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Moduli spaces of residueless meromorphic differentials and the KP hierarchy

Alexandr Buryak, Paolo Rossi, Dimitri Zvonkine

Abstract

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.

Moduli spaces of residueless meromorphic differentials and the KP hierarchy

Abstract

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.

Paper Structure

This paper contains 13 sections, 14 theorems, 93 equations.

Table of Contents

  1. Moduli spaces of meromorphic differentials with residue conditions
  2. Meromorphic differentials with residue conditions
  3. Multiscale differentials with residue conditions
  4. The class $[\overline{\mathcal{H}}_g^{\mathop{\mathrm{res}}\nolimits} (a_1,\ldots,a_n)]$ as a partial cohomological field theory
  5. The DR hierarchy for the cycle of residueless meromorphic differentials
  6. A reduction to meromorphic differentials with two zeros and the KP hierarchy
  7. A reduction of the DR hierarchy
  8. The Miura transformation
  9. The KP hierarchy
  10. The main result
  11. Step 1 of the proof: more properties of the DR hierarchy
  12. Step 2 of the proof: more properties of the KP hierarchy
  13. Step 3 of the proof: a limited amount of data determines the hierarchies uniquely

Key Result

Proposition 1.4

CMZ20

Theorems & Definitions (36)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Proposition 1.8
  • proof
  • Remark 2.1
  • ...and 26 more