Topological Quantum Field Theory for Calabi-Yau threefolds and G_2 manifolds

TL;DR

This paper develops a topological quantum field theory framework for Calabi-Yau threefolds and G_2 manifolds, introducing a homology theory that counts anti-self-dual bundles over co-associative submanifolds, and explores Fukaya-Floer categories in this context.

Contribution

It introduces a novel homology theory and TQFT framework linking G_2 manifolds, Calabi-Yau threefolds, and special Lagrangian submanifolds, expanding the mathematical tools for these geometries.

Findings

Euler characteristic counts ASD bundles over co-associative submanifolds

Fukaya-Floer category of Lagrangians in CY threefolds

Establishment of a TQFT for these geometries

Abstract

We introduce a homology theory whose Euler characteristics counts ASD bundles over four dimensional co-associative submanifolds in (almost) G_2 manifolds. As a TQFT, in relative situations, we have the Fukaya-Floer category of Lagrangians intersection in the moduli space of special Lagrangian submanifolds in CY threefolds.