On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections

This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau Ginsburg model (X-can, W-q). Here X-can is the complement of an anticanonical divisor in a Langlands dual quadric. The superpotential Wq is a regular function on X-can and is written in terms of coordinates which are naturally identified with a cohomology basis of the original quadric. This superpotential is shown to extend the earlier Landau-Ginsburg model of Givental, and to be isomorphic to the Lie-theoretic mirror introduced in [36]. We also introduce a Laurent polynomial superpotential which is the restriction of W-q to a particular torus in X-can. Together with results from [31] for odd quadrics, we obtain a combinatorial model for the Laurent polynomial superpotential in terms of a quiver, in the vein of those introduced in the 1990's by Givental for type A full flag varieties. These Laurent polynomial superpotentials form a single series, despite the fact that our mirrors of even quadrics are defined on dual quadrics, while the mirror to an odd quadric is naturally defined on a projective space. Finally, we express flat sections of the (dual) Dubrovin connection in a, natural way in terms of oscillating integrals associated to (X-can, W-q) and compute explicitly a particular flat section. (C) 2016 Elsevier Inc. All rights reserved.