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

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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. Now Z0 corresponds to possible values of theta in the SUð2Þ case are θ ¼ 0; π. the periodic boundary conditions, as in the case of pure Since w2 only depends on the topological type of the SUð2Þ, and we already know the answer: bundle, the path integral splits into two parts, correspond- X∞ ing to the trivial and nontrivial SOð3Þ-bundles over the −amðmþ2Þ=4 Z0 ¼ e : ð3:10Þ torus. The trivial bundle is defined by the boundary m¼0 conditions (3.4) with Ω0ð0Þ¼Ω0ðLÞ, and the value of w2 is 0. The nontrivial bundle is defined by the boundary To compute Z1, we use the boundary conditions to derive conditions (3.4) with Ω1ð0Þ¼Ω1ðLÞz1, and the value of Z w2 is 1. In this way we get the following answer: 1 ¼ Ω ð0Þ ð 0Þ dx σ Ω−1ð Þ U2 1 Pexp A1 x; σ d 2 1 L 1 X∞ 0 d 2 SW {π −að2kþ1Þð2kþ3Þ=4 Z ðθ ¼ πÞ¼ ðZ0 þ e Z1Þ¼ e ; −1 −1 2 ¼ Ω1ð0ÞU1 Ω1 ðLÞ: ð3:11Þ k¼0 ð3:18Þ Then the partition function for the cylinder is X where SW stands for Stiefel-Whitney. −aC2ðRÞ Z1ða; U1;U2Þ¼ e χRðU1ÞχRðU2Þ: ð3:12Þ R IV. SU(N)=ZN GAUGE THEORY

Applying the gluing procedure and integrating over U1,we For the group SUðNÞ=ZN,wehaveN nonequivalent arrive at periodic boundary conditions (3.4) with

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Ωkð0Þ¼ΩkðLÞzk;zk ∈ ZN;k¼ 0; 1; …;N− 1: we derive for the partition function ð4:1Þ X∞ 2 2 Z ¼ e−aðn þm þnmþ3nþ3mÞ=3δð½n þ 2mmod 3Þ; Repeating the steps from the previous section, we write the n;m¼0 partition function as ð4:10Þ 1 XN−1 Z ¼ Z ; ð4:2Þ where due to the Kronecker delta function the only nonzero k þ 2 ≡ 0ð 3Þ N k¼0 terms are those that have n m mod . Adding u2 with θ ¼ θκ changes the argument of the delta where function by κ, and we get

X X∞ χ ðz Þ SW −aðn2þm2þnmþ3nþ3mÞ=3 −aC2ðRÞ R k Zκ ¼ e δð½κ þ n þ 2mmod 3Þ: Zk ¼ e : ð4:3Þ ¼0 R dim R n;m ð4:11Þ We can add the θ term to the Lagrangian as in (3.17) with w2 replaced by an invariant of PSUðNÞ bundles u2 ∈ 2ðT2; Z Þ H N [18]. Allowing theta to take more values inside B. General case: SU(N)=ZN the ½0; 2πÞ interval and labeling these values by κ, we get Labeling representations of SUðNÞ by the Dynkin 2πκ coefficients ðq1; …;qN−1Þ ≡ q and using the fundamental θκ ¼ ; κ ¼ 0; …;N− 1: ð4:4Þ N weights, we derive for the characters of zk in the representation ðq1; …;qN−1Þ, Thus, each Z acquires the factor of e{θκk, and the k 2π ð þ2 þþð −1Þ Þ χ ð Þ¼ {k q1 q2 N qN−1 =N corresponding partition function is q zk dim Rqe ; k ¼ 0; 1; …;N− 1: ð4:12Þ 1 XN−1 SW SW 2π{κk=N Zκ ≡ Z ðθ ¼ θκÞ¼ e Z : ð4:5Þ k Then with the help of the simple identity N k¼0

XN−1 e2π{kn=N ¼ Nδðn mod NÞ; ð4:13Þ A. Example: SU(3)=Z3 k¼0 Irreducible representations of SUð3Þ can be labeled by the Dynkin coefficients ðn; mÞ. The two fundamental we get for the partition function weights of SUð3Þ are X∞ XN−1 ¼ −aC2ðRqÞδ ð Þ 1 1 1 Z e jqj mod N ; 4:14 1 2 … ¼0 ¼1 μ ¼ ; pffiffiffi ; μ ¼ 0; pffiffiffi : ð4:6Þ q1; ;qN−1 j 2 2 3 3 Pwhere the only nonzero terms are those that have −1 For the characters of zk in the representation ðn; mÞ, this N ≡ 0ð Þ j¼1 jqj mod N . The eigenvalues of the quadratic gives Casimir operator in (4.14) are given by [52]

