Abstract
We show how to exactly calculate the refined indices of \( \mathcal{N}=4\;\mathrm{U}(1)\times \mathrm{U}(N) \) supersymmetric quiver quantum mechanics in the Coulomb branch by using the localization technique. The Coulomb branch localization is discussed from the viewpoint of both non-linear and gauged linear sigma models. A classification of fixed points in the Coulomb branch differs from one in the Higgs branch, but the derived indices completely agree with the results which were obtained by the localization in the Higgs branch. In the Coulomb branch localization, the refined indices can be written as a summation over different sets of the Coulomb branch fixed points. We also discuss a space-time picture of the fixed points in the Coulomb branch.
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References
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
F. Denef, Quantum quivers and Hall/hole halos, JHEP 10 (2002) 023 [hep-th/0206072] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
I. Bena, M. Berkooz, J. de Boer, S. El-Showk and D. Van den Bleeken, Scaling BPS Solutions and pure-Higgs States, JHEP 11 (2012) 171 [arXiv:1205.5023] [INSPIRE].
S.-J. Lee, Z.-L. Wang and P. Yi, Quiver Invariants from Intrinsic Higgs States, JHEP 07 (2012) 169 [arXiv:1205.6511] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP 07 (2015) 063 [arXiv:1406.6793] [INSPIRE].
C. Cordova and S.-H. Shao, An Index Formula for Supersymmetric Quantum Mechanics, arXiv:1406.7853 [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
K. Ohta and Y. Sasai, Exact Results in Quiver Quantum Mechanics and BPS Bound State Counting, JHEP 11 (2014) 123 [arXiv:1408.0582] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, Wall Crossing from Boltzmann Black Hole Halos, JHEP 07 (2011) 059 [arXiv:1011.1258] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, A Fixed point formula for the index of multi-centered N = 2 black holes, JHEP 05 (2011) 057 [arXiv:1103.1887] [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press, Princeton U.S.A. (1992).
E.A. Ivanov and A.V. Smilga, Supersymmetric gauge quantum mechanics: Superfield description, Phys. Lett. B 257 (1991) 79 [INSPIRE].
D.-E. Diaconescu and R. Entin, A Nonrenormalization theorem for the D = 1, N = 8 vector multiplet, Phys. Rev. D 56 (1997) 8045 [hep-th/9706059] [INSPIRE].
S.J. Gates, M.T. Grisaru, M. Roček and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry, Front. Phys. 58 (1983) 1 [hep-th/0108200] [INSPIRE].
N.A. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
K. Ohta and Y. Yoshida, Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold, Phys. Rev. D 86 (2012) 105018 [arXiv:1205.0046] [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].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N}=2 \) Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
M.R. Douglas, D.N. Kabat, P. Pouliot and S.H. Shenker, D-branes and short distances in string theory, Nucl. Phys. B 485 (1997) 85 [hep-th/9608024] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
O. Lunin and S.D. Mathur, Metric of the multiply wound rotating string, Nucl. Phys. B 610 (2001) 49 [hep-th/0105136] [INSPIRE].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1–D5 system with angular momentum, hep-th/0212210 [INSPIRE].
O. Lunin, Adding momentum to D1–D5 system, JHEP 04 (2004) 054 [hep-th/0404006] [INSPIRE].
S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates, Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [INSPIRE].
S. Giusto and S.D. Mathur, Geometry of D1–D5–P bound states, Nucl. Phys. B 729 (2005) 203 [hep-th/0409067] [INSPIRE].
I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [INSPIRE].
P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [INSPIRE].
M. Taylor, General 2 charge geometries, JHEP 03 (2006) 009 [hep-th/0507223] [INSPIRE].
I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, JHEP 11 (2006) 042 [hep-th/0608217] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
S.D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
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Ohta, K., Sasai, Y. Coulomb branch localization in quiver quantum mechanics. J. High Energ. Phys. 2016, 106 (2016). https://doi.org/10.1007/JHEP02(2016)106
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DOI: https://doi.org/10.1007/JHEP02(2016)106