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 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 with 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.
Highlights
The AGT correspondence [1] implies a relation between the canonical partition function of a β-ensemble and the grand canonical partition function of a generalized matrix model
The cluster expansion of Mayer and Montroll [26] has been successfully employed to derived an effective action relevant to the NS limit [15]. Can we relate this cluster expansion to the topological expansion of a generalized matrix model? Is there an equivalent of the loop equations technique on the grand canonical side? And more generally, how do canonical and grand canonical coupled integrals relate to each other? These are the issues we propose to address in these notes
The cluster expansion was introduced by Mayer and Montroll as a way to compute the free energy knowing the form of the interaction between particles [26]
Summary
The AGT correspondence [1] implies a relation between the canonical partition function of a β-ensemble and the grand canonical partition function of a generalized matrix model The former represents a correlator of Liouville theory, according to the proposal of Dijkgraaf and Vafa [2], further investigated in [3, 4, 5, 6, 7, 8, 9, 10, 11]. We go on with the study of the canonical loop equations We show that they relate to graphical identities between generating functions of rooted clusters.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
