Notes on Mayer Expansions and Matrix Models

Abstract

Mayer cluster expansion is an important tool in statistical physics to evaluate grand canonical partition functions. It has recently been applied to the Nekrasov instanton partition function of $\mathcal{N}=2$ 4d gauge theories. The associated canonical model involves coupled integrations that take the form of a generalized matrix model. It can be studied with the standard techniques of matrix models, in particular collective field theory and loop equations. In the first part of these notes, we explain how the results of collective field theory can be derived from the cluster expansion. The equalities between free energies at first orders is explained by the discrete Laplace transform relating canonical and grand canonical models. In a second part, we study the canonical loop equations and associate them to similar relations on the grand canonical side. It leads to relate the multi-point densities, fundamental objects of the matrix model, to the generating functions of multi-rooted clusters. Finally, a method is proposed to derive loop equations directly on the grand canonical model.

24 pages, 8 figures, v2: references added, published in NPB

References in corpus (10)

• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Deforming SW curve
• Gauge theories on Omega-backgrounds from non commutative Seiberg-Witten curves
• Matrix model version of AGT conjecture and generalized Selberg integrals
• Nekrasov prepotential with fundamental matter from the quantum spin chain
• Quantum Hitchin Systems via beta-deformed Matrix Models
• 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
• Large N limit of beta-ensembles and deformed Seiberg-Witten relations

Cited by in corpus (9)

• M-theoretic matrix models
• Hexagonal Wilson Loops in Planar $\mathcal{N}=4$ SYM Theory at Finite Coupling
• Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
• A slow review of the AGT correspondence
• Mayer-Cluster Expansion of Instanton Partition Functions and Thermodynamic Bethe Ansatz
• Finite ${\epsilon }_2$-corrections to the $\mathcal{N}=2$ SYM prepotential
• Spherical Hecke algebra in the Nekrasov-Shatashvili limit
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Mayer expansion of the Nekrasov pre potential: the subleading ${\epsilon }_2$-order