Toric reduction and a conjecture of Batyrev and Materov
Andras Szenes, Michele Vergne
TL;DR
The paper proves the Toric Residue Mirror Conjecture of Batyrev and Materov by deriving a new integration formula for intersection numbers on toric quotients and connecting A- and B-side generating functions through cycle constructions. It combines a novel iterated Jeffrey–Kirwan residue formula with Morrison–Plesser toric moduli spaces and a tropical degeneration analysis to express toric residues as cycle integrals, enabling a direct comparison of A- and B-side expressions. The main result shows that the A-side generating function expands to the B-side toric-residue expansion in a suitable domain, thereby validating the conjecture in a general integral-vertex toric setting beyond reflexive polytopes. A key technical advance is the explicit construction of a cycle representing the JK-residue and the use of Gale duality to bridge the A- and B-model data in toric mirror symmetry.
Abstract
We present a new integration formula for intersection numbers on toric quotients, extending the results of Witten, Jeffrey and Kirwan on localization. Our work was motivated by the Toric Residue Mirror Conjecture of Batyrev and Materov; as an application of our integration formula, we obtain a proof of this conjecture in a generalized setting.