Counting Monster Potentials

Abstract: The monster potentials were introduced by  Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that:

1) The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.

2) The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me  to associate to each partition of N a unique monster potential with N roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer N, the number of monster potentials with N roots coincide with the number of integer partitions of N, which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

The talk is based on the preprint arxiv.org/abs/2009.14638 , written in collaboration with Riccardo Conti (Group of Mathematical Physics of Lisbon University).

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

( video )

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