A series of four lectures will be given on the following dates:
Monday, November 9, 2020, 16:00, online (joint with the Master Class Mathematical Physics seminar)
Thursday, November 12, 2020, 16:15 - online
Monday, November 16, 2020, 16:15 - online
Thursday, November 19, 2020, 16:15 - online
Abstract:
A celebrated result of Reshetikhin-Turaev states that any semisimple
modular tensor category gives rise to a topological field theory,
which assigns vector spaces to surfaces and linear maps to cobordisms.
As a geometric example, the Drinfeld double of a finite group G gives
rise to the Dijkgraaf-Witten topological field theory, which assigns to
surfaces the space of G-bundles. For non-semisimple modular tensor
categories, the result does not hold in this form, but Lyubaschenko
succeeded in constructing a modular functor, which assigns vector
spaces to surfaces and at least an action of the mapping class group of
the surface. For example, the modular group SL2(Z) acts on the space
assigned to the torus. Accordingly, in conformal quantum field theory
the chiral conformal blocks on the torus are modular forms.
There will be two introductory talks, aimed at students of the Master
Program with no previous contact to the topic. In the first talk , I
will given an brief introduction to modular tensor categories and
topological field theories, and I will also sketch the connection to
conformal field theory. In the second talk, we review Lyubaschenko's
construction of a modular functor for non-semisimple modular tensor
categories and discuss some examples, again of physical relevance. In
two subsequent talks, I will discuss my recent joint work on a derived
modular functor and thoroughly discuss some examples and ongoing work.
Despite being carried out in a nonsemisimple category, Lyubaschenko's
construction still only uses of the Hom-spaces of the category and
is not an exact functor in its arguments. So the question arises: Is
there a derived version of Lyubschenko's modular functor? Does it
contain more information, such as new mapping class group
representations on the Ext-spaces? In terms of physics: Are there
derived spaces of conformal blocks in a logarithmic conformal field
theory? Recent joint work with Mierach, Sommerhäuser, Schweigert and
Woike has established such a derived modular functor.