Abstract
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.
Highlights
The n × n matrix a is Hermitian, and the m × m matrix b is Hermitian; the matrices α and αare rectangular, respectively n × m and m × n and consist of Grassmanian variables
Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique
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
Summary
The same technique allows one to compute correlation functions such U(t1, t2) =< streit1Mstreit2M >. (n + m) terms which can all be computed with the help of the identity (2.10) through an appropriate shift of the eigenvalues of the source matrix such as ra → ra + t1δai + t2δaj (3.3). In which both contours encircle the poles ri. Enough these four terms recombine nicely into one single compact expression. After appropriate shifs zi → zi ± ti/2 the four integrands become identical and their sum is obtained by taking the residues at all the poles in the z1, z2 plane. The final expression for the connected correlation function is . It is clear that this may be generalized to a k-point function as in the usual GUE case [1]
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