Abstract
In this paper we revisit the S1 reduction of 4d \( \mathcal{N} \) = 1 gauge theories, considering a double scaling on the radius of the circle and on the real masses arising from the global symmetries in the compactification. We discuss the implication of this double scaling for SQCD with gauge algebra of ABCD type. We then show how our prescription translates in the reduction of the 4d superconformal index to the 3d squashed three sphere partition function. This allows us to derive the expected integral identities for the 3d dualities directly from the four dimensional ones. This is relevant for the study of orthogonal SQCD, where the derivation from the 4d index is not possible in absence of the double scaling, because of a divergence due to a flat direction in the Coulomb branch of the effective theory on the circle. Furthermore, we obtain, for the even orthogonal case, a 3d duality with a quadratic fundamental monopole superpotential already discussed in the literature, that receives in this way an explanation from 4d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
A. Gadde and W. Yan, Reducing the 4d Index to the S3 Partition Function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].
V. Niarchos, Seiberg dualities and the 3d/4d connection, JHEP 07 (2012) 075 [arXiv:1205.2086] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities for orthogonal groups, JHEP 08 (2013) 099 [arXiv:1307.0511] [INSPIRE].
N.M. Davies, T.J. Hollowood, V.V. Khoze and M.P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B 559 (1999) 123 [hep-th/9905015] [INSPIRE].
N.M. Davies, T.J. Hollowood and V.V. Khoze, Monopoles, affine algebras and the gluino condensate, J. Math. Phys. 44 (2003) 3640 [hep-th/0006011] [INSPIRE].
S. Kim, K.-M. Lee, H.-U. Yee and P. Yi, The N = 1* theories on R1+2 × S1 with twisted boundary conditions, JHEP 08 (2004) 040 [hep-th/0403076] [INSPIRE].
O. Aharony, IR duality in d = 3 N = 2 supersymmetric USp(2Nc) and U(Nc) gauge theories, Phys. Lett. B 404 (1997) 71 [hep-th/9703215] [INSPIRE].
A. Karch, Seiberg duality in three-dimensions, Phys. Lett. B 405 (1997) 79 [hep-th/9703172] [INSPIRE].
F. Benini, S. Benvenuti and S. Pasquetti, SUSY monopole potentials in 2+1 dimensions, JHEP 08 (2017) 086 [arXiv:1703.08460] [INSPIRE].
A. Amariti, I. Garozzo and N. Mekareeya, New 3d \( \mathcal{N} \) = 2 dualities from quadratic monopoles, JHEP 11 (2018) 135 [arXiv:1806.01356] [INSPIRE].
A. Amariti, L. Cassia, I. Garozzo and N. Mekareeya, Branes, partition functions and quadratic monopole superpotentials, Phys. Rev. D 100 (2019) 046001 [arXiv:1901.07559] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
E.M. Rains, Limits of elliptic hypergeometric integrals, Ramanujan J. 18 (2007) 257 [math/0607093] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
A. Amariti et al., 4D/3D reduction of dualities: mirrors on the circle, JHEP 10 (2015) 048 [arXiv:1504.02783] [INSPIRE].
L. Di Pietro and M. Honda, Cardy Formula for 4d SUSY Theories and Localization, JHEP 04 (2017) 055 [arXiv:1611.00380] [INSPIRE].
C. Hwang, S. Lee and P. Yi, Holonomy Saddles and Supersymmetry, Phys. Rev. D 97 (2018) 125013 [arXiv:1801.05460] [INSPIRE].
A. Amariti et al., The braneology of 3D dualities, J. Phys. A 48 (2015) 265401 [arXiv:1501.06571] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices, Commun. Math. Phys. 325 (2014) 421 [arXiv:1107.5788] [INSPIRE].
C. Callias, Index Theorems on Open Spaces, Commun. Math. Phys. 62 (1978) 213 [INSPIRE].
E.J. Weinberg, Fundamental Monopoles and Multi-Monopole Solutions for Arbitrary Simple Gauge Groups, Nucl. Phys. B 167 (1980) 500 [INSPIRE].
