On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles

TL;DR

The work investigates how degenerate insertions in conformal blocks realize surface operator data within AGT, mapping to beta-ensemble resolvents and yielding a Schödinger/Baxter perspective on Seiberg–Witten prepotentials. It shows that in the ε2→0 limit the conformal blocks exponentiate to generate SW-like free energies, with the generating differential S(x;ε1) governing Bohr–Sommerfeld periods and connecting to Harish-Chandra functions in the perturbative regime. The methodology unifies CFT differential equations, free-field and beta-ensemble realizations, and topological recursion, enabling explicit evaluation of one-parameter deformed prepotentials and the corresponding spectral problems for various gauge theories (including SU(2) with fundamental matter and adjoint cases). The framework highlights a deep link between degeneracy constraints, matrix-model resolvents, and the spectral-curve quantization that underlies Nekrasov–Okounkov-type structures, while suggesting wide avenues for generalization to higher rank and genus.

Abstract

We present a summary of current knowledge about the AGT relations for conformal blocks with additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, when the conformal blocks satisfy hypergeometric-type differential equations in position of the degenerate operator. A special attention is devoted to representation of conformal block through the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr-Sommerfeld representation of the one-parameter deformed Seiberg-Witten prepotentials (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish-Chandra function. The Shrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This provides a powerful method for evaluation of the single-deformed prepotentials, and even for the Seiberg-Witten prepotentials themselves. We mostly concentrate on the representative case of the insertion into the four-point block on sphere and one-point block on torus.

Figures (2)

• Figure 1: Here $z_{1,2,3,4} = (0,q,1,\infty)$ and $q\ll x \ll 1$. In conformal theory, the structure constant for degenerate primary vanishes unless $\alpha' =\alpha \pm {1/2b}$CFT. In free field representation of MMS1MMS for $\alpha' \neq \alpha \pm {1/2b}$, there are additional screening insertions in the matrix model ($\beta$-ensemble) representation, with open integration contours stretching from $0$ to $x$. As explained in s.\ref{['3.2.2']} such insertions violate differential equations naively following from the equation (\ref{['degdifeq']}) for the degenerate field, (\ref{['degeq']}). Therefore, in this paper we consider only the case of $\alpha'=\alpha \pm {1/2b}$. The relation to the $a$-parameter in Yang-Mills theory is $\alpha = a+\epsilon/2$. In the limit of $\epsilon_2\rightarrow 0$ the difference $1/2b = \frac{1}{2}\sqrt{-\frac{\epsilon_2}{\epsilon_1}}$ between $\alpha$ and $\alpha'$ gets negligible and $a$ in the corresponding Nekrasov function $F(\epsilon_1)$ at this limit can be considered as related to either $\alpha$ or $\alpha'$, thus restoring the symmetry of the diagram in application to the AGT relation.
• Figure 2: Topology of the tree diagram implies certain ordering of pairings in the definition of the conformal block. From each OPE only the contribution of one particular Verma module is picked up, thus, the associativity of OPE is restored only after sums are taken over the intermediate dimensions. This diagram corresponds to the ordering different from Fig.\ref{['conf41']}: $x\gg q\gg 1$. Here the intermediate dimension $\alpha_{11}=\alpha_1\pm1/2b$. The two diagrams are connected by a duality transformation.