Abstract
We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S5 and the superconformal index of a large number of 6 dimensional (2, 0) and (1, 0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3,Z) modular transformations acting by inverting the coupling constants of the refined topological string play a key role.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Vafa, Supersymmetric partition functions and a string theory in 4 dimensions, arXiv:1209.2425 [INSPIRE].
A. Iqbal and C. Vafa, BPS degeneracies and superconformal index in diverse dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].
H.-C. Kim and S. Kim, M 5-branes from gauge theories on the 5-sphere, JHEP 05 (2013) 144 [arXiv:1206.6339] [INSPIRE].
J. Kallen, J.A. Minahan, A. Nedelin and M. Zabzine, N 3 -behavior from 5D Yang-Mills theory, JHEP 10 (2012) 184 [arXiv:1207.3763] [INSPIRE].
D.L. Jafferis and S.S. Pufu, Exact results for five-dimensional superconformal field theories with gravity duals, JHEP 05 (2014) 032 [arXiv:1207.4359] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, del Pezzo surfaces and type-I’ theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories: 1, Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
S. Cecotti, C. Cordova and C. Vafa, Braids, walls and mirrors, arXiv:1110.2115 [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
T. Dimofte, Holomorphic blocks for 3d SCFT’s, talk given at the 10th Simons workshop on math and physics, based on joint work with C. Beem and S. Pasquetti, Stony Brook, NY, U.S.A., (2012).
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].
C. Closset, T.T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, Comments on Chern-Simons contact terms in three dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
M. Jimbo and T. Miwa, Quantum KZ equation with |q| = 1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A 29 (1996) 2923 [hep-th/9601135] [INSPIRE].
M. Nishizawa, An elliptic analogue of the multiple gamma function, J. Phys. A 34 (2001) 7411.
N. Kurokawa and S.-Y. Koyama, Multiple sine functions, Forum Math. 15 (2003) 839.
A. Narukawa, The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004) 247 [math.QA/0306164].
P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].
R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [INSPIRE].
R. Gopakumar and C. Vafa, M-theory and topological strings. 1, hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M-theory and topological strings. 2, hep-th/9812127 [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
M. Aganagic and M. Yamazaki, Open BPS wall crossing and M-theory, Nucl. Phys. B 834 (2010) 258 [arXiv:0911.5342] [INSPIRE].
N.C. Leung and C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) 91 [hep-th/9711013] [INSPIRE].
T.T. Dumitrescu, G. Festuccia and N. Seiberg, Exploring curved superspace, JHEP 08 (2012) 141 [arXiv:1205.1115] [INSPIRE].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for superconformal field theories in 3, 5 and 6 dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
A. Gadde and W. Yan, Reducing the 4d index to the S 3 partition function, JHEP 12 (2012) 003 [arXiv:1104.2592] [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].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [INSPIRE].
J.A. Minahan, D. Nemeschansky and N.P. Warner, Investigating the BPS spectrum of noncritical E n strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].
J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, E-strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
S. Hosono, M.H. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 [hep-th/9901151] [INSPIRE].
A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, Nucl. Phys. Proc. Suppl. 58 (1997) 177 [hep-th/9607139] [INSPIRE].
M. Alim and E. Scheidegger, Topological strings on elliptic fibrations, Commun. Num. Theor. Phys. 08 (2014) 729 [arXiv:1205.1784] [INSPIRE].
F. Bonetti and T.W. Grimm, Six-dimensional (1, 0) effective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE].
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
D.R. Morrison and W. Taylor, Toric bases for 6D F-theory models, Fortsch. Phys. 60 (2012) 1187 [arXiv:1204.0283] [INSPIRE].
A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE].
T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2∗ super Yang-Mills on S 4, JHEP 03 (2012) 017 [arXiv:1004.1222] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
A. Iqbal, C. Kozcaz and T. Sohail, Periodic Schur process, cylindric partitions and N = 2∗ theory, arXiv:0903.0961 [INSPIRE].
N.A. Nekrasov, Instanton partition functions and M-theory, Jpn. J. Math. 4 (2009) 63.
F. Bonetti, T.W. Grimm and S. Hohenegger, Non-Abelian tensor towers and (2, 0) superconformal theories, JHEP 05 (2013) 129 [arXiv:1209.3017] [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozcaz, G. Lockhart and C. Vafa, M-strings, work in progress.
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].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
B. Haghighat and S. Vandoren, Five-dimensional gauge theory and compactification on a torus, JHEP 09 (2011) 060 [arXiv:1107.2847] [INSPIRE].
Y. Imamura, Perturbative partition function for squashed S 5, PTEP 2013 (2013) 073B01 [arXiv:1210.6308] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M 5-branes, arXiv:1211.0144 [INSPIRE].
V.P. Spiridonov, Modified elliptic gamma functions and 6d superconformal indices, Lett. Math. Phys. 104 (2014) 397 [arXiv:1211.2703] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1210.5909
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Lockhart, G., Vafa, C. Superconformal partition functions and non-perturbative topological strings. J. High Energ. Phys. 2018, 51 (2018). https://doi.org/10.1007/JHEP10(2018)051
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)051