Gauge theories as matrix models

Abstract

The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. The matrix models of Eguchi-Yang type are derived quasiclassically as describing the instantonic contribution to the deformed partition functions of supersymmetric gauge theories. The constructed explicitly exact solution for the case of conformal four-dimensional theory is studied in detail, and some aspects of its relation with the recently proposed logarithmic beta-ensembles are considered. We discuss also the "quantization" of this picture in terms of twodimensional conformal theory with extended symmetry, and stress its difference from common picture of perturbative expansion a la matrix models. Instead, the representation for Nekrasov functions in terms of conformal blocks or Whittaker vector suggests some nontrivial relation with Teichmüller spaces and quantum integrable systems.

Matrix model from $ \mathcal{N} = 2 $ orbifold partition function

“…can be rewritten as Then integrating (5.35) by parts, we have Therefore analysis of this matrix model is equivalent to that of [3]. But in this section we reconsider the result of the gauge theory from the viewpoint of the matrix model (similar approach is found in [67]).…”

“…Here w should be the smaller root with the boundary condition as Note that it is shown in [68,67] we have to take the vanishing fraction limit to obtain the Coulomb moduli from the matrix model contour integral. This is the essential difference between the profile function method and the matrix model description.…”

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

“…In [98,99,100,101] a different matrix model which describes the Nekrasov partition function has been found. While the form of the potential in particular is quite different, it would be interesting to see the relation with the model in this review.…”

Gauge Theory, Combinatorics, and Matrix Models

“…This restriction is considered in order to study the four dimensional supersymmetric gauge theory on orbifold ℝ 4 / ℤ r ≅ 2 / ℤ r [37], [38], [16], thus we call this orbifold partition function.…”