# Core Research Area T: Three-dimensional topological quantum field theories

### Leading Researchers

Topological quantum field theories (TFTs) represent a prominent example of a fruitful interplay between mathematics and physics. Its basic concepts formalize properties one can expect for a quantum field theory defined by some path integral. Using structures from geometry and representation theory, examples can be constructed in a mathematically rigorous way. This stimulated new areas of mathematics, but also lead to unexpected interactions between traditional areas, in particular between algebra and topology.

The research topics in CRA T can be organised into three areas:

- Topological defects in 3d TFTs: classification and applications,
- Non-semisimple generalizations of 3d TFTs,
- Non-compact generalizations of 3d TFTs.

**1.**It has become increasingly clear that defects of various codimensions are crucial structures in quantum field theories. A distinguished class of defects are codimension-one topological defects. Their importance for the description of symmetries and dualities has been realized in the context of two-dimensional rational conformal field theories already some time ago [FFRS]. A model independent analysis of topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type has been carried out recently [FSV1]. It confirms a close link to the theory of module categories over monoidal categories and thus to (a categorified version of) representation theory. These results can be rather explicitly confronted to gauge theoretic computations in the case of Dijkgraaf-Witten theories [FSV2]. Further research topics include the generalization of the results of [FSV2] to larger classes of models, as well as using defects to gauge symmetries by an orbifold construction along the lines of [FFRS2] [CR].

**2.**The best-understood 3d TFTs are those arising via the Reshetikhin-Turaev or via the Turaev-Viro construction. Both start from a finitely semisimple monoidal category with appropriate extra data. In trying to obtain new types of 3d TFTs (as 3-2 or extended 3-2-1 theories) one can attempt to weaken these requirements by dropping semisimplicity or by dropping finiteness (point 3). In both cases, a qualitatively new feature as compared to the finitely simisimple case is the appearance of continuous parameters in the description of representations. On the side of 2d CFT, non-semisimplicity amounts to studying logarithmic CFTs. Progress in understanding the relevant representation categories, and in combining chiral into full CFTs in a model-independent way has been made in [R][FSS]. Possible research projects are to obtain more examples of non-semisimple braided tensor categories from simple chiral logarithmic CFTs and to study the resulting link invariants via [L][CGPM], and to extend the study of correlation functions in logarithmic CFTs in [FSS] to include boundaries and defects.

**3.**Further interesting examples of 3d TFTs arise from Chern-Simons theories having a non-compact gauge group like SL(2,C) or SL(2,R). The quantum Teichmüller theory can be interpreted as a sector of the quantum theory resulting from the canonical quantization of the Chern-Simons theory with non-compact gauge group SL(2,R). Very recently there has been progress towards the rigorous construction of quantum Chern-Simons theory with gauge group SL(2,C) [AK][D]. The associated three-manifold invariants generalize the Reshetikhin-Turaev invariants considerably, depending on a continuous and a discrete parameter. Two directions of future research in this area appear to be particularly promising: It should, on the one hand, be very interesting to identify the non-rational CFT appearing in noncompact generalizations of the relations between Chern-Simons theory and CFT discovered by Witten [W1]. So far, the only known example is provided by the relation between the quantum Teichmüller theory and Liouville theory [TV]. Studying limits of the new invariants defined by the non-compact Chern-Simons theories should, on the other hand, help us to understand the relations to known invariants as interesting and subtle as the hyperbolic volume of knot complements.

### References

[AK]
J. Andersen and R. Kashaev,
*Complex Quantum Chern-Simons*,
arXiv:1409.1208.

[CGPM]
F. Costantino, N. Geer and B. Patureau-Mirand,
*Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories*,
J. Topology **7** (2014) 1005–1053.

[CR]
N. Carqueville and I. Runkel,
*Orbifold completion of defect bicategories*,
arXiv:1210.6363.

[D]
T. Dimofte,
*omplex Chern-Simons theory at level k via the 3d-3d
correspondence*,
arXiv:1409.0857.

[FFRS1]
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert,
*Duality and defects in rational conformal field theory*,
Nucl. Phys. B **763** (2007) 354–430.

[FFRS2]
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert,
*Defect lines, dualities, and generalised orbifolds*,
Proceedings of the XVI International Congress on Mathematical Physics, Prague, 2009.
arXiv:0909.5013.

[FSS]
J. Fuchs, C. Schweigert and C. Stigner,
*From non-semisimple Hopf algebras to correlation functions for logarithmic CFT*,
J. Phys. A **46** (2013) 494008.

[FSV1]
J. Fuchs, C. Schweigert and A. Valentino,
*Bicategories for boundary conditions and for surface defects in 3-d TFT*,
Commun. Math. Phys. **331** (2013) 543.

[FSV2]
J. Fuchs, C. Schweigert and A. Valentino,
*A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories*,
Commun. Math. Phys. **332** (2014) 981–1015.

[L]
V. V. Lyubashenko,
*Invariants of three manifolds and projective representations of mapping class groups via quantum groups at roots of unity*,
Commun. Math. Phys. **172** (1995) 467–516

[R]
I. Runkel,
*A braided monoidal category for free super-bosons*,
J. Math. Phys. **55** (2014) 041702.

[RT] N. Reshetikhin and V. G. Turaev. *Invariants of 3-manifolds via link polynomials and quantum groups*, Invent. Math. **103** (1991) 547–597.

[TV]
J. Teschner and G. Vartanov,
*Supersymmetric gauge theories, quantization of M _{flat}, and
conformal field theory*,
arXiv:1302.3778.

[W] E. Witten. * Quantum field theory and the Jones polynomial*, Commun. Math. Phys. **121** (1989) 351–399.