Abstract
We study the partition functions of topologically twisted 3d \( \mathcal{N} \) = 2 gauge theories on a hemisphere spacetime with boundary HS2 × S1. We show that the partition function may be localised to either the Higgs branch or the Coulomb branch where the contributions to the path integral are vortex or monopole configurations respectively. Turning to \( \mathcal{N} \) = 4 supersymmetry, we consider partition functions for exceptional Dirichlet boundary conditions that yield a complete set of ‘IR holomorphic blocks’. We demonstrate that these correspond to vertex functions: equivariant Euler characteristics of quasimap moduli spaces. In this context, we explore the geometric interpretation of both the Higgs and Coulomb branch localisation schemes in terms of the enumerative geometry of quasimaps and discuss the action of mirror symmetry.
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References
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
M. Inglese, D. Martelli and A. Pittelli, The Spindle Index from Localization, arXiv:2303.14199 [INSPIRE].
F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N} \) = (2, 2) Gauge Theories on S2 and Vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d \( \mathcal{N} \) = 2 theories, PTEP 2014 (2014) 123B02 [arXiv:1312.3627] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
M. Blau and G. Thompson, Aspects of NT ≥ 2 topological gauge theories and D-branes, Nucl. Phys. B 492 (1997) 545 [hep-th/9612143] [INSPIRE].
L. Rozansky and E. Witten, HyperKahler geometry and invariants of three manifolds, Selecta Math. 3 (1997) 401 [hep-th/9612216] [INSPIRE].
M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry, and Symplectic Duality in 3d \( \mathcal{N} \) = 4 Gauge Theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, [hep-th/0005247] [INSPIRE].
D. Gaiotto, G.W. Moore and E. Witten, Algebra of the Infrared: String Field Theoretic Structures in Massive \( \mathcal{N} \) = (2, 2) Field Theory In Two Dimensions, arXiv:1506.04087 [INSPIRE].
M. Bullimore and D. Zhang, 3d \( \mathcal{N} \) = 4 Gauge Theories on an Elliptic Curve, SciPost Phys. 13 (2022) 005 [arXiv:2109.10907] [INSPIRE].
M. Dedushenko and N. Nekrasov, Interfaces and quantum algebras, I: Stable envelopes, J. Geom. Phys. 194 (2023) 104991 [arXiv:2109.10941] [INSPIRE].
M. Bullimore, S. Crew and D. Zhang, Boundaries, Vermas, and Factorisation, JHEP 04 (2021) 263 [arXiv:2010.09741] [INSPIRE].
Y. Yoshida and K. Sugiyama, Localization of three-dimensional \( \mathcal{N} \) = 2 supersymmetric theories on S1 × D2, PTEP 2020 (2020) 113B02 [arXiv:1409.6713] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Walls, Lines, and Spectral Dualities in 3d Gauge Theories, JHEP 05 (2014) 047 [arXiv:1302.0015] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, Prog. Math. 319 (2016) 155 [arXiv:1306.4320] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir Energy and the Anomaly Polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
S. Crew, N. Dorey and D. Zhang, Factorisation of 3d \( \mathcal{N} \) = 4 twisted indices and the geometry of vortex moduli space, JHEP 08 (2020) 015 [arXiv:2002.04573] [INSPIRE].
A. Cabo-Bizet, Factorising the 3D Topologically Twisted Index, JHEP 04 (2017) 115 [arXiv:1606.06341] [INSPIRE].
M. Bullimore, A. Ferrari and H. Kim, Twisted indices of 3d \( \mathcal{N} \) = 4 gauge theories and enumerative geometry of quasi-maps, JHEP 07 (2019) 014 [arXiv:1812.05567] [INSPIRE].
M. Bullimore, A.E.V. Ferrari and H. Kim, The 3d twisted index and wall-crossing, SciPost Phys. 12 (2022) 186 [arXiv:1912.09591] [INSPIRE].
M. Bullimore, A.E.V. Ferrari and H. Kim, Supersymmetric Ground States of 3d \( \mathcal{N} \) = 4 Gauge Theories on a Riemann Surface, SciPost Phys. 12 (2022) 072 [arXiv:2105.08783] [INSPIRE].
A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE].
B. Kim, Stable quasimaps to holomorphic symplectic quotients, arXiv e-prints (2010) arXiv:1005.4125 [arXiv:1005.4125].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants, Commun. Math. Phys. 333 (2015) 717 [arXiv:1307.5997] [INSPIRE].
