Emergence in String Theory and Fermi Gases

TL;DR

This paper explores how the prepotential in string theory can emerge from integrating out Fermi gas degrees of freedom, linking non-perturbative effects to Swampland criteria in compact Calabi-Yau manifolds.

Contribution

It demonstrates that the cubic term of the prepotential can be derived from a Fermi gas model in compact Calabi-Yau manifolds, extending previous non-compact results.

Findings

Exact cubic term coefficient recovered from Fermi gas integration.

Established a map between the Grand Canonical potential and periods for one-parameter models.

Showed emergence of the cubic prepotential term in compact Calabi-Yau cases.

Abstract

The Emergence Proposal suggests that some Swampland criteria, in particular on large field distances, are a consequence of the emergent nature of dynamics for fields in the infrared. In the context of type II string theory compactified on Calabi-Yau manifolds, it proposes that the cubic tree-level piece of the genus-zero prepotential is emergent from integrating out massive non-perturbative states. For a certain special non-compact Calabi-Yau, the blown-up conifold, it is known that the full all-genus prepotential can be matched onto the Grand Canonical potential of a two-dimensional Fermi gas. We propose here that this should be understood in the context of emergence: the prepotential is induced by integrating out the Fermi gas degrees of freedom. To make contact with the Swampland we need dynamical gravity, so compact Calabi-Yau manifolds. We show that for specifically the cubic term, an integrating out calculation also works for compact cases. In particular, the exact cubic term coefficient can be recovered from integrating out a Fermi gas for any compact Calabi-Yau that is an elliptic fibration over a reflexive toric base. We also propose a general map, for any one-parameter Calabi-Yau, between the Grand Canonical potential of the ultraviolet non-perturbative system and the period. In particular, this map leads to an emergent cubic term in the genus-zero prepotential for any such one-parameter model.