Which 4d TQFTs and 4-manifold invariants detect the Gluck twist? Guided by questions like this, we will look for new invariants of smooth 4-manifolds and knotted surfaces in 4-manifolds. Traditionally, a construction of most such invariants and TQFTs involves a choice of certain algebraic structure, so that one can talk about "invariants for $SU(2)$" or a "TQFT defined by a Frobenius algebra." Surprisingly, recent developments lead to an opposite phenomenon, where algebraic structures are labeled by 3-manifolds and 4-manifolds, so that one can speak of VOA-valued invariants of 4-manifolds or MTC-valued invariants of 3-manifolds. Explaining these intriguing connections between topology and algebra will be the main goal of this talk.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/9096