Advances in Theoretical and Mathematical Physics

Volume 24 (2020)

Number 1

Cubic hypergeometric integrals of motion in affine Gaudin models

Pages: 155 – 187



Sylvain Lacroix (Laboratoire de Physique, École normale supérieure de Lyon, France)

Benoît Vicedo (Department of Mathematics, University of York, United Kingdom)

Charles Young (School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, United Kingdom)


We construct cubic Hamiltonians for quantum Gaudin models of affine types $\widehat{\mathfrak{sl}}_M$. They are given by hypergeometric integrals of a form we recently conjectured in [LVY]. We prove that they commute amongst themselves and with the quadratic Hamiltonians. We prove that their vacuum eigenvalues, and their eigenvalues for one Bethe root, are given by certain hypergeometric functions on a space of affine opers.

Published 22 May 2020