Quadratic double ramification integrals and the noncommutative KdV hierarchy

Alexandr Buryak, Paolo Rossi

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1 Citation (Scopus)


In this paper we compute the intersection number of two double ramification (DR) 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 DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite-dimensional partial cohomological field theory given by (Formula presented.), where (Formula presented.) is a parameter and (Formula presented.) is Hain's theta class, appearing in Hain's formula for the DR cycle on the moduli space of curves of compact type. This infinite rank DR 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 Korteweg-de-Vries (KdV) hierarchy on a noncommutative Moyal torus.

Original languageEnglish
Pages (from-to)843-854
Number of pages12
JournalBulletin of the London Mathematical Society
Issue number3
Early online date21 Jan 2021
Publication statusPublished - Jun 2021


  • 14H10
  • 37K10 (primary)



State classification of scientific and technological information

  • 29.05 Particle Physics. The theory of fields. High energy physics


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