Higher-equivariant (G-equivariant) TMF cocycle conjecture

Construct, for a compact Lie group G acting on a manifold X and a level [ℓ]∈H^4(BG;Z), a cocycle map from degree-ℓ twisted 2|1-dimensional Euclidean field theories over the stack X_G to ℓ-twisted G-equivariant TMF of X, and establish the equivariant analog of the commuting triangle for families of G-equivariant string manifolds.

Background

This conjecture refines the main TMF–field theory conjecture to incorporate background gauge symmetries and higher equivariance. It aligns anticipated field-theoretic anomalies and line bundles over moduli of G-bundles with Lurie’s 2-equivariant elliptic cohomology, aiming to produce a universal comparison map via a higher-equivariant structure.

Its resolution would tightly connect twisted, equivariant TMF with sections of theta-lines on the derived moduli of flat G-bundles on elliptic curves, giving a robust framework for cocycles in equivariant elliptic cohomology.