Determinant formulas for matrix model free energy

The paper contains a new non-perturbative representation for subleading contribution to the free energy of multicut solution for hermitian matrix model. This representation is a generalisation of the formula, proposed by Klemm, Marino and Theisen for two cut solution, which was obtained by comparing the cubic matrix model with the topological B-model on the local Calabi-Yau geometry $\widehat{II}$ and was checked perturbatively. In this paper we give a direct proof of their formula and generalise it to the general multicut solution.