Embedding and partial resolution of complex cones over Fano threefolds

TL;DR

This paper explores embeddings of toric Calabi-Yau fourfolds over Fano threefolds, aiming to discover new dualities and theories, but finds limited consistent quiver theories, while providing an alternative derivation of a known Chern-Simons theory.

Contribution

It investigates embeddings and partial resolutions of Fano threefolds, offering new diagrams and an alternative method to derive a known quiver Chern-Simons theory.

Findings

Many embedding diagrams found, but none yield consistent quiver theories.

Successfully derived a known quiver Chern-Simons theory through an alternative approach.

Limited new dualities identified due to consistency issues.

Abstract

This work deals with the study of embeddings of toric Calabi-Yau fourfolds which are complex cones over the smooth Fano threefolds. In particular, we focus on finding various embeddings of Fano threefolds inside other Fano threefolds and study the partial resolution of the latter in hope to find new toric dualities. We find many diagrams possible for many of these Fano threefolds, but unfortunately, none of them are consistent quiver theories. We also obtain a quiver Chern-Simons theory which matches a theory known to the literature, thus providing an alternate method of obtaining it.