let us examine the physics of this (free) theory at finite temperature. To first approximation the low energy 3d theory consists of a free photon and a real neutral (periodic) scalar arising from the gauge field holonomy. However, we need to add to this theory a monopole operator to reflect the fact that the 4d 1-form symmetry is Z2 rather than U(1).20 In addition, since the electric one-form symmetry is completely broken, this shift symmetry of the holonomy scalar is broken and we expect it to have a potential and be massive. As in the famous Polyakov mechanism, the photon acquires a mass due to the monopole operator. Since the monopole operator is Z2 invariant, there are two such gapped vacua and the Z2 0-form symmetry is spontaneously broken. The 4d 1-form symmetry (which is spontaneously broken at zero temperature) therefore leads to a spontaneously broken Z 0-form global symmetry and an unbroken Z us elaborate on the latter point. The Z 2 1-form symmetry in three dimensions. Let 2 1-form symmetry in four dimensions splits to a 0-form symmetry and a 1-form symmetry. The former corresponds to the topological The 1-form symmetry acts on lines in the 3d theory. Confinement in the 3d theory means that these lines have an area law and this symmetry is unbroken. since it is in the same universality class as the high temperature phase — see figure 9. In this example we see that a compactification on a very large circle can lead to a rather dramatic change in the pattern of symmetry breaking and in the spectrum. This is of course only possible if there are massless modes to start with. It could happen that the two deconfinement transition lines touch the zero temperature would be no Coulomb phase in this case. This picture is reminiscent of the O(3) model in (see [61, 62] and references therein). No Higgs phase at zero temperature. We would like to close by making some comments about the possibility that these deconfinement lines do not simply curve down, but there is a more complicated pattern. For example, we can have a phase diagram like figure 7 with the Higgs region extended all the way to zero temperature. It can touch the 19We do not specify the CFTs at the transitions. Note that the simplest option, namely scalar QED with the scalar being a monopole or a dyon of the microscopic degrees of freedom, is ruled out because this would be a first-order transition due to the Coleman-Weinberg effect [59]. 20An analogous situation was encountered in [60]. There the proper reduction of a 4d theory to 3d needed the addition of a monopole operator in the 3d effective Lagrangian. spontaneously broken Z2 one-form symmetry. scenario can actually be ruled out. Since the 0-form center symmetry is unrbroken inside the blob, it would be also unbroken in four dimensions and hence such a scenario contradicts the fact that the anomaly forces either the center symmetry or CP to be broken at We would like to thank O. Aharony, F. Benini, C. Cordova, M. Dine, J. Gomis, M.B. Green, T. Johnson-Freyd, M. Metlitski, A. Schwimmer, S. Shenker, and E. Witten for useful discussions, and especially Y. Tachikawa for collaboration at the early stage of this work. The work of D.G. was supported by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. A.K. is supported by the Simons Investigator Award and in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632 Z.K. is supported in part by an Israel Science Foundation center for excellence grant and by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). Z.K. is also supported by the ERC STG grant 335182 and by the United States-Israel BSF grant 2010/629. NS was supported in part by DOE grant DE-SC0009988. NS thanks the Hanna Visiting Professor Program and the Stanford Institute for Theoretical Physics for support and hospitality during the completion of this work. A modification of the Z2 gauge theory The standard BF theory in two dimensions with Z2 symmetry has the following continuum S = 1 Z b ∧ da . between these vacua. This theory can be extended to unorientable space-times in two different ways. One of them is obvious and the other is reminiscent of Dijkgraaf-Witten-type gauge theories [43]. It is easiest to describe it on the lattice. Then instead of continuum fields b and a we use cochains b ∈ C0(Σ, Z2) and a ∈ C1(Σ, Z2). The action for this theory is write an action as follows: The key point is that the cup product is not supercommutative on the cochain level, so a ∪ a is a nonzero 2-cochain, in general. To see that on orientable manifolds this theory is equivalent to the (4.7) , but not in general, let us integrate over b, so that a is constrained to be closed modulo 2. For such an 2 gauge