Abstract
We sharpen the duality between open and closed topological string partition functions for topological gravity coupled to matter. The closed string partition function is a generalized Kontsevich matrix model in the large dimension limit. We integrate out off-diagonal degrees of freedom associated to one source eigenvalue, and find an open/closed topological string partition function, thus proving open/closed duality. We match the resulting open partition function to the generating function of intersection numbers on moduli spaces of Riemann surfaces with boundaries and boundary insertions. Moreover, we connect our work to the literature on a wave function of the KP integrable hierarchy and clarify the role of the extended Virasoro generators that include all time variables as well as the coupling to the open string observable.
Highlights
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
The closed versus open/closed duality in the case of pure gravity was derived in a 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 [19]
Amongst the many insights that string theory has provided into topological gravity is the fact that the integration variables in the Kontsevich matrix model correspond to open strings
Summary
The generalized Kontsevich model is a matrix generalization of the higher Airy function [9]. When the matrix model integration variable tends to infinite size, it becomes a generating function for the p-spin intersection numbers on moduli spaces of Riemann surfaces [9, 28,29,30,31,32,33]. The idea of open/closed string duality can at least be traced back to the advent of D-branes in string theory [37]. The fact that it takes a simple form in the case of pure topological gravity was understood in [16]. The more rigorous reference [19] follows almost the same technical route we described above to render the physical intuition in [16] mathematically precise, but there are important differences that we will highlight
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