Abstract
We apply the method of Higgs branch localization to the \( \mathcal{N} \) = 1 supersymmetric partition function on S 3 × S 1. As a result, we show that it can be written as the product of an elliptic vortex and anti-vortex partition function summed over a finite number of Higgs vacua.
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References
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
E. Witten, Mirror manifolds and topological field theory, in Mirror Symmetry, S.T. Yau eds., AMS press, New York U.S.A. (1991), pg. 121 [hep-th/9112056] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
F. Benini and S. Cremonesi, Partition functions of calN = (2,2) gauge theories on S 2 and vortices, arXiv:1206.2356 [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].
A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, arXiv:1308.4896 [INSPIRE].
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [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].
Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev. D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
L.F. Alday, M. Fluder and J. Sparks, The large-N limit of M2-branes on Lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].
L.F. Alday, D. Martelli, P. Richmond and J. Sparks, Localization on three-manifolds, JHEP 10 (2013) 095 [arXiv:1307.6848] [INSPIRE].
S. Nawata, Localization of N = 4 Superconformal Field Theory on S 1 × S 3 and Index, JHEP 11 (2011) 144 [arXiv:1104.4470] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [Addendum ibid. 1210 (2012) 051] [arXiv:1206.6359] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N} \) = 1 Chiral Multiplet on T 2 × S 2 and Supersymmetric Localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric Gauge Theories on the Five-Sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].
H.-C. Kim and S. Kim, M5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
Y. Imamura, Supersymmetric theories on squashed five-sphere, PTEP 2013 (2013) 013B04 [arXiv:1209.0561] [INSPIRE].
S. Pasquetti, Factorisation of calN = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, arXiv:1211.1986 [INSPIRE].
M. Taki, Holomorphic Blocks for 3d Non-abelian Partition Functions, arXiv:1303.5915 [INSPIRE].
C. Krattenthaler, V.P. Spiridonov and G.S. Vartanov, Superconformal indices of three-dimensional theories related by mirror symmetry, JHEP 06 (2011) 008 [arXiv:1103.4075] [INSPIRE].
C. Hwang, H.-C. Kim and J. Park, Factorization of the 3d superconformal index, arXiv:1211.6023 [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d N = 2 theories, arXiv:1312.3627 [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
D. Cassani and D. Martelli, The gravity dual of supersymmetric gauge theories on a squashed S 1 × S 3, arXiv:1402.2278 [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].
C. Romelsberger, Calculating the Superconformal Index and Seiberg Duality, arXiv:0707.3702 [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].
S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic Hypergeometry of Supersymmetric Dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [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].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, arXiv:1211.3409 [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
S.S. Razamat and B. Willett, Global Properties of Supersymmetric Theories and the Lens Space, arXiv:1307.4381 [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge Theories and Macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
D. Sen, Supersymmetry in the Space-time R × S 3, Nucl. Phys. B 284 (1987) 201 [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring Curved Superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
T.T. Dumitrescu and G. Festuccia, Exploring Curved Superspace (II), JHEP 01 (2013) 072 [arXiv:1209.5408] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, The Geometry of Supersymmetric Partition Functions, JHEP 01 (2014) 124 [arXiv:1309.5876] [INSPIRE].
M.F. Atiyah, Elliptic operators and compact groups,” in Lecture Notes in Mathematics. Vol. 401, Springer-Verlag, Berlin Germany (1974).
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [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].
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Peelaers, W. Higgs branch localization of \( \mathcal{N} \) = 1 theories on S 3 × S 1 . J. High Energ. Phys. 2014, 60 (2014). https://doi.org/10.1007/JHEP08(2014)060
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DOI: https://doi.org/10.1007/JHEP08(2014)060