Abstract
In this paper we study the contribution of monopole bubbling to the expectation value of supersymmetric ’t Hooft defects in Lagrangian theories of class S on ℝ3× S1. This can be understood as the Witten index of an SQM living on the world volume of the ’t Hooft defect that couples to the bulk 4D theory. The computation of this Witten index has many subtleties originating from a continuous spectrum of scattering states along the non-compact vacuum branches. We find that even after properly dealing with the spectral asymmetry, the standard localization result for the ’t Hooft defect does not agree with the result obtained from the AGT correspondence. In this paper we will explicitly show that one must correct the localization result by adding an extra term to the standard Jeffrey-Kirwan residue formula. This extra term accounts for the contribution of ground states localized along the non-compact branches. This extra term restores both the ex- pected symmetry properties of the line defect expectation value and reproduces the results derived using the AGT correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Akhoury and A. Comtet, Anomalous behavior of the Witten index: exactly soluble models, Nucl. Phys.B 246 (1984) 253 [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].
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, JHEP01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
L. Álvarez-Gaumé, S. Della Pietra and G.W. Moore, Anomalies and odd dimensions, Annals Phys.163 (1985) 288 [INSPIRE].
N. Arkani-Hamed, A.G. Cohen, D.B. Kaplan, A. Karch and L. Motl, Deconstructing (2, 0) and little string theories, JHEP01 (2003) 083 [hep-th/0110146] [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 N = 2 gauge theories, Commun. Math. Phys.333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
T.D. Brennan and G.W. Moore, A note on the semiclassical formulation of BPS states in four-dimensional N = 2 theories, PTEP2016 (2016) 12C110 [arXiv:1610.00697] [INSPIRE].
T.D. Brennan, A. Dey and G.W. Moore, On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics, JHEP09 (2018) 014 [arXiv:1801.01986] [INSPIRE].
T.D. Brennan, Monopole bubbling via string theory, JHEP11 (2018) 126 [arXiv:1806.00024] [INSPIRE].
T.D. Brennan, G.W. Moore and A.B. Royston, Wall crossing from Dirac zeromodes, JHEP09 (2018) 038 [arXiv:1805.08783] [INSPIRE].
T.D. Brennan and G.W. Moore, Index-like theorems from line defect vevs, JHEP09 (2019) 073 [arXiv:1903.08172] [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].
D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys.B 503 (1997) 220 [hep-th/9608163] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, JHEP05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
F. Denef, Quantum quivers and Hall/hole halos, JHEP10 (2002) 023 [hep-th/0206072] [INSPIRE].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP09 (2009) 031 [arXiv:0907.2593] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys.299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS states, Adv. Theor. Math. Phys.17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
D. Gaiotto, G.W. Moore and E. Witten, Algebra of the infrared: string field theoretic structures in massive N = (2, 2) field theory in two dimensions, arXiv:1506.04087 [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact results for ’t Hooft loops in gauge theories on S4 , JHEP05 (2012) 141 [arXiv:1105.2568] [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].
S. Hellerman, D. Orlando and S. Reffert, String theory of the Ω deformation, JHEP01 (2012) 148 [arXiv:1106.0279] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
C. Hwang, J. Kim, S. Kim and J. Park, General instanton counting and 5d SCFT, JHEP07 (2015) 063 [Addendum ibid.04 (2016) 094] [arXiv:1406.6793] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett.B 387 (1996) 513 [hep-th/9607207] [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].
Y. Ito, T. Okuda and M. Taki, Line operators on S1× R3and quantization of the Hitchin moduli space, JHEP04 (2012) 010 [Erratum ibid.03 (2016) 085] [arXiv:1111.4221] [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/9507012.
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev.D 74 (2006) 025005 [hep-th/0501015] [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].
P.B. Kronheimer, Monopoles and Taub-NUT metrics, M.Sc. thesis, Oxford, U.K. (1985).
P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Annalen288 (1990) 263.
S.-J. Lee and P. Yi, Witten index for noncompact dynamics, JHEP06 (2016) 089 [arXiv:1602.03530] [INSPIRE].
G.W. Moore and N. Seiberg, Lectures on RCFT, RU-89-32, (1989) [YCTP-P13-89] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys.209 (2000) 97 [hep-th/9712241] [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].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Parameter counting for singular monopoles on R3 , JHEP10 (2014) 142 [arXiv:1404.5616] [INSPIRE].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Brane bending and monopole moduli, JHEP10 (2014) 157 [arXiv:1404.7158] [INSPIRE].
G.W. Moore, A.B. Royston and D. Van den Bleeken, Semiclassical framed BPS states, JHEP07 (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].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [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].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys.252 (2004) 359 [hep-th/0404225] [INSPIRE].
A.J. Niemi and G.W. Semenoff, Axial anomaly induced fermion fractionization and effective gauge theory actions in odd dimensional space-times, Phys. Rev. Lett.51 (1983) 2077 [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].
A.N. Redlich, Gauge noninvariance and parity violation of three-dimensional fermions, Phys. Rev. Lett.52 (1984) 18 [INSPIRE].
A.N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three-dimensions, Phys. Rev.D 29 (1984) 2366 [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] [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. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys.B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
S. Sethi and M. Stern, A comment on the spectrum of H monopoles, Phys. Lett.B 398 (1997) 47 [hep-th/9607145] [INSPIRE].
S. Sethi and M. Stern, D-brane bound states redux, Commun. Math. Phys.194 (1998) 675 [hep-th/9705046] [INSPIRE].
G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys.B 138 (1978) 1 [INSPIRE].
D. Tong, The holographic dual of AdS3× S3× S3× S1 , JHEP04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys.B 300 (1988) 360 [INSPIRE].
E. Witten, Branes, instantons, and Taub-NUT spaces, JHEP06 (2009) 067 [arXiv:0902.0948] [INSPIRE].
P. Yi, Witten index and threshold bound states of D-branes, Nucl. Phys.B 505 (1997) 307 [hep-th/9704098] [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: 1810.07191
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
Brennan, T.D., Dey, A. & Moore, G.W. ’t Hooft defects and wall crossing in SQM. J. High Energ. Phys. 2019, 173 (2019). https://doi.org/10.1007/JHEP10(2019)173
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)173