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

Abstract

Abstract. In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes through a matrix model and is based on a Kontsevich type combinatorial formula for the intersection numbers that was found by the second author.

Refined open intersection numbers and the Kontsevich-Penner matrix model

“…When g = 0 it coincides with M R 0,k,l . Canonical boundary conditions are then constructed for the line bundles L i , and again it is proven that one can define 6) where e(E, s) is the relative Euler with respect to the canonical boundary conditions. As in g = 0, generic choices of canonical boundary conditions give rise to the same integrals.…”

“…A combinatorial formula for the open intersection numbers was found in [21]. The conjecture of R. Pandharipande, J. P. Solomon and the third author was proved in [6].…”

“…Then we can expand the function The integral on the right-hand side of (3.3) is understood as the term-wise integral of (3.4) with respect to our Gaussian probability measure on H M . We refer the reader to [6] for a more detailed discussion.…”

Topological open/closed string dualities: matrix models and wave functions

“…Of particular interest to us here is the open/closed duality which was understood fairly well in the string theory literature [13][14][15][16][17], and which was rigorously derived, including important additional details, in the more recent mathematical physics literature [18][19][20][21] in the case of pure two-dimensional gravity. This duality relates an open/closed string partition function to a purely closed string partition function; to be precise, the addition of a D-brane in topological string theory is transmuted into a shift of the background, a renormalization of the partition function, an operator insertion and an integral transform.…”

“…This matrix model depends only on the closed string sources, and can be viewed as a closed string matrix model in that sense. The closed versus open/closed duality in the case of pure gravity was derived in a particularly clear manner by integrating out off-diagonal degrees of freedom of a (N + 1) × (N + 1) matrix model to obtain a N × N matrix model depending on one extra eigenvalue which is the integral transform of an open string coupling (or D-brane modulus) [19]. In the large N limit, this then gives rise to a duality between a closed string theory and an open/closed string theory.…”