Extended SL(2,ℝ)/U(1) characters, or modular properties of a simple non-rational conformal field theory

We define extended SL(2,ℝ)/U(1) characters which include a sum over winding sectors. By embedding these characters into similarly extended characters of N = 2 algebras, we show that they have nice modular transformation properties. We calculate the modular matrices of this simple but non-trivial non-rational conformal field theory explicitly. As a result, we show that discrete SL(2,ℝ) representations mix with continuous SL(2,ℝ) representations under modular transformations in the coset conformal field theory. We comment upon the significance of our results for a general theory of non-rational conformal field theories.