field a ∈ Z1(Σ, Z2), and its square integrates to 1. In general, a ∪ a is cohomologous to w1 ∪ a, where w1 ∈ Z1(Σ, Z2) is a representative of the 1st Stiefel-Whitney class [w1] ∈ H1(Σ, Z2) [63]. Since [w1] = 0 if and Let us discuss some properties of the theory (A.2). It is convenient to reinstate b and This is not the same b as in (A.2) . In particular, if we change a representative w1 by exchanges the two vacua. This is precisely what happens on the center-symmetry domain wall discussed in section 3.3. This is in fact required by the anomaly of the Yang-Mills theory (3.5) if we interpret the 2d TQFT (A.2) as describing the physics of the center-symmetry domain wall. Indeed, on the domain wall the center Z2 is restored. Placing the domain wall on an unorientable background means gauging the CP symmetry, and the anomaly then requires the 1-form symmeand we can gauge it. In the gauged theory the Wilson loop for a is not an observable since it is the generator of the 1-form symmetry. Thus the gauged theory has a unique vacuum and no nontrivial observables, i.e. it is trivial. This is consistent with the fact that in the mixed theory the center symmetry domain wall does not carry any topological degrees of freedom. Another derivation of D8 symmetry For the discussion in this appendix we need to introduce a gauge field for CP symmetry in 3d. Indeed, from the 3d viewpoint it is an ordinary global symmetry, and hence we can couple it to a standard one-form Z2-valued gauge field a on Y . The anomaly inflow was given in (3.5). Now we define our mixed gauge theory by turning the 2-form gauge field B into a dynamical field and coupling to it the 1-form Z2 gauge field b. The coupling iπ RY b∪B is invariant under B → B +δχ if δb = 0, but the rest of the the theory has an anomaly under the 1-form symmetry transformation, and the constraint on b should be chosen to cancel it. Recall that the anomaly is described by the 4d action (3.5), symmetry transformation as above, this 4d action varies by a boundary term To cancel the anomaly we must choose Here a and A are 1-cocycles with values in Z2, while b is a 1-cochain with values in Z2. We want to interpret this equation as a 1-cocycle condition for 1-cochain with values in some group G. In general, when G is an extension of some G0 by an Abelian group H, the G gauge field can be described by a pair (b, b), where b is a G0 gauge field (i.e. a 1-cocycle where ω ∈ Z2(G0, H) is a 2-cocycle describing the extension, and ω(b) ∈ Z2(Y, H) is its evaluation on b. Applying this to equation (B.2) , we see that G must be an extension of generators of H1(Z2a, Z2) and H1(Z2A, Z2). This is the usual description of the D8 group. Z2A by Zb2, and that the cohomology class describing the extension is the product of the The (absence of ) anomalies for D8 symmetry A three-dimensional theory with D8 global symmetry could, in principle, have global ’t Hooft anomalies classified by H5(D8, Z) ≃ H4(D8, U(1)) ≃ Z 2 ⊕ Z2 . In three-dimensional theories that arise from some four-dimensional theory, the anomaly could be in principle even larger because not all three-dimensional counter-terms arise from four-dimensional counter-terms. If the anomaly does not vanish, trivial phases with unbroken D8 are forbidden. It is very useful to start from a seemingly different problem, concerning the 1-form a discrete analog of the Green-Schwarz mechanism [64], where the two-form background connection, B, for the 1-form center symmetry is not invariant under some 0-form gauge transformations and hence B satisfies a modified Bianchi identity: 2-form gauge field B is closed if a is set to zero. If the gauge field a is nontrivial, although B was closed in four dimensions, its three-dimensional counterpart may not be closed. However, it still has to be homogenous linear in (A, B) and therefore we only have to discuss the possible modification of the Bianchi identity of the form Let the 4d manifold be X, and suppose it is a circle fibration over a three-dimensional base Y . We want Y to be orientable, but the total space X would be unorientable because we want the orientation of the circle to undergo a flip as we travel along non-contractible closed curves on Y .21 This class is precisely the 3d Z2 gauge field for the CP symmetry, a. To detect whether there is a modification of the Bianchi identity of the form (C.3), we must ensure that a2 is nonzero. (In general, a2 could be trivial even if a is a nontrivial element of H1(Y, Z2). For example, on a torus every Z2 gauge field squares to zero.) This restricts the choice of Y . The simplest possible choice is RP 3. It is orientable but it has field on RP 3 squares to the generator of