Mathematical Physics -Physical Mathematics
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The seminar is taking a break and will resume on Tuesday 26th January 2021.
Tuesday 26th January 4.30 pm UK time
Claudia Scheimbauer (TU Munich)
The AKSZ construction as a fully extended topological field theory
Classical Poincaré-Lefschetz duality is a starting point of understanding what an orientation on a derived stack given by a homotopy type of a cobordism M is. If furthermore we have an (n-)symplectic “target’’ X, we obtain a (shifted) symplectic on Map(M,X) which is given by pulling back along an evaluation and then integrating. This explains the Atiyah-Bott symplectic form on G-local systems, arising from the Killing form on the Lie algebra of a reductive group G, via derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi. In this talk I will sketch these ideas and explain how this leads to a fully extended oriented topological field theory with values in a suitable higher category of Lagrangians, explaining the difficulties with coherence in defining such a functor. In essence, the difficulty with coherence stems from integration of differential forms not being functorial for cobordisms on the nose, but only up to homotopy. In mathematical physics, this TFT is a reinterpretation/analog of the classical AKSZ construction for certain $\sigma$-models and describes “semi-classical TFTs”. This is joint work with Damien Calaque and Rune Haugseng and fully extends a construction by the former.