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

Abstract

This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau–Ginzburg model (Xˇcan,Wq). Here Xˇcan is the complement of an anticanonical divisor in a Langlands dual quadric. The superpotential Wq is a regular function on Xˇcan 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–Ginzburg 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 Wq to a particular torus in Xˇcan. 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 (Xˇcan,Wq) and compute explicitly a particular flat section.