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Invariants of algebraic curves and topological expansion
For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties.…
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Matrices coupled in a chain: I. Eigenvalue correlations
The general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.
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Topological expansion for the 1-hermitian matrix model correlation functions
We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the…
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Invariants of spectral curves and intersection theory of moduli spaces of complex curves
To any spectral curve S, we associate a topological class {\Lambda}(S) in a moduli space M^b_{g,n} of "b-colored" stable Riemann surfaces of given topology (genus g, n boundaries), whose integral…
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Torus Knots and Mirror Symmetry
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants.…
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Hermitian matrix model free energy: Feynman graph technique for all genera
We present the diagrammatic technique for calculating the free energy of the Hermitian one-matrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans…
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Matrix eigenvalue model: Feynman graph technique for all genera
We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power β by the Vandermonde determinant) to all orders of 1/N expansion…
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Breakdown of universality in multi-cut matrix models
We solve the puzzle of the disagreement between orthogonal polynomials methods and mean-field calculations for random N×N matrices with a disconnected eigenvalue support. We show that the difference…
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Exact solution of the O(n) model on a random lattice
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Algebraic methods in random matrices and enumerative geometry
We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one…
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