Orthogonal Polynomials with Complex Densities and Quantum Minimal Surfaces
Giovanni Felder, Jens Hoppe
TL;DR
The paper develops a framework in which discrete Painlevé-type equations governing quantum minimal surfaces arise as recurrence relations for monic orthogonal polynomials defined by indefinite Hermitian inner products with complex densities. Focusing on a cubic potential, the authors derive explicit recurrences for the ratio $V_n=h_{n+1}/h_n$ of squared norms, showing it satisfies $V_n\left(1+\frac{9t^2}{a^2}(V_{n-1}+V_{n+1})\right)=\frac{n+1}{a}$, with initial data tied to $h(a)$ expressed in terms of a modified Bessel function. They generalize to monomial potentials of degree $d$, introducing a symmetry under $d$-th roots of unity and constructing orthogonal polynomials via explicit recurrences, where the initial conditions are governed by a solution $h(a)$ of a linear differential equation of order $d-1$ and representable through generalized hypergeometric functions or Bessel-type integrals. The work then connects these polynomials to quantum minimal surfaces through operator realizations $M$ and $L$ built from shift operators $W$ and $W^{\dagger}$, yielding a concrete algebraic relation $[L,M]=1$ that encodes the minimal-surface structure. The results suggest a broader picture linking non-definite orthogonal-polynomial theory, discrete Painlevé equations, and matrix models in complex settings, with conjectural extensions to curves $w_1^p = w_2^q$ and associated normal-matrix models, potentially connecting to integrable hierarchies such as the 2D Toda system.
Abstract
We show that the discrete Painlevé-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula for the initial conditions leading to positive solutions.