Anomaly constraints on deconfinement and chiral phase transition

We study constraints on thermal phase transitions of ${\rm SU}(N_c)$ gauge theories by using the 't Hooft anomaly involving the center symmetry and chiral symmetry. We consider two cases of massless fermions: (i) adjoint fermions, and (ii) $N_f$ flavors of fundamental fermions with a nontrivial greatest common divisor ${\rm gcd}(N_c,N_f) \neq 1$. For the first case (i), we show that the chiral symmetry restoration in terms of the standard Landau-Ginzburg effective action is impossible at a temperature lower than that of deconfinement. For the second case (ii), we introduce a modified version of the center symmetry which we call center-flavor symmetry, and draw similar conclusions under a certain definition of confinement. Moreover, at zero temperature, our results give a partial explanation of the appearance of dual magnetic gauge group in (supersymmetric) QCD when ${\rm gcd}(N_c,N_f) \neq 1$.


I. INTRODUCTION AND SUMMARY
Thermal phase transition in gauge theories is a very interesting and important subject. Theoretically, it is related to the mystery of how strong dynamics works in confinement and chiral symmetry breaking. Phenomenologically, the nature of phase transition affects cosmological observables such as dark matter abundance. It might even provide the dark matter itself via QCD effects [1].
The standard way to study chiral symmetry restoration is as follows [2]. The quark bilinear Φ ∼ ψψ, where ψ represents left-handed fermions, is believed to be the most relevant order parameter for the chiral symmetry breaking. This operator Φ is treated as the effective degrees of freedom near the critical temperature T chiral , and the phase transition is described by a Landau-Ginzburg effective Lagrangian where the coefficients depend on temperature T and in particular c 0 ∝ (T −T chiral ), and t R is the Dynkin index of of the left-handed fermion representation. However, because of the strong coupling, it is not easy to see whether such a scenario is likely or not. It is conceivable that deconfinement happens at a lower temperature. If so, we lose intuitive reasons for treating the composite Φ as the effective elementary degrees of freedom. These questions may be rigorously asked in theories where center symmetry is well-defined, such as adjoint QCD, QCD with imaginary baryon chemical potential [3], or QCD in the large N c limit.
Recently, a very remarkable paper [4] appeared which studied phase transitions in pure Yang-Mills. They used a mixed 't Hooft anomaly between the CP symmetry and the 1-form center symmetry to constrain possible phase transitions at the theta angle θ = π. Under reasonable assumptions about the dynamics of pure Yang-Mills, the CP symmetry cannot be restored below the temperature at which the center symmetry is broken, i.e., deconfinement. See the original paper for more careful discussions.
In this paper, we point out that a similar discussion is possible for chiral symmetry. The mixed 't Hooft anomaly between chiral and center symmetry is also known [5], so we can repeat the argument of [4] when the center symmetry exists, such as SU(N c ) gauge theories with n f fermions in the adjoint representation. We will see that chiral symmetry restoration by (1) cannot happen below the deconfinement temperature.
When there are N f fermions in the fundamental representation of SU(N c ), the center symmetry no longer exists. However, we argue that there is a more subtle "symmetry" which mixes the center symmetry and flavor symmetry, by using the fact that the fermions are in the representation of [SU(N c ) × SU(N f ) V ]/Z n where n := gcd(N c , N f ) is the greatest common divisor. The division by Z n leads to what we call "center-flavor symmetry". Then, we get similar constraints as in the case of adjoint fermions, under a technical definition of confinement in terms of the quantum fluctuations of the gauge field in confining phase. This has implications even at zero temperature. If the chiral symmetry is not broken, we need dynamical gauge fields to match the anomaly of the center-flavor symmetry. This partially explains the reason why there appears dual magnetic gauge group in Seiberg's description of supersymmetric QCD [6,7]. In a sense, we can directly see the existence of gauge fields via the 't Hooft anomaly.

II. SU(Nc) WITH ADJOINT FERMIONS
In this section, we consider SU(N c ) gauge theories with n f massless Weyl fermions in the adjoint representation.

A. 't Hooft anomaly of chiral and center symmetry
Here we describe the mixed 't Hooft anomaly of chiral and center symmetry [5].
a. Chiral symmetry. Classically the theory has U(n f ) = [U(1) A × SU(n f )]/Z n f chiral symmetry acting on the fermions. The U(1) A is quantum mechanically broken to the anomaly-free subgroup Z axial 2Ncn f , whose gen-erator acts on fermions ψ via Thus the chiral symmetry of the theory is reduced to [SU(n f ) × Z axial 2Ncn f ]/Z n f . The order parameter of the breaking is Φ ab = ψ a ψ b (a, b = 1, · · · , n f ) which behave as Assuming it is nonzero, there are N c distinct connected components in the moduli space of vacua. The generator of Z axial 2Ncn f transformation is implemented by the theta angle rotation θ → θ +2π which permutes the N c connected components.
The continuous part SU(n f ) of the chiral symmetry is also broken by the vacuum expectation values of the matrix (Φ ab ), which produce Goldstone bosons at each connected component. However, the details of this breaking do not play any role in the following discussion.
b. Center symmetry. Since adjoint fermions transform trivially under the center Z Nc ⊂ SU(N c ), the theory possesses the Z Nc center symmetry. The center symmetry is a typical example of 1-form symmetry [5], which acts on line operators in the present theory.
The 1-form center symmetry can be coupled to a 2form background B ∈ H 2 (X, Z Nc ), where X is spacetime, as follows. For a topologically nontrivial gauge bundle on a manifold X, we first take open covers {U a } a∈A of X such that the bundle is trivialized on each of U a . They are glued by transition functions g ab on U a ∩ U b which we take to be N c × N c matrices in the fundamental representation. For SU(N c ) (as opposed to SU(N c )/Z Nc ) bundles, they satisfy the standard consistency condition g ab g bc g ca = 1 if U a ∩ U b ∩ U c = ∅. However, when we consider SU(N c )/Z Nc bundles, we have This is allowed because exp(2πiw gauge abc /N c ) is in the center Z Nc ⊂ SU(N c ) and hence it is trivial in SU(N c )/Z Nc . These w gauge abc give an element of cohomology group which gives the obstruction to uplifting an SU(N c )/Z Nc bundle to an SU(N c ) bundle. Including the background field B ∈ H 2 (X, Z Nc ) for the center 1-form symmetry corresponds to considering gauge bundles which satisfy B = w gauge 2 . Namely, we perform path integral under this topological condition for the gauge field.
c. Mixed 't Hooft anomaly. Let us describe the mixed 't Hooft anomaly between the axial symmetry Z axial 2Ncn f and the center symmetry [5]. Under the axial rotation (2), the standard Fujikawa's argument tells us that the path integral measure Z(X) changes as where F is the gauge field strength and the trace is in the fundamental representation. If B = w gauge 2 = 0, the above phase factor is trivial because the instanton number is integral. However, when we turn on the background field B = w gauge This represents the mixed 't Hooft anomaly. d. Low energy behavior. There is an immediate consequence of the above mixed 't Hooft anomaly. It is impossible that the low energy limit have a trivial gapped vacuum with both the chiral and center symmetries unbroken.
By looking at the two-loop beta function, a likely scenario is as follows [9]. When n f ≤ 2, the chiral symmetry is spontaneously broken. When n f = 5, it flows to a conformal fixed point. The cases n f = 3, 4 are unclear, but n f = 4 may have a conformal fixed point.

B. Constraints on phase transition
Next we use the mixed 't Hooft anomaly of the Z axial 2Ncn f and the Z Nc center symmetry to constrain the thermal phase transition. We reduce the theory along the thermal circle S 1 T and obtain the effective theory on R 3 .The center symmetry now splits into two global symmetries. One is the 0-form center symmetry Z 0-form Nc acting on the Polyakov loop L = tr Nc P exp(i S 1 T A). The other is the 1-form center symmetry Z 1-form Nc acting on space-like Wilson loops extending along R 3 .
