Quantization of Integrable Systems and Four Dimensional Gauge Theories
Nikita Nekrasov,Samson L. Shatashvili +1 more
- pp 265-289
TLDR
In this article, the authors studied four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N = 2 super-Poincare invariance and explained how this gauge theory provided the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional n=2 theory.Abstract:
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.read more
Citations
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Gauge Theories Labelled by Three-Manifolds
TL;DR: In this article, a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories is proposed, which can be seen as a way of describing boundary conditions and duality walls in four-dimensional SCFTs.
Journal ArticleDOI
Vortex Counting and Lagrangian 3-manifolds
TL;DR: In this paper, the relation between vortex counting in two-dimensional supersymmetric field theories and the refined BPS invariants of the dual geometries was studied, which can also be mapped to the computation of degenerate conformal blocks in 2-dimensional CFTs.
Book
Quantum Groups and Quantum Cohomology
Davesh Maulik,Andrei Okounkov +1 more
TL;DR: In this article, a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of Nakajima quiver varieties was constructed, and a formula for quantum multiplication by divisors in terms of this Yangian action was proved.
Posted Content
R-Twisting and 4d/2d Correspondences
TL;DR: In this paper, the q-deformed Kontsevich-Soibelman monodromy operator of N = 2 CFTs is shown to have finite order if the R-charges are rational.
Journal ArticleDOI
Holomorphic blocks in three dimensions
TL;DR: In this paper, the authors decompose sphere partition functions and indices of three-dimensional = 2 gauge theories into a sum of products involving a universal set of holomorphic blocks, which are in one-to-one correspondence with the theory's massive vacua.
References
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Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
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Exactly solved models in statistical mechanics
TL;DR: In this article, exactly solved models of statistical mechanics are discussed. But they do not consider exactly solvable models in statistical mechanics, which is a special issue in the statistical mechanics of the classical two-dimensional faculty of science.
Journal ArticleDOI
On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain
Journal ArticleDOI
Seiberg-Witten Prepotential from Instanton Counting
TL;DR: In this article, a two-parameter generalization of the Seiberg-Witten prepotential is presented, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
Book ChapterDOI
Seiberg-Witten theory and random partitions
Nikita Nekrasov,Andrei Okounkov +1 more
TL;DR: In this paper, the authors investigated various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.