Isomonodromic deformations and conformal field theory with W-symmetry

The correspondence between isomonodromic deformations and conformal field theories with W-symmetry relates isomonodromic tau-functions to linear combinations of conformal blocks of the W-algebras. In my thesis I conjecture the existence of such a relation in the general case, thus generalizing the result of Gamayun-Iorgov-Lisovyy from the Virasoro algebra to an arbitrary W-algebra. Two different proofs of this relation are presented. One of the main results is the representation of the general isomonodromic tau-function in the form of a Fredholm determinant with matrix kernel. One part of this thesis is devoted to study of cases with a particular monodromy given by quasi-permutation matrices. In these cases I present explicit formulas for conformal blocks given in terms of branched covers of the Riemann sphere. Also I present the computation of the characters of the corresponding representations of the W-algebra.