Abstract
We study 2d \( \mathcal{N} \) = (2, 2) quiver gauge theories without flavor nodes. There is a special class of quivers whose gauge group ranks stay positive in any duality frame. We illustrate this with the Abelian Kronecker quiver and the Abelian Markov quiver as the simplest examples. In the geometric phase, they engineer an infinite sequence of projective spaces and hypersurfaces in Calabi-Yau spaces, respectively. We show that the Markov quiver provides an Abelianization of SU(3) SQCD. Turning on the FI parameters and the θ angles for the Abelian quiver effectively deform SQCD by such parameters. For an Abelian necklace quiver corresponding to SU(k) SQCD, we find evidence for singular loci supporting non-compact Coulomb branches in the Kähler moduli space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
R. Donagi and E. Sharpe, GLSM’s for partial flag manifolds, J. Geom. Phys. 58 (2008) 1662 [arXiv:0704.1761] [INSPIRE].
H. Jockers et al., Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties, JHEP 11 (2012) 166 [arXiv:1205.3192] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants, Commun. Math. Phys. 333 (2015) 717 [arXiv:1307.5997] [INSPIRE].
F. Benini, D.S. Park and P. Zhao, Cluster Algebras from Dualities of 2d \( \mathcal{N} \) = (2, 2) Quiver Gauge Theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].
S. Franco et al., 2d (0, 2) Quiver Gauge Theories and D-Branes, JHEP 09 (2015) 072 [arXiv:1506.03818] [INSPIRE].
C. Closset, J. Guo and E. Sharpe, B-branes and supersymmetric quivers in 2d, JHEP 02 (2018) 051 [arXiv:1711.10195] [INSPIRE].
J. Guo, Quantum Sheaf Cohomology and Duality of Flag Manifolds, Commun. Math. Phys. 374 (2019) 661 [arXiv:1808.00716] [INSPIRE].
C. Closset, S. Franco, J. Guo and A. Hasan, Graded quivers and B-branes at Calabi-Yau singularities, JHEP 03 (2019) 053 [arXiv:1811.07016] [INSPIRE].
J. Guo and H. Zou, Quantum cohomology of symplectic flag manifolds, J. Phys. A 55 (2022) 275401 [arXiv:2107.09880] [INSPIRE].
D. Galakhov, W. Li and M. Yamazaki, Gauge/Bethe correspondence from quiver BPS algebras, JHEP 11 (2022) 119 [arXiv:2206.13340] [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
K. Hori and D. Tong, Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N = (2, 2) Theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].
F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N} \) = (2, 2) Gauge Theories on S2 and Vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
D. Berenstein and M.R. Douglas, Seiberg duality for quiver gauge theories, hep-th/0207027 [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
C. Closset, Seiberg duality for Chern-Simons quivers and D-brane mutations, JHEP 03 (2012) 056 [arXiv:1201.2432] [INSPIRE].
D. Xie, Three dimensional Seiberg-like duality and tropical cluster algebra, arXiv:1311.0889 [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
H. Jockers et al., Two-Sphere Partition Functions and Gromov-Witten Invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].
Y. Ruan, Nonabelian gauged linear sigma model, Chin. Ann. Math. B 38 (2017) 963.
Y. Zhang, Gromov-Witten Theory of An type quiver varieties and Seiberg Duality, arXiv:2112.11812 [INSPIRE].
M.R. Douglas, B. Fiol and C. Romelsberger, The Spectrum of BPS branes on a noncompact Calabi-Yau, JHEP 09 (2005) 057 [hep-th/0003263] [INSPIRE].
F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].
I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: Duality cascades and χ-SB resolution of naked singularities, JHEP 08 (2000) 052 [hep-th/0007191] [INSPIRE].
B. Jia, E. Sharpe and R. Wu, Notes on nonabelian (0, 2) theories and dualities, JHEP 08 (2014) 017 [arXiv:1401.1511] [INSPIRE].
M.R. Douglas, B.R. Greene and D.R. Morrison, Orbifold resolution by D-branes, Nucl. Phys. B 506 (1997) 84 [hep-th/9704151] [INSPIRE].
