Quantum Integrability and Supersymmetric Vacua

By N. Nekrasov, S. Shatashvili. 2009

322 citations recorded by the source corpus.

Supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspon- dence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU(2) XXX spin chain which is mapped to the two dimensional U(N )t heory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model which are related to the four dimensional theory compactified on T 2 . A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as T ∗ Gr(N, L) and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, represen- tations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrodinger models as well as the dynamical spin chains like Hubbard model. These more general spin chains correspond to quiver gauge theories with twisted masses, with classical gauge groups. We give the gauge-theoretic interpretation of Drinfeld polynomials and Baxter operators. In the classical weak coupling limit our results make contact with Nakajima con- structions. Toric compactifications of four dimensional N = 2 theories lead to the instanton corrected Bethe equations. To Prof. T. Eguchi on the occasion of his 60th anniversary §1. Gauge theories and integrable systems The dynamics of gauge theory is a subject of long history and the ever growing importance. In the last fifteen years or so it has become clear that the gauge theory dynamics in the vacuum sector is related to that of quantum many-body systems. A classic example is the equivalence of the pure Yang-Mills theory with gauge group U (N ) in two dimensions to the system of N free non-relativistic fermions on a circle. The same theory embeds as a supersymmetric vacuum sector of a (deformation of) N = 2 super-Yang-Mills theory in two dimensions. A bit less trivial example found in Ref. 1) is that the vacuum sector of a certain supersymmetric two dimensional U (N ) gauge theory with massive adjoint matter is described by the solutions of Bethe ansatz equations for the quantum Nonlinear Schrodinger equation (NLS) in the N -particle sector. The model of Ref. 1) describes the U (1)-equivariant intersection theory on the moduli space of solutions to Hitchin's equations, 2) just as the pure Yang-Mills theory describes the intersection theory on moduli space of flat connections on a two dimensional Riemann surface. This subject was revived in Refs. 3) and 4) by showing that the natural interpretation of the results of Ref. 1) is in terms of the equivalence of the vacua of the U (N ) Yang-Mills- Higgs theory in a sense of Ref. 3) and the energy eigenstates of the N -particle Yang

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d403bac585d94799ea15af8bce287f244b6bca88

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In Geometry and Topology

1. C. Vafa
2. N. Nekrasov
3. A. Tseytlin
4. Marcos Mariño
5. R. Roiban
6. S. Frolov
7. D. Gaiotto
8. E. Witten

In Nuclear and High Energy Physics

1. A. Tseytlin
2. S. Frolov
3. R. Roiban
4. N. Beisert
5. G. Arutyunov
6. L. Alday
7. D. Gaiotto
8. J. Maldacena

In Statistical and Nonlinear Physics

1. C. Vafa
2. Marcos Mariño
3. E. Witten
4. Mina Aganagic
5. A. Tseytlin
6. D. Gaiotto
7. S. Frolov
8. N. Beisert

Citations in the corpus

• Defects and quantum Seiberg-Witten geometry (2014, methodology): “At finite Q we will consider the Nekrasov-Shatashvili limit 2 → 0 of the normalized expectation value of the defect D(i)ρ (~a,Q, 1, 2,~t) = lim 2→0 Z(i)ρ (~a,Q, 1, 2,~t) Z(a,Q, 1, 2) , (4.5) 8Some work is this direction is now being done in [41].”
• Framed BPS states (2010, unknown): “Upon choosing certain real slices and specific quantization schemes we might expect to make some contact with the work of Nekrasov and Shatashvili [68, 69] and Nekrasov and Witten [44].”
• Comments on twisted indices in 3d supersymmetric gauge theories (2016, methodology): “1) was first computed in [9] in the context of the Bethe/gauge correspondence [10, 11], using slightly different topological field theory methods.”
• Quantum Groups and Quantum Cohomology (2012, background): “ul correspondence between quantum integrablesystems andmoduli of vacua in supersymmetric gauge theories (of which Nakajima varieties are examples) was proposed in the work of Nekrasov and Shatashvili [89, 90, 91]. In particular, quantum group actions on their cohomology or K-theory constructed by Va”
• Review of AdS/CFT Integrability: An Overview (2010, background): “Furthermore, supersymmetric vacua in D = 2, N = 4 gauge theories with matter can be described by Bethe ansätze [20,21].”
• Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory (2008, background): “Another relation between fourdimensional super Yang-Mills theory and the TBA has recently been discussed by Nekrasov and Shatashvili [46].”
• Quantum Geometry and Quiver Gauge Theories (2013, background): “3) which can be identified with the Yang-Yang function of some quantum integrable system [4], [5], [6], [7].”
• Knot Invariants from Four-Dimensional Gauge Theory (2011, background): “It has been argued [63] that the S-duality between these two branes is important in understanding the AGT correspondence [64] as well as recent developments relating supersymmetric gauge theory and integrable systems [65].”
• Quantization of Integrable Systems and Four Dimensional Gauge Theories (2009, methodology): “15) HkΨλ = Ek(λ)Ψλ Most of these models of [4, 5] are intrinsically quantum.”
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, background): “It was proposed in [89, 97, 98] that there are symmetries acting between different quantum field theories, for example changing the gauge groups.”
• The omega deformation, branes, integrability and Liouville theory (2010, background): “This correspondence has recently been extended [4, 5] to a much wider class of examples.”
• Quantum geometry of refined topological strings (2011, background): “Indeed, in [7] Nekrasov and Shatashvili observed that the gauge theory in the Ω-background in this limit is connected to various quantum integrable systems in a highly non-trivial way, a feature shared by various two-dimensional supersymmetric gauge systems [49, 50, 51, 52, 53, 26].”
• Supersymmetric vacua and Bethe ansatz (2009, unknown): “47 44 v2 [ he p- th ] 4 F eb 2 00 9 IHES-P/09/09 TCD-MATH-09-04 HMI-09-01 NSF-KITP-09-11 SUPERSYMMETRIC VACUA AND BETHE ANSATZ Nikita A. Nekrasova,1, and Samson L. Shatashvili1,2,3”