Quantization of Integrable Systems and Four Dimensional Gauge Theories

By N. Nekrasov, S. Shatashvili. 2009

926 citations recorded by the source corpus.

We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/a8727d9b11295ece153c23af98f2fbe4f2b9af3e

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In Geometry and Topology

1. C. Vafa
2. Sergei Gukov
3. Marcos Mariño
4. Tudor Dimofte
5. E. Witten
6. Mina Aganagic
7. D. Gaiotto
8. N. Nekrasov

In Nuclear and High Energy Physics

1. Marcos Mariño
2. C. Vafa
3. Sergei Gukov
4. E. Witten
5. D. Gaiotto
6. Tudor Dimofte
7. Mina Aganagic
8. A. Klemm

Citations in the corpus

• N=2 Supersymmetric Dynamics for Pedestrians (2013, background): “In a certain limiting case, it is found in [87] that it gives rise to a quantized integrable system which is a quantized version of the Donagi-Witten integrable system.”
• Gauge Theories Labelled by Three-Manifolds (2011, background): “3) with W̃ a sum of dilogarithm functions, is highly reminiscent of recent work relating effective 2d field theories to quantum integrable systems [36, 37, 38].”
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, methodology): “As a by-product we get the explicit realization of the Bethe/gauge correspondence [89, 96] with an additional bonus: the gauge theory produces not only the equations, characterizing the spectrum of the quantum integrable system, but also gives an expression for the common eigenfunction of the full s”
• Nekrasov functions and exact Bohr-Sommerfeld integrals (2009, methodology): “A piece of such analysis is presented in [ 1 ] in terms of advanced integrability theory a la [60–63], but it is desirable to convert it into a much simpler form, close to the one in the present paper.”
• An operator product expansion for polygonal null Wilson loops (2010, background): “In addition, we will find that the expression for the area can be written as the critical value of the associated Yang-Yang functional [31, 32 ].”
• The Ω Deformed B-model for Rigid N = 2 Theories (2011, methodology): “With the methods described in this paper, we are able to check these equations and thus the conjecture of [24] exactly (27)This paper establishes the equivalence of the Nekrasov partition function in the NS limit with the spectrum of an associated Schroedinger operator with Toda potential and genera”
• R-Twisting and 4d/2d Correspondences (2010, result): “It would also be interesting to see if there is any relation between the way TBA arises here as compared to the setup of [16] Even though there are a number of open questions we feel that the basic explanation we have found for our findings is on the right track.”
• Non-perturbative effects and the refined topological string (2013, background): “In these cases, the topological string free energy in the NS limit has been related [1] to the quantization of the classical integrable system associated to the N = 2 theory.”
• The omega deformation, branes, integrability and Liouville theory (2010, methodology): “In section 3, where we discuss the relation [6] of the Ω-deformation to quantization of Hitchin’s integrable system, X is not affine in complex structure I (but instead is a fibration by abelian varieties over an affine base B).”
• Wild quiver gauge theories (2011, background): “This should be obtained by analyzing the irregular conformal block in the ǫ2 → 0 limit [79, 80].”
• Five-dimensional AGT conjecture and the deformed Virasoro algebra (2009, background): “The AGT relation is also closely related with the current active subjects such as loop operators [11]–[13], wall crossing [14]–[16], integrability structures (see, e.g., [17, 18 ] and references therein) and so on.”
• The String Landscape, the Swampland, and the Missing Corner (2017, methodology): “There is also a way to use this picture and more advanced topics in the topological B-model to derive the work of [133] which relates theories of class S to integrable models.”