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Intersection theory on moduli of disks, open KdV and Virasoro

Rahul Pandharipande, Jake P. Solomon, Ran J. Tessler

Abstract

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $τ$-function of an open KdV heirarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.

Intersection theory on moduli of disks, open KdV and Virasoro

Abstract

We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the -function of an open KdV heirarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.

Paper Structure

This paper contains 55 sections, 25 theorems, 454 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Moduli of closed Riemann surfaces
  3. Witten's conjectures
  4. Moduli of Riemann surfaces with boundary
  5. Descendents
  6. Construction of the descendent theory of pointed disks
  7. Differential equations
  8. Partition functions
  9. Virasoro constraints
  10. String and dilaton equations
  11. KdV equations
  12. Formulas in genus 0
  13. Context and motivation
  14. Boundary conditions
  15. Mathematical and physical corroboration
  16. ...and 40 more sections

Key Result

Theorem 1.1

The operators $\mathcal{L}_n$ annihilate the genus zero partition function up to terms of higher genus. That is, for $n \geq -1,$ the coefficient of $u^{-1}$ in vanishes.

Figures (6)

  • Figure 1: A nodal disk with $3$ disk components, one sphere component, $5$ internal marked points and $6$ boundary marked points.
  • Figure 2: The hexagon represents ${\overline{\mathcal{M}}}^{main}_{0,3,1}$ and the line segment represents ${\overline{\mathcal{M}}}^{main}_{0,2,1}.$ The dotted line represents a typical fiber of the forgetful map. Stable marked disks are shown that represent a typical point of each space as well as a typical point of each boundary component.
  • Figure 3: We show the stable graphs from Example \ref{['ex:edgelabels']}. Boundary labels are shown as half-edges and interior labels are shown as double half-edges.
  • Figure 4: A canonical multisection at different boundary points.
  • Figure 5: (a) shows $\Gamma,$ (b) shows $v,$ (c) shows $\Gamma',$ (d) shows $\Lambda,$ and (e) shows $v'.$
  • ...and 1 more figures

Theorems & Definitions (105)

  • Conjecture 1
  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.6
  • Definition 2.7
  • ...and 95 more