Stolz–Teichner TMF–field theory conjecture

Establish the existence of the quantization map sending families of n-dimensional string manifolds M→X to fully extended, degree −n, 2-dimensional supersymmetric Euclidean field theories over X, construct the cocycle map sending such degree −n field theories to classes in TMF^{-n}(X), and prove commutativity of the resulting triangle equating the analytic and topological indices; for n<0, construct the cocycle map alone.

Background

This is the central conjecture connecting topological modular forms (TMF) with 2-dimensional supersymmetric quantum field theories. It posits a geometric construction of TMF whose cocycles are 2D field theories and a TMF-index theorem identifying analytic and topological indices. The conjecture’s verification would provide a geometric description of TMF and unlock applications across topology, geometry, and physics.

The statement specifies two dashed arrows in a commuting triangle: a quantization map from families of string manifolds to field theories, and a cocycle map from field theories to TMF, with their composition agreeing with the string orientation. For negative degrees, only the cocycle map is asserted.

References

In Stolz and Teichner's framework, the basic conjecture can be summarized as follows. For each n and smooth manifold X there exist dashed arrows making the triangle commute, where "topological index" is the string orientation, "quantize" is quantization of the supersymmetric σ-model on the fibers of M→X, and "cocycle_n" sends a degree n field theory over X to a class in TMFn(X). For negative n, the empty manifold is the only string manifold and the content of (1.1) is the existence of a cocycle map.

— Elliptic cohomology and quantum field theory (2408.07693 - Berwick-Evans, 14 Aug 2024) in Conjecture 1.1, Section 1 (Introduction)