Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

Details Nonperturbative effects and nonperturbative definitions in matrix models and topological strings Nonperturbative effects and nonperturbative definitions in matrix models and topological strings

2008-05-20

arxiv.org/abs/0805.3033v2

JHEP 0812:114,2008

Physics

High Energy Physics

High Energy Physics - Theory

50 pages, 12 figures, comments and references added, small corrections

Scientific paper

10.1088/1126-6708/2008/12/114

We develop techniques to compute multi-instanton corrections to the 1/N expansion in matrix models described by orthogonal polynomials. These techniques are based on finding trans-series solutions, i.e. formal solutions with exponentially small corrections, to the recursion relations characterizing the free energy. We illustrate this method in the Hermitian, quartic matrix model, and we provide a detailed description of the instanton corrections in the Gross-Witten-Wadia (GWW) unitary matrix model. Moreover, we use Borel resummation techniques and results from the theory of resurgent functions to relate the formal multi-instanton series to the nonperturbative definition of the matrix model. We study this relation in the case of the GWW model and its double-scaling limit, providing in this way a nice illustration of various mechanisms connecting the resummation of perturbative series to nonperturbative results, like the cancellation of nonperturbative ambiguities. Finally, we argue that trans-series solutions are also relevant in the context of topological string theory. In particular, we point out that in topological string models with both a matrix model and a large N gauge theory description, the nonperturbative, holographic definition involves a sum over the multi-instanton sectors of the matrix model

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