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 $\tau$-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.

98 pages, 5 figures; v3: minor corrections, expanded introduction, added details, updated references

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• Matrix models and a proof of the open analog of Witten's conjecture
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• The combinatorial formula for open gravitational descendents
• Higher Airy structures and topological recursion for singular spectral curves
• Refined open intersection numbers and the Kontsevich-Penner matrix model
• The hypergeometric functions of the Faber-Zagier and Pixton relations
• Higher symplectic capacities
• Open intersection numbers and the wave function of the KdV hierarchy
• Open intersection numbers and free fields
• Differential forms, Fukaya $A_{\infty }$ algebras, and Gromov-Witten axioms
• Open KdV hierarchy and minimal gravity on disk
• Combinatorial models for moduli spaces of open Riemann surfaces
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• Extended $r$-spin theory in all genera and the discrete KdV hierarchy
• Spectral Form Factor for Time-dependent Matrix model
• Open ${\mathbb{CP}}^1$ descendent theory I: The stationary sector
• Fixed-point Localization for ${\mathbb{RP}}^{2m}\subset {\mathbb{CP}}^{2m}$
• Punctures and p-spin curves from matrix models II
• From minimal gravity to open intersection theory
• On a conjectural solution to open KdV and Virasoro
• Open r-spin theory I: Foundations
• Holography for BCFTs with Multiple Boundaries: Multi-Splitting Quenches
• Punctures and p-spin curves from matrix models
• Random Matrix, Singularities and Open/Close Intersection Numbers
• Open KdV hierarchy of 2d minimal gravity of Lee-Yang series
• Closed extended $r$-spin theory and the Gelfand-Dickey wave function
• Noncommutativity in two-matrix model extension of one-dimensional topological gravity
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds
• On affine coordinates of the tau-function for open intersection numbers