Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

TL;DR

This paper computes the Floer cohomology of the Clifford torus in complex projective space, analyzing how spin structures influence orientations and establishing classifications of holomorphic discs, revealing dimension-dependent properties.

Contribution

It introduces a classification of holomorphic discs with boundary on the Clifford torus and analyzes the impact of spin structures on Floer cohomology, including explicit computations and mirror symmetry verification.

Findings

In odd dimensions, two spin structures yield non-vanishing Floer cohomology.

In even dimensions, only one spin structure results in non-vanishing Floer cohomology.

Non-vanishing Floer cohomology is isomorphic to the singular cohomology of the torus.

Abstract

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in even dimensions, there is only one such spin structure. When the Floer cohomology is non-vanishing, it is isomorphic to the singular cohomology of the torus (with a Novikov ring as its coefficients). As a corollary, we prove that any Hamiltonian deformation of the Clifford torus intersects with it at least at $2^n$ distinct intersection points, when the intersection is transversal. We also compute the Floer cohomology of the Clifford torus with flat line bundles on it and verify the prediction made by Hori using a mirror symmetry calculation.