Quantum Periods for 3-Dimensional Fano Manifolds

TL;DR

This paper computes the quantum periods of all 3-dimensional Fano manifolds, linking their mirror symmetry to Laurent polynomial classifications and providing a new approach to Fano manifold classification.

Contribution

It reworks the Mori-Mukai classification of 3D Fano manifolds and connects quantum periods with mirror symmetry via Laurent polynomials.

Findings

Quantum periods computed for all 3D Fano manifolds.

Fano manifolds with very ample anticanonical bundle have Minkowski polynomial mirrors.

New method for classifying Fano manifolds via mirror symmetry.

Abstract

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. Our methods are likely to be of independent interest. We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V/G, where G is a product of groups of the form GL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses the Fano 3-fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the Quantum Lefschetz Hyperplane Theorem of Coates-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah.