Logarithmic Painleve tau Functions and Logarithmic Representations of the Virasoro Algebra

Painleve equation were introduced more 100 years in the study of second order differential equations without movable branching points. They occupy special place in the theory of differential equations due to their huge symmetry (Backlund transformations). They have several applications for example in random matrix theory and in the Ising model. Gamayun, Iorgov and Lisovyy suggested the relation between Painleve equations the conformal field theory (CFT). Namely the explicit formula of the expansion of tau-function in terms of so called Virasoro conformal blocks. This formula is now proven by different methods. But the original formula works only for generic values of integration constants, in special cases this formula has singularities. On the other hand it is know that in such cases solution and tau-function exist but has logarithmic behavior. Note also that such logarithmic solutions appear in the application to the Ising model. The formulas for the logarithmic case can be obtained from the generic formula using certain limit procedure. But the complete CFT interpretation of such tau-function was unclear. This interpretation is the main aim of this diploma work. The answer is given in terms certain logarithmic (not completely reducible) representation of the Virasoro algebra.