Confinement and Mayer cluster expansions

Abstract

In these notes, we study a class of grand-canonical partition functions with a kernel depending on a small parameter $\epsilon$. This class is directly relevant to Nekrasov partition functions of $\mathcal{N}=2$ SUSY gauge theories on the 4d $\Omega$-background, for which $\epsilon$ is identified with one of the equivariant deformation parameter. In the Nekrasov-Shatashvili limit $\epsilon \to 0$, we show that the free energy is given by an on-shell effective action. The equations of motion take the form of a TBA equation. The free energy is identified with the Yang-Yang functional of the corresponding system of Bethe roots. We further study the associated canonical model that takes the form of a generalized matrix model. Confinement of the eigenvalues by the short-range potential is observed. In the limit where this confining potential becomes weak, the collective field theory formulation is recovered. Finally, we discuss the connection with the alternative expression of instanton partition functions as sums over Young tableaux.

24 pages, 4 figures

References in corpus (13)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• Deforming SW curve
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Instanton calculus in R-R background and the topological string
• Seiberg-Witten theory and matrix models
• Large spin corrections in $\mathcal{N}=4$ SYM sl(2): still a linear integral equation
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Matrix models for 2* theories
• Generalized matrix models and AGT correspondence at all genera
• W(1+infinity) algebra as a symmetry behind AGT relation
• Virasoro constraint for Nekrasov instanton partition function
• The high spin expansion of twist sector dimensions: the planar N=4 super Yang-Mills theory

Cited by in corpus (14)

• Exact quantization conditions for the relativistic Toda lattice
• Clustering and the Three-Point Function
• Coherent states in quantum ${\mathcal{W}}_{1+\infty }$ algebra and qq-character for 5d Super Yang-Mills
• Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops
• A slow review of the AGT correspondence
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Finite ${\epsilon }_2$-corrections to the $\mathcal{N}=2$ SYM prepotential
• Fermions and scalars in $\mathcal{N}=4$ Wilson loops at strong coupling and beyond
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Strong Wilson polygons from the lodge of free and bound mesons
• Mayer expansion of the Nekrasov pre potential: the subleading ${\epsilon }_2$-order
• Quantum integrability of $\mathcal{N}=2$ 4d gauge theories
• Seiberg-Witten period relations in Omega background
• Solution of quantum integrable systems from quiver gauge theories