Abstract
We study complex Chern–Simons theory on a Seifert manifold M 3 by embedding it into string theory. We show that complex Chern–Simons theory on M 3 is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern–Simons theory on \({\Sigma\times S^1}\) and (4) index of a spinc Dirac operator on the moduli space of flat connections to a new set of relations between (1) the “equivariant Verlinde algebra” for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern–Simons theory on \({\Sigma \times S^1}\) and (4) the equivariant index of a spinc Dirac operator on the moduli space of Higgs bundles.
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Verlinde E.P.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988)
Gukov S., Witten E.: Branes and quantization. Adv. Theor. Math. Phys. 13, 1. (2009) arXiv:0809.0305
Dijkgraaf R., Hollands L., Sulkowski P.: Quantum curves and D-modules. JHEP 0911, 047. (2009) arXiv:0810.4157
Nekrasov N., Witten E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092. arXiv:1002.0888 (2010)
Gukov S.: Quantization via mirror symmetry. Jpn. J. Math. 6, 65. (2011) arXiv:1011.2218
Yagi J.: \({\Omega}\) -Deformation and quantization. JHEP 1408, 112. (2014) arXiv:1405.6714
Schwarz, A.: New topological invariants arising in the theory of quantized fields. In: Baku International Topological Conference, Abstracts (Part 2), Baku (1987)
Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989)
Gerasimov, A.: Localization in GWZW and Verlinde formula. arXiv:hep-th/9305090
Witten, E.: The Verlinde algebra and the cohomology of the Grassmannian. arXiv:hep-th/9312104
Alday L.F., Bullimore M., Fluder M.: On S-duality of the superconformal index on lens spaces and 2d TQFT. JHEP 1305, 122. (2013) arXiv:1301.7486
Razamat S.S., Yamazaki M.: S-duality and the N = 2 lens space index. JHEP 1310, 048. (2013) arXiv:1306.1543
Dimofte T., Gukov S., Hollands L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287. arXiv:1006.0977 (2011)
Bershadsky M., Vafa C., Sadov V.: D-branes and topological field theories. Nucl. Phys. B 463, 420–434. (1996) arXiv:hep-th/9511222
Blau M., Thompson G.: Aspects of \({N_T \geq 2}\) topological gauge theories and D-branes. Nucl. Phys. B 492, 545–590. (1997) arXiv:hep-th/9612143
Blau M., Thompson G.: Euclidean SYM theories by time reduction and special holonomy manifolds. Phys. Lett. B 415, 242–252. (1997) arXiv:hep-th/9706225
Festuccia G., Seiberg N.: Rigid supersymmetric theories in curved superspace. JHEP 1106, 114. (2011) arXiv:1105.0689
Imamura Y., Yokoyama D.: N = 2 supersymmetric theories on squashed three-sphere. Phys. Rev. D 85, 025015. (2012) arXiv:1109.4734
Cordova, C., Jafferis, D.L.: Complex Chern–Simons from M5-branes on the squashed three-sphere. arXiv:1305.2891
Lee S., Yamazaki M.: 3d Chern–Simons theory from M5-branes. JHEP 1312, 035. (2013) arXiv:1305.2429
Dimofte T., Gaiotto D., Gukov S.: Gauge theories labelled by three-manifolds. Commun. Math. Phys. 325, 367–419. (2014) arXiv:1108.4389
Terashima Y., Yamazaki M.: SL(2,R) Chern–Simons, Liouville, and gauge theory on duality walls. JHEP 1108, 135. (2011) arXiv:1103.5748
Cecotti, S., Cordova, C., Vafa, C.: Braids, walls, and mirrors. arXiv:1110.2115
Dimofte T., Gaiotto D., Gukov S.: 3-Manifolds and 3d indices. Adv. Theor. Math. Phys. 17, 975–1076. (2013) arXiv:1112.5179
Yagi J.: 3d TQFT from 6d SCFT. JHEP 1308, 017. (2013) arXiv:1305.0291
Dimofte, T.: Complex Chern–Simons theory at level k via the 3d–3d correspondence. Commun. Math. Phys. 339(2), 619–662. arXiv:1409.0857
Dimofte T., Gukov S., Lenells J., Zagier D.: Exact results for perturbative Chern–Simons theory with complex gauge group. Commun. Number Theor. Phys. 3, 363–443. (2009) arXiv:0903.2472
Dimofte T.: Quantum Riemann surfaces in Chern–Simons theory. Adv. Theor. Math. Phys. 17, 479–599. (2013) arXiv:1102.4847
Gukov S., Sulkowski P.: A-polynomial, B-model, and quantization. JHEP 1202, 070. (2012) arXiv:1108.0002
Chung H.-J., Dimofte T., Gukov S., Sulkowski P.: 3d–3d correspondence revisited. JHEP 1604, 140. (2016) arXiv:1405.3663
Gadde, A., Gukov, S., Putrov, P.: Fivebranes and 4-manifolds. In: Ballmann, W. et al. (ed.) Arbeitstagung Bonn, Progress in Mathematics 319, pp. 155–245. Springer, Berlin (2016) arXiv:1306.4320
Harvey J.A., Moore G.W., Strominger A.: Reducing S duality to T duality. Phys. Rev. D 52, 7161–7167. (1995) arXiv:hep-th/9501022
