Landau-Ginzburg/Calabi-Yau correspondence for the complete intersections X_{3,3} and X_{2,2,2,2}

TL;DR

This paper introduces a generalized hybrid model linking Fan-Jarvis-Ruan-Witten theory to Gromov-Witten theory for Calabi-Yau complete intersections, proving the correspondence in genus zero for certain three-dimensional cases.

Contribution

It extends the Landau-Ginzburg/Calabi-Yau correspondence to new complete intersections and establishes the genus zero case for three-dimensional varieties in ordinary projective space.

Findings

Proves genus zero correspondence for X_{3,3} and X_{2,2,2,2}

Generalizes previous results for the quintic threefold

Suggests higher-genus relations via Givental's quantization

Abstract

We define a generalization of Fan-Jarvis-Ruan-Witten theory, a "hybrid" model associated to a collection of quasihomogeneous polynomials of the same weights and degree, which is expected to match the Gromov-Witten theory of the Calabi-Yau complete intersection cut out by the polynomials. In genus zero, we prove that the correspondence holds for any such complete intersection of dimension three in ordinary, rather than weighted, projective space. These results generalize those of Chiodo-Ruan for the quintic threefold, and as in that setting, Givental's quantization can be used to yield a conjectural relation between the full higher-genus theories.