Non-relativistic string monodromies

TL;DR

This paper explores the spectral curve and monodromy properties of non-relativistic strings in AdS$_5\times S^5$, revealing unique spectral parameter behaviors and proposing generalized quasi-momenta concepts.

Contribution

It initiates the spectral curve analysis for non-relativistic strings, identifying cases with spectral parameter independence and introducing generalized quasi-momenta for non-diagonalisable monodromy matrices.

Findings

Eigenvalues of monodromy matrix are spectral parameter independent for certain solutions.

Monodromy matrix can be diagonalisable or non-diagonalisable, affecting spectral analysis.

Proposes generalized quasi-momenta for non-diagonalisable cases.

Abstract

Spectral curve methods proved to be powerful techniques in the context of relativistic integrable string theories, since they allow to derive the semiclassical spectrum from the minimal knowledge of a Lax pair and a classical string solution. In this paper we initiate the study of the spectral curve for non-relativistic strings in AdS$_5\times S^5$. First we show that for string solutions whose Lax connection is independent of $\sigma$, the eigenvalues of the monodromy matrix do not have any spectral parameter dependence. We remark that this particular behaviour also appears for relativistic strings in flat space. Second, for some simple non-relativistic string solutions where the path ordered exponential of the Lax connection can be computed, we show that the monodromy matrix is either diagonalisable with quasi-momenta independent of the spectral parameter, or non-diagonalisable. For the latter case, we propose a notion of generalised quasi-momenta, based on maximal abelian subalgebras, which retain a dependence on the spectral parameter.