Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.

See "J. Fuchs, I. Runkel, C. Schweigert: TFT construction of RCFT correlators I-V " for the full story or, for a summary, Schweigert's ICM talk Categorification and correlation functions in conformal field theory.

Jürgen Fuchs, Ingo Runkel and Christoph Schweigert have developed a complete treatment of Rational Conformal Field Theory based on algebra in braided tensor categories. They have applications to string theory as well as to statistical physics, most importantly to conformal defects and so-called Kramers-Wannier-dualities.

See J. Fuchs, I. Runkel, C. Schweigert: TFT construction of RCFT correlators I, II, III, IV, V for the full story or, for a summary, Schweigert's 2006 ICM talk Categorification and correlation functions in conformal field theory.