[Paper Review] Simple Lie algebras and topological ODEs

Marco Bertola, Boris Dubrovin+1 more|arXiv (Cornell University)|Aug 15, 2015

Algebraic Geometry and Number Theory30 references3 citations

TL;DR

This paper introduces a system of linear ODEs with polynomial coefficients—called the topological ODE—associated with any simple Lie algebra 𝔤, whose regular solutions at infinity form a vector space of dimension equal to the rank of 𝔤. The authors construct generalized Airy resolvents as a basis for this solution space, extending the known Airy function connection in the 𝔰𝔩₂ case to all simple Lie algebras, and apply them to compute Witten–Kontsevich and Fan–Jarvis–Ruan invariants of moduli spaces of curves.

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.

Motivation & Objective

• To define a system of linear ODEs with polynomial coefficients, called the topological ODE, for any simple Lie algebra 𝔤.
• To characterize the space of solutions regular at infinity, showing its dimension equals the rank of 𝔤.
• To construct a basis of solutions—generalized Airy resolvents—for this space using the structure of the Lie algebra and its Weyl generators.
• To apply the generalized Airy resolvents to compute intersection numbers in the Deligne–Mumford moduli spaces of stable curves.
• To extend the known connection between Airy functions and the 𝔰𝔩₂ case to arbitrary simple Lie algebras via a uniform framework.

Proposed method

• Define the cyclic element Λ = I₊ + λE₋ᵣₕₑₜₐ in the Lie algebra 𝔤, where I₊ is a principal nilpotent and E₋ᵣₕₑₜₐ is the lowest root vector.
• Formulate the topological ODE as M′ = [M, Λ], where M is a 𝔤-valued function of λ.
• Study the asymptotic behavior of solutions at infinity, focusing on those growing at most polynomially—called regular solutions.
• Construct generalized Airy resolvents as a basis for the space of regular solutions using the root system and Cartan–Killing form.
• Use Laplace–Borel transforms and contour integral representations (Pearcey-type integrals) to express solutions for classical and exceptional Lie algebras.
• Derive differential equations for the Laplace–Borel transforms of components of the solution, leading to hypergeometric-type ODEs for exceptional cases like E₆.

Experimental results

Research questions

• RQ1 What is the structure of the solution space of the topological ODE for a given simple Lie algebra 𝔤?
• RQ2 How can one explicitly construct a basis of regular solutions (generalized Airy resolvents) at infinity for any simple Lie algebra?
• RQ3 To what extent can the generalized Airy resolvents be used to compute intersection numbers in the moduli space of stable curves?
• RQ4 How does the topological ODE generalize the known Airy function connection in the 𝔰𝔩₂ case to higher-rank Lie algebras?
• RQ5 What is the role of the cyclic element Λ = I₊ + λE₋ᵣₕₑₜₐ in encoding the algebraic and asymptotic structure of the solutions?

Key findings

• The space of regular solutions to the topological ODE of 𝔤-type has dimension equal to the rank of 𝔤.
• For 𝔰𝔩₂(ℂ), the regular solution is expressible in terms of products of Airy functions and their derivatives, matching a known construction used in computing Witten–Kontsevich tau-function derivatives.
• Generalized Airy resolvents are constructed as a basis for the solution space of the topological ODE for arbitrary simple Lie algebras.
• For classical Lie algebras (Aₙ, Bₙ, Cₙ, Dₙ), solutions are represented via Pearcey-type integrals involving x^{4n-2} or x^{4n+2} in the exponent.
• For E₆, the solution components satisfy a system of second-order ODEs involving hypergeometric functions ₀F₁, with explicit solutions in terms of Bessel-type functions.
• The asymptotic expansions of the generalized Airy resolvents can be derived via saddle-point methods on the integral representations, enabling computation of intersection numbers.