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Spontaneous Symmetry Breaking in Pure 2D Yang-Mills Theory

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Spontaneous Symmetry Breaking in Pure 2D Yang-Mills Theory PHYSICAL REVIEW D 101, 105017 (2020) Spontaneous symmetry breaking in pure 2D Yang-Mills theory G. Aminov * Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA and ITEP NRC KI, Moscow 117218, Russia (Received 10 February 2020; accepted 11 May 2020; published 26 May 2020) 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 ¼ π there exists a class of SUð2NÞ=Z2 (N>1) gauge theories with a twofold degenerate vacuum, which spontaneously breaks the time reversal and charge conjugation symmetries. The corresponding order parameter is given by the generator O of the ZN one-form symmetry. DOI: 10.1103/PhysRevD.101.105017 I. INTRODUCTION be a trivial nondegenerate gapped state. Another example of the ‘t Hooft anomaly constraining the vacuum of the The possibility of having a number of degenerate vacua theory is related to the 2D CPn−1 model [15], where for called θ-vacua in two dimensional gauge theories was n>2 the mixed anomaly between time reversal symmetry studied in the 1970s by a number of authors [1–6]. Both and the global PSUðnÞ symmetry at θ ¼ π leads to the Abelian and non-Abelian theories were considered, and the spontaneous breaking of time reversal symmetry with a existence of the multiple vacua was shown to be indepen- twofold degeneracy of the vacuum [28]. The list of dent of the spontaneous symmetry breaking of the gauge examples could be longer, but we will conclude by symmetry. Instead, the presence of some matter fields, mentioning the works [12,13], where the ‘t Hooft anoma- either fermionic or scalar, was required. lies for discrete global symmetries in bosonic theories were In this paper we consider purely topological 2D Yang- studied in two, three, and four dimensions. Although, in Mills theory on a torus with the second Stiefel-Whitney this paper, we are not going to discuss the possible relation class added to the Lagrangian in the form of a θ term. It will of the spontaneous symmetry breaking to the anomaly, one be shown that at θ ¼ π there exists a class of SUð2NÞ=Z2 1 could hypothesize the existence of the mixed anomaly (N> ) gauge theories with a twofold degenerate vacuum. between ð−1Þ-form symmetry and the charge conjugation These two vacuum states are related by the time reversal or in the theories under consideration.1 the charge conjugation and thus indicate the spontaneous Recently, we became aware of the paper by Kapec, symmetry breaking. The corresponding order parameter is O Z Mahajan, and Stanford [25], which has partial overlap with given by the generator of the N one-form symmetry our results for the partition functions of PSUðNÞ gauge with the following action of the charge conjugation on it: theories. In [25] the higher genus partition functions were −1 −1 computed and utilized in the context of random matrix COC ¼ O : ð1:1Þ ensembles. As we will show in the main text of the paper, there is no spontaneous symmetry breaking for the case of The motivation to consider such theories comes from the PSUðNÞ gauge theories, and thus, the main results of our recent developments in generalized global symmetries and study are not covered in [25]. Also, the paper by Sharpe ‘t Hooft anomalies [7–27]. In particular, authors of [16] [29] discussing one-form symmetries in the various 2D considered SUðNÞ gauge theory in four dimensions and theories appeared recently. This paper studies the connec- showed that at θ ¼ π there is the discrete ‘t Hooft anomaly tion with the cluster decomposition and is based on a involving time reversal and the center symmetry. As a number of previous results (to name a few [30–32]). consequence of this anomaly, the vacuum at θ ¼ π cannot The paper is organized as follows. In Sec. II, we review the Hamiltonian approach for computing the partition *[email protected] functions of the pure gauge theories in two dimensions. This method originates from the work of Migdal [33] and Published by the American Physical Society under the terms of was extensively developed in the 1980s and 1990s along- the Creative Commons Attribution 4.0 International license. side other approaches for studying the 2D Yang-Mills Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. 