This paper proves certain facts concerning the equivariance of quantization of pi-finite spaces. We argue that these facts establish an analogy between this quantization procedure and the geometric quantization of a symplectic vector space. Specifically, we observe that symmetries of a given polarization/Lagrangian always induce coherent symmetries of the quantization. On the other hand, symmetries of the entire phase space a priori only induce projective symmetries. For certain three-dimensional theories, this projectivity appears via a twice-categorified analogue of Blattner-Kostant-Sternberg kernels in geometric quantization and the associated integral transforms.
We survey Ozsv\'ath-Szab\'o's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts ($\mathcal{A}_\infty$-modules, type $D$-structures, box tensor, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsv\'ath and Szab\'o's analytic construction of the type $D$-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the $DA$-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in $S^3$. The boundary DGAs $\mathcal{B}(n,k)$ and $\mathcal{A}(n,k)$ of [7] are replaced here by an associative algebra $\mathcal{C}(n)$. These are the notes of two lecture series delivered by Peter Ozsv\'ath and Zolt\'an Szab\'o at Princeton University during the summer of 2018.
Quantum field theory has various projective characteristics which are captured by what are called anomalies. This paper explores this idea in the context of fully-extended three-dimensional topological quantum field theories (TQFTs). Given a three-dimensional TQFT (valued in the Morita 3-category of fusion categories), the anomaly identified herein is an obstruction to gauging a naturally occurring orthogonal group of symmetries, i.e. we study 't Hooft anomalies. In other words, the orthogonal group almost acts: There is a lack of coherence at the top level. This lack of coherence is captured by a "higher (central) extension" of the orthogonal group, obtained via a modification of the obstruction theory of Etingof-Nikshych-Ostrik-Meir [ENO10]. This extension tautologically acts on the given TQFT/fusion category, and this precisely classifies a projective (equivalently anomalous) TQFT. We explain the sense in which this is an analogue of the classical spin representation. This is an instance of a phenomenon emphasized by Freed [Fre23]: Quantum theory is projective. In the appendices we establish a general relationship between the language of projectivity/anomalies and the language of topological symmetries. We also identify a universal anomaly associated with any theory which is appropriately "simple".
We survey Ozsv\'ath-Szab\'o's bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts ($\mathcal{A}_\infty$-modules, type $D$-structures, box tensor, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsv\'ath and Szab\'o's analytic construction of the type $D$-structure associated to an upper diagram. Finally we give an explicit description of the structure maps of the $DA$-bimodules of some elementary partial diagrams. These can be used to perform explicit computations of the knot Floer differential of any knot in $S^3$. The boundary DGAs $\mathcal{B}(n,k)$ and $\mathcal{A}(n,k)$ of [7] are replaced here by an associative algebra $\mathcal{C}(n)$. These are the notes of two lecture series delivered by Peter Ozsv\'ath and Zolt\'an Szab\'o at Princeton University during the summer of 2018.
