abstract duality: opposite category,
The quantum geometric Langlands correspondence is a conjectured equivalence between the derived category of certain twisted D-modules on the moduli stack of -principal bundles over some complex curve for some reductive group , and that of -modules with a dual twist on the stack of -bundles, for the Langlands dual group.
In a certain limit of the twisting parameter the -modules on one side of this correspondence become equivalent to just -modules, but on the moduli space of local systems. This limit of the quantum correspondence reproduces the statement of the (ordinary) geometric Langlands correspondence.
Slightly more precisely, writing for the connected component of the moduli stack of -bundles, there is a line bundle denoted on and dually a line bundle on and these serve to defined sheaves of twisted differential operators on and dually sheaves of twisted differential operators on , where the parameters lie in
Writing then and , respectively, for the derived categories of modules over these sheaves, the conjectured statement is:
Quantum geometric Langlands correspondence
There is an equivalence
(In fact and here should further be shifted by the dual Coxeter number? of and , respectively.)
Where the plain geometric Langlands correspondence is meant to be a shadow of the 6d (2,0)-superconformal field theory, the quantum version is meant to correspondingly relate to 6d little string theory (Aganagic-Frenkel-Okounkov 17) (which turns into the 6d QFT in the limit that the strings shrink to points).
The twisted sheaves of differential operators have the following limits:
For and this is the sheaf of differential operators acting on ;
for and this is the pushforward of the sheaf of -modules along the canonical that sends a local system to its underlying bundle;
for and arbitrary this is .
The statement of the quantum geometric Langlands correspondence is surveyed on page 70-71 of
The construction of the correspondence in the abelian case, where it is given by a Fourier-Mukai transform, is given in
An interpretation of the quantum Langlands correspondence in terms of the B-model is given in
A general quantum geometric Langlands correspondence is produced in