2π{kðnþ2mÞ=3 χðn;mÞðzkÞ¼dim Rðn;mÞe ;k¼ 0; 1; 2: ð4:7Þ XN−1 ð Þ¼ ð þ 2Þ kj ð Þ C2 Rq qj qk G ; 4:15 Since j;k¼1

X2 where Gij is the inverse of the symmetrized Cartan matrix 2π{kðnþ2mÞ=3 e ¼ 3δð½n þ 2mmod 3Þ; δðnÞ ≡ δ 0 n; Gij [53], k¼0 ð Þ 4:8 8ðαi; αjÞ Gij ≡ ð4:16Þ ðαi; αiÞðαj; αjÞ and

2 2 and we are using the normalization, which provides the C2ðRðn;mÞÞ¼ðn þ m þ nm þ 3n þ 3mÞ=3; ð4:9Þ 1 Killing metric of the form gab ¼ 2 δab and ðαi; αiÞ¼2.

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Adding the usual θ term with θ ¼ θκ, we obtain In this case the two vacua contributions are given by q ¼ ð1 0 0Þ q ¼ð0 0 1Þ ; ; and ; ; . X∞ XN−1 SW −aC2ðRqÞ Zκ ¼ e δ κ þ jqj mod N ; B. General case of SU(2N)=Z2 with N > 1 q1;…;qN−1¼0 j¼1 κ ¼ 1 … − 1 ð Þ It is easy to show that the first nontrivial example of 2D ; ;N : 4:17 theory with two vacua discussed earlier is just one of the infinite series of SUð2NÞ=Z2 theories with N>1.We SWj κ ¼ 1 V. LOOKING FOR TWO VACUA IN SU(N)=Γ, again consider the partition function Zκ Γ¼Z2 with θ ¼ π Γ ⊂ ZN GAUGE THEORY or, equivalently, : In this section we will consider the more general case X∞ 2XN−1 SW −aC2ðRqÞ when the factor group is taken with respect to the subgroup Z1 jΓ¼Z ¼ e δ 1 þ jqj mod 2 : Γ ð Þ 2 of the center of SU N . By this point, we have gone q1;…;q2N−1¼0 j¼1 through several derivations of the partition functions, and it ð5:5Þ is clear what the generalization is of (4.17) for Γ ≠ ZN.If Γ the order of is n, then we have n nonequivalent periodic To show that these theories have two vacua, we need the boundary conditions (3.4), and the corresponding partition following facts about the inverse Cartan matrix Gij. The function is first fact is that all elements of this matrix are strictly positive: X∞ XN−1 SW −aC2ðRqÞ Zκ ¼ e δ κ þ jq mod n ; j ∀ i; j∶Gij > 0: ð5:6Þ q1;…;qN−1¼0 j¼1 κ ¼ 1 … − 1 ð Þ ; ;n : 5:1 Second, the diagonal elements Gii are given by [53,54]

