Numerical invariants of normed matrix factorizations

TL;DR

This paper introduces a framework for defining and classifying numerical invariants of matrix factorizations associated with Delzant polytopes, linking them to open Gromov-Witten invariants via mirror symmetry.

Contribution

It constructs a normed matrix factorization category for Delzant polytopes and identifies specific objects called Dirac factorizations, establishing their relation to Lagrangian submanifolds and mirror symmetry.

Findings

Dirac factorizations are spherical for odd-dimensional simplices.

Numerical invariants match open Gromov-Witten invariants for real projective spaces.

Computer calculations verify invariants correspondence in low degrees.

Abstract

We define a normed matrix factorization category and a notion of bounding cochains for objects of this category. We classify bounding cochains up to gauge equivalence for spherical objects and use this classification to define numerical invariants. These invariants are expected to correspond under mirror symmetry to the open Gromov-Witten invariants with only boundary constraints of Lagrangian rational cohomology spheres defined by the second author and Tukachinsky. For each Delzant polytope, we construct a normed matrix factorization category. For Delzant polytopes satisfying a combinatorial relative spin condition, we construct an object of this category called the Dirac factorization. The Dirac factorization is expected to correspond under mirror symmetry to the Lagrangian submanifold given by the real locus of the toric symplectic manifold associated to the Delzant polytope. In the case of the $n$-simplex for $n$ odd, we show that the Dirac factorization is spherical, mirroring the fact that $\mathbb{R} P^n$ is a rational cohomology sphere. For $n = 1,$ we show the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten invariants of $\mathbb{R} P^1 \subset \mathbb{C} P^1.$ For $n = 3$ in low degrees, computer calculations verify that the numerical invariants of the Dirac factorization coincide with the open Gromov-Witten-Welschinger invariants of $\mathbb{R}P^3 \subset \mathbb{C} P^3.$ Although $\mathbb{R} P^n$ is trivial in the Fukaya category of $\mathbb{C} P^n$ over any field of characteristic zero, the above results can be seen as a manifestation of mirror symmetry over a Novikov ring.