Simple Lie Algebras and Topological ODEs

Abstract

For a simple Lie algebra |$\mathfrak{g}$| we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of |$\mathfrak{g}$|-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example |$\mathfrak{g}=sl_2(\mathbb C)$| the regular solution can be expressed via products of Airy functions and their derivatives; this matrix-valued function was used in our previous work [4] for computing logarithmic derivatives of the Witten–Kontsevich tau-function. For an arbitrary simple Lie algebra we construct a basis in the space of regular solutions to the topological equation called generalized Airy resolvents. We also outline applications of the generalized Airy resolvents for computing the Witten and Fan–Jarvis–Ruan invariants of the Deligne–Mumford moduli spaces of stable algebraic curves.