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Random supermatrices with an external source. (English) Zbl 1396.81188

Summary: In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich’s Airy matrix models and generalizations. The techniques relied on explicit computations of the \(k\)-point functions for arbitrary \(N\) (the size of the matrices) and on an \(N\)-\(k\) duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the \(k\)-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.

MSC:

81T45 Topological field theories in quantum mechanics
62P35 Applications of statistics to physics
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
83C80 Analogues of general relativity in lower dimensions
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