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 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. It is crucial for our considerations that f is independent of . In this way, we exclude a class of models more relevant to the study of Nekrasov partition functions. For instance, setting = γ 2 , one recovers the model proposed by J. Hoppe in [27]. This model is a one-parameter version of the Nekrasov partition function that depends on two Ω-background equivariant deformation parameters 1 and 2 [28,29]. As → 0, it exhibits a phenomenon referred 1 Except for the first (planar) equation, which is a Ricatti equation, therefore non-linear. It is equivalent to a Schrödinger equation, i.e. a linear differential equation of second order.2 Such a structure should be related to the invariance of Nekrasov partition functions under transformations representing the SHc algebra uncovered in [23] (see also [24,25]). 3 In the case of real singularities, a prescription should be given to move away the poles from the contour by a small imaginary shift.

M-theoretic matrix models

“…This partition function can be regarded as the grand canonical partition function of a classical gas [42], and the number of particles is the instanton number of the underlying gauge theory. In certain limits, this grand canonical partition function can be evaluated in closed form, as shown in [42,43,44]. The M-theory expansion of a matrix model can be regarded as a direct thermodynamic limit, in which N → ∞ but the other parameters are kept fixed.…”

Confinement and Mayer cluster expansions

“…The latter is a statistical physics technique employed in [16] to treat the gauge theory partition functions. An investigation of this possible connection was performed in [22]. There, the Mayer expansion of a grand-canonical (generalized) matrix model is compared to the canonical model at large N .…”

Mayer expansion of the Nekrasov prepotential: The subleading ε 2 -order

“…In fact, the application of the path-integral techniques developed in [30] (for amplitudes) to the present case would simplify the procedure by first summing up all the leading contributions to the Yang-Yang functional (2.16) and then derive (2.12) as associated equation of motion. In specific, this functional approach gives immediately the Yang-Yang functional as in [32] (first line of (2.13)) without bound states or short-range potential p(x): this is interestingly the TBA for Maxwell-Boltzmann statistics. On the contrary, the addition of bound states, i.e.…”

“…The coefficient Γ 0 involves a sum over connected p-clusters of l vertices ∆ l . It is convenient to re-write it as a sum over rooted clusters ∆ x l by exploiting the combinatorial property (B.3) of [32] (without short-range potential):…”

“…Since the difference is proportional to the delta function δ(x−y), it only affects the degenerate terms in which the two roots coincide. As in the case α = 0 treated in [32], the original dressed propagator Y (x, y) has a singularity Y (x, y) ∼ −2iπY (x)δ(x − y) as x → y. This singularity is dressed by the presence of p-links inȲ (x) where nowȲ (x, y) ∼ −2iπ∆(x)δ(x−y).…”