Bulk-deformations, Floer complex bordism, and Grothendieck-Riemann-Roch

TL;DR

This paper links Floer homotopy invariants of Liouville manifolds to classical symplectic cohomology, providing explicit models and criteria for non-trivial cobordism classes, advancing understanding in symplectic topology.

Contribution

It introduces an explicit model for complexified Floer homotopy groups and establishes a homotopy coherent Grothendieck-Riemann-Roch theorem, connecting Floer invariants with cobordism theory.

Findings

Computed the Floer-homotopical invariant in terms of symplectic cohomology.

Established a cohomological criterion for non-base change of the MU lift.

Identified examples of non-trivial higher-dimensional cobordism classes.

Abstract

Given a Liouville manifold, we compute a Floer-homotopical invariant -- the complexification of the lift of symplectic cohomology to complex cobordism -- in terms of a classical Floer-theoretic invariant, namely, symplectic cohomology bulk-deformed by the Chern character. We do this by giving an explicit model for the complexified homotopy groups of the MU-module spectrum associated to a complex-oriented flow category and proving a ``homotopy coherent'' version of the classical Grothedieck-Riemann-Roch theorem. Using the aforementioned relation, we establish a computable cohomological criterion, in terms of the pair-of-pants product and the BV operator on symplectic cohomology, for when this MU lift cannot be obtained via base change from the sphere spectrum; moreover, we give examples where this holds. Finally, we use this non-base change criterion to detect examples of non-trivial higher-dimensional complex cobordism classes of relative Gromov-Witten type moduli spaces in the context of a smooth complex projective variety relative to an ample smooth divisor.