Spontaneous symmetry breaking in pure 2D Yang-Mills theory

We consider purely topological $2$d Yang-Mills theory on a torus with the second Stiefel-Whitney class added to the Lagrangian in the form of a $\theta$-term. It will be shown, that at $\theta=\pi$ there exists a class of $SU(2 N)/\mathbb{Z}_2$ ($N>1$) gauge theories with a two-fold degenerate vacuum, which spontaneously breaks the time reversal and charge conjugation symmetries. The corresponding order parameter is given by the generator $\mathcal{O}$ of the $\mathbb{Z}_N$ one-form symmetry.


Introduction
The possibility of having a number of degenerate vacua called θ-vacua in two dimensional gauge theories was studied in the 70's by a number of authors [1,2,3,4,5,6]. Both abelian and non-abelian theories were considered and the existence of the multiple vacua was shown to be independent of the spontaneous symmetry breaking of the gauge symmetry. Instead, the presence of some matter fields, either fermionic or scalar, was required.
In this note we consider purely topological 2d Yang-Mills theory on a torus with the second Stiefel-Whitney class added to the Lagrangian in the form of a θ-term. It will be shown, that at θ = π there exists a class of SU (2N )/Z2 (N > 1) gauge theories with a two-fold degenerate vacuum. These two vacuum states are related by the time reversal or the charge conjugation and thus indicate the spontaneous symmetry breaking. The corresponding order parameter is given by the generator O of the ZN one-form symmetry with the following action of the charge conjugation on it: (1.1) The motivation to consider such theories comes from the recent developments in generalized global symmetries and 't Hooft anomalies [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. In particular, authors of [16] considered SU (N ) gauge theory in 4 dimensions and showed that at θ = π there is the discrete 't Hooft anomaly involving time reversal and the center symmetry. As a consequence of this anomaly, the vacuum at θ = π cannot be a trivial non-degenerate gapped state. Another example of 't Hooft anomaly constraining the vacuum of the theory is related to the 2d CP n−1 model [15], where for n > 2 the mixed anomaly between time reversal symmetry and the global P SU (n) symmetry at θ = π leads to the spontaneous breaking of time reversal symmetry with a two-fold degeneracy of the vacuum [26]. The list of the examples could be made longer, but we will conclude by mentioning the works [12,13], where the 't Hooft anomalies for discrete global symmetries in bosonic theories were studied in 2, 3 and 4 dimensions. Although in this note we are not going to discuss possible relation of the spontaneous symmetry breaking to the anomaly, but one could hypothesize the existence of the mixed anomaly between (−1)-form symmetry and the charge conjugation in the theories under consideration 1 . While the note was in preparation, we became aware of the paper by D. Kapec, R. Mahajan and D. Stanford [25], which has partial overlap with our results. In [25] the higher genus partition functions were computed and utilized in the context of random matrix ensembles. Also, the paper by E. Sharpe [27] discussing 1-form symmetries in the various 2d theories appeared soon after the first draft of this note. This paper studies the connection with the cluster decomposition and is based on a number of previous results (to name a few [28,29,30]).
The note is organized as follows. In section 2 we review the Hamiltonian approach for computing the partition functions of the pure gauge theories in two dimensions. This method originates from the work of A. Migdal [31] and was extensively developed in the 80's and 90's alongside other approaches for studying the 2d Yang-Mills theories [32,33,34,35,36,37,38,39,40,41,42,43,44]. We would also like to mention the path integral approach by M. Blau and G. Thompson [45,46,47], which leads to the same results, but requires more involved mathematical structures. In section 3 we use G. 't Hooft's twisted boundary conditions [48] to compute the partition function of the SU (2)/Z2 gauge theory. This computation is equivalent to the approach used by E. Witten [38] to compute the SO (3) partition function starting from the SU (2) gauge theory. We conclude section 3 by introducing the θ-term to the Lagrangian and computing the partition function at θ = π, which repeats one of the results of [49] and [30]. In section 4 we extend all the previous arguments to the case of SU (N )/ZN theory. However, since there is no spontaneous symmetry breaking in P SU (N ) theory for any N , we switch in section 5 to the more general case of SU (N ) /Γ, where Γ is the subgroup of the center of SU (N ). Indeed, we find out that there exists a class of SU (2N )/Z2, N > 1 theories with two vacuum states given by the fundamental and antifundamental representations of SU (2N ). Additionally, we argue that there exists a broader class of SU (2N m)/Z2m theories with degenerate vacuum. Finally, in section 6 we relate the two-fold degeneracy of the vacuum to the spontaneous breaking of C and T symmetries.

