Research

Current Interests

• The role of TQFTs in 3d quantum gravity:

One can use TQFTs and defects as a mathematical tool when studying non-topological theories like 2d CFT. In particular, chiral CFT can be described by the modular functor part of a 3d TQFT, i.e. its mapping class group representations. Defects and boundary conditions provide a construction of correlators (at least for rational CFTs). Another surprising appearance of such mapping class group actions is in the context of 3d quantum gravity, where mapping class group averages provide candidate gravity partition functions (see my Thesis and arXiv:2309.14000 ).

• Mapping class group representations: 

Motivated by expectations in quantum gravity, I am interested (for Reshetikhin-Turaev TQFTs) in questions of: 

• Irreducibility: See arXiv:2106.01454 for a result (with I. Runkel) that uses irreducibility of mapping class group representations to imply that there are no non-trivial surface defects. This result can be related to the no global symmetry conjecture in quantum gravity.  

• Finiteness: This requires that the representation image of the mapping class group is finite. Note that questions on the size of the representation image are also relevant in Topological Quantum Computation (TQC). 

• Skein modules: 

For a given ribbon category, one can associate skein algebras  to surfaces and skein modules on 3-manifolds. In fact, these data can be collected into a 3d (once-categorified) TQFT. Even though skein modules have existed and been studied for a long time (particularly sl(2)-skein modules), there have been exciting recent developments and results like the proof of Witten's finiteness conjecture for skein modules on closed 3-manifolds for q generic. 

Papers

1. CFT correlators and mapping class group averages with I. Runkel,  Commun. Math. Phys. 405, 247 (2024), arXiv:2309.14000.

2. Mapping class group representations and Morita classes of algebras with I. Runkel, Quantum Topol. (2023), arXiv:2106.01454.

3. Mapping class group actions and their applications to 3D gravity, Doctoral dissertation, Staats und Universitätsbibliothek Hamburg  2022. [pdf

Others

• Master Thesis: Permutation Orbifolds in Reshetikhin-Turaev TQFT (2019) [pdf]