We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 matrices. This generalizes Suleimanov's result on the Painlevé equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.
Systems of linear equations accompanying the Painlevé andSchlesinger equations
The Schlesinger system [1] for the matriceswhere i, j = 1, m (the second group of equations can be replaced with the condition that the matrix A 1 + · · · + A m = A ∞ is constant in t 1 , . . . , t m ), was discovered as the compatibility condition for the system(2)All algebraic integrals of motion of system (1) are known. They are the traces tr A k i , k = 1, 2, . . . (equivalently, the characteristic polynomials of the matrices A i ). A closed form ω = H i dt i was found in [2],and the τ -function (log τ ) ti = H i was also determined.Here, we consider only one aspect of the problem of the complete integrability of system (1) in terms of special functions. Namely, in the case of 2×2 matrices A i , we seek a second-order linear equation for
We show that under the Euler integral transformation with the kernel (x − z) −α , some solutions of the Fuchs equations (the original pair for the Painlevé VI equation) pass into solutions of a system of the same form with the parameters changed according to the Okamoto transformation.
АннотацияWe consruct solutions of analogues of the nonstationary Schrödinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin Polyakov Zamolodchikov equations with four time variables and two space variables. *
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.