After some reading, I have an inuitive idea what topological phases of matter are. But where is the connection to modular tensor categories? Is there fundamental literature where this is covered?

Edit: A topological phase is characterized by a TQFT as low-energy effective theory. Furthermore, every modular tensor category leads to a TQFT, as shown by Turaev. However, according to Wang, "Topological Quantum Computation" (CBMS, Vol. 112, 2010), the converse is only a conjecture. Is it already proven that a strict fusion category of a TQFT can be extended uniquely to a modular tensor category compatible with the TQFT? Even if it is: Is there a more illustrative explanation why modular tensor categories are studied as mathematical models for topological phases?

  • $\begingroup$ Have you read the relevant This Week's Finds? $\endgroup$
    – AHusain
    Jul 28 '16 at 18:53
  • $\begingroup$ @AHusain What do you mean? Please add a link. $\endgroup$
    – PhCrx
    Jul 28 '16 at 20:05
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    $\begingroup$ John Baez Week 137 $\endgroup$
    – AHusain
    Jul 28 '16 at 20:33
  • $\begingroup$ this this one physics.stackexchange.com/q/5029/12961 $\endgroup$ Jul 28 '16 at 22:00
  • $\begingroup$ @AHusain : Thanks, this gives a bit of context, but I'm afraid this does not completely answer my question. There still seems to be a "missing link" between the physicist's and the mathematician's viewpoint. $\endgroup$
    – PhCrx
    Jul 29 '16 at 6:43

Modular tensor categories only describe the non-abelian statistics of the point-like topological excitations in 2+1D bosonic topologically ordered phases. So every topologically ordered phase gives rise to a modular tensor category. But the inverse is not true. Every modular tensor category correspond to infinite many 2+1D bosonic topologically ordered phases, and those phases differ by E8 states.

  • $\begingroup$ Thank you very much, that already helps a lot! Could you please explain in more detail or give sources (papers, articles...) how a topologically ordered phase gives rise to a modular tensor category (especially: in an anyon model, where does the modularity condition come into play) and how this difference by E8 states arises? $\endgroup$
    – PhCrx
    Sep 17 '16 at 7:45

There are several approaches connecting physics with the topology of mathematical objects — and this is the task of proving that a certain physical effect is of topological origin. For homogeneous or periodic systems such as crystalline solids or photonic crystals, you can use classification theory of vector bundles endowed with symmetries or twisted equivariant K-theory (which can also deal with random perturbations). A priori it is not clear whether all of these have go give rise to the same classification.

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    $\begingroup$ Sorry, but this is not an answer to my question. Your post touches the topic only superficially. $\endgroup$
    – PhCrx
    Sep 16 '16 at 15:46
  • $\begingroup$ Your question was very vague and quite general, and you claimed to only have an “intuitive idea” of what topological phases are. Coming from both, the physics and the mathematics side, topological insulators/field theories/superconducters have a huge literature with many approaches. Modular tensor categories are only one of several ways to a topological classification of physical systems. $\endgroup$
    – Max Lein
    Sep 18 '16 at 1:28
  • $\begingroup$ @MaxLein I do not see at which point my question should be "very vague". My question is very concrete, and it should be obvious that (for a specific reason) I'm only interested in the connection between topological phases and modular tensor categories. Your answer mentions several other approaches, but my question not in the slightest. $\endgroup$
    – PhCrx
    Sep 25 '16 at 17:41

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