Liouville theory, $ \mathcal{N} = 2 $ gauge theories and accessory parameters

Abstract

The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Ω-deformed) U(2) superYang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function.

Classical conformal blocks and accessory parameters from isomonodromic deformations

“…Many other special limits remain to be explored, for example, the classical, algebraic and Riccati solutions of PVI [34,43]. Moreover, the connection between the isomonodromic and the AGT approaches [25,72] can give further insights on all these applications.…”

“…We present the next two terms in the expansion above in appendix C. We implemented the algorithm of this section in a computer algebra program, as we do not have the explicit series solution for the constraint (4.33). We tested the accessory parameter expansion up to order x 2 analytically by comparing with the semiclassical limit of the CFT conformal block [3,25] using the inverse gram matrix approach [43,59]. We also checked this algorithm numerically, substituting the θ's with some fixed numbers from the beginning, up to order x 5 .…”

“…One can show that the so-called Nekrasov-Shatashvili limit [23] is equivalent to the large central charge limit of Liouville theory. This fact was used in [24,25] to obtain an expression for the classical conformal block in terms of the N = 2 twisted superpotential, calculated at the saddle-point of the partition function. The saddle-point condition then has to be solved order-by-order in x.…”

“…The crucial step is to solve a special initial condition to the isomonodromic flow. We can only solve this condition order-by-order in x, in a similar fashion to [3] or the AGT approach of [25]. This constrains the general moduli space of the monodromy group to a subspace with only one parameter, the composite monodromy σ.…”

Quantum periods and prepotential in N = 2 $$ \mathcal{N}=2 $$ SU(2) SQCD

“…Then we compare the prepotential with the NS limit of the Nekrasov partition function [17]. Note that the deformed prepotentials for N f = 1, 2, 4 are obtained from the classical limit of the conformal blocks of two dimensional conformal field theories [39][40][41]. The SW periods (2.6) around u = ∞ have been given by (2.20) and (2.21) [35].…”