# Hamiltonian spectral invariants, symplectic spinors and Frobenius structures II

## TL;DR

This paper extends the study of Frobenius structures and symplectic spectral invariants in symplectic spinors, introducing algebraic and geometric frameworks for dual pairs and matrix factorizations, with applications to Hamiltonian systems.

## Contribution

It introduces a novel algebraic approach linking dual pairs of Frobenius structures to matrix factorizations and establishes a Riemann-Roch type theorem relating these to cohomological data.

## Key findings

- Established a Hopf-algebra-like structure on Frobenius structures.
- Defined conditions for dual pairs involving sections and almost complex structures.
- Proved a Riemann-Roch type theorem connecting matrix factorizations to cohomology.

## Abstract

In this article, we continue our study of 'Frobenius structures' and symplectic spectral invariants in the context of symplectic spinors. By studying the case of $C^1$-small Hamiltonian mappings on symplectic manifolds $M$ admitting a metaplectic structure and a parallel $\hat O(n)$-reduction of its metaplectic frame bundle we derive how the construction of 'singularly rigid' resp. 'self-dual' pairs of irreducible Frobenius structures associated to this Hamiltonian mapping $\Phi$ leads to a Hopf-algebra-type structure on the set of irreducible Frobenius structures. We then generalize this construction and define abstractly conditions under which 'dual pairs' associated to a given $C^1$-small Hamiltonian mapping emerge, these dual pairs are essentially pairs $(s_1, J_1), (s_2, J_2)$ of closed sections of the cotangent bundle $T^*M$ and (in general singular) compatible almost complex structures on $M$ satisfying certain integrability conditions involving a Koszul bracket. In the second part of this paper, we translate these characterizing conditions for general 'dual pairs' of Frobenius structures associated to a $C^1$-small Hamiltonian system into the notion of matrix factorization. We propose an algebraic setting involving modules over certain fractional ideals of function rings on $M$ so that the set of 'dual pairs' in the above sense and the set of matrix factorizations associated to these modules stand in bijective relation. We prove, in the real-analytic case, a Riemann Roch-type theorem relating a certain Euler characteristic arising from a given matrix factorization in the above sense to (integral) cohomological data on $M$ using Cheeger-Simons-type differential characters, derived from a given pair $(s_1, J_1), (s_2, J_2)$. We propose extensions of these techniques to the case of 'geodesic convexity-smallness' of $\Phi$ and to the case of general Hamiltonian systems on $M$.