Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A group action on Losev-Manin cohomological field theories

Sergey Shadrin, Dimitri Zvonkine

Abstract

We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus 0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained via dressing transformations technique.

A group action on Losev-Manin cohomological field theories

Abstract

We discuss an analog of the Givental group action for the space of solutions of the commutativity equation. There are equivalent formulations in terms of cohomology classes on the Losev-Manin compactifications of genus 0 moduli spaces; in terms of linear algebra in the space of Laurent series; in terms of differential operators acting on Gromov-Witten potentials; and in terms of multi-component KP tau-functions. The last approach is equivalent to the Losev-Polyubin classification that was obtained via dressing transformations technique.

Paper Structure

This paper contains 16 sections, 20 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. The commutativity equation
  3. Pencils of flat connections
  4. The Losev-Manin moduli spaces
  5. The Losev-Manin cohomological field theories
  6. Gromov-Witten potentials
  7. Acknowledgements
  8. The upper triangular group
  9. The lower triangular group
  10. Group action orbits
  11. The commutativity equation and the loop space
  12. A special family of linear maps
  13. Symplectic framework
  14. A link to the Losev-Polyubin action
  15. Interpretation of the Losev-Polyubin action
  16. ...and 1 more sections

Key Result

Proposition 1.4

$\alpha_n = (p_n)_* (\beta_n)$ is a Losev-Manin CohFT with $T=V$.

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Proposition 1.6
  • proof
  • Definition 1.7
  • Definition 1.8
  • ...and 40 more