In preparation (april 2016)
The topological recursion
The topological recursion was first discovered in random matrices in 2004 [AMM04, Eynard04].
It became a mathematical theory in 2007 [EO07].
The Topological recursion takes as
input: a spectral curve S (an analytical plane curve with some structure)
output: a sequence of invariants (the T.R. invariants) of the spectral curve, denoted Wg,n(S). The n=0 invariants are denoted Wg,0(S) = Fg(S).
- The Witten-Kontsevich intersection numbers
(curve S: x=y^2 )
- The Mirzakhani relation among hyperbolic volumes of moduli spaces
(curve S: y = sin( x^(1/2)) )
- The Gromov-Witten invariants of Calabi-Yau manifolds
(curve S: = mirror curve)
- the large size expansion of random matrix laws
(curve S: = equilibrium spectral density at large N)
- the enumeration of maps of all topologies
(curve S: = counting function of rooted planar maps)
- the WKB asymptotic expansion of Hitchin systems
( curve S: = Hitchin's spectral curve)
- The Knot polynomials (Jones, Homfly,...)
( curve S: = A-polynomial of the knot)
The T.R. invariants have many beautiful mathematical properties:
- symplectic invariance
- modular properties, automorphicity
- form-cycle dualities for their deformations
- nice behavior under limits
Study the statistical properties of eigenvalues of large random matrices.
of Harish-Chandra- Itzykson-Zuber type.
Let G be a semi-simple Lie group. Let X and Y be matrices in the Lie Algebra of G.
For any polynomial function f(x,y) of 2 variables, I found in [E-PF05] [dFEPFZ] that:
where the product is over positive roots in a Cartan
subalgebra, and the last integral is over the strictly positive Borel
subalgebra, and is a Gaussian integral.
(for G=U(N), the integral is over strictly upper triangular matrices).
The case f=1, is Harish-Chandra formula.
Example of application: