Description
What are the possible anomalies of four-dimensional topological field theories?
Abstract: Whether a given quantum field theory (QFT) can be approximated by a topological field theory (TFT) at long distances is an important qualitative aspect of the QFT. Since the anomaly of a QFT is invariant under such an approximation, we can attack this question by asking, "of all possible anomalies, which ones are anomalies of TFTs?", which is a rigorous mathematical question. In joint work with Weicheng Ye and Matthew Yu, we solve this question for four-dimensional Spin x_{\pm 1} G TFTs: the bottommost layer of the Atiyah--Hirzebruch spectral sequence is a complete obstruction to being the anomaly of a TFT. Moreover, when this obstruction vanishes, there is an algorithmic construction of a TFT realizing the anomaly. Our work builds on Décoppet--Yu's symmetry extension construction for fusion 2-categories as well as Córdova--Ohmori's answer to this question for cyclic G.