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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?

Xiao-Gang Wen
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kolaka
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2 Answers2

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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.

Xiao-Gang Wen
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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.

Max Lein
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