Bruguières modularity theorem

In mathematics, the Bruguières modularity theorem is a theorem about modular tensor categories. It asserts that two different formulations of the modularity condition of a modular tensor category are equivalent. The Bruguières modularity theorem was introduced by mathematician Alain Bruguières in the year 2000.^[1] The first notion of modularity used in the theorem statement is in terms of the non-degeneracy of the braid statistics of the simple objects, and the other is in terms of the non-degeneracy of the modular S-matrix. Historically, the non-degeneracy condition for modular tensor categories was originally stated in terms of the invertibility of the ${\displaystyle S}$-matrix.^[2] Nowadays, it is common to define modular category in terms of the non-degeneracy of its braiding statistics, especially in the condensed matter physics literature.^[3]

The Bruguières modularity theorem is stated in terms of pre-modular tensor categories, a notion introduced by Bruguières in the same paper in which he proved the modularity theorem.^[1] A category ${\displaystyle {\mathcal {C}}}$ is called pre-modular if it is equipped with all of the structures of a modular tensor category, and satisfies all of the axioms except possibly for non-degeneracy. Bruguières' modularity theorem asserts that a pre-modular tensor category has non-degenerate braiding if and only if its modular S-matrix is invertible.

Let ${\displaystyle {\mathcal {C}}}$ denote a pre-modular tensor category. One direction of the Bruguières modularity theorem is straightforward - if the braiding is degenerate, then there will be a simple object ${\displaystyle A\in {\mathcal {C}}}$ which braids trivially with all the other simple objects, and so the column of the modular ${\displaystyle S}$-matrix corresponding to the simple object ${\displaystyle A\in {\mathcal {C}}}$ will be proportional to the column of the modular ${\displaystyle S}$-matrix associated to the tensor unit ${\displaystyle {\bf {1}}}$. Thus, the ${\displaystyle S}$-matrix will be degenerate. The content of the theorem is that if the braiding is non-degenerate, then the modular ${\displaystyle S}$-matrix will be invertible.

If modular tensor categories are defined in terms of the non-degeneracy of their braiding, then the Bruguières modularity theorem is a necessary ingredient for the existence of the modular group representation. In the modular group representation the ${\displaystyle S}$-matrix is the image, up to scaling, of one of the generators of ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$. As such, for the modular group representation to be a valid representation it is necessary for the ${\displaystyle S}$-matrix to be invertible.^[4]