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From Three Dimensional Manifolds to Modular Tensor Categories

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Abstract

Using M-theory in physics, Cho et al. (JHEP 2020:115 (2020) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of \(\text {SL}(2,{\mathbb {C}})\)-flat connections and \(\text {SL}(2,{\mathbb {C}})\)-adjoint Reidemeister torsions of a three manifold can be packaged together to produce a \((2+1)\)-topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T-matrix and the quantum dimensions of a candidate modular data. The modular S-matrix follows from essentially a trial-and-error procedure. We find premodular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our examples. Our main result is a mathematical construction of the modular data of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The modular data of premodular categories from Seifert fibered spaces can be realized using Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a \({\mathbb {Z}}_2\)-homology sphere, and condensation of bosons in the resulting properly premodular categories leads to either modular or super-modular tensor categories.

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Notes

  1. We omit the irrelevant base point.

  2. Here all maps involved are isomorphisms, so the notion of direct limit and inverse limit do not make a difference.

  3. The sign and hence the negative sign in front of \(\text {CS}\) invariant below is not important and the choice is made to be the same as in [4].

  4. A terminology due to C. Delaney: distinct MTCs with the same MD are called modular isotopes of each other.

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Acknowledgements

Y. Q. and Z.W. are partially supported by NSF grant FRG-1664351 and CCF 2006463. S.-X. C. is partially supported by NSF CCF 2006667. The research is also partially supported by ARO MURI contract W911NF-20-1-0082. The third author thanks Dongmin Gang for helpful communications, who pointed out that the Seifert fibered spaces (3, 3, r) would give rise to modular tensor categories related to \(SU(2)_k\). The first author thanks Eric Rowell for clarifying some facts about \({\mathcal {C}}({{\mathfrak {s}}}{{\mathfrak {o}}},q,l)\).

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Authors and Affiliations

  1. Department of Mathematics and Department of Physics and Astronomy, Purdue University, West Lafayette, IN, 47906, USA

    Shawn X. Cui

  2. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    Yang Qiu

  3. Microsoft Station Q and Dept. of Math., University of California, Santa Barbara, CA, 93106, USA

    Zhenghan Wang

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  1. Shawn X. Cui
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  2. Yang Qiu
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  3. Zhenghan Wang
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Correspondence to Zhenghan Wang.

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Communicated by Y. Kawahigashi.

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Cui, S.X., Qiu, Y. & Wang, Z. From Three Dimensional Manifolds to Modular Tensor Categories. Commun. Math. Phys. 397, 1191–1235 (2023). https://doi.org/10.1007/s00220-022-04517-4

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