# Quantisation as a functor

## About the project

**Quantisation as a functor**

* Goal:* Abstraction of the quantisation procedure as an algebraic (hilbert) realisation of a distinguished geometric category (of cobordisms) with additional structure which represents histories of elementary objects (pointlike particles, paths and loops, membranes, Cauchy hypersurfaces etc.).

**Logical scheme:**

- Category-theoretic requisites – from the rudiments to Yoneda’s Lemma, adjoint functors and universal objects. (after Leinster and S’s monographic lecture)
- Categories with additional structure (monoidal, semi-simple, braided, modular, generators etc.). (after Kassel, Bakalov and Kirillov Jr. and Turaev)
- Introduction to the geometric category of (decorated) cobordisms. (after Kock, with elements of differential topology after Hirsch)
- The Segal-Atiyah-Turaev axioms. (after Atiyah and Turaev)
- Functorial quantisation and topological invariants. Knot polynomials. (after Turaev and Hu)
- A case study: Prelude to functorial quantisation of 2d (Rational) Conformal Field Theory. Classification of (undecorated, oriented) 2d tQFTs. (after Kock)

* Perspective:* Conceptualising quantisation in abstraction from the technicalities of its various standard concretisations, insights into the topological phase of a field theory, fully fledged functorial quantisation of 2d CFT in the spirit of Fröhlich, Fuchs, Runkel and Schweigert, systematic construction of topological and cohomological invariants of low-dimensional manifolds, cobordism hypothesis by Baez and Dolan.

* Study team: *Aleksander Syrewicz, Hubert Ziajka

*Rafał R. Suszek*

**Supervision:****Literature:**

- Leinster, T., “Basic Category Theory”, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, 2014.
- Kassel, C., “Quantum Groups”, Graduate Texts in Mathematics, Vol. 155, Springer, 1994.
- Bakalov, B. and Kirillov, Jr., A., “Lectures on Tensor Categories and Modular Functors”, University Lecture Series, Vol. 21, American Mathematical Society, 2000.
- Turaev, V.G., “Quantum Invariants of Knots and 3-Manifolds”, de Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter, 2016.
- Kock, J., “Frobenius Algebras and 2-D Topological Quantum Field Theories”, London Mathematical Society Student Texts, Vol. 59, Cambridge University Press, 2003.
- Hirsch, M.W., “Differential Topology”, Graduate Texts in Mathematics, Vol. 33, Springer, 1976.
- Atiyah, M.F., “Topological Quantum Field Theory”, Publications mathématiques de l’I.H.É.S., Vol. 68 (1988) 175-186.
- Hu, S., “Lecture Notes on Chern-Simons-Witten Theory”, World Scientific, 2001.