Decomposition and framing of F-bundles and applications to quantum cohomology
Thorgal Hinault, Tony Yue Yu, Chi Zhang, Shaowu Zhang
TL;DR
The paper develops F-bundles as formal/non-archimedean analogs of nc-Hodge variations, and proves both formal and non-archimedean spectral decomposition theorems along generalized Euler-field eigenspaces $K_b= abla_{u^2d}ig|_{u=0}$. It then establishes extension results for framings (logarithmic formal and non-archimedean), under a strong framing condition that yields unique, explicit extensions via solving recursive PDEs. These structural results are applied to the A-model F-bundle and quantum D-module of projective bundles and blowups, proving the existence and uniqueness of the decomposition map in those settings and complementing prior work by Iritani–Koto. The work connects decomposition theory for F-bundles to quantum cohomology through mirror-map–like reconstructions and provides tools for birational invariants via decomposition data, framed by a rigorous formal/analytic framework.
Abstract
F-bundle is a formal/non-archimedean version of variation of nc-Hodge structures which plays a crucial role in the theory of atoms as birational invariants from Gromov-Witten theory. In this paper, we establish the spectral decomposition theorem for F-bundles according to the generalized eigenspaces of the Euler vector field action. The proof relies on solving systems of partial differential equations recursively in terms of power series, and on estimating the size of the coefficients for non-archimedean convergence. The same technique allows us to establish the existence and uniqueness of the extension of framing for logarithmic F-bundles. As an application, we prove the uniqueness of the decomposition map for the A-model F-bundle (hence quantum D-module and quantum cohomology) associated to a projective bundle, as well as to a blowup of an algebraic variety. This complements the existence results by Iritani-Koto and Iritani.