# Exotic matter on singular divisors in F-theory

## TL;DR

This paper investigates exotic matter representations on singular divisors in F-theory, providing a framework for their analysis and explicit models, revealing new insights into geometric constraints and the swampland.

## Contribution

It introduces a general method for analyzing exotic matter in F-theory on singular divisors, including explicit descriptions for symmetric representations and their algebraic structures.

## Key findings

- Explicit models for symmetric matter representations in F-theory.
- Identification of geometric constraints affecting low-energy spectra.
- Examples of theories in the swampland with no F-theory realization.

## Abstract

We analyze exotic matter representations that arise on singular seven-brane configurations in F-theory. We develop a general framework for analyzing such representations, and work out explicit descriptions for models with matter in the 2-index and 3-index symmetric representations of SU($N$) and SU(2) respectively, associated with double and triple point singularities in the seven-brane locus. These matter representations are associated with Weierstrass models whose discriminants vanish to high order thanks to nontrivial cancellations possible only in the presence of a non-UFD algebraic structure. This structure can be described using the normalization of the ring of intrinsic local functions on a singular divisor. We consider the connection between geometric constraints on singular curves and corresponding constraints on the low-energy spectrum of 6D theories, identifying some new examples of apparent "swampland" theories that cannot be realized in F-theory but have no apparent low-energy inconsistency.