Abstract
While studying supersymmetric G-gauge theories, one often observes that a zero-radius limit of the twisted partition function ΩG is computed by the partition function \( {\mathcal{Z}}^G \) in one less dimensions. We show how this type of identification fails generically due to integrations over Wilson lines. Tracing the problem, physically, to saddles with reduced effective theories, we relate ΩG to a sum of distinct \( {\mathcal{Z}}^G \)’s and classify the latter, dubbed H-saddles. This explains why, in the context of pure Yang-Mills quantum mechanics, earlier estimates of the matrix integrals \( {\mathcal{Z}}^G \) had failed to capture the recently constructed bulk index \( {\mathrm{\mathcal{I}}}_{\mathrm{bulk}}^G \). The purported agreement between 4d and 5d instanton partition functions, despite such subtleties also present in the ADHM data, is explained.
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References
E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
S.-J. Lee and P. Yi, Witten index for noncompact dynamics, JHEP 06 (2016) 089 [arXiv:1602.03530] [INSPIRE].
S.-J. Lee and P. Yi, D-particles on orientifolds and rational invariants, arXiv:1702.01749 [INSPIRE].
L. Álvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Commun. Math. Phys. 90 (1983) 161 [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
E.A. Ivanov and A.V. Smilga, Supersymmetric gauge quantum mechanics: superfield description, Phys. Lett. B 257 (1991) 79 [INSPIRE].
U.H. Danielsson, G. Ferretti and B. Sundborg, D particle dynamics and bound states, Int. J. Mod. Phys. A 11 (1996) 5463 [hep-th/9603081] [INSPIRE].
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
P. Yi, Witten index and threshold bound states of D-branes, Nucl. Phys. B 505 (1997) 307 [hep-th/9704098] [INSPIRE].
S. Sethi and M. Stern, D-brane bound states redux, Commun. Math. Phys. 194 (1998) 675 [hep-th/9705046] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [INSPIRE].
M. Staudacher, Bulk Witten indices and the number of normalizable ground states in supersymmetric quantum mechanics of orthogonal, symplectic and exceptional groups, Phys. Lett. B 488 (2000) 194 [hep-th/0006234] [INSPIRE].
V. Pestun, N = 4 SYM matrix integrals for almost all simple gauge groups (except E 7 and E 8 ), JHEP 09 (2002) 012 [hep-th/0206069] [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].
M.B. Green and M. Gutperle, D particle bound states and the D instanton measure, JHEP 01 (1998) 005 [hep-th/9711107] [INSPIRE].
V.G. Kac and A.V. Smilga, Normalized vacuum states in N = 4 supersymmetric Yang-Mills quantum mechanics with any gauge group, Nucl. Phys. B 571 (2000) 515 [hep-th/9908096] [INSPIRE].
A. Hanany, B. Kol and A. Rajaraman, Orientifold points in M-theory, JHEP 10 (1999) 027 [hep-th/9909028] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [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].
W. Krauth and M. Staudacher, Yang-Mills integrals for orthogonal, symplectic and exceptional groups, Nucl. Phys. B 584 (2000) 641 [hep-th/0004076] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [Addendum ibid. 04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
Y. Hwang, J. Kim and S. Kim, M5-branes, orientifolds and S-duality, JHEP 12 (2016) 148 [arXiv:1607.08557] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
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Hwang, C., Yi, P. Twisted partition functions and H-saddles. J. High Energ. Phys. 2017, 45 (2017). https://doi.org/10.1007/JHEP06(2017)045
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DOI: https://doi.org/10.1007/JHEP06(2017)045