Bershadsky M., Johansen A., Sadov V., Vafa C.: Topological reduction of 4-d SYM to 2-d sigma models. Nucl. Phys. B 448, 166–186. (1995) arXiv:hep-th/9501096
Hitchin N.J.: The selfduality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–131 (1987)
Witten E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)
Kapustin, A., Willett, B.: Wilson loops in supersymmetric Chern–Simons-matter theories and duality. arXiv:1302.2164
Hanany A., Tong D.: Vortices, instantons and branes. JHEP 0307, 037. (2003) arXiv:hep-th/0306150
Callan C.G., Harvey J.A.: Anomalies and fermion zero modes on strings and domain walls. Nucl. Phys. B 250, 427 (1985)
Buchbinder E.I., Gomis J., Passerini F.: Holographic gauge theories in background fields and surface operators. JHEP 0712, 101 (2007)
Atiyah M., Bott R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A 308, 523–615 (1982)
Souriau J.-M.: Quantification g om trique. Commun. Math. Phys. 1(5), 374–398 (1966)
Atiyah M.F., Bott R.: The moment map and equivariant cohomology. Topology 23(1), 1–28 (1984)
Moore G.W., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121. (2000) arXiv:hep-th/9712241
Gerasimov A.A., Shatashvili S.L.: Higgs bundles, gauge theories and quantum groups. Commun. Math. Phys. 277, 323–367. (2008) arXiv:hep-th/0609024
Gerasimov, A.A., Shatashvili, S.L.: Two-dimensional gauge theories and quantum integrable systems. arXiv:0711.1472
Kallen J.: Cohomological localization of Chern–Simons theory. JHEP 1108, 008. (2011) arXiv:1104.5353
Ohta K., Yoshida Y.: Non-Abelian localization for supersymmetric Yang–Mills–Chern–Simons theories on Seifert manifold. Phys. Rev. D 86, 105018. (2012) arXiv:1205.0046
Kao H.-C., Lee K.-M., Lee T.: The Chern–Simons coefficient in supersymmetric Yang–Mills Chern–Simons theories. Phys. Lett. B 373, 94–99. (1996) arXiv:hep-th/9506170
Blau M., Thompson G.: Equivariant Kahler geometry and localization in the G/G model. Nucl. Phys. B 439, 367–394. (1995) arXiv:hep-th/9407042
Okuda S., Yoshida Y.: G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and commutative Frobenius algebra. JHEP 1403, 003. (2014) arXiv:1308.4608
Blau M., Thompson G.: Derivation of the Verlinde formula from Chern–Simons theory and the G/G model. Nucl. Phys. B 408, 345–390. (1993) arXiv:hep-th/9305010
Korff C.: Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra. Commun. Math. Phys. 318, 173–246. (2013) arXiv:1110.6356
Teleman, C.: K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra. ArXiv Mathematics e-prints (June, 2003). arXiv:math/0306347
Teleman, C., Woodward, C.T.: The index formula on the moduli of G-Bundles, ArXiv Mathematics e-prints (Dec, 2003). arXiv:math/0312154
Nekrasov N.A., Shatashvili S.L.: Supersymmetric vacua and Bethe ansatz. Nucl. Phys. Proc. Suppl. 192(193), 91–112. (2009) arXiv:0901.4744
Nekrasov N.A., Shatashvili S.L.: Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl. 177, 105–119. (2009) arXiv:0901.4748
Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052
Gadde A., Gukov S., Putrov P.: Walls, lines, and spectral dualities in 3d gauge theories. JHEP 1405, 047. (2014) arXiv:1302.0015
Gaiotto D., Koroteev P.: On three dimensional quiver gauge theories and integrability. JHEP 1305, 126. (2013) arXiv:1304.0779
Nekrasov N.A., Shatashvili S.L.: Bethe/gauge correspondence on curved spaces. JHEP 1501, 100. (2015) arXiv:1405.6046
Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A.: Spectral dualities in XXZ spin chains and five dimensional gauge theories. JHEP 1312, 034. (2013) arXiv:1307.1502
Gukov S., Stosic M.: Homological algebra of knots and BPS states. Geom. Topol. Monogr. 18, 309–367. (2012) arXiv:1112.0030
Fuji H., Gukov S., Stosic M., Sulkowski P.: 3d analogs of Argyres-Douglas theories and knot homologies. JHEP 1301, 175. (2013) arXiv:1209.1416
Gukov, S., Witten, E.: Gauge theory, ramification, and the geometric Langlands program. arXiv:hep-th/0612073
Gukov S.: Gauge theory and knot homologies. Fortschr. Phys. 55, 473–490. (2007) arXiv:0706.2369
Boden, H.U., Yokogawa, K.: Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves: I. eprint, p. 10014 (Oct, 1996) arXiv:alg-geom/9610014
Alekseev A.: Notes on equivariant localization. Lect. Notes Phys. 543, 1–24 (2000)
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Gukov, S., Pei, D. Equivariant Verlinde Formula from Fivebranes and Vortices. Commun. Math. Phys. 355, 1–50 (2017). https://doi.org/10.1007/s00220-017-2931-9
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DOI: https://doi.org/10.1007/s00220-017-2931-9