The hyperplane string, RCFTs, and the swampland

TL;DR

This paper investigates BPS strings in 6d supergravity, especially hyperplane strings, to distinguish geometric theories from non-geometric ones and explores their relation to the swampland and quantum gravity landscape.

Contribution

It provides an exact RCFT description of hyperplane strings in certain non-geometric 6d models and connects these to geometric models via Higgsing transitions.

Findings

Identified five 6d models with non-geometric hyperplane strings.

Constructed the left-moving sector of the hyperplane string worldsheet as a rational CFT.

Matched elliptic genera between non-geometric and geometric models in the unflavored limit.

Abstract

Six dimensional $\mathcal{N}=(1,0)$ supergravity features BPS strings whose properties encode highly nontrivial information about the parent 6d theory. We focus on a distinguished set of theories whose string charge lattice is one-dimensional. In geometric theories, the generator of the lattice arises from a D3 brane wrapping the hyperplane class in $\mathbb{P}^2$. This hyperplane string is expected to remain stable even when one ventures beyond the geometric regime where it becomes challenging to verify which candidate 6d theories belong to the swampland. We identify five 6d models which from the perspective of the hyperplane string deviate the most from being geometric. For these theories we are able to provide an exact description of the left-moving sector of the hyperplane string worldsheet theory in terms of a rational conformal field theory and provide evidence for their consistency. In one instance, using RCFT methods we are able to determine the elliptic genus and find that in the unflavored limit it matches with the elliptic genus of geometric models. We argue that the non-geometric model is connected to geometric ones via a sequence of Higgsing transitions. These results lead us to formulate a proposal relating the quantum corrected moduli space of the hyperplane string CFT with a region of the landscape of 6d $(1,0)$ quantum gravity.