Abstract
We calculate the partition function and correlation functions in A-twisted 2d \( \mathcal{N} \) = (2, 2) U(N) gauge theories and topologically twisted 3d \( \mathcal{N} \) = 2 U(N) gauge theories containing an adjoint chiral multiplet with particular choices of R-charges and the magnetic fluxes for flavor symmetries. According to the Gauge-Bethe correspondence, they correspond to the Heisenberg XXX1/2 and XXZ1/2 spin chain models, respectively. We identify the partition function with the inverse of the norm of the Bethe eigenstate. Correlation functions are identified to coefficients of the expectation value of Baxter Q-operator. In addition, we consider correlation functions of 2d \( \mathcal{N} \) = (2, 2)* theories and their relations to the equivariant integration of the equivariant quantum cohomology classes of the cotangent bundle of Grassmann manifolds and the equivariant quantum cohomology ring. Also, we study the twisted chiral ring relations of supersymmetric Wilson loops in 3d \( \mathcal{N} \) = 2* theories and the Bethe subalgebra of the XXZ1/2 spin chain models.
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N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [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].
C. Closset and I. Shamir, The N = 1 chiral multiplet on T 2 × S 2 and supersymmetric localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
T. Nishioka and I. Yaakov, Generalized indices for N = 1 theories in four-dimensions, JHEP 12 (2014) 150 [arXiv:1407.8520] [INSPIRE].
M. Honda and Y. Yoshida, Supersymmetric index on T 2 × S 2 and elliptic genus, arXiv:1504.04355 [INSPIRE].
V. Gorbounov, R. Rimányi, V. Tarasov and A. Varchenko, Cohomology of a flag variety as a Yangian Bethe algebra, J. Geom. Phys. 74 (2013) 56 [arXiv:1204.5138] [INSPIRE].
R. Rimányi, V. Tarasov and A. Varchenko, Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015) 81 [arXiv:1411.0478].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, 3–8 August 2009, World Scientific, Singapore (2010), pg. 265 [arXiv:0908.4052] [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, volume 3, Cambridge University Press, Cambridge, U.K. (1997).
L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, 26 September–6 October 1995, pg. 149 [hep-th/9605187] [INSPIRE].
H.J. de Vega, Families of commuting transfer matrices and integrable models with disorder, Nucl. Phys. B 240 (1984) 495 [INSPIRE].
V.E. Korepin, Calculation of norms of Bethe wave functions, Commun. Math. Phys. 86 (1982) 391 [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW model and Bethe ansatz for the phase model, JHEP 11 (2012) 146 [arXiv:1209.3800] [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW-matter model, Bethe ansatz for q-boson model and commutative Frobenius algebra, JHEP 03 (2014) 003 [arXiv:1308.4608] [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
S. Okuda and Y. Yoshida, Gauge/Bethe correspondence on S 1 × Σh and index over moduli space, arXiv:1501.03469 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
D.R. Morrison and M.R. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys. B 440 (1995) 279 [hep-th/9412236] [INSPIRE].
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [AMS/IP Stud. Adv. Math. 1 (1996) 143] [hep-th/9301042] [INSPIRE].
F. Benini, D.S. Park and P. Zhao, Cluster algebras from dualities of 2d N = (2,2) quiver gauge theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and quantum Seiberg-Witten geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
C. Korff, Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Commun. Math. Phys. 318 (2013) 173 [arXiv:1110.6356] [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].
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Chung, HJ., Yoshida, Y. Topologically twisted SUSY gauge theory, gauge-Bethe correspondence and quantum cohomology. J. High Energ. Phys. 2019, 52 (2019). https://doi.org/10.1007/JHEP02(2019)052
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DOI: https://doi.org/10.1007/JHEP02(2019)052