Abstract
In this note we study IR limits of pure two-dimensional supersymmetric gauge theories with semisimple non-simply-connected gauge groups including SU(k)/ℤk, SO(2k)/ℤ2, Sp(2k)/ℤ2, E6/ℤ3, and E7/ℤ2 for various discrete theta angles, both directly in the gauge theory and also in nonabelian mirrors, extending a classification begun in previous work. We find in each case that there are supersymmetric vacua for precisely one value of the discrete theta angle, and no supersymmetric vacua for other values, hence supersymmetry is broken in the IR for most discrete theta angles. Furthermore, for the one distinguished value of the discrete theta angle for which supersymmetry is unbroken, the theory has as many twisted chiral multiplet degrees of freedom in the IR as the rank. We take this opportunity to further develop the technology of nonabelian mirrors to discuss how the mirror to a G gauge theory differs from the mirror to a G/K gauge theory for K a subgroup of the center of G. In particular, the discrete theta angles in these cases are considerably more intricate than those of the pure gauge theories studied in previous papers, so we discuss the realization of these more complex discrete theta angles in the mirror construction. We find that discrete theta angles, both in the original gauge theory and their mirrors, are intimately related to the description of centers of universal covering groups as quotients of weight lattices by root sublattices. We perform numerous consistency checks, comparing results against basic group-theoretic relations as well as with decomposition, which describes how two-dimensional theories with one-form symmetries (such as pure gauge theories with nontrivial centers) decompose into disjoint unions, in this case of pure gauge theories with quotiented gauge groups and discrete theta angles.
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References
O. Aharony, S. S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].
W. Gu and E. Sharpe, A proposal for nonabelian mirrors, arXiv:1806.04678 [INSPIRE].
Z. Chen, W. Gu, H. Parsian and E. Sharpe, Two-dimensional supersymmetric gauge theories with exceptional gauge groups, Adv. Theor. Math. Phys. 24 (2020) 67 [arXiv:1808.04070] [INSPIRE].
K. Hori, Duality in two-dimensional (2, 2) supersymmetric non-Abelian gauge theories, JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].
K. Hori, On global aspects of gauged Wess-Zumino-Witten model, hep-th/9402019 [INSPIRE].
K. Hori, Global aspects of gauged Wess-Zumino-Witten models, Commun. Math. Phys. 182 (1996) 1 [hep-th/9411134] [INSPIRE].
E. Witten, θ vacua in two-dimensional quantum chromodynamics, Nuovo Cim. A 51 (1979) 325 [INSPIRE].
D. Gaiotto, G. W. Moore and A. Neitzke, Framed BPS states, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
T. Pantev and E. Sharpe, Notes on gauging noneffective group actions, hep-th/0502027 [INSPIRE].
T. Pantev and E. Sharpe, GLSM’s for gerbes (and other toric stacks), Adv. Theor. Math. Phys. 10 (2006) 77 [hep-th/0502053] [INSPIRE].
S. Hellerman, A. Henriques, T. Pantev, E. Sharpe and M. Ando, Cluster decomposition, T-duality, and gerby CFT’s, Adv. Theor. Math. Phys. 11 (2007) 751 [hep-th/0606034] [INSPIRE].
J. Halverson, V. Kumar and D. R. Morrison, New methods for characterizing phases of 2D supersymmetric gauge theories, JHEP 09 (2013) 143 [arXiv:1305.3278] [INSPIRE].
A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe, Non-birational twisted derived equivalences in abelian GLSMs, Commun. Math. Phys. 294 (2010) 605 [arXiv:0709.3855] [INSPIRE].
H. Parsian, E. Sharpe and H. Zou, (0, 2) versions of exotic (2, 2) GLSMs, Int. J. Mod. Phys. A 33 (2018) 1850113 [arXiv:1803.00286] [INSPIRE].
E. Andreini, Y. Jiang and H. H. Tseng, Gromov-Witten theory of etale gerbes, I: root gerbes, arXiv:0907.2087.
H. H. Tseng, On degree zero elliptic orbifold Gromov-Witten invariants, arXiv:0912.3580.
A. Gholampour and H. H. Tseng, On Donaldson-Thomas invariants of threefold stacks and gerbes, arXiv:1001.0435.
E. Sharpe, Decomposition in diverse dimensions, Phys. Rev. D 90 (2014) 025030 [arXiv:1404.3986] [INSPIRE].
E. Sharpe, Undoing decomposition, Int. J. Mod. Phys. A 34 (2020) 1950233 [arXiv:1911.05080] [INSPIRE].
Y. Tanizaki and M. Ünsal, Modified instanton sum in QCD and higher-groups, JHEP 03 (2020) 123 [arXiv:1912.01033] [INSPIRE].
K. Rietsch, A mirror symmetry construction for \( {qH}_T^{\ast }{\left(G/P\right)}_q \), Adv. Math. 217 (2008) 2401 [math/0511124].
C. Teleman, The role of Coulomb branches in 2D gauge theory, arXiv:1801.10124 [INSPIRE].
N. A. Nekrasov and S. L. Shatashvili, Bethe/Gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
C. Closset, S. Cremonesi and D. S. Park, The equivariant A-twist and gauged linear sigma models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
C. Closset, N. Mekareeya and D. S. Park, A-twisted correlators and Hori dualities, JHEP 08 (2017) 101 [arXiv:1705.04137] [INSPIRE].
K. Hori and M. Romo, Exact results in two-dimensional (2, 2) supersymmetric gauge theories with boundary, arXiv:1308.2438 [INSPIRE].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
W. Gu, J. Guo and E. Sharpe, A proposal for nonabelian (0, 2) mirrors, arXiv:1908.06036 [INSPIRE].
W. Gu, H. Parsian and E. Sharpe, More non-Abelian mirrors and some two-dimensional dualities, Int. J. Mod. Phys. A 34 (2019) 1950181 [arXiv:1907.06647] [INSPIRE].
W. Gu, Correlation functions in massive Landau-Ginzburg orbifolds and tests of dualities, JHEP 12 (2020) 180 [arXiv:2001.10562] [INSPIRE].
T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer, Germany (1985).
J. P. Ang, K. Roumpedakis and S. Seifnashri, Line operators of gauge theories on non-spin manifolds, JHEP 04 (2020) 087 [arXiv:1911.00589] [INSPIRE].
H. Kim, S. Kim and J. Park, 2D Seiberg-like dualities for orthogonal gauge groups, JHEP 10 (2019) 079 [arXiv:1710.06069] [INSPIRE].
H. Georgi, Lie algebras in particle physics, second edition, Perseus Books, Reading U.S.A. (1999).
W. Fulton and J. Harris, Representation theory: a first course, Graduate texts in mathematics 129, Springer, U.S.A. (1991).
J. Distler and E. Sharpe, Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys. 14 (2010) 335 [hep-th/0701244] [INSPIRE].
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Gu, W., Sharpe, E. & Zou, H. Notes on two-dimensional pure supersymmetric gauge theories. J. High Energ. Phys. 2021, 261 (2021). https://doi.org/10.1007/JHEP04(2021)261
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DOI: https://doi.org/10.1007/JHEP04(2021)261