In preparation (mid 2017)
The topological recursion
The topological recursion was first discovered in random matrices in 2004 [Eynard04, AMM04].
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).
Fg(S) is a complex number, and Wg,n(S) is a differential n-form.
- The Witten-Kontsevich intersection numbers (curve S: x=y^2 )
- The Mirzakhani relation among hyperbolic volumes of moduli spaces (curve S: y = sin( √x) )
- The Gromov-Witten invariants of toric Calabi-Yau 3-manifolds (curve S: = mirror curve)
where GWg,n(X,L;ß,µ) is the open Gromov-Witten invariant of the Calabi-Yau 3-fold X,
i.e. the "number" of holomorphic immersions in X of surfaces of genus
g, with n boundaries, so that boundaries are immersed on a special
Lagrangian submanifold L, with winding number µi, and of relative homology class ß.
- 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
A main question in random matrix theory is the large size asymptotic behavior of the eigenvalues distribution.
Below: plots of histograms for eigenvalues af a randdom Hermitian
matrix, in a Gaussian model (semi-circle law), and non Gaussian. In the
large size (N) limit, the eigenvalue density tends to an algebraic curve. The deviation, of order 1/N2, is also an algebraic function, and more generally, all the coefficients in the 1/N2 expandion are algebraic functions.
- See here : my GOE online generator.
- See here : spectral curve of a quartic matrix model.
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 dual 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:
I found the following results