The three-dimensional effective theory still has the mixed "triangle" 't Hooft anomaly among the three sym- 2Ncn f ) obtained by dimensional reduction of (7). The anomaly forbids the three symmetries to be simultaneously preserved.
a. High/Low temperature phases. We summarize the symmetry breaking in the high/low temperature limit. First, we note that the fermions have the antiperiodic boundary condition along the thermal circle and have no zero-modes on R 3 . Then, when the temperature is sufficiently high, they can be safely integrated out. The axial Z axial 2Ncn f is unbroken. The remaining degrees of freedom consists of the 3d Yang-Mills and the periodic scalars coming from the gauge field in the direction S 1 T . The scalars get the effective potential at the one-loop level such that the Z 0-form Nc is broken. The 3d Yang-Mills is expected to confine with the area law for space-like Wilson loops. Therefore, at extremely high temperature, the Z 0-form Nc is broken, while the Z 1-form Nc × Z axial 2Ncn f are unbroken. At very low temperature, the theory can be regarded as four-dimensional. We focus on the case in which the theory confines and the chiral symmetry is broken. Then Z axial 2Ncn f is broken, while Z 0-form b. Inequality for T deconf and T chiral . We can define at least two critical temperatures: deconfinement temperature T deconf for Z 0-form Nc , and chiral symmetry restoration temperature T chiral for Z axial 2Ncn f . We don't consider the cases with more than two critical temperatures. Now, suppose that the chiral symmetry is restored by the Landau-Ginzburg effective action (1). Then the mixed 't Hooft anomaly implies the inequality between the two temperatures, The reason is as follows. Suppose (8)  and Z axial 2Ncn f are unbroken. If the physics near T chiral is described by (1), there is no way to break the 1-form symmetry Z 1-form Nc , and also there is no gapless degrees of freedom in T chiral < T < T deconf . This contradicts with the anomaly because all the symmetries are unbroken and there is no degrees of freedom to match the anomaly.
If (8) holds, both Z 0-form Nc and Z axial 2Ncn f are broken in T deconf < T < T chiral . This is consistent with the mixed 't Hooft anomaly. But it is a little counter-intuitive to use (1) because, intuitively, the gluons and quarks are liberated in the deconfining phase T ∼ T chiral > T deconf .
Finally, let us mention two alternative scenarios without assuming (1). They do not require (8) .
1. There is a single first-order phase transition at T c = T chiral = T deconf . When we cross the temperature T c , the decofinement transition and the chiral symmetry restoration occurs at the same time.
2. We allow a phase with broken Z 1-form Nc in T chiral < T < T deconf . Namely, we have a Higgs phase for the effective 3d Yang-Mills in the intermediate temperatures as discussed in [4]. However, this scenario seems difficult in the presence of the order parameter Φ ab .

III. SU(Nc) WITH FUNDAMENTAL FERMIONS
In this section, we consider SU(N c ) gauge theories with massless fermions in the fundamental and antifundamental representations N c + N c . We assume that the flavor number N f and the color number N c have a nontrivial greatest common divisor n := gcd(N f , N c ) = 1 which includes the case N c = N f , such as the SU(3) QCD in the massless limit of up, down and strange quarks.

A. Center-flavor symmetry
First, we explain a way to introduce non-trivial background fields to detect the anomaly.
When matter fields are in the fundamental representation, it does not make mathematical sense to take transition functions as in (4) with w gauge abc = 0. However, we can avoid this problem by the following trick. 1 The matter fields are in the bifundamental representations Under this condition, the fermions are put on X because the total transition functions (g ⊗ h † ) ab satisfy (g ⊗ h † ) ab · (g ⊗ h † ) bc · (g ⊗ h † ) ca = 1. We call the "symmetry" corresponding to this background as center-flavor symmetry, although we do not give Hilbert space interpretation. 1 The idea similar to here appeared in [20] where spinors are put on non-spin 4-manifolds by considering spin c structure. The gravitational background there corresponds to the flavor background here, and the U(1) gauge field there corresponds to the SU(N ) gauge field here. Analogous interplay between global and gauge symmetries have also appeared in recent discussions of topological phases of matter, see e.g., [21][22][23]. Formally, we are going to use the fact that the flavor symmetry acting on gauge invariant operators is SU(N f ) V /Zn, and it has the extension The ideas very close to ours have appeared also in [24,25].
