The Bethe/Gauge Correspondence

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70 Chapter 4. Bethe/gauge• describe the relation between twisted masses and the isometries of supersymmetric sigmamodels, and take a look at the Omega deformation;• treat the calculation of ˜W eff in more detail in the nonabelian case, especially for the adjointmatter superfields;• consider quiver gauge theories;• discuss (twisted) chiral rings, topological field theory, and topological twisting;• look at the Nekrasov partition function and instantons; and• understand the link with string theory and branes.We also plan to include the more recent work of Nekrasov and Shatashvili [4, 60].In addition, there are several papers of others that are related to the Bethe/gauge correspondenceand are worth looking into. We mention the conjecture of Alday, Gaiotto andTachikawa [71], the work of Orlando and Reffert, starting with [72], and the papers of Dorey etal [73].Further directions. Of course there are several aspects of the Bethe/gauge correspondencethat are not yet well-understood; see the discussions of [3, 4]. For example, in our schematic representationof the Bethe/gauge correspondence, the top right is conspicuously empty. It wouldbe interesting to try and lift the Bethe/gauge correspondence beyond the vacuum structure ofthe gauge theory:BetheGaugeN = (2, 2) sym? ? ? ←→ with massive matterlow energy limitquantum integrable model ←→ vacua on Coulomb branchAnother interesting possible direction of future research is to investigate the relation betweenthe Bethe/gauge correspondence and integrability in AdS/CFT.
References[1] N. Nekrasov and S. Shatashvili, “Supersymmetric vacua and Bethe ansatz,” Nucl. Phys.Proc. Suppl., vol. 192–193, pp. 91–112, 2009. arXiv:0901.4744v2 [hep-th].[2] N. Nekrasov and S. Shatashvili, “Quantum Integrability and Supersymmetric Vacua,” Prog.Theor. Phys. Suppl., vol. 177, pp. 105–119, 2009. arXiv:0901.4748v2 [hep-th].[3] N. Nekrasov and S. Shatashvili, “Quantization of Integrable Systems and Four DimensionalGauge Theories,” in XVIth International Congress on Mathematical Physics. Prague, August2009 (P. Exner, ed.), pp. 265–289, World Scientific, 2010. arXiv:0908.4052v1 [hep-th].[4] N. Nekrasov, A. Rosly, and S. Shatashvili, “Darboux coordinates, Yang-Yang functional,and gauge theory,” Nucl. Phys. Proc. Suppl., vol. 216, pp. 69–93, 2011. arXiv:1103.3919v1[hep-th].[5] J. Figueroa-O’Farrill, “BUSSTEPP Lectures on Supersymmetry.” arXiv:hep-th/0109172,2001.[6] J. Wess and J. Bagger, Supersymmetry And Supergravity. Princeton University Press,second ed., 1992.[7] I. Aitchison, Supersymmetry in Particle Physics. An Elementary Introduction. CambridgeUniversity Press, 2007. See also arXiv: hep-ph/0505105.[8] E. Witten, “Phases of N = 2 theories in two-dimensions,” Nucl. Phys. B, vol. 403, pp. 159–222, 1993. arXiv:hep-th/9301042.[9] M. Karbach and G. Müller, “Introduction to the Bethe ansatz I.” arXiv:cond-mat/9809162v1, 1998.[10] G. Arutyunov, “Student Seminar: Classical and Quantum Integrable Systems.” Lecturenotes for lectures delivered at Utrecht University, 20 September 2006 – 24 January 2007.[11] L. Faddeev, “How Algebraic Bethe Ansatz works for integrable model.” arXiv:hepth/9605187v1,1995.[12] J. Caux, “Lecture Notes on the Bethe Ansatz.” Draft, October 2010.[13] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, et al., Mirror Symmetry.American Mathematical Society, 2003.[14] E. Witten, “The Verlinde Algebra and the Cohomology of the Grassmannian,” in Geometry,Topology, and Physics (S.-T. Yau, ed.), pp. 357–422, 1993. arXiv:hep-th/9312104v1.[15] A. Hanany and K. Hori, “Branes and N = 2 theories in two-dimensions,” Nucl. Phys. B,vol. 513, pp. 119–174, 1998. arXiv:hep-th/9707192v2.[16] N. Dorey, “The BPS spectra of two-dimensional supersymmetric gauge theories with twistedmass terms,” JHEP, vol. 9811, p. 005, 1998. arXiv:hep-th/9806056v2.71
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Institute for Theoretical PhysicsMa
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4 Bethe/gauge 614.1 The corresponde
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viPrefaceA note on notationMany asp
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Chapter 1OverviewIn quantum field t
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1.1. Bethe: quantum integrable mode
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1.2. Gauge: supersymmetry 9providin
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1.2. Gauge: supersymmetry 11Because
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1.2. Gauge: supersymmetry 13The def
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1.3. Physics in two dimensions 15be
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1.4. Bethe/gauge: the general idea
Page 27 and 28: Chapter 2BetheIn this chapter we ta
Page 29 and 30: 2.1. Algebraic Bethe Ansatz 21on ou
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Page 33 and 34: 2.1. Algebraic Bethe Ansatz 25Accor
Page 35 and 36: 2.2. Generalizations of the xxx s m
Page 37 and 38: 2.3. Yang-Yang function 292.2.3 Fur
Page 39: 2.3. Yang-Yang function 31To see th
Page 42 and 43: 34 Chapter 3. GaugeThe Poincaré gr
Page 44 and 45: 36 Chapter 3. Gaugeexpansion we wil
Page 46 and 47: 38 Chapter 3. Gaugewhere we can in
Page 48 and 49: 40 Chapter 3. GaugeHere K is a real
Page 50 and 51: 42 Chapter 3. GaugeVector superfiel
Page 52 and 53: 44 Chapter 3. GaugeThus, L r is pre
Page 54 and 55: 46 Chapter 3. Gaugeunder U(1) G as
Page 56 and 57: 48 Chapter 3. Gauge˜σ k¯f= ˜m k
Page 58 and 59: 50 Chapter 3. GaugeVacuum manifolds
Page 60 and 61: 52 Chapter 3. GaugeThis (almost) is
Page 62 and 63: 54 Chapter 3. GaugeLooking at the t
Page 64 and 65: 56 Chapter 3. Gaugethe effective tw
Page 66 and 67: 58 Chapter 3. GaugeMore matter repr
Page 69 and 70: Chapter 4Bethe/gaugeNow that we hav
Page 71 and 72: 4.1. The correspondence 63the spin
Page 73 and 74: 4.2. Further topics 654.1.3 Diction
Page 75: 4.2. Further topics 67N = 2 symin f
Page 80 and 81: [17] N. Dorey, T. Hollowood, and D.
Page 82: [56] U. Lindström, “Supersymmetr
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The Bethe/Gauge Correspondence

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