Abstract
We revisit the localization computation of the expectation values of ’t Hooft operators in \( \mathcal{N} \) = 2* SU(N) theory on ℝ3 × S1. We show that the part of the answer arising from “monopole bubbling” on ℝ3 can be understood as an equivariant integral over a Kronheimer-Nakajima moduli space of instantons on an orbifold of ℂ2. It can also be described as a Witten index of a certain supersymmetric quiver quantum mechanics with \( \mathcal{N} \) = (4, 4) supersymmetry. The map between the defect data and the quiver quantum mechanics is worked out for all values of N. For the SU(2) theory, we compute several examples of these line defect expectation values using the Witten index formula and confirm that the expressions agree with the formula derived by Okuda, Ito and Taki [16]. In addition, we present a Type IIB construction — involving D1-D3-NS5-branes — for monopole bubbling in \( \mathcal{N} \) = 2* SU(N) SYM and demonstrate how the quiver quantum mechanics arises in this brane picture.
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S.A. Cherkis, A. Larrain-Hubach and M. Stern, Instantons on multi-Taub-NUT Spaces I: Asymptotic Form and Index Theorem, arXiv:1608.00018 [INSPIRE].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Semiclassical framed BPS states, JHEP 07 (2016) 071 [arXiv:1512.08924] [INSPIRE].
H. Nakajima and Y. Takayama, Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A, arXiv:1606.02002 [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].
U. Bruzzo, F. Sala and R.J. Szabo, \( \mathcal{N} \) = 2 Quiver Gauge Theories on A-type ALE Spaces, Lett. Math. Phys. 105 (2015) 401 [arXiv:1410.2742] [INSPIRE].
M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d \( \mathcal{N} \) = 4 Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [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].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Brane bending and monopole moduli, JHEP 10 (2014) 157 [arXiv:1404.7158] [INSPIRE].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Parameter counting for singular monopoles on ℝ3, JHEP 10 (2014) 142 [arXiv:1404.5616] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
N. Mekareeya and D. Rodriguez-Gomez, 5d gauge theories on orbifolds and 4d ‘t Hooft line indices, JHEP 11 (2013) 157 [arXiv:1309.1213] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
Y. Ito, T. Okuda and M. Taki, Line operators on S 1 × R 3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [Erratum ibid. 03 (2016) 085] [arXiv:1111.4221] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
C.D.A. Blair and S.A. Cherkis, Singular Monopoles from Cheshire Bows, Nucl. Phys. B 845 (2011) 140 [arXiv:1010.0740] [INSPIRE].
S.A. Cherkis, Instantons on Gravitons, Commun. Math. Phys. 306 (2011) 449 [arXiv:1007.0044] [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].
N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl. 216 (2011) 69 [arXiv:1103.3919] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
S.A. Cherkis, Instantons on the Taub-NUT Space, Adv. Theor. Math. Phys. 14 (2010) 609 [arXiv:0902.4724] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
S.A. Cherkis and B. Durcan, The ’t Hooft-Polyakov monopole in the presence of an ’t Hooft operator, Phys. Lett. B 671 (2009) 123 [arXiv:0711.2318] [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].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
E. Witten, Branes, Instantons, And Taub-NUT Spaces, JHEP 06 (2009) 067 [arXiv:0902.0948] [INSPIRE].
S.A. Cherkis and B. Durcan, Singular monopoles via the Nahm transform, JHEP 04 (2008) 070 [arXiv:0712.0850] [INSPIRE].
J. Martens, Equivariant volumes of non-compact quotients and instanton counting, Commun. Math. Phys. 281 (2008) 827 [math/0609841] [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
E.J. Weinberg and P. Yi, Magnetic Monopole Dynamics, Supersymmetry and Duality, Phys. Rept. 438 (2007) 65 [hep-th/0609055] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
S. Fujii and S. Minabe, A Combinatorial study on quiver varieties, SIGMA 13 (2017) 052 [math/0510455] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) 313 [math/0306198] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, Ph.D. Thesis, Orsay, LPT (2005) [hep-th/0502180] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Multi instanton calculus on ALE spaces, Nucl. Phys. B 703 (2004) 518 [hep-th/0406243] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
S. Shadchin, Saddle point equations in Seiberg-Witten theory, JHEP 10 (2004) 033 [hep-th/0408066] [INSPIRE].
A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004) 453 [math/0306311].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 525 (1998) 215 [hep-th/9711145] [INSPIRE].
A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108] [INSPIRE].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [INSPIRE].
D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, in Strings, branes and dualities. Proceedings, NATO Advanced Study Institute, Cargese, France, May 26–June 14, 1997, pp. 359–372 (1997) [hep-th/9801061] [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, alg-geom/9608029.
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson. Proceedings, Conference, Saclay, France, June 5–7, 1996, pp. 333–366 (1996) [hep-th/9607163] [INSPIRE].
E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135] [INSPIRE].
S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2-D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [INSPIRE].
L. Jeffrey and F. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291.
H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, alg-geom/9507012.
E. Prato and S. Wu, Duistermaat-Heckman measures in a non-compact setting, alg-geom/9307005.
A. Degeratu and T. Walpuski, Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three, SIGMA 12 (2016) 017 [arXiv:1207.6938] [INSPIRE].
G.W. Moore and N. Seiberg, Lectures on RCFT, in 1989 Banff NATO ASI: Physics, Geometry and Topology Banff, Canada, August 14–25, 1989, Springer (1990) [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
P. Kronheimer, Monopoles and Taub-NUT Metrics, MSc Thesis, Oxford (1985).
W. Nahm, The construction of all selfdual multi-monopoles by the ADHM method, in Monopoles in Quantum Field Theory, Trieste, Italy, December 11-15, 1981, pp. 87–94 [INSPIRE].
G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories, Nucl. Phys. B 190 (1981) 455 [INSPIRE].
W. Nahm, All selfdual multi-monopoles for arbitrary gauge groups, in 12th NATO Advanced Summer Institute on Theoretical Physics: Structural Elements in Particle Physics and Statistical Mechanics, Freiburg, Germany, August 31–September 11, 1981, pp. 301 [INSPIRE].
W. Nahm, A Simple Formalism for the BPS Monopole, Phys. Lett. B 90 (1980) 413 [INSPIRE].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
D. Gang, E. Koh and K. Lee, Superconformal Index with Duality Domain Wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].
D. Gang, E. Koh and K. Lee, Line Operator Index on S 1 × S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [INSPIRE].
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Brennan, T.D., Dey, A. & Moore, G.W. On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics. J. High Energ. Phys. 2018, 14 (2018). https://doi.org/10.1007/JHEP09(2018)014
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DOI: https://doi.org/10.1007/JHEP09(2018)014