a. Anomaly. We can see the existence of the anomaly of center-flavor symmetry by the following concrete setup. Compactify the spacetime to X = S 1 T × S 1 A × S 1 B × R C , where S 1 T will be the temporal direction (i.e., thermal circle) and S 1 A × S 1 B × R C are the spatial directions. The radii of S 1 A,B are taken to be much larger than that of S 1 T . We introduce the flavor background along S T,A,B as follows [8,27,28]. In the direction S 1 A ×S 1 B , we introduce the flavor Wilson lines Ω A and Ω B given as where I m means the unit m × m matrix, and ω A and ω B are n × n matrices with the commutation relation ω A ω B = e 2πi/n ω A ω B . Explicitly, ω A = diag(1, e 2πi/n , e 4πi/n , · · · , e 2(n−1)πi/n ) (13) We take the flavor Wilson line in the direction S 1 T to be an imaginary baryonic chemical potential µ B [3,29] The flavor background is a flat connection. For the gauge field, we have a freedom to choose their boundary conditions. Let x C ∈ R C be the coordinate. We impose boundary conditions at x C → ±∞ such that the gauge field approaches flat connections represented by gauge Wilson lines where ω A and ω B are the same ones as in the flavor background introduced above, and (V A , V B , V ±∞ ) are N c /n × N c /n unitary matrices such that det W A = det W B = det W T = 1, and det(V ±∞ ) = exp(2πim ±∞ /n) (17) for m ±∞ ∈ Z n . The V A , V B and V ±∞ commute with each other so that the gauge field is flat at infinity. The above configurations give a nontrivial w 2 as Due to this w 2 , there are fractional instantons in X with instanton charge (m +∞ − m −∞ )/n mod 1 [30] (see also [31]). We remark that explicit instanton solutions are not at all necessary for our discussion, and only the topological data are important. Then, Atiyah-Patodi-Singer (APS) index theorem states (for generic V A , V B , V ±∞ , µ B ) that there are fermion zero modes such that under the Z axial 2N f axial rotation the path integral measure Z(X) gets a phase factor This is the key anomaly for our purposes.

B. Constraints by anomaly
We would like to discuss some consequences of the anomaly (19).
a. Thermal phase transition. Because of the anomaly (19), there are constraints on phase transitions. First, let us discuss the case of a specific value of µ B . The fundamental fermions are coupled to the total Wilson line W T ⊗ Ω † T . Now, the effect of center symmetry action W T → e −2πi/Nc W T can be compensated by the shift µ B → µ B − 2π. By combining parity in the S T direction µ B → −µ B , we find that there is a Z 2 symmetry if µ B = π [3,29]. This Z 2 acts on the Polyakov loop L = tr Nc W T as L → (e −2πi/Nc L) * where e −2πi/Nc comes from the center symmetry action and the complex conjugate comes from the parity flip on S 1 T . Thus this is a symmetry whose order parameter is the Polyakov loop, and as we discuss below, it is broken at high temperature while it is unbroken at low temperature. Therefore, this Z 2 can be used for a rigorous definition of deconfinement/confinement phases at µ B = π, just as the Z 0-form Nc symmetry of the adjoint fermion theory.
At high temperature, the Z 2 is spontaneously broken just by standard perturbative computation at finite temperature, and the minima of the effective potential are at W T = I Nc and e 2πi/Nc I Nc which are related by Z 2 . Now let us take the boundary conditions as m −∞ = 0 and m +∞ = 1. These two values are in the two vacua related by the spontaneously broken Z 2 . Then m −∞ = 0 and m +∞ = 1 means that the gauge configuration approaches to these two vacua at x C → ±∞. The domain wall interpolating the two vacua is the fractional instanton.
At very low temperature, we can see that Z 2 is unbroken as follows. If it were broken, then by changing the value of µ B from π − to π + for an infinitesimal , there would be a phase transition from one phase to another which are related by Z 2 . However, the µ B is coupled to the Baryon number and all the particles having nonzero baryon numbers are heavy. Therefore, the change of µ B from π − to π + cannot change the dynamics of low energy physics (i.e., the effective theory of pions) if the temperature is significantly lower than the lowest baryon mass, and hence there are no phase transitions associated to the assumed spontaneously broken Z 2 symmetry. Therefore, the Z 2 should not be broken. This consideration is in agreement with the numerical results in [29] (see Figure 3 of that paper).
