Laurent Inversion

TL;DR

This paper introduces a method to reconstruct Fano toric complete intersections from their Laurent polynomial mirrors, enabling the direct construction of the original geometric object and leading to new examples.

Contribution

The paper presents a novel technique for inverting mirror symmetry to construct Fano manifolds directly from Laurent polynomials.

Findings

Constructed a new four-dimensional Fano manifold.

Developed a method for inverting the mirror correspondence.

Demonstrated the technique on known Laurent polynomials.

Abstract

There are well-understood methods, going back to Givental and Hori--Vafa, that to a Fano toric complete intersection X associate a Laurent polynomial f that corresponds to X under mirror symmetry. We describe a technique for inverting this process, constructing the toric complete intersection X directly from its Laurent polynomial mirror f. We use this technique to construct a new four-dimensional Fano manifold.