[Paper Review] On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two

V. N. Davletshina, Andrey E. Mironov|arXiv (Cornell University)|Jun 4, 2016

Advanced Algebra and Geometry4 references3 citations

TL;DR

This paper constructs a new family of commuting ordinary differential operators with polynomial coefficients corresponding to genus two spectral curves, proving that for generic coefficients, the set of orbits under the automorphism group of the first Weyl algebra is infinite. The construction uses a specific fourth-order operator $ L_4^{lat} $ and its higher-order companion $ L_{10}^{lat} $, with coefficients depending on six parameters, and demonstrates the infinitude of orbits via a dominant morphism from $ \mathbb{C}^6 $ to the coefficient space $ \mathbb{C}^5 $.

ABSTRACT

The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

Motivation & Objective

• To investigate the structure of orbits under the action of $ \mathrm{Aut}(A_1) $ on solutions of commuting differential operator equations with polynomial coefficients.
• To determine whether the set of orbits is finite or infinite for generic spectral curves of genus two.
• To construct an explicit, simple family of commuting differential operators in the first Weyl algebra satisfying a genus two hyperelliptic spectral curve equation.
• To provide evidence supporting Berest's conjecture on the finiteness of orbits depending on the genus of the spectral curve.

Proposed method

• Construct a fourth-order differential operator $ L_4^{lat} $ with variable coefficients depending on six complex parameters $ \alpha_i $.
• Explicitly compute a tenth-order operator $ L_{10}^{lat} $ that commutes with $ L_4^{lat} $, using direct algebraic computation.
• Derive the spectral curve equation $ Y^2 = X^5 + c_4X^4 + \cdots + c_0 $, where the coefficients $ c_i $ are polynomial functions of the $ \alpha_i $.
• Establish a morphism $ \mathbb{C}^6 \to \mathbb{C}^5 $ mapping parameters $ \alpha_i $ to coefficients $ c_i $, and prove its differential is surjective at a generic point.
• Use the dominance of the morphism to deduce that for generic $ c_i $, there exists a 1-parameter family of solutions.
• Show that operators from this family lie in distinct $ \mathrm{Aut}(A_1) $-orbits by analyzing the action of automorphisms on the parameter space.

Experimental results

Research questions

• RQ1 For a generic genus two spectral curve defined by $ Y^2 = X^5 + c_4X^4 + \cdots + c_0 $, is the set of $ \mathrm{Aut}(A_1) $-orbits finite or infinite?
• RQ2 Can a simple, explicit family of commuting differential operators in the first Weyl algebra be constructed that realizes such a spectral curve?
• RQ3 Does the existence of a 1-parameter family of solutions imply infinitely many distinct orbits under the Weyl algebra automorphism group?
• RQ4 Can the methods used for self-adjoint operators be extended to non-self-adjoint cases in the context of commuting differential operators?

Key findings

• The operator $ L_4^{lat} = \left( (\alpha_1 x^2 + 1)\partial_x^2 + (\alpha_2 x + \alpha_3)\partial_x + \alpha_4 x + \alpha_5 \right)^2 + \alpha_1 \alpha_4 g(g+1)x + \alpha_6 $ commutes with a tenth-order operator $ L_{10}^\flat $ for $ g = 2 $, for all complex $ \alpha_i $.
• The pair $ (L_4^\flat, L_{10}^\flat) $ satisfies the spectral curve equation $ Y^2 = X^5 + c_4X^4 + \cdots + c_0 $, with coefficients $ c_i $ depending polynomially on the $ \alpha_i $.
• The morphism $ \mathbb{C}^6 \to \mathbb{C}^5 $ sending $ \alpha_i \mapsto c_i $ is dominant, as its differential is surjective at a generic point.
• For generic $ c_i $, the solution space contains a 1-parameter family of operators, implying the existence of infinitely many distinct $ \mathrm{Aut}(A_1) $-orbits.
• The constructed family provides the first known simple example of non-self-adjoint commuting operators in $ A_1 $ with polynomial coefficients corresponding to a genus two curve.
• The authors conjecture that $ L_4^\flat $ commutes with an operator of order $ 4g+2 $ for all $ g $, extending the construction beyond $ g=2 $.