From Link Homology to Topological Quantum Field Theories
Paul Wedrich
TL;DR
This work surveys how link homology theories, notably $\mathfrak{gl}_N$-type homologies, extend to skein-theoretic invariants of 4-manifolds and integrate into extended TQFTs via braided monoidal 2-categories. Central to this framework are skein lasagna modules, which encode labels from link homology on 4-manifolds with boundary links and admit computation along handle decompositions; they also provide invariants for embedded and immersed surfaces and exhibit sensitivity to exotic smooth structures. The paper also outlines chain-level, homotopy-coherent approaches toward a fully fledged chain-level TQFT in both 4D and 3D, linking Soergel bimodule technology to higher-categorical foundations. Collectively, these developments advance a categorified, computable 4D TQFT program that parallels classical skein theory while benefiting from the richer structure of link homology and higher categories, with potential connections to Donaldson–Seiberg–Witten-type invariants and exotic surface phenomena.
Abstract
This survey reviews recent advances connecting link homology theories to invariants of smooth 4-manifolds and extended topological quantum field theories. Starting from joint work with Morrison and Walker, I explain how functorial link homologies that satisfy additional invariance conditions become diagram-independent, give rise to braided monoidal 2-categories, extend naturally to links in the 3-sphere, and globalize to skein modules for 4-manifolds. Later developments show that these skein lasagna modules furnish invariants of embedded and immersed surfaces and admit computation via handle decompositions. I then survey structural properties, explicit computations, and applications to exotic phenomena in 4-manifold topology, and place link homology and skein lasagna modules within the framework of extended topological quantum field theories.