# Little String Theories on Curved Manifolds

## TL;DR

This paper investigates the behavior of 6d Little String Theory on curved manifolds using holography, revealing constraints on the types of spaces it can inhabit and analyzing its thermodynamics on spheres.

## Contribution

It demonstrates that LST can only be defined on spaces with zero or positive curvature via holography and explores its thermodynamics and holographic renormalization on such spaces.

## Key findings

- LST cannot be placed on negatively curved spaces without extra fields.
- Holography yields multiple solutions for LST on the same space, contributing to the partition function.
- The spectrum's Hagedorn behavior is modified on curved spaces.

## Abstract

In this paper, we study the 6d Little String Theory (LST) (the decoupled theory on the worldvolume of $N$ NS5-branes) on curved manifolds, by using its holographic duality to Type II string theory in asymptotically linear dilaton backgrounds. We focus on backgrounds with a large number of Killing vectors (namely, products of maximally symmetric spaces), without requiring supersymmetry (we do not turn on any background fields except the metric). LST is non-local so it is not obvious which spaces it can be defined on; we show that holography implies that the theory cannot be put on negatively curved spaces, but only on spaces with zero or positive curvature. For example, one cannot put LST on a product of an anti-de Sitter space times another space, without turning on extra background fields. On spaces with positive curvature, such as $S^6$, $\mathbb{R}^2\times S^4$, $S^3\times S^3$, etc., we typically find (for large $N$) dual holographic backgrounds which are weakly coupled and weakly curved everywhere, so that they can be well-described by Type II supergravity. In some cases more than one smooth solution exists for LST on the same space, and they all contribute to the partition function. We also study the thermodynamical properties of LST compactified on spheres, finding the leading correction to the Hagedorn behavior of the spectrum, which is different on curved space than on flat space. We discuss the holographic renormalization procedure, which must be implemented in order to get a finite free energy for the LST; we do not know how to implement it for general spaces, but we can (and we do) implement it for the theory compactified on $S^4$.