ið2N − iÞ Gii ¼ ;i≤ 2N − 1: ð5:7Þ A. N =4 4N We start with the explicit answer for the case of And finally, the following relations hold: SUð4Þ=Z4: X∞ 2XN−1 2XN−1 − ð Þ i1 ij SWj ¼ aC2 Rðq1;q2;q3Þ ∀ j ¼ 2; …;N∶ G < G ; ð5:8Þ Zκ Γ¼Z4 e ¼1 ¼1 q1;q2;q3¼0 i i 3 X 2XN−1 2XN−1 δ κ þ 4 κ ¼ 1 … 3 − þ × jqj mod ; ; ; ; ∀ j ¼ 1; …;N− 1∶ Gi;N j ¼ Gi;N j: ð5:9Þ ¼1 j i¼1 i¼1 ð5:2Þ Thus, the two vacua contributions are coming from where q ¼ð1; 0; …; 0Þ and q ¼ð0; …; 0; 1Þ: ð5:10Þ 1 2 2 2 C2ðRðq ;q ;q ÞÞ¼ ð3q1 þ 4q2 þ 3q3 þ 4q1q2 1 2 3 8 Notice that due to the superselection rules þ 2q1q3 þ 4q2q3 þ 12q1 þ 16q2 þ 12q3Þ: Z h j i¼ χ ð Þχ ð −1Þ¼δ ð Þ ð5:3Þ R2 R1 dU R1 U R2 U R1;R2; 5:11 As it can be checked directly, there is no such value of κ that we indeed have two different ground states that indicate would produce two vacua. However, we can also consider spontaneous symmetry breaking. the case of SUð4Þ=Z2 with κ ¼ 1 and the following partition function: C. Γ ≠ Z2 X∞ − ð Þ By studying some particular examples with low enough SWj ¼ aC2 Rðq1;q2;q3Þ Z1 Γ¼Z2 e values of N, one can check that the following theories q ;q ;q ¼0 1 2 3 with Γ ≠ Z2 have two vacuum states: SUð8Þ=Z4 with 3 X κ ¼ 2, SUð12Þ=Z4 with κ ¼ 2, SUð12Þ=Z6 with κ ¼ 3, δ 1 þ 2 ð Þ jqj mod : 5:4 SUð16Þ=Z4 with κ ¼ 2, and SUð16Þ=Z8 with κ ¼ 4. j¼1 Basically, any SUð2NmÞ=Z2m theory with κ ¼ m and

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1 1 2π{kðq1þ2q2þþð2N−1Þq2 −1Þ=ð2NÞ N> , m> is a candidate for having a twofold degen- χqðzkÞ¼dim Rqe N ; erate vacuum. The only obstacle to making this statement ¼ 0 1 … 2 − 1 ð Þ true, in general, is that for higher values of N and m there k ; ; ; N : 6:5 could be, in principle, states with energies lower than the Z Z energy of the following two states: After factoring out 2, the generator of N corresponds to z1, and its action on the wave functions is simply qm ¼ 1; ∀ j ≠ m∶qj ¼ 0 and qð2N−1Þm ¼ 1; OjFi¼eπ{=NjFi; OjF¯ i¼e−π{=NjF¯ i; ð6:6Þ ∀ j ≠ ð2N − 1Þm∶qj ¼ 0: ð5:12Þ ON ¼ 1 ð−1Þ ∈ Z However, explicit computations for a number of different where due to the fact that 2 in funda- values of N and m suggest that the above states are always mental and antifundamental representations. Here, we also the lowest energy states of the theory with κ ¼ m.Ifwe assume that adding a second Stiefel-Whitney class only Z Z assume that there are states with even lower energies, then affects the 2-charges of the states and N-charges remain they will also come in pairs. This allows us to conclude that the same. In this way the relation between the two expectation values reads the vacuum of the SUð2NmÞ=Z2m theory with κ ¼ m is at least twofold degenerate. Moreover, the lack of discrete hOi ¼ 2π{=NhOi ð Þ symmetries with orders higher than 2 hints that the twofold F e F¯ ; 6:7 degeneracy is the only option. which fixes the phase factor picked by O upon crossing the VI. SPONTANEOUS SYMMETRY BREAKING IN domain wall to be e2π{=N. Then the action of charge O SUð2NÞ=Z2 THEORIES WITH θ = π, N > 1 conjugation on can be inferred from If we look at the Dynkin coefficients corresponding to COC−1j i¼ −π{=Nj i COC−1j ¯ i¼ π{=Nj ¯ i the two vacuum states, we see that these states are given by F e F ; F e F : the fundamental and antifundamental representations of ð6:8Þ SUð2NÞ. Hence, the question is what transformation brings us from one representation to its complex conjugate. Since The latter implies the wave functions in the propagator (2.1) are given by COC−1 ¼ O−1: ð6:9Þ hUjRi¼χ ðUÞ¼Tr ðUÞ; R Z R L U ¼ Pexp A1ðx; tÞdx ; ð6:1Þ VII. CONCLUSION 0 In this paper we described a new mechanism of sponta- T −1 T the transformation A1 → −A1 yields U → ðU Þ ¼ U neous symmetry breaking in pure two dimensional Yang- and χRðUÞ → χR ðUÞ. From the Gauss law constraint, one Mills theories. Using the well-developed methods for could derive the following transformation rules for the C, computing the 2D partition functions on compact mani- P, and T operators: folds [33,40,45] and ‘t Hooft’s idea of twisted boundary conditions [50], we analyzed the wide range of systems and T C∶e → −e; A1 → −A1 ; ð6:2Þ presented the corresponding partition functions. We observed that there is no spontaneous symmetry breaking ð Þ P∶x → −x; A1 → −A1; ð6:3Þ in PSU N theories for any N, which led us to consider a more general case of SUðNÞ=Γ theories with Γ being the T ð Þ T∶t → −t; A1 → −A1 : ð6:4Þ subgroup of the center of SU N . Within this class of systems we found many examples of theories with a Thus, the C-symmetry (as well as T) is spontaneously degenerate vacuum state, and in the particular case of broken, and the overall CPT-symmetry is conserved since SUð2NÞ=Z2, N>1, we proved that spontaneous sym- both CT and P act trivially on the wave functions χRðUÞ. metry breaking occurs. Additionally, we argued that the Spontaneously broken C- and T-symmetries lead to the corresponding order parameter is given by the generator O Z domain wall between the two vacuum states χFðUÞ and of the N one-form symmetry. There are still a number of χ ð Þ F¯ U . In the theories under consideration there is a questions left to answer. In particular, it will be interesting discrete one-form symmetry ZN, generated by a local to prove that the same mechanism of spontaneous sym- unitary operator O [8,55]. This local operator picks up a metry breaking takes place in SUð2NmÞ=Z2m theories. As phase when crossing the domain wall. To figure out the we mentioned in the introduction, the spontaneous sym- phase, we consider the Z2N subgroup before factoring out metry breaking could also imply the existence of the mixed Z2. As before, the corresponding characters are given by anomaly, which may be a topic of a separate study.

