On the homological mirror symmetry conjecture for pairs of pants

TL;DR

This paper constructs a specific Lagrangian sphere in the n-dimensional pair of pants, computes its algebraic structure, and provides evidence supporting the homological mirror symmetry conjecture relating it to a Landau-Ginzburg model.

Contribution

It introduces an immersed Lagrangian sphere in the pair of pants and explicitly computes its endomorphism algebra, linking it to the mirror Landau-Ginzburg model.

Findings

The endomorphism algebra is an exterior algebra with n+2 generators.

The algebra is not formal; higher products are computed.

Results support the homological mirror symmetry conjecture for pairs of pants.

Abstract

The n-dimensional pair of pants is defined to be the complement of n+2 generic hyperplanes in CP^n. We construct an immersed Lagrangian sphere in the pair of pants and compute its endomorphism A_{\infty} algebra in the Fukaya category. On the level of cohomology, it is an exterior algebra with n+2 generators. It is not formal, and we compute certain higher products in order to determine it up to quasi-isomorphism. This allows us to give some evidence for the homological mirror symmetry conjecture: the pair of pants is conjectured to be mirror to the Landau-Ginzburg model (C^{n+2},W), where W = z_1 ... z_{n+2}. We show that the endomorphism A_{\infty} algebra of our Lagrangian is quasi-isomorphic to the endomorphism dg algebra of the structure sheaf of the origin in the mirror. This implies similar results for finite covers of the pair of pants, in particular for certain affine Fermat hypersurfaces.