@article { ,
title = {Quantum periods for 3-dimensional Fano manifolds},
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.},
doi = {10.2140/gt.2016.20.103},
eissn = {1364-0380},
issn = {1465-3060},
issue = {1},
journal = {Geometry \& Topology},
publicationstatus = {Published},
publisher = {Mathematical Sciences Publishers},
url = {https://nottingham-repository.worktribe.com/output/774714},
volume = {20},
year = {2016},
author = {Coates, Tom and Corti, Alessio and Galkin, Sergey and Kasprzyk, Alexander M.}
}