Non-oriented maps and the BKP hierarchy, with applications to the O(N)-BGW integral and recurrence formulas

par Valentin Bonzom (LIPN - Institut Galilée - Université Paris 13)

mercredi 6 avr. 2022, 14:00 → 15:00 Europe/Paris

https://greenlight.virtualdata.cloud.math.cnrs.fr/b/fab-49u-gkt

The Hermitian 1-matrix model, or equivalently the generating series of all oriented maps, satisfies the equations of the KP hierarchy.  This means that its expansion in a special basis, the Schur functions, has minor determinants as coefficients. I will recall how to prove it from the matrix integral itself and also from the celebrated Frobenius formula in the more general case of constellations (i.e. weighted Hurwitz numbers). A nice application of this formalism is recurrence formulas for a few families of maps, which are the most efficient formulas to compute the numbers of maps, and which completely bypass Tutte's resolution method of the usual loop equations.
Then I will motivate the generalization to non-oriented maps through a beautiful conjecture of Goulden and Jackson, known as the b-conjecture. I will explain the difficulties in proving integrability for non-oriented maps, in particular for matrix models with an external matrix. In addition to recurrence formulas counting the numbers of non-oriented maps (like in the oriented case), we found another application, which is a Pfaffian expression for the O(N)-BGW matrix integral in the case of even multiplicities.