1This possibility was pointed out by Komargodski. 2470-0010=2020=101(10)=105017(8) 105017-1 Published by the American Physical Society G. AMINOV PHYS. REV. D 101, 105017 (2020) Z theories [34–46]. We would also like to mention the path −1 Z ¼ dU1Zða; U1;U Þ integral approach by Blau and Thompson [47–49], which 1 Z leads to the same results but requires more involved X ¼ −aC2ðRÞ χ ð Þχ ð −1Þ ð Þ mathematical structures. In Sec. III we use ‘t Hooft’s e dU1 R U1 R U1 : 2:2 twisted boundary conditions [50] to compute the partition R function of the SUð2Þ=Z2 gauge theory. This computation Using the identity is equivalent to the approach used by Witten [40] to compute the SOð3Þ partition function starting from the Z SUð2Þ gauge theory. We conclude Sec. III by introducing −1 χRðVWÞ dUχ ðVUÞχ ðU WÞ¼ ; ð2:3Þ the θ term to the Lagrangian and computing the partition R R dim R function at θ ¼ π, which repeats one of the results of [51,32]. In Sec. IV we extend all of the previous arguments we get ð Þ Z to the case of SU N = N theory. However, since there is no X spontaneous symmetry breaking in PSUðNÞ theory for any − ð Þ Z ¼ e aC2 R : ð2:4Þ N, in Sec. V we focus on the more general case of R SUðNÞ=Γ, where Γ is the subgroup of the center of SUðNÞ. Indeed, we find out that there exists a class of For SUð2Þ, we have C2ðRÞ¼jðj þ 1Þ with half-integer j SUð2NÞ=Z2, N>1 theories with two vacuum states given and hence by the fundamental and antifundamental representations of SUð2NÞ. Additionally, we argue that there exists a broader X∞ Z ¼ e−amðmþ2Þ=4: ð2:5Þ class of SUð2NmÞ=Z2m theories with degenerate vacuum. Finally, in Sec. VI we relate the twofold degeneracy of the m¼0 vacuum to the spontaneous breaking of C and T symmetries. III. SU(2)=Z2 GAUGE THEORY II. REVIEW: SU(2) GAUGE THEORY Now we consider the cylinder as a rectangular plaquette with one pair of opposite sides being glued together. To derive the answer for the partition function on the According to [50] we can introduce the following boundary torus, we consider the canonical quantization of the theory conditions for the vector potential Aμðx; tÞ: on a cylinder (see Fig. 1). The corresponding propagator [33,40,45] is given by ˜ AμðL; tÞ¼Ω1ðtÞAμð0;tÞ; ð Þ X ˜ 3:1 Aμðx; TÞ¼Ω2ðxÞAμðx; 0Þ; −aC2ðRÞ Zða; U1;U2Þ¼ χRðU1ÞχRðU2Þe ; ð2:1Þ R Ω ¼ Ω Ω−1 þ { Ω∂ Ω−1 with the notation Aμ Aμ g μ . However, 2 since we are using the A0 ¼ 0 gauge, we are left with time where a ¼ e LT=2 is proportional to the surface area of the independent gauge transformations: cylinder. The final answer for the partition function on the torus comes from gluing together the opposite sides of the ˜ ˜ Ω1ðtÞ¼Ω1ð0Þ: ð : Þ cylinder: 3 2 ˜ Now making a constant gauge transformation Aμ → ΩAμ with ˜ ˜ ˜ −1 ˜ ˜ ˜ −1 ΩΩ1Ω ¼ Id; ΩðxÞ ≡ ΩΩ2ðxÞΩ ; ð3:3Þ we arrive at A1ðL; tÞ¼A1ð0;tÞ; ð3:4Þ A1ðx; TÞ¼ΩðxÞA1ðx; 0Þ; and the consistency condition for Ω is Ωð0Þ¼Ωð Þ ∈ Z ð Þ FIG. 1. Holonomies on the cylinder. L z; z 2: 3:5 105017-2 SPONTANEOUS SYMMETRY BREAKING IN PURE 2D … PHYS. REV. D 101, 105017 (2020) X Z −aC2ðRÞ −1 Z1 ¼ e dU1χRðU1ÞχRðz1U1 Þ R X χ ðz1Þ ¼ e−aC2ðRÞ R : ð3:13Þ R dim R Using the Weyl character formula for the SUð2Þ case FIG. 2. Holonomy around the boundary of the plaquette. e{ϕ 0 sinðnϕÞ χR ¼ ;n¼ dim R; ð3:14Þ 0 e−{ϕ sinðϕÞ Now we should be more accurate with the definition of the nþ1 holonomy around the boundary (see Fig. 2). For each Ui, we get χRðz1Þ¼nð−1Þ and we have X∞ Z m −amðmþ2Þ=4 L Z1 ¼ ð−1Þ e : ð3:15Þ U1 ¼ Pexp A1ðx; 0Þdx ; ð3:6Þ m¼0 0 Z Thus, the answer for the total partition function is T ¼ ð Þ ð Þ U Pexp A1 L; t dt ; 3:7 ∞ 0 X Z ¼ e−akðkþ1Þ; ð3:16Þ Z ¼0 1 dx k U2 ¼ Pexp A1ðx; TÞ dσ2 0 dσ2 which coincides with the general answer (2.4) for the ð3Þ with xðσ2 ¼ 0Þ¼L; xðσ2 ¼ 1Þ¼0: ð3:8Þ group SO . Since we have two kinds of vector potentials defined A. Adding w2 to SU(2)=Z2 by the boundary conditions with Ω0ð0Þ¼Ω0ðLÞ and Following [40,51], we use a topological approach to the Ω1ð0Þ¼Ω1ðLÞz1, z1 ≠ Id, the total partition function calculation of the partition function with the second Stiefel- can be represented as the following sum: Whitney class w2 added to the Lagrangian: 1 Z Z ¼ ðZ0 þ Z1Þ; ð3:9Þ þ θ 2 ZSW ¼ DAe{SYM { w2 ; ð3:17Þ where the factor 1=2 comes from the normalization of the where the dependence on θ is 2π-periodic and different Haar measure to give volume one.
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