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

Abstract

We review the $\beta$-deformed matrix model approach to the correspondence between four-dimensional $\mathcal {N}=2$ gauge theories and two-dimensional conformal field theories. The $\beta$-deformed 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 $\epsilon _{1}, \epsilon _{2} \rightarrow 0$. We show that the large N analysis produces the purely gauge theory results. Furthermore we discuss the Nekrasov-Shatashvili limit ($\epsilon _{2} \rightarrow 0$) 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.

A citation of the form [V:x] refers to article number x in this volume.

Appendix: Integral Formulas

Let us define the following multiple integral

$$\begin{aligned} \langle \!\langle x^{Y}\rangle \!\rangle _{N}= & {} \prod _{I=1}^{N} \int _{0}^{1} d x_{I} \prod _{I=1}^{N} x_{I}^{\alpha } (1-x_{I})^{\beta } \prod _{1\le I <J\le N} (x_{I} - x_{J})^{2\gamma } x^{Y} \end{aligned}$$

(6.1)

where supposing that $\mathfrak {R}\beta >0, \ldots$ for convergence of the integrals. When $Y=\emptyset$ and $Y = [1^{k}]$, this is the Selberg integral [98] and Aomoto integral [99]

$$\begin{aligned} \langle \!\langle 1\rangle \!\rangle _{N}= & {} \prod _{j=0}^{N-1}\frac{\Gamma (\alpha + 1 + j\gamma ) \Gamma (\beta + 1 + j \gamma ) \Gamma (1 + (j+1)\gamma )}{\Gamma (\alpha + \beta + 2 + (N + j-1)\gamma ) \Gamma (1 + \gamma )}, \nonumber \\ \langle \!\langle x^{Y=[1^{k}]}\rangle \!\rangle _{N}= & {} \langle \!\langle 1\rangle \!\rangle _{N} \prod _{j=1}^{k} \frac{\alpha + 1 +(N-j)\gamma }{\alpha + \beta + 2 + (2N - j -1)\gamma }. \end{aligned}$$

(6.2)

Another multiple integral which appeared in the main text is involving the Jack polynomial $P_{Y}(x)$. This is a polynomial of $(x_{1},x_{2},\ldots ,x_{N})$ and written as

$$\begin{aligned} P_{Y}(x)= m_{Y}(x) + \sum _{Y'<Y} a_{Y,Y'}m_{Y'}(x), \end{aligned}$$

(6.3)

where $m_{Y}(x)$ is the monomial symmetric polynomial. Then the following integral is given by [100, 101]

$$\begin{aligned} \langle \!\langle P_{Y}(x) \rangle \!\rangle _{N}= & {} \prod _{I=1}^{N} \int _{0}^{1} d x_{I} \prod _{I=1}^{N} x_{I}^{\alpha } (1-x_{I})^{\beta } \prod _{1\le I <J\le N} (x_{I} - x_{J})^{2\gamma } P_{Y}(x) \\= & {} \prod _{i\ge 1}\prod _{j=0}^{y_{i}-1} \frac{\alpha + 1 +j+(N-i)\gamma }{\alpha + \beta + 2 + j+ (2N - i -1)\gamma } \frac{\prod _{i\ge 1}\prod _{j=0}^{y_{i}-1} (N+1-i)\gamma + j}{\prod _{(i,j)\in Y} (y_{i}-j+(\tilde{y}_{j} - i+1)\gamma )}, \nonumber \end{aligned}$$

(6.4)

where $Y=[y_{1},y_{2},\ldots ]$ with $y_{1}\ge y_{2}\ge \ldots$ and $\tilde{Y} = [\tilde{y}_{1} \ge \tilde{y}_{2} \ge \ldots ]$ is the transpose of Y.