The curious case of large-N expansions on a (pseudo)sphere

TL;DR

This paper investigates the large-N behavior of one-dimensional sigma models on spherical and hyperbolic spaces, revealing dualities, new operators, and unexpected zero modes affecting the 1/N expansion.

Contribution

It introduces a duality between Lagrange multipliers and angular momentum, proposes new operators for hyperbolic models, and explores zero modes impacting the large-N expansion.

Findings

Discovered a duality between Lagrange multiplier and angular momentum.

Proposed a new class of operators based on hyperbolic space representations.

Identified zero modes causing double scaling in the 1/N expansion.

Abstract

We elucidate the large-N dynamics of one-dimensional sigma models with spherical and hyperbolic target spaces and find a duality between the Lagrange multiplier and the angular momentum. In the hyperbolic model we propose a new class of operators based on the irreducible representations of hyperbolic space. We also uncover unexpected zero modes which lead to the double scaling of the 1/N expansion and explore these modes using Gelfand-Dikiy equations.