D.-E. Diaconescu and J. Gomis, Fractional branes and boundary states in orbifold theories, JHEP 10 (2000) 001 [hep-th/9906242] [INSPIRE].
F. Cachazo et al., A Geometric unification of dualities, Nucl. Phys. B 628 (2002) 3 [hep-th/0110028] [INSPIRE].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [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].
S.-J. Lee, Z.-L. Wang and P. Yi, Abelianization of BPS Quivers and the Refined Higgs Index, JHEP 02 (2014) 047 [arXiv:1310.1265] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Am. Math. Soc. 15 (2001) 497 [math/0104151].
S.R. Coleman, More About the Massive Schwinger Model, Annals Phys. 101 (1976) 239 [INSPIRE].
F. Benini and B. Le Floch, Supersymmetric localization in two dimensions, J. Phys. A 50 (2017) 443003 [arXiv:1608.02955] [INSPIRE].
D.S. Park, Recent developments in 2d \( \mathcal{N} \) = (2, 2) supersymmetric gauge theories, Int. J. Mod. Phys. A 31 (2016) 1630045 [arXiv:1608.03607] [INSPIRE].
H. Derksen, J. Weyman and A. Zelevinsky, Quivers with potentials and their representations I: Mutations, Selecta Math. 14 (2008) 59 [arXiv:0704.0649].
V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 [math/0311149].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, The long flow to freedom, JHEP 02 (2017) 056 [arXiv:1611.02763] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N} \) = 2 Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
K. Ohmori, N. Seiberg and S.-H. Shao, Sigma Models on Flags, SciPost Phys. 6 (2019) 017 [arXiv:1809.10604] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003) 63.
D. Galakhov et al., Wild Wall Crossing and BPS Giants, JHEP 11 (2013) 046 [arXiv:1305.5454] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, Generalized quiver mutations and single-centered indices, JHEP 01 (2014) 050 [arXiv:1309.7053] [INSPIRE].
C. Cordova and S.-H. Shao, Asymptotics of Ground State Degeneracies in Quiver Quantum Mechanics, Commun. Num. Theor. Phys. 10 (2016) 339 [arXiv:1503.03178] [INSPIRE].
H. Kim, Scaling Behaviour of Quiver Quantum Mechanics, JHEP 07 (2015) 079 [arXiv:1503.02623] [INSPIRE].
M. Alim et al., \( \mathcal{N} \) = 2 quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014) 27 [arXiv:1112.3984] [INSPIRE].
H. Kim, S.-J. Lee and P. Yi, Mutation, Witten Index, and Quiver Invariant, JHEP 07 (2015) 093 [arXiv:1504.00068] [INSPIRE].
S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes, Acta Math. 201 (2008) 83.
B. Feng, A. Hanany and Y.-H. He, D-brane gauge theories from toric singularities and toric duality, Nucl. Phys. B 595 (2001) 165 [hep-th/0003085] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
K. Hori, Duality In Two-Dimensional (2, 2) Supersymmetric Non-Abelian Gauge Theories, JHEP 10 (2013) 121 [arXiv:1104.2853] [INSPIRE].
G. Sivek, On Vanishing Sums of Distinct Roots of Unity, Integers 10 (2010) 365.
T.Y. Lam and K.H. Leung, On Vanishing Sums of Roots of Unity, J. Algebra 224 (2000) 91.
Acknowledgments
We thank Yingchun Zhang for raising a question on unframed quivers that led to this note. We thank Jin Chen, Wei Cui, Wei Gu, Babak Haghighat, Taro Kimura, Sungjay Lee, Mauricio Romo, Eric Sharpe, Fengjun Xu, Junya Yagi, and especially Prajna Lin for discussions. We thank Hongfei Shu and Ruidong Zhu for collaboration on a related project. PZ is supported in part by a China Postdoctoral Science Foundation Special Fund, grant number Y9Y2231. HZ is supported by the China Postdoctoral Science Foundation (Grant No. 2022M720509).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2206.14793
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Zhao, P., Zou, H. Remarks on 2d unframed quiver gauge theories. J. High Energ. Phys. 2023, 121 (2023). https://doi.org/10.1007/JHEP05(2023)121
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2023)121