H2(Y, Z2), which is exactly what we need for a2 Now we consider again our potential modification of the Bianchi identity (C.3). For a solution to exist, the cohomology class of a2A must be trivial. But on RP 3, if a and A are both nontrivial, a2A is also nontrivial. So there is no solution for B, if A is nontrivial (and for a nontrivial). We are forced to set A = 0. original unorientable four-dimensional space X, the background two-form gauge field must not have components along the Kaluza-Klein circle.) However, we will now show that this conclusion is false and the correct answer is H2(X, Z2) = Z 2 ⊕ Z2. The problem boils down to computing the cohomology of a concrete 4-manifold X. One can either compute this cohomology directly or use the Gysin exact sequence [63]. The latter can be used to show that for an arbitrary circle bundle over an arbitrary oriented 3-manifold Y , if the total space X of the fibration is a P in+ 4-manifold, then Hp(X, Z2) = Hp−1(Y, Z2) + Hp(Y, Z2) . That is, as far as cohomology is concerned, such a circle bundle X (even though it is unorientable) behaves as if it were the Cartesian product of S1 and Y . H2(X, Z2) = Z 2 ⊕ Z2 and the Bianchi identity is not modified. Having shown that the three-dimensional two-form B is closed (and hence a 2-cocycle), we can now proceed to the anomalies of the D8 symmetry in the mixed theory. We obtain due to the anomaly (3.5), this is only possible if we assume that b is not a cocycle and hence the symmetry group is D 8 rather than Z3. This is explained in detail in appendix 2 B. But we should also consider Z2 gauge transformations of b. Since B is a 2-cocycle, the system is perfectly invariant under such gauge transformations (and the partition function 21Mathematically, this means that the circle bundle is a unit circle bundle of a two-dimensional real is also invariant under the standard Z2 gauge transformations of a and A) and hence D8 The lack of anomalies is crucially important for the consistency of the phase diagram An alternative proof There is an alternative way to prove the absence of anomalies for the D8 symmetry in the mixed gauge theory: we can deform our theory to another theory in which the D symmetry is manifestly anomaly-free. As long as the D8 symmetry is preserved along the deformation, ’t Hooft anomaly matching will imply the desired result. A simple way to accomplish our objective is to add an adjoint Higgs scalar field to the four-dimensional SU(2) gauge theory. This preserves the 1-form center symmetry and CP invariance of the theory. The original theory is recovered when the Higgs field is very massive. On the other hand, in a Higgs phase one obtains a U(1) gauge theory. The U(1) gauge theory is coupled to massive particles with even electric charges and general magnetic charges, so that the naive U(1) × U(1) 1-form symmetries are broken to the Z2 of the underlying SU(2) gauge We can set our conventions so that the non-Abelian Wilson loop in the fundamental representation goes to Abelian loops of electric charge ±1 and the non-Abelian (0, 1) and (1, 1) loops go to Abelian loops of magnetic charge ± 21 and appropriate electric charge. shifts the Abelian electric charges by twice the Abelian magnetic charges. The Abelian Upon compactification on the thermal circle, a U(1) gauge theory can be dualized to a sigma model of two scalar fields valued on a two-torus, whose modular parameter coincides with the complexified 4d gauge coupling. The (exponentiated) scalar fields are simply Abelian ’t Hooft loop. The 0-form symmetries which descend from the U(1) × U(1) 1-form symmetries of the U(1) gauge theory coincides with the translations of the torus. We can imagine that the low energy effective action inherited from the SU(2) gauge theory induces a metric and potential on the torus which break translations to a single Z2 subgroup acting on the 2 1-form symmetry acts non-trivially on twist line defects around which 2 1-form symmetry to go to the mixed gauge theory has Notice that the CP , h and c generators satisfy the expected D8 relations. This is continuously connected to the D8 symmetry of the original compactified, mixed SU(2) model and is thus not anomalous. Furthermore, D8 acts in a completely geometric way on the scalar fields of the sigma ’t Hooft anomalies in quantum mechanics ’t Hooft anomalies manifest themselves in quantum mechanics as a projective representation of the global symmetry group. We start with a classical symmetry G. Often the quantum system realizes a central extension of it Gb. The added central element P obviously commutes with all the group elements in Gb. But the situation here is more specific than merely saying