Conformal Field Theory and the Modular Group
Conformal field theory has been one of the stellar success stories for both theoretical physics and pure mathematics. Remarkably, it describes the scaling limits of statistical lattice theories such as percolation and the Ising model as well as providing the means to carry out quantum computations in string theory. Mathematically, its (partial) axiomatisation, the theory of vertex operator algebras, has been used to prove (some of) the Conway-Norton monstrous moonshine conjectures and plays a central role in the geometric Langlands correspondence.
One of the core consistency requirements of conformal field theory involves the modular group SL(2;Z). This leads to many deep and beautiful relationships between the underlying algebraic structures and number theory. The aim of this talk is to introduce some of these relationships through examples and discuss how recent research is forcing us to rethink the paradigms that these examples have suggested.