Abstract
We study the supersymmetric partition function of a 2d linear σ-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d \( \mathcal{N} \) = 4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T2 fibered over S1) times a circle with an SL(2, ℤ) duality wall inserted on S1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2, ℤ), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Bak, M. Gutperle and S. Hirano, A dilatonic deformation of AdS5 and its field theory dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].
A. Clark and A. Karch, Super Janus, JHEP 10 (2005) 094 [hep-th/0506265] [INSPIRE].
E. D’Hoker, J. Estes and M. Gutperle, Interface Yang-Mills, supersymmetry, and Janus, Nucl. Phys. B 753 (2006) 16 [hep-th/0603013] [INSPIRE].
D. Gaiotto and E. Witten, Janus configurations, Chern-Simons couplings, and the theta-angle in N = 4 super Yang-Mills theory, JHEP 06 (2010) 097 [arXiv:0804.2907] [INSPIRE].
C. Kim, E. Koh and K.-M. Lee, Janus and multifaced supersymmetric theories, JHEP 06 (2008) 040 [arXiv:0802.2143] [INSPIRE].
C. Kim, E. Koh and K.-M. Lee, Janus and multifaced supersymmetric theories II, Phys. Rev. D 79 (2009) 126013 [arXiv:0901.0506] [INSPIRE].
Y. Terashima and M. Yamazaki, SL(2, ℝ) Chern-Simons, Liouville, and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The virtue of defects in 4D gauge theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
J.A. Minahan and U. Naseer, Gauge theories on spheres with 16 supercharges and non-constant couplings, J. Phys. A 52 (2019) 235401 [arXiv:1811.11652] [INSPIRE].
D. Gaiotto, Surface operators in N = 2 4d gauge theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
K. Goto and T. Okuda, Janus interface in two-dimensional supersymmetric gauge theories, JHEP 10 (2019) 045 [arXiv:1810.03247] [INSPIRE].
A. Dabholkar and J.A. Harvey, String islands, JHEP 02 (1999) 006 [hep-th/9809122] [INSPIRE].
O.J. Ganor, U duality twists and possible phase transitions in (2+1)-dimensions supergravity, Nucl. Phys. B 549 (1999) 145 [hep-th/9812024] [INSPIRE].
S. Hellerman, J. McGreevy and B. Williams, Geometric constructions of nongeometric string theories, JHEP 01 (2004) 024 [hep-th/0208174] [INSPIRE].
A. Dabholkar and C. Hull, Duality twists, orbifolds, and fluxes, JHEP 09 (2003) 054 [hep-th/0210209] [INSPIRE].
A. Dabholkar and C. Hull, Generalised T-duality and non-geometric backgrounds, JHEP 05 (2006) 009 [hep-th/0512005] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Duality defects, arXiv:1404.2929 [INSPIRE].
O.J. Ganor, N.P. Moore, H.-Y. Sun and N.R. Torres-Chicon, Janus configurations with SL(2, ℤ)-duality twists, strings on mapping tori and a tridiagonal determinant formula, JHEP 07 (2014) 010 [arXiv:1403.2365] [INSPIRE].
B. Assel and A. Tomasiello, Holographic duals of 3d S-fold CFTs, JHEP 06 (2018) 019 [arXiv:1804.06419] [INSPIRE].
L. Martucci, Topological duality twist and brane instantons in F-theory, JHEP 06 (2014) 180 [arXiv:1403.2530] [INSPIRE].
C.-T. Hsieh, Y. Tachikawa and K. Yonekura, Anomaly of the electromagnetic duality of Maxwell theory, Phys. Rev. Lett. 123 (2019) 161601 [arXiv:1905.08943] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
D. Gang, N. Kim, M. Romo and M. Yamazaki, Taming supersymmetric defects in 3d–3d correspondence, J. Phys. A 49 (2016) 30LT02 [arXiv:1510.03884] [INSPIRE].
D. Gang, N. Kim, M. Romo and M. Yamazaki, Aspects of defects in 3d–3d correspondence, JHEP 10 (2016) 062 [arXiv:1510.05011] [INSPIRE].
S. Chun, S. Gukov, S. Park and N. Sopenko, 3d–3d correspondence for mapping tori, JHEP 09 (2020) 152 [arXiv:1911.08456] [INSPIRE].
E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abhan. Math. Sem. Univ. Hamburg 5 (1927) 353.
