Abstract
We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional \( \mathcal{N} \) = 1 supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [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. Eichler and D. Zagier, The theory of Jacobi forms, Progress in mathematics, Birkhäuser (1985).
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP 06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal field theory, Nucl. Phys. B 414 (1994) 191 [hep-th/9306096] [INSPIRE].
E. Witten, Global Gravitational Anomalies, Commun. Math. Phys. 100 (1985) 197 [INSPIRE].
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and riemannian geometry, Bull. London Math. Soc. 5 (1973) 229.
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian Geometry 1, Math. Proc. Cambridge Phil. Soc. 77 (1975) 43 [INSPIRE].
E. Witten, Global Anomalies in String Theory, in Symposium on Anomalies, Geometry, Topology Argonne, Illinois, March 28–30, 1985, (1985).
X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 (2013) 155114 [arXiv:1106.4772] [INSPIRE].
E. Witten and K. Yonekura, Anomaly Inflow and the η-Invariant, in The Shoucheng Zhang Memorial Workshop Stanford, CA, U.S.A., May 2–4, 2019, (2019), [arXiv:1909.0877].
G. Felder and A. Varchenko, The elliptic gamma function and sl(3, ℤ) 1> ℤ3, Adv. Math. 156 (2000) 44.
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
W. Peelaers, Higgs branch localization of \( \mathcal{N} \) = 1 theories on S3 × S1 , JHEP 08 (2014) 060 [arXiv:1403.2711] [INSPIRE].
Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
M. Dedushenko and M. Fluder, Chiral Algebra, Localization, Modularity, Surface defects, And All That, J. Math. Phys. 61 (2020) 092302 [arXiv:1904.02704] [INSPIRE].
S.S. Razamat and B. Willett, Global Properties of Supersymmetric Theories and the Lens Space, Commun. Math. Phys. 334 (2015) 661 [arXiv:1307.4381] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N} \) = 1 Chiral Multiplet on T2 × S2 and Supersymmetric Localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [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 M. Yamazaki, S-duality and the N = 2 Lens Space Index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].
A.P. Kels and M. Yamazaki, Elliptic hypergeometric sum/integral transformations and supersymmetric lens index, SIGMA 14 (2018) 013 [arXiv:1704.0315].
P. Longhi, F. Nieri and A. Pittelli, Localization of 4d \( \mathcal{N} \) = 1 theories on D2 × T2 , JHEP 12 (2019) 147 [arXiv:1906.02051] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d N = 1 Cardy Formula, arXiv:1904.0345.
R. Dijkgraaf and E. Witten, Topological Gauge Theories and Group Cohomology, Commun. Math. Phys. 129 (1990) 393 [INSPIRE].
A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP 07 (2016) 025 [arXiv:1512.03376] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The asymptotic growth of states of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 08 (2019) 120 [arXiv:1904.05865] [INSPIRE].
G. Lockhart and C. Vafa, Superconformal Partition Functions and Non-perturbative Topological Strings, JHEP 10 (2018) 051 [arXiv:1210.5909] [INSPIRE].
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004) 247.
E. Shaghoulian, Modular invariance of conformal field theory on s1 × s3 and circle fibrations, Phys. Rev. Lett. 119 (2017) 131601 [arXiv:1612.05257] [INSPIRE].
M. Nishizawa, An elliptic analogue of the multiple gamma function, J. Phys. A 34 (2001) 7411.
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: 2004.13490
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
Gadde, A. Modularity of supersymmetric partition functions. J. High Energ. Phys. 2021, 181 (2021). https://doi.org/10.1007/JHEP12(2021)181
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)181