E.J. Weinberg, Fundamental Monopoles in Theories With Arbitrary Symmetry Breaking, Nucl. Phys. B 203 (1982) 445 [INSPIRE].
J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].
E. Poppitz and M. Unsal, Index theorem for topological excitations on R3 × S1 and Chern-Simons theory, JHEP 03 (2009) 027 [arXiv:0812.2085] [INSPIRE].
V. Borokhov, A. Kapustin and X.-K. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
V. Borokhov, A. Kapustin and X.-K. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
C. Cordova, P.-S. Hsin and N. Seiberg, Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups, SciPost Phys. 4 (2018) 021 [arXiv:1711.10008] [INSPIRE].
O. Aharony and I. Shamir, On O(Nc) d = 3 N = 2 supersymmetric QCD Theories, JHEP 12 (2011) 043 [arXiv:1109.5081] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
B. Assel et al., The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
L.D. Faddeev, R.M. Kashaev and A.Y. Volkov, Strongly coupled quantum discrete Liouville theory. 1. Algebraic approach and duality, Commun. Math. Phys. 219 (2001) 199 [hep-th/0006156] [INSPIRE].
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004) 247.
J.F. Van Diejen and V.P. Spiridonov, Unit Circle Elliptic Beta Integrals, Ramanujan J. 10 (2005) 187.
A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP 07 (2016) 025 [arXiv:1512.03376] [INSPIRE].
K.A. Intriligator and N. Seiberg, Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(Nc) gauge theories, Nucl. Phys. B 444 (1995) 125 [hep-th/9503179] [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].
C. Hwang, K.-J. Park and J. Park, Evidence for Aharony duality for orthogonal gauge groups, JHEP 11 (2011) 011 [arXiv:1109.2828] [INSPIRE].
A. Kapustin, Seiberg-like duality in three dimensions for orthogonal gauge groups, arXiv:1104.0466 [INSPIRE].
S. Benvenuti and G. Lo Monaco, A toolkit for ortho-symplectic dualities, JHEP 09 (2023) 002 [arXiv:2112.12154] [INSPIRE].
R.G. Leigh and M.J. Strassler, Duality of Sp(2Nc) and SO(Nc) supersymmetric gauge theories with adjoint matter, Phys. Lett. B 356 (1995) 492 [hep-th/9505088] [INSPIRE].
K.A. Intriligator, New RG fixed points and duality in supersymmetric SP (Nc) and SO(Nc) gauge theories, Nucl. Phys. B 448 (1995) 187 [hep-th/9505051] [INSPIRE].
K.A. Intriligator, R.G. Leigh and M.J. Strassler, New examples of duality in chiral and nonchiral supersymmetric gauge theories, Nucl. Phys. B 456 (1995) 567 [hep-th/9506148] [INSPIRE].
J.H. Brodie, Duality in supersymmetric SU(Nc) gauge theory with two adjoint chiral superfields, Nucl. Phys. B 478 (1996) 123 [hep-th/9605232] [INSPIRE].
J.H. Brodie and M.J. Strassler, Patterns of duality in N = 1 SUSY gauge theories, or: Seating preferences of theater going nonAbelian dualities, Nucl. Phys. B 524 (1998) 224 [hep-th/9611197] [INSPIRE].
Acknowledgments
We are grateful to Simone Rota for discussions. The work of the authors has been supported in part by the Italian Ministero dell’Istruzione, Università e Ricerca (MIUR), in part by Istituto Nazionale di Fisica Nucleare (INFN) through the “Gauge Theories, Strings, Supergravity” (GSS) research project.
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: 2402.00613
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
Amariti, A., Zanetti, A. A double scaling for the 4d/3d reduction of \( \mathcal{N} \) = 1 dualities. J. High Energ. Phys. 2024, 158 (2024). https://doi.org/10.1007/JHEP07(2024)158
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2024)158