K. Ireland and M. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics 84, Springer, Germany (1990).
L. Glasser and M.S. Milgram, On quadratic Gauss sums and variations thereof, Cogent Math. 2 (2015) 1021187 [arXiv:1405.3194].
M. Schaar, Mémoire sur la théorie des résidus quadratiques, Mém. Acad. Sci. Lett. Beaux-Arts Belgique 24 (1850) 5.
G. Landsberg, Zur Theorie der Gaussschen Summen und der linearen Transformation der Thetafunctionen, J. Reine Angew. Math. 111 (1893) 234.
H. Dym and H.P. McKean, Fourier series and integrals, Academic Press, U.S.A. (1972).
J.M. Borwein and P.B. Borwein, Pi and the AGM: a study in analytic number theory and computational complexity, Wiley, U.S.A. (1987).
B. Moore, A proof of the Landsberg-Schaar relation by finite methods, Ramanujan J. (2020) 1 [arXiv:1810.06172].
B.C. Berndt and R.J. Evans, The determination of Gauss sums, Bull. Amer. Math. Soc. 5 (1981) 107.
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
A. Giveon, M. Porrati and E. Rabinovici, Target space duality in string theory, Phys. Rept. 244 (1994) 77 [hep-th/9401139] [INSPIRE].
K.S. Narain, M.H. Sarmadi and C. Vafa, Asymmetric orbifolds, Nucl. Phys. B 288 (1987) 551 [INSPIRE].
A. Kapustin and M. Tikhonov, Abelian duality, walls and boundary conditions in diverse dimensions, JHEP 11 (2009) 006 [arXiv:0904.0840] [INSPIRE].
O.J. Ganor, S. Jue and S. McCurdy, Ground states of duality-twisted sigma-models with K3 target space, JHEP 02 (2013) 017 [arXiv:1211.4179] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
Y. Lozano, S duality in gauge theories as a canonical transformation, Phys. Lett. B 364 (1995) 19 [hep-th/9508021] [INSPIRE].
O.J. Ganor, A note on zeros of superpotentials in F-theory, Nucl. Phys. B 499 (1997) 55 [hep-th/9612077] [INSPIRE].
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, in From fields to strings, M. Shifman et al. eds., World Scientific, Singapore (2003), hep-th/0307041 [INSPIRE].
O.J. Ganor and Y.P. Hong, Selfduality and Chern-Simons theory, arXiv:0812.1213 [INSPIRE].
O.J. Ganor, Y.P. Hong and H.S. Tan, Ground states of S-duality twisted N = 4 super Yang-Mills theory, JHEP 03 (2011) 099 [arXiv:1007.3749] [INSPIRE].
E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].
F. Cooper and B. Freedman, Aspects of supersymmetric quantum mechanics, Ann. Phys. 146 (1983) 262 [INSPIRE].
F. Cooper, A. Khare and U. Sukhatme, Supersymmetry and quantum mechanics, Phys. Rept. 251 (1995) 267 [hep-th/9405029] [INSPIRE].
S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
M.C.N. Cheng, S. Chun, F. Ferrari, S. Gukov and S.M. Harrison, 3d modularity, JHEP 10 (2019) 010 [arXiv:1809.10148] [INSPIRE].
C.T.C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963) 281.
C. Galindo and N.J. Torres, Solutions of the hexagon equation for abelian anyons, Rev. Colomb. de Mat. 50 (2016) 277 [arXiv:1606.01414].
M.S. Zini, Lecture notes on vector-valued modular forms, by Z. Wang at UC Santa Barbara (Math 227C, Spring 2019), unpublished.
D. Belov and G.W. Moore, Classification of Abelian spin Chern-Simons theories, [INSPIRE].
D. Delmastro and J. Gomis, Symmetries of Abelian Chern-Simons Theories and arithmetic, JHEP 03 (2021) 006 [arXiv:1904.12884] [INSPIRE].
Y. Lee and Y. Tachikawa, A study of time reversal symmetry of abelian anyons, JHEP 07 (2018) 090 [arXiv:1805.02738] [INSPIRE].
F. Deloup, Topological quantum field theory, reciprocity and the Weil representation, available at http://www.math.univ-toulouse.fr/~deloup/Weil-book10.pdf.
R.J. Milgram, Surgery with coefficients, Ann. Math. 100 (1974) 194.
B. Williams, Computing modular forms for the Weil representation, Ph.D. thesis, University of California, Berkeley, U.S.A. (2018).
