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.

Simple Lie Algebras and Topological ODEs / Bertola, M.; Dubrovin, B.; Yang, D.. - In: INTERNATIONAL MATHEMATICS RESEARCH NOTICES. - ISSN 1073-7928. - 2018:5(2018), pp. 1368-1410. [10.1093/imrn/rnw285]

Simple Lie Algebras and Topological ODEs

Bertola, M.;Dubrovin, B.;Yang, D.
2018

Abstract

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.
2018
5
1368
1410
rnw285
https://doi.org/10.1093/imrn/rnw285
https://arxiv.org/abs/1508.03750
Bertola, M.; Dubrovin, B.; Yang, D.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/50169
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