H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Two-Sphere Partition Functions and Gromov-Witten Invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [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 [hep-th/9407087] [Erratum ibid. 430 (1994) 485] [INSPIRE].
S. Shadchin, On certain aspects of string theory/gauge theory correspondence, [hep-th/0502180] [INSPIRE].
H. Dinkins, Exotic Quantum Difference Equations and Integral Solutions, Ph.D. thesis, University of North Carolina at Chapel Hill, Chapel Hill, U.S.A. (2022), https://doi.org/10.17615/4h4e-sj63 [arXiv:2205.01596] [INSPIRE].
M. Aganagic and A. Okounkov, Elliptic stable envelopes, J. Am. Math. Soc. 34 (2021) 79 [arXiv:1604.00423] [INSPIRE].
K. Hori, H. Kim and P. Yi, Witten Index and Wall Crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
T. Dimofte, N. Garner, M. Geracie and J. Hilburn, Mirror symmetry and line operators, JHEP 02 (2020) 075 [arXiv:1908.00013] [INSPIRE].
M. Dedushenko, Gluing. Part I. Integrals and symmetries, JHEP 04 (2020) 175 [arXiv:1807.04274] [INSPIRE].
M. Dedushenko, Gluing II: boundary localization and gluing formulas, Lett. Math. Phys. 111 (2021) 18 [arXiv:1807.04278] [INSPIRE].
P. Longhi, F. Nieri and A. Pittelli, Localization of 4d \( \mathcal{N} \) = 1 theories on 𝔻2 × 𝕋2, JHEP 12 (2019) 147 [arXiv:1906.02051] [INSPIRE].
A. Pittelli, Supersymmetric localization of refined chiral multiplets on topologically twisted H2 × S1, Phys. Lett. B 801 (2020) 135154 [arXiv:1812.11151] [INSPIRE].
S. Crew, Geometric aspects of three dimensional N = 4 gauge theories, Ph.D. thesis, University of Cambridge, Cambridge, U.K. (2022), https://doi.org/10.17863/CAM.80877 [INSPIRE].
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [arXiv:0903.4172] [Erratum ibid. 864 (2012) 884] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
J. Gomis and S. Lee, Exact Kahler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
A. Tanaka, H. Mori and T. Morita, Superconformal index on ℝℙ2 × 𝕊1 and mirror symmetry, Phys. Rev. D 91 (2015) 105023 [arXiv:1408.3371] [INSPIRE].
N.S. Manton and B. Zhao, Neumann boundary condition for Abelian BPS vortices, JHEP 09 (2023) 181 [arXiv:2305.16069] [INSPIRE].
I. Ciocan-Fontanine, B. Kim and D. Maulik, Stable quasimaps to GIT quotients, J. Geom. Phys. 75 (2014) 17 [arXiv:1106.3724].
T. Graber and R. Pandharipande, Localization of virtual classes, alg-geom/9708001.
M. Aganagic and A. Okounkov, Quasimap counts and Bethe eigenfunctions, Moscow Math. J. 17 (2017) 565 [arXiv:1704.08746] [INSPIRE].
K. McGerty and T. Nevins, Kirwan surjectivity for quiver varieties, Invent. Math. 212 (2018) 161 [arXiv:1610.08121].
M. Aganagic, E. Frenkel and A. Okounkov, Quantum q-Langlands Correspondence, Trans. Moscow Math. Soc. 79 (2018) 1 [arXiv:1701.03146] [INSPIRE].
H. Dinkins, Symplectic Duality of T*Gr(k, n), arXiv:2008.05516 [INSPIRE].
H. Dinkins, 3d mirror symmetry of the cotangent bundle of the full flag variety, Lett. Math. Phys. 112 (2022) 100 [arXiv:2011.08603] [INSPIRE].
Acknowledgments
It is a pleasure to thank Mathew Bullimore, Adam Chalabi, Cyril Closset, Hunter Dinkins, Nick Dorey, Andrea Ferrari, Mark Gross, Nick Manton, Ivan Smith, Michael Walter, Claude Warnick, and Yutaka Yoshida for helpful discussions. SC and DZ are grateful to ICMAT for their hospitality while part of this work was completed. BZ is supported by a Trinity College internal graduate studentship. DZ is supported by a Junior Research Fellowship from St. John’s College, Oxford.
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Crew, S., Zhang, D. & Zhao, B. Boundaries & localisation with a topological twist. J. High Energ. Phys. 2023, 93 (2023). https://doi.org/10.1007/JHEP10(2023)093
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DOI: https://doi.org/10.1007/JHEP10(2023)093