J. Cano, M. Cheng, M. Mulligan, C. Nayak, E. Plamadeala and J. Yard, Bulk-edge correspondence in (2 + 1)-dimensional Abelian topological phases, Phys. Rev. B 89 (2014) 115116 [arXiv:1310.5708] [INSPIRE].
X.G. Wen, Vacuum degeneracy of chiral spin states in compactified space, Phys. Rev. B 40 (1989) 7387 [INSPIRE].
X.G. Wen and Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces, Phys. Rev. B 41 (1990) 9377 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Diff. Geom. 33 (1991) 787 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
L.C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys. 147 (1992) 563 [INSPIRE].
L.C. Jeffrey, Symplectic quantum mechanics and Chern-Simons gauge theory II: mapping tori of tori, J. Math. Phys. 54 (2013) 052305 [arXiv:1210.6635] [INSPIRE].
M.F. Atiyah, On framings of 3-manifolds, Topology 29 (1990) 1.
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, Prog. Math. 319 (2016) 155 [arXiv:1306.4320] [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
X.G. Wen and A. Zee, A classification of Abelian quantum Hall states and matrix formulation of topological fluids, Phys. Rev. B 46 (1992) 2290 [INSPIRE].
E. Witten, Fermion path integrals and topological phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
F. Haldane and E. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Phys. Rev. B 31 (1985) 2529 [INSPIRE].
M. Spera, Quantization on Abelian Varieties, Rend. Semin. Mat. Univ. Politec. Torino 44 (1986) 383.
J.R. Klauder and E. Onofri, Landau levels and geometric quantization, Int. J. Mod. Phys. A 4 (1989) 3939 [INSPIRE].
F.D.M. Haldane, O(3) nonlinear sigma model and the topological distinction between integer- and half-integer-spin antiferromagnets in two dimensions, Phys. Rev. Lett. 61 (1988) 1029 [INSPIRE].
A. Altland and B. Simons, Condensed matter field theory, Cambridge University Press Cambridge U.K. (2010).
K. Conrad, SL2(ℤ), available at https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf.
P. Bonderson, E.C. Rowell, Z. Wang and Q. Zhang, Congruence subgroups and super-modular categories, Pacific J. Math. 296 (2018) 257 [arXiv:1704.02041].
A. Krazer, Zur Theorie der mehrfachen Gaußschen Summen, in Festschrift Heinrich Weber zu seinem siebzigsten Geburtstag am 5. März 1912, B.G. Teubner, Germany (1912).
S. Alaca and G. Doyle, Explicit evaluation of double Gauss sums, J. Comb. Number Theory 8 (2016) 47 [arXiv:1609.03919].
V. Turaev, Reciprocity for Gauss sums on finite abelian groups, Math. Proc. Camb. Phil. Soc. 124 (1998) 205.
H. Braun, Geschlechter quadratischer formen, J. reine angew. Math. 182 (1940) 32.
C.L. Siegel, Über das quadratische Reziprozitätsgesetz algebraischen Zahlkörpern, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, Vandenhoeck & Ruprecht, Germany (1960).
F. Deloup and V. Turaev, On reciprocity, J. Pure Appl. Alg. 208 (2007) 153 [math/0512050].
F. Deloup, Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds, Trans. Amer. Math. Soc. 351 (1999) 1895.
F. Deloup, On abelian quantum invariants of links in 3-manifolds, Math. Ann. 319 (2001) 759.
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, J. Knot Theor. Ramifications 29 (2020) 2040003 [arXiv:1701.06567] [INSPIRE].
F. Lemmermeyer, Reciprocity laws: from Euler to Eisenstein, Springer, Germany (2000).
G. Doyle, Quadratic form Gauss sums, Ph.D. thesis, University of California, Berkeley U.S.A. (2016).
V. Armitage and A. Rogers, Gauss sums and quantum mechanics, J. Phys. A 33 (2000) 5993 [quant-ph/0003107].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
J.E. Tener and Z. Wang, On classification of extremal non-holomorphic conformal field theories, J. Phys. A 50 (2017) 115204 [arXiv:1611.04071] [INSPIRE].
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: 1912.11471
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
Ganor, O.J., Sun, HY. & Torres-Chicon, N.R. Double-Janus linear sigma models and generalized reciprocity for Gauss sums. J. High Energ. Phys. 2021, 227 (2021). https://doi.org/10.1007/JHEP05(2021)227
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)227