M-theoretic matrix models

Abstract

Some matrix models admit, on top of the usual 't Hooft expansion, an Mtheory-like expansion, i.e. an expansion at large N but where the rest of the parameters are fixed, instead of scaling with N . These models, which we call M-theoretic matrix models, appear in the localization of Chern-Simons-matter theories, and also in two-dimensional statistical physics. Generically, their partition function receives non-perturbative corrections which are not captured by the 't Hooft expansion. In this paper, we discuss general aspects of these type of matrix integrals and we analyze in detail two different examples. The first one is the matrix model computing the partition function of N = 4 supersymmetric Yang-Mills theory in three dimensions with one adjoint hypermultiplet and N f fundamentals, which has a conjectured M-theory dual, and which we call the N f matrix model. The second one, which we call the polymer matrix model, computes form factors of the 2d Ising model and is related to the physics of 2d polymers. In both cases we determine their exact planar limit. In the N f matrix model, the planar free energy reproduces the expected behavior of the M-theory dual. We also study their M-theory expansion by using Fermi gas techniques, and we find non-perturbative corrections to the 't Hooft expansion.

Partition functions of superconformal Chern-Simons theories from Fermi gas approach

“…earlier works [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. Especially it is worthwhile to mention that, according to [15] the expression of the coefficient C was given a geometrical interpretation as the classical volume inside the Fermi surface.…”

The short pulse equation by a Riemann–Hilbert approach

“…Therefore in this correspondence the parameters describing the canonical ensemble of the gas, namely the number of particles N and the strength of the potential T , get mapped to the dual periodã D and the graviphoton strength . Similarly by specializing the general formula for F 1 of [69,79] to the potential (3.51) we find…”

“…This is the so-called polymer matrix model studied in [28,[69][70][71] (see chapter 20 of [72] for a survey on the subject). It was found in [69] that this matrix model describes an ideal Fermi gas whose density matrix is…”

“…In this section we show that the free energies F g emerge when we study the so-call 't Hooft expansion of (3.50). The 't Hooft regime of (3.50) is defined as [69] N, T → ∞,…”

“…without using tools such as the holomorphic anomaly. However, for the O(2) matrix models, exact formulae for F 0 and F 1 exist in the literature [69,[79][80][81]. For the particular potential (3.51) one can show that (3.50) is a one cut matrix model with endpoints (a, a −1 ) where a is related to λ as [69] λ…”