Abstract
We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E 8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E 8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.
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References
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, in La Londe les Maures 1996, Advanced quantum field theory, G. Itzykson ed., North-Holland, The Netherlands (1996), hep-th/9607139 [INSPIRE].
O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
J. Minahan, D. Nemeschansky, C. Vafa and N. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
O.J. Ganor, Toroidal compactification of heterotic 6D noncritical strings down to four-dimensions, Nucl. Phys. B 488 (1997) 223 [hep-th/9608109] [INSPIRE].
W. Lerche, P. Mayr and N. Warner, Noncritical strings, Del Pezzo singularities and Seiberg-Witten curves, Nucl. Phys. B 499 (1997) 125 [hep-th/9612085] [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Investigating the BPS spectrum of noncritical E n strings, Nucl. Phys. B 508 (1997) 64 [hep-th/9705237] [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Partition functions for BPS states of the noncritical E 8 string, Adv. Theor. Math. Phys. 1 (1998) 167 [hep-th/9707149] [INSPIRE].
K. Mohri, Exceptional string: Instanton expansions and Seiberg-Witten curve, Rev. Math. Phys. 14 (2002) 913 [hep-th/0110121] [INSPIRE].
T. Eguchi and K. Sakai, Seiberg-Witten curve for the E string theory, JHEP 05 (2002) 058 [hep-th/0203025] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485-486] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
K. Sakai, Topological string amplitudes for the local half K3 surface, arXiv:1111.3967 [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
S. Hosono, M. 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].
S. Hosono, Counting BPS states via holomorphic anomaly equations, in Calabi-Yau varieties and mirror symmetry, N. Yui and J. Lewis eds., American Mathematical Society, U.S.A. (2003), hep-th/0206206 [INSPIRE].
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ArXiv ePrint: 1203.2921
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Sakai, K. Seiberg-Witten prepotential for E-string theory and random partitions. J. High Energ. Phys. 2012, 27 (2012). https://doi.org/10.1007/JHEP06(2012)027
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DOI: https://doi.org/10.1007/JHEP06(2012)027