All genus correlation functions for the hermitian 1-matrix model

@inproceedings{Eynard2004AllGC,
  title={All genus correlation functions for the hermitian 1-matrix model},
  author={Bertrand Eynard},
  year={2004}
}
  • B. Eynard
  • Published 29 July 2004
  • Mathematics, Physics
We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be represented diagrammaticaly as Feynmann graphs of a cubic interaction field theory on the curve. 

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References

SHOWING 1-10 OF 51 REFERENCES
1/N2 Correction to Free Energy in Hermitian Two-Matrix Model
Using the loop equations we find an explicit expression for genus 1 correction in hermitian two-matrix model in terms of holomorphic objects associated to spectral curve arising in large N limit. Our
Large N expansion of the 2-matrix model, multicut case
We present a method, based on loop equations, to compute recursively, all the terms in the large $N$ topological expansion of the free energy for the 2-hermitian matrix model, in the case where the
Complex curve of the two-matrix model and its tau-function
We study the Hermitian and normal two-matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general
Matrix model calculations beyond the spherical limit
Combinatorial solution of the two-matrix model
The τ‐function of the universal whitham hierarchy, matrix models and topological field theories
The universal Witham hierarchy is considered from the point of view of topological field theories. The �-function for this hierarchy is defined. It is proved that the algebraic orbits of Whitham
LOOP EQUATIONS AND NON-PERTURBATIVE EFFECTS IN TWO-DIMENSIONAL QUANTUM GRAVITY
We present the loop equations of motion which define the correlation functions for loop operators in two-dimensional quantum gravity. We show that non-perturbative correlation functions constructed
Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation
Abstract:We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a
...
...