2025/09/15 by Sergei Gukov, Vyacheslav Krushkal, Gukov, Sergei +5 · 1 citation
Mathematics · #Commutative Algebra and Its Applications
paper · pdf · doi:10.48550/arxiv.2509.12402
In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of π_*(\rm TMF)-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in π_*(\rm TMF), conjecturally generalizing the theta function of a lattice. We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions.