Categorification: tangle invariants and TQFTs. (English) Zbl 1536.57014
Beliaev, Dmitry (ed.) et al., International congress of mathematicians 2022, ICM 2022, Helsinki, Finland, virtual, July 6–14, 2022. Volume 2. Plenary lectures. Berlin: European Mathematical Society (EMS). 1312-1353 (2023).
Topological Quantum Field Theories are the study of an interaction between mathematics and physics. “A first mathematical formulation of TQFTs goes back to M. Atiyah [Publ. Math., Inst. Hautes Étud. Sci. 68, 175–186 (1988; Zbl 0692.53053)], influenced by G. Segal [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 359, No. 1784, 1389–1398 (2001; Zbl 1041.81094)] and E. Witten [J. Differ. Geom. 17, 661–692 (1982; Zbl 0499.53056)].”
“Based on different views on the Jones polynomial, we review representation-theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them through the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups, on the one hand, and monoidal 2-categories of Soergel bimodules, on the other hand, might lead to new interesting 4-dimensional TQFTs.”
For the entire collection see [Zbl 1532.00036].
“Based on different views on the Jones polynomial, we review representation-theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them through the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups, on the one hand, and monoidal 2-categories of Soergel bimodules, on the other hand, might lead to new interesting 4-dimensional TQFTs.”
For the entire collection see [Zbl 1532.00036].
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 18N25 | Categorification |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 05E10 | Combinatorial aspects of representation theory |
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
Keywords:
Jones polynomial; knot invariants; categorification; Topological Quantum Field Theory; TQFT; Khovanov homology; category \(\mathcal O\); skew Howe duality; quantum groupsReferences:
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