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, 2024) in Conjecture 1.1, Section 1 (Introduction)

The extent to which the value theorem holds beyond simply connected spin manifolds is also unresolved.

— The hyperbolic class and vanishing laws in a TMF-valued four-manifold invariant  (2609.09819 - Li et al., 9 Sep 2026) in Section 6, “Interpretation and further questions”

It is tempting to speculate on extensions of such operations to include quantum corrections. It is natural to conjecture that this would involve replacing $M$ with the space of maps $T2 \rightarrow M$. It is then natural to speculate that the gauge anomaly constraint on $c_3$ might be related to a String constraint on moduli spaces via transgression. We leave such matters for future work.

— Towards quantum elliptic cohomology from GLSMs  (2609.09277 - Cao et al., 8 Sep 2026) in Section 5, Surface defects and elliptic genera

The identification of a general interacting holomorphic $\mathcal N=1$ SCFT with a TMF class remains conjectural, although the predicted divisibility has been verified in nontrivial classes of examples.

— Topological Modular Forms Constrain Threshold States in Extremal $\mathcal{N}=1$ AdS$_3$ Gravity  (2609.17359 - Yoshida, 15 Sep 2026) in Introduction

This determines the class-level KO_{\mathrm{MF} lift. Constructing a pair-level KO_{\mathrm{MF} orientation, a local 2|1 field-theory representative, and its identification with the interacting sigma model remain open problems, which will be discussed in \cref{sec:discussion}.

— Chiral Spin(7) Sigma Models and Topological Modular Forms  (2609.11856 - Yu, 10 Sep 2026) in Section 3, subsection “The degree-one KO_MF class,” immediately before Section 4; also discussed in Section 7, Discussion