The fact that Z 2 is unbroken at low temperatures means that W T is well-fluctuating and the vacuum state has overlap with any value of m ±∞ . So the boundary conditions m −∞ = 0 and m +∞ = 1 become irrelevant at low temperature.
Therefore we have the following situation; Z axial Comparison with the case of adjoint fermions is that Nc corresponds to the center-flavor background Ω A,B described above. The anomaly (19) excludes the possibility that the phase transition is simply described by (1) below the deconfinement temperature at which the Z 2 symmetry is broken.
Next we give a speculative discussions on the case of 0 ≤ µ B < π. Even though there is no Z 2 symmetry, we may still define confinement in the following technical sense. Our spacetime is Euclidean, so we can regard R C as a time direction and find a ground state |Ω on S 1 (This idea is familiar in 2d CFT.) The boundary conditions m ±∞ also define physical states |m ±∞ . 2 We define confinement as the statement that m = 1|Ω = 0 as well as m = 0|Ω = 0. The |m = 0 is expected to always have overlap with |Ω for |µ B | ≤ π.
In the presence of Z 2 we have m = 1|Ω = m = 0|Ω if Z 2 is unbroken in |Ω . So this condition m = 1|Ω = 0 is a generalization of the above case of µ B = π to any value of µ B . Deconfinement means that m|Ω = 0 for m = 0. This criterion of (de)confinement might be supported by analytic picture of confinement [30]. Intuitively, confinement means (see e.g., [8]) that the gauge field and in particular W T is quantum mechanically well-fluctuating. Then |Ω is a superposition of states with all possible values of W T . Because m is related to the values of W T (see (17)), we expect m|Ω = 0 for all m in confining phase. On the other hand, it is localized near W T = I Nc in deconfining phase. We leave it as a future work to study more details on this criterion.
If the theory confines in the above technical sense, we do not have a domain wall interpolating x C = +∞ and x C = −∞. Thus the anomaly cannot be matched by a domain wall (i.e., fractional instanton) and hence the chiral symmetry must be broken. Then the anomaly constraint (19) works in the same way as in the case of µ B = π. Again, (1) is impossible below the deconfinement temperature.
Let us make another remark about the effects of µ B . Because the baryon charge of quarks is 1/N c , the µ B is coupled to quarks via µ B /N c . Therefore, in large N c counting, the effect of µ B /N c is a sub-leading effect. The inequality T deconf ≤ T chiral is indeed satisfied [33] in a holographic QCD model [34]. For recent numerical studies at N c = 3, see e.g., [35,36].
b. Dual magnetic gauge group. The anomaly (19) has implications at zero temperature. Let us consider supersymmetric QCD. One of the most remarkable phenomena is the appearance of dual magnetic gauge group in Seiberg's description of those theories [6,7]. The 't Hooft anomaly of the usual chiral symmetries was important for those results.
We would like to add one more evidence which may give some new insight. Suppose that the chiral symmetry is unbroken. Now, if the theory contains only scalars and fermions in low energy after confinement, it cannot match the anomaly (19). The key fact here is that the anomaly exists even when the flavor background is flat and that the flat background itself does not produce any fermion zero modes. Therefore, the anomaly (19) detects the existence of dual magnetic gauge fields. Now let us apply anomaly matching of (19) to supersymmetric QCD.
• N f ≤ N c : the chiral symmetry is broken. 3 • N f = N c + 1: the chiral symmetry is unbroken, and the theory confines. In this case, we have gcd(N c , N c + 1) = 1, so the anomaly (19) vanishes. Thus dual magnetic gauge group need not appear.
• N c + 2 ≤ N f < 3N c : the chiral symmetry is unbroken and there appears dual magnetic gauge group SU(N f − N c ). This satisfies the anomaly matching of (19) because gcd(N c , N f ) = gcd(N f − N c , N f ). 4 Our anomaly argument does not rely on supersymmetry at all, so it can also constrain magnetic gauge group in non-supersymmetric QCD. It would be interesting to apply these constraints to ideas such as hidden local symmetry [37] (see also [38]).
Note added: On the same day we submitted our paper to arXiv, two closely related papers [39,40] appeared. In [39], the authors obtained the inequality (8) for adjoint QCD by using the same anomaly as ours. In [40], the authors introduced the new order-parameter for QCD by using the mixing of the center and flavor symmetry.