Review: SU (2) gauge theory
To derive the answer for the partition function on the torus we consider the canonical quantization of the theory on a cylinder: The corresponding propagator [31,38,43] is given by where a = e 2 LT /2 is proportional to the surface area of the cylinder. The final answer for the partition function on the torus comes from gluing together the opposite sides of the cylinder: Using the identity 3 SU (2)/Z 2 gauge theory Now, we consider the cylinder as a rectangular plaquette with one pair of opposite sides being glued together. According to [48] we can introduce the following boundary conditions for the vector potential Aµ (x, t): with the notation ΩAµ = ΩAµΩ −1 + ı g Ω∂µΩ −1 . However, since we are using the A0 = 0 gauge, we are left with time-independent gauge transformations: Now, making a constant gauge transformation Aµ → Ω Aµ with 4) and the consistency condition for Ω is Now we should be more accurate with the definition of the holonomy around the boundary: Since we have two kinds of vector potentials defined by the boundary conditions with Ω0 (0) = Ω0 (L) and Ω1 (0) = Ω1 (L) z1, z1 = Id, the total partition function can be represented as the following sum: where the factor 1/2 comes from the normalization of the Haar measure to give volume one. Now, Z0 corresponds to the periodic boundary conditions as in the case of pure SU (2) and we already know the answer: To compute Z1 we use the boundary conditions to derive Then the partition function for the cylinder is Applying the gluing procedure and integrating over U1 we arrive at Using the Weyl character formula for the SU (2) case χR e ıφ 0 0 e −ıφ = sin (n φ) sin (φ) , n = dim R, (3.14) we get χR (z1) = n (−1) n+1 and Thus, the answer for the total partition function is where the dependence on θ is 2π-periodic and different possible values of theta in the SU (2) case are θ = 0, π. Since w2 only depends on the topological type of the bundle, the path integral splits into two parts, corresponding to the trivial and nontrivial SO (3) Repeating the steps from the previous section we write the partition function as where Adding the θ term to the Lagrangian as in (3.17) will affect the partition function in the similar way as before, but in the SU (N ) case theta can take more values inside the [0, 2π) interval. Labeling these possible values by κ, we get This gives for the characters of z k in the representation (n, m) χ (n,m) (z k ) = dim R (n,m) e 2πı k(n+2 m)/3 , k = 0, 1, 2.   we get for the partition function where the only non-zero terms are those that have N−1 j=1 j qj ≡ 0 (mod N ). The eigenvalues of the quadratic Casimir operator in (4.14) are given by [50] C2 (Rq) = where G ij is the inverse of the symmetrized Cartan matrix Gij [51]: and we are using the normalization, which provides the Killing metric of the form g ab = 1 2 δ ab and (αi, αi) = 2. Adding the usual θ term with θ = θκ, we obtain j qj mod N , κ = 1, . . . , N − 1. 5 Looking for two vacua in SU (N ) /Γ, Γ ⊂ Z N gauge theory In this section we will consider the more general case, when the factor group is taken with respect to the subgroup Γ of the center of SU (N ). By now we went through several derivations of the partition functions and it is clear what is the generalization of (4.17) for Γ = ZN . If the order of Γ is n, then we have n non-equivalent periodic boundary conditions (3.4) and the corresponding partition function is j qj mod n , κ = 1, . . . , n − 1. (5.1)
As it can be checked directly, there is no such value of κ that would produce two vacua. However, we can also consider the case of SU (4)/Z2 with κ = 1 and the following partition function: In this case the two vacua contributions are given by q = (1, 0, 0) and q = (0, 0, 1).

General case of SU(2N)/Z 2 with N > 1
It is easy to show, that the first non-trivial example of 2d theory with two vacua discussed earlier is just one of the infinite series of SU (2N )/Z2 theories with N > 1. We again consider the partition function Z SW κ Γ=Z 2 with κ = 1 or, equivalently, θ = π: To show that these theories have two vacua we need the following facts about the inverse Cartan matrix G ij . The first fact is that all elements of this matrix are strictly positive: Second, the diagonal elements G ii are given by [52,51]: And finally, the following relations hold: ∀j = 2, . . . , N : However, explicit computations for a number of different values of N and m suggest that the above states are always the lowest energy states of the theory with κ = m. If we assume that there are states with even lower energies, then they will also come in pairs. This allows us to conclude that the vacuum of the SU (2N m)/Z2m theory with κ = m is at least two-fold degenerate. Moreover, the lack of discrete symmetries with order higher than 2 hints that the two-fold degeneracy is the only option.
6 Spontaneous symmetry breaking in SU (2N )/Z 2 theories with θ = π, N > 1 If we look at the Dynkin coefficients corresponding to the two vacuum states, we see that these states are given by the fundamental and antifundamental representations of SU (2N ). Hence the question is what transformation brings us from one representation to its complex conjugate. Since the wave functions in the propagator (2.1) are given by . From the Gauss Law constraint one could derive the following transformation rules for the C, P and T operators: Thus, the C-symmetry (as well as T) is spontaneously broken and the overall CPT-symmetry is conserved, since both CT and P act trivially on the wave functions χR (U ). Spontaneously broken C-and T-symmetries lead to the domain wall between the two vacuum states χF (U ) and χF (U ). In the theories under consideration there is a discrete one-form symmetry ZN , generated by a local unitary operator O [8,53]. This local operator picks up a phase when crossing the domain wall. To figure out the phase, we consider the Z2N subgroup before factoring out Z2. As before, the corresponding characters are given by χq (z k ) = dim Rqe 2πı k(q 1 +2 q 2 +···+(2N−1)q 2N −1) /(2N) , k = 0, 1, . . . , 2N − 1. (6.5) After factoring out Z2 the generator of ZN corresponds to z1 and its action on the wave functions is simply O|F = e πı/N |F , O|F = e −πı/N |F , (6.6) where O N = 1 due to the fact that (−1) ∈ Z2 in fundamental and antifundamental representations. Here we also assume that adding second Stiefel-Whitney class only affects the Z2-charges of the states and ZN -charges remain the same. In this way the relation between the two expectation values reads