The combinatorial formula for open gravitational descendents

Abstract

In recent works, [20,21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals satisfies the open KdV equations. In this paper we prove a formula of these integrals in terms of sums over weighted graphs. Based on this formula, the conjecture of [20] was proved in [5]. Contents 2.2.4. A comment about the alternative definition in the stable case 2.2.5. Spin graphs 2.2.6. M g,k,l 2.3. The line bundles L i 2.4. Boundary conditions and intersection numbers 2.5. The orientation of M g,k,l 3. Sphere bundles and relative Euler class 4. Symmetric Jenkins-Strebel stratification 4.1. JS stratification for the closed moduli 4.1.1. JS differential and the induced graph 4.1.2. Combinatorial moduli 4.1.3. Tautological line bundles and associated forms 4.2.

Matrix Models and A Proof of the Open Analog of Witten’s Conjecture

“…The construction of the higher genus moduli and intersection theory was found by J. Solomon and R.T. in [STa]. The details of these constructions also appear in [Tes15], Section 2. A combinatorial formula for the open intersection numbers in all genera was found in [Tes15].…”

“…The details of these constructions also appear in [Tes15], Section 2. A combinatorial formula for the open intersection numbers in all genera was found in [Tes15].…”

Open $\mathbb{CP}^1$ descendent theory I: The stationary sector

“…Although Theorem 6.2 guarantees the existence of such an orientation to each moduli separately, in order to define invariants we must choose a specific orientation for each moduli, and our calculations required to understand the relations between different orientations. Our approach resembles the approach of [33], where the orientation of the moduli space of graded surfaces with boundary is constructed using Strebel's stratification and its boundary behaviour is explored. We assume ⃗ d ≠ ⃗ 0.…”

“…The open Gromov-Witten theory of a point has been defined and solved in a series of papers [6,31,32,33] resulting in open analogues of the KdV equations and Virasoro constraints.…”

On affine coordinates of the tau-function for open intersection numbers

“…The Virasoro constraints and Wconstraints for τ N were also constructed in his works. See Buryak-Tessler [15] for a proof of the conjecture of [35] (where the construction of the higher genus intersection theory was announced by Solomon and Tessler, see also [41]). For more about the open intersection theory, see [4, 7, 8, 10-12, 29, 32, 39, 40].…”