$$\beta $$-Deformed Matrix Models and 2d/4d Correspondence

Abstract

We review the β-deformed matrix model approach to the correspondence between fourdimensional N = 2 gauge theories and two-dimensional conformal field theories. The βdeformed matrix model equipped with the log-type potential is obtained as a free field (Dotsenko-Fateev) representation of the conformal block of chiral conformal algebra in two dimensions, with the precise choice of integration contours. After reviewing various matrix models related to the conformal field theories in two-dimensions, we study the large N limit corresponding to turning off the Omega-background 1 , 2 → 0. We show that the large N analysis produces the purely gauge theory results. Furthermore we discuss the Nekrasov-Shatashvili limit ( 2 → 0) by which we see the connection with the quantum integrable system. We then perform the explicit integration of the matrix model. With the precise choice of the contours we see that this reproduces the expansion of the conformal block and also the Nekrasov partition function. This is a contribution to the special volume on the 2d/4d correspondence, edited by J. Teschner. * k.maruyoshi@imperial.ac.uk * The matrix model with a logarithmic potential was first studied by Penner [20] related to the Eular characteristic of a Riemann surface.

Argyres-Douglas theories, Painlevé II and quantum mechanics

“…In this section, we show that perturbatively the two-cut phase of the β deformed Hermitian cubic matrix model can be identified with the H 1 Argyres-Douglas theory in the magnetic frame coupled to the Ω background. The 't Hooft expansion of the Hermitian cubic matrix has been discussed for instance in [11][12][13]19], while its β deformation was studied in [14], see also [20,21] for more details on the β deformed models. Some results for the free energies of the H 1 theory can be found for instance in [8,22].…”

Non-perturbative studies of N=2 conformal quiver gauge theories

“…Various approaches have been pursued: the geometric description of the low-energy effective actionà la Seiberg-Witten (SW) [2,3], the exact computation of instanton corrections by means of localization techniques [4,5], the relations to integrable models [6], the 2d/4d correspondence also known as the AGT correspondence [7,8], the use of β-ensembles and matrix model techniques [9,10]. Moreover, the string embedding of such theories via geometric engineering has led to the possibility of expressing some relevant observables via topological string amplitudes [11,12] 1 , and further insights 1 We refer to the series of recent review articles [13][14][15][16][17] for an extensive discussion of these topics.…”

Exact partition functions for deformed N = 2 $$ \mathcal{N}=2 $$ theories with N f = 4 $$ {\mathcal{N}}_f=4 $$ flavours

“…2 This is a map between deformed N = 2 instanton partition functions and conformal blocks of a suitable CFT with certain worldsheet genus and operator insertions. AGT correspondence may be checked perturbatively in the number of instantons [34][35][36] and it has been partially proved in [37][38][39][40][41][42], see also [43]. For the N = 2 * ε-deformed SU(2) gauge theory the relevant CFT quantity is the one-point conformal block on the torus.…”

“…Alternatively, one may exploit SO(8) symmetry of the gauge theory to reduce to integer exponents, but then one needs to evaluate the DF integral with generic number of FD integration variables. The Jack polynomial machinery developed in[108,43] is useful at low instanton numbers, but becomes increasingly unwieldy at higher orders and is ultimately equivalent to the Nekrasov determinant expressions and basically unuseful to obtain closed sums over all instantons.…”