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?
Best Answer
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.