Topological modular forms and conformal nets

TL;DR

This paper explores how conformal nets can provide a geometric framework for topological modular forms (TMF), linking conformal field theory, boundary conditions, and string structures to cohomology theories.

Contribution

It proposes a novel approach connecting conformal nets and TMF, including a conjecture on algebraic periodicity and a lower bound derived from homotopy invariants.

Findings

Conjecture of 576-fold periodicity for free fermion nets.

Construction of fermionic boundary condition bundles for TMF classes.

Establishment of a lower bound of 24 on periodicity using homotopy invariants.

Abstract

We describe the role conformal nets, a mathematical model for conformal field theory, could play in a geometric definition of the generalized cohomology theory TMF of topological modular forms. Inspired by work of Segal and Stolz-Teichner, we speculate that bundles of boundary conditions for the net of free fermions will be the basic underlying objects representing TMF-cohomology classes. String structures, which are the fundamental orientations for TMF-cohomology, can be encoded by defects between free fermions, and we construct the bundle of fermionic boundary conditions for the TMF-Euler class of a string vector bundle. We conjecture that the free fermion net exhibits an algebraic periodicity corresponding to the 576-fold cohomological periodicity of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we establish a lower bound of 24 on this periodicity of the free fermions.