that the symmetry group is Gb. Since P does not exist in the classical theory, it commutes with all the operators in the theory. It is central not only with respect to the other symmetry elements but with respect to all the operators. However, it can act nontrivially on the states in the system. But as a central element it must act in the same way on all the states in the system. The purpose of this appendix is to discuss two examples demonstrating it and to present them in the spirit of the ’t Hooft anomalies in the bulk of the paper. A particle on a circle We consider the Lagrangian (in Minkowski space, so that there is an i in front of the action) with compact q ≃ q + 2π. Since for all field configurations on a Euclidean circle R q˙ ∈ 2πZ, The conjugate momentum is Πq = q˙ + 21π θ and the Hamiltonian is just L = H = formally that they are equivalent. The appropriate differential operator to diagonalize is En = under this transformation, which is equivalent to adding a background gauge field for the phase of the wave degenerate. We will soon see that these two states are related by a discrete symmetry, i.e. there is a spontaneously broken discrete symmetry (even though we are talking about quantum mechanics, where, naively, discrete symmetries cannot be broken). Since the particle is free, the spectrum is in representations of U(1), corresponding to to the constant wave function, invariant under all the symmetries. This is the standard the circle). The instantons have alternating signs and they can partially cancel each other. One can convince oneself that the action of the U(1) symmetry and of charge conjugation q → −q : |ni → | − n + 1i . | − n + 1i , | − n + 1i It is also useful to note now that expected based on classical considerations) since the above generator does not coincide element and consider (D.7) modulo a scalar action on the Hilbert space, we get a projective representation of O(2). Therefore, O(2) is realized projectively. We therefore see that 23As in our discussion in footnote 7, we can use time reversal and charge conjugation interchangeably since their product is always a manifest symmetry. should be viewed as a central extension of O(2) by P . Central extensions are common manifestations of anomalies (this is very familiar in two dimensions, where the Kac-Moody and Virasoro algebras are centrally extended). One should therefore conclude that there is a ’t Hooft anomaly in the O(2) symmetry. We will see that this anomaly has many of the usual consequences of anomalies, including anomaly matching in a nontrivial ground state, inability to couple to classical gauge fields, anomaly ground U(1) gauge field A0 Another perspective on the system is obtained by coupling (D.1) to a classical backL = where kA0 can be thought of as a “Chern-Simons” counter term for A0. Clearly for We can compactify the Euclidean time direction and compute the partition function in the presence of a U(1) chemical potential µ = H A ∼ µ + 2π. Then the partition function due to the vacuum Qubit is We neglect the contributions from the excited states, as they would not affect the discussion below. Note that (D.12) is not charge conjugation invariant. The Z2 charge conjugation → −µ . This is clearly broken by the partition function. We can say that under a charge conjugation the partition function transforms by This can be interpreted as adding a counter-term which is a correctly quantized 1d Chern-Simons term. To make the partition function charge conjugation invariant we could attempt to add a clearly see that there is an O(2) anomaly. Here are some possible additional points of view on this theory. 1. We can interpret (D.15) as giving the vacuum half-integer charge. This would mean that the symmetry group is not O(2) but its double cover, which is precisely what we found through the computation of the central extension. 2. There is a way to write a consistent partition function invariant under all the symme tries, but we have to add a two-dimensional bulk. This is similar to the mechanism of anomaly inflow (topological insulator). Indeed, the half-integer Chern-Simons term δS = 12 R A is not a well defined object in one dimension, but we can view it as resulting from the bulk integral 1 Z where the two-manifold M2 ends on our the “time line” of our quantum system. This depends on the choice of M2, but once we choose such an M2 it is well defined. We conclude that as long as we restrict the theory to be one-dimensional, O(2) is a projectively realized symmetry and the vacuum breaks it spontaneously. The O(2) is field charge conjugation is necessarily broken. This conclusion can be avoided if we couple the system to a two-dimensional