Quadratic double ramification integrals and the noncommutative KdV hierarchy

Abstract

In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by $\exp ({\mu }^2\Theta )$ where $\mu$ is a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank $1$ integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.

arXiv commentv3: updated affiliations, minor change at the beginning of section 4, updated references; 12 pages

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Cited by in corpus (6)

• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Higher genus relative Gromov--Witten theory and DR-cycles
• The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra
• A conjectural formula for $D{R}_g(a,-a){\lambda }_g$
• Moduli spaces of residueless meromorphic differentials and the KP hierarchy
• A generalization of Witten's conjecture for the Pixton class and the noncommutative KdV hierarchy