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. Research partially supported by the EEC under the contracts HPRN-CT-2000-00122, HPRN-CT-200000131. Unité mixte du CNRS et de l’Ecole Normale Supérieure, UMR 8549.