Simple Lie Algebras and Topological ODEs

For a simple Lie algebra g we define a system of linear ODEs with polynomial coeffi- cients, which we call the topological equation of 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 g = sl2(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 applica- tions of the generalized Airy resolvents for computing the Witten and Fan–Jarvis–Ruan invariants of the Deligne–Mumford moduli spaces of stable algebraic curves.