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Another interesting question is how to perturb the and q3. In the perturbed system the energy acquires theories under consideration so that the vacuum is no nonzero contributions proportional to longer degenerate. As we discussed earlier, the two vacuum states are given by the fundamental and antifundamental 3 C3ðRqÞ¼ ðq − q Þðq þ q þ 2Þðq þ 2q þ q þ 4Þ; representations. The reason is that the energy of the states is 16 1 3 1 3 1 2 3 proportional to the eigenvalue of the quadratic Casimir operator, which does not distinguish between any given which is clearly antisymmetric in q1 and q3. Hence, the representation and its complex conjugate. However, if we energies of the two original vacuum states with q ¼ find a way to modify the theory such that the Hamiltonian ð1; 0; 0Þ and q ¼ð0; 0; 1Þ will get different corrections, will include higher order Casimir operators, we will lift the and the resulting system will have a single vacuum state. So degeneracy. Indeed, such modifications exist and were far, we described one possible approach to lifting the discussed in [40,45]. Below we will briefly repeat the degeneracy in the SUð2NÞ=Z2 theories with θ ¼ π and arguments from [40,45] and show how to perturb the N>1. The existence of any other approaches and detailed Hamiltonian by the cubic Casimir operator. calculations for the case of general N are left for In 2D it is possible to define the Lie algebra valued scalar future work. f ¼F. Thus, we can add an irrelevant operator TrðfkÞ with any k>2 as a perturbation to the original theory. This ACKNOWLEDGMENTS perturbation will affect the Hamiltonian by introducing new Casimirs of order less than or equal to k [45]. Since our goal G. A. would like to thank Zohar Komargodski for is to distinguish between the fundamental and antifunda- initiating the work and many valuable discussions and mental representations of SUðNÞ, it is enough to consider Konstantinos Roumpedakis for useful remarks and dis- k ¼ 3. Then the Hamiltonian is a linear combination of the cussions. The work was supported by the Foundation for quadratic and cubic Casimirs (in the representation basis). the Advancement of Theoretical Physics and Mathematics For example, consider the case of the SUð4Þ=Z2 theory. “BASIS” Grant No. 18-1-1-50-3 and in part by RFBR Without the perturbation the energy is proportional to Grants No. 18-02-01081-A, No. 18-31-20046-mol_a_ved, C2ðRqÞ (5.3) and symmetric under the permutation of q1 No. 18-51-05015-Arm_a, and No. 19-51-18006-Bolg_a.

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