The quantum geometric Langlands correspondence compares categories of twisted D-modules on the principal-bundle stacks of an algebraic group and its Langlands
dual group at reciprocal shifted levels. Let 
be a smooth projective connected complex curve, let 
be a connected simple complex algebraic
group, and let 
be defined by the dual of the full root datum. The global
form of 
is retained. Write 
for the dual Coxeter number
and 
for the adjoint determinant line, whose fiber at a bundle
is 
. Use the convention
 |
where 
is the cocomplete derived differential
graded category of all twisted D-modules
for that complex exponent. It is a twist of differential operators rather than a
chosen root of 
.
