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arXiv · 1508.03750

Simple Lie algebras and topological ODEs

Marco Bertola·Boris Dubrovin·Di Yang
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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 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 to computing the Witten and Fan - Jarvis - Ruan invariants of the Deligne - Mumford moduli spaces of stable algebraic curves.

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Marco Bertola, Boris Dubrovin, Di Yang. 2015-08-15. Simple Lie algebras and topological ODEs. https://arxiv.org/abs/1508.03750

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