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.).

Study team: Aleksander Syrewicz, Hubert Ziajka, Daniel-Grigore Chelaru

Supervisor: Rafał R. Suszek

Logical scheme:

1. Category-theoretic requisites – from the rudiments to Yoneda’s Lemma, adjoint functors and universal objects. (after Leinster and S’s monographic lecture) (slides)
2. Categories with additional structure (monoidal, semi-simple, braided, modular, generators etc.). (after Kassel, Bakalov and Kirillov Jr. and Turaev)
3. Introduction to the geometric category of (decorated) cobordisms. (after Kock, with elements of differential topology after Hirsch)
4. The Segal-Atiyah-Turaev axioms. (after Atiyah and Turaev)
5. Functorial quantisation and topological invariants. Knot polynomials. (after Turaev and Hu)
6. 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.

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.