bulk. It is worth noting that the situation for a massless fermion in three dimensions is We can either couple it to a U(1) gauge field and thus break time reversal invariance (this is what happens when we choose the fermion path integral phase A generic potential on the circle, which is some function of sin(q) and cos(q), would explicitly break the O(2) symmetry completely and there would be generically one ground state. It is interesting to leave some subgroup of O(2) unbroken. Let us consider a potential which is only a function of cos(2q), V = V (cos(2q)) . One can have in mind some polynomial in cos(2q). This preserves reflections and rotations 2 × Z2. Such a potential generically has two or four minima, depending on the relative coefficients of the polynomial. degeneracy. So the symmetry is spontaneously broken to Z2. The group is therefore rank 2. Hence we identify this with the D8 group (we identify this as D8 since it has 6 elements of rank 2). nontrivial representation proving our assertion above.24 The instantons that are normally expected to remove the degeneracy and leave one symmetric vacuum cancel here because of the signs with which we multiply instanton factors. One place where we can find this model is by thinking of q as the sphaleron degree of freedom in Yang-Mills theory, q = 41π RΣ3 A ∧ dA + 32 A3. Then CP becomes q → −q and the 1-form Z2 symmetry acting on A becomes q → q + π. The identification q ≃ q + 2π is just due to the usual gauge transformations in the homotpoy classes of S3 → SU(N ). Equivalently, only the exponential of the Chern-Simons term is well defined. So we can interpret our quantum mechanical model as the sphaleron theory. It preserves a Z symmetry. The two vacua and the anomaly are indicative of the anomaly and spontaneous breaking of CP in the original theory. For a connection between the above-discussed model with hidden supersymmetry in purely bosonic systems see [66, 67]. We now describe the connection to 1+1 dimensional Connecting with QED2. Consider free Abelian gauge theory in two dimensions This model has no degrees of freedom on R but let us study it a space manifold which is a circle of size L. Then we can pick the gauge A0 = 0 and remain with S = d2x −1 F 2 + S = for the electric field. model has a one-form Z The mass of the quantum mechanical particle is therefore ∼ e2L. Now let us add massive charged particles and suppose that the two-dimensional QED2 2 symmetry. In other words, all the charges are even. Then we have no dynamical charge 1 particles and the two ground states cannot mix even in finite volume. This is the twofold degeneracy found above. If we integrate out the even charge particles, we remain with a theory that has charge conjugation and Z2 symmetry and a potential like (D.17), which is some polynomial in cos(2q). elements would have to be represented by some phases and hence we would get a contradiction. mixed charge-conjugation/1-form anomaly which upon a circle compactification manifests itself as the quantum mechanical model above. leading to S3 ∼= SU(2). This system has an O(4) ∼= Z gauging the global symmetry is Z symmetry. Next, we gauge the Z2 symmetry generated by the nontrivial central element 2 × SO(3)L × SO(3)R. All the operators transform under with half-integer isospin in the two SU(2) factors. The target space of the system is not simply connected and therefore there is a discrete the SU(2)L × SU(2)R symmetry. Before the Z2 gauging we had all integer and half-integer between states with integer j and states with half-integer j. This is consistent with the The nontrivial element in the Z2 orbifold, P acts as +1 on H0 and as −1 on Hπ. Now we want to couple the system to background gauge fields. In quantum mechanics we compactify Euclidean time and all the information in the background field is in the holonomy U around that circle. The effect of this holonomy is that we compute traces with an insertion of the group element U . Let us start with SO(4) gauge fields. There is Next, we try to repeat this with background SO(3)L × SO(3)R gauge fields. Let us SO(4), which are identified in SO(3)L × SO(3)R. When working in H0 these traces are the insertion of U ′. This means that the answers with background SO(3)L × SO(3)R fields are 25It is often stated that ’t Hooft anomalies are diagnosed by making the gauge field dynamical and checking whether the Hilbert space is empty, or equivalently checking whether the partition function vanishes. If ’t Hooft anomalies exist this conclusion is right. 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Theta, time reversal and temperature, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP05(2017)091