abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
The quantum geometric Langlands correspondence is a conjectured equivalence between the derived category of certain twisted D-modules on the moduli stack of $G$-principal bundles over some complex curve for some reductive group $G$, and that of $D$-modules with a dual twist on the stack of ${}^L G$-bundles, for ${}^L G$ the Langlands dual group.
In a certain limit of the twisting parameter the $D$-modules on one side of this correspondence become equivalent to just $\mathcal{O}$-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 $Bun_G^\circ$ for the connected component of the moduli stack of $G$-bundles, there is a line bundle denoted $\mathcal{L}^{\otimes k}$ on $Bun_G^\circ$ and dually a line bundle $\hat \mathcal{L}^{\otimes k}$ on $Bun_{{}^L G}^\circ$ and these serve to defined sheaves $\mathcal{D}^{(k,\lambda)}$ of twisted differential operators on $Bun_{G}^\circ$ and dually sheaves $\hat \mathcal{G}^{(k,\lambda)}$ of twisted differential operators on $Bun_{{}^L G}^\circ$, where the parameters lie in
Writing then $\mathcal{D}^{k,\lambda} Mod(Bun_{G})$ and $\hat \mathcal{D}^{k,\lambda} Mod(Bun_{{}^L G})$, respectively, for the derived categories of modules over these sheaves, the conjectured statement is:
Quantum geometric Langlands correspondence
There is an equivalence
(In fact $k$ and $\frac{1}{k}$ here should further be shifted by the dual Coxeter number? of $G$ and ${}^L G$, respectively.)
The twisted sheaves of differential operators $\mathcal{D}^{k,\lambda}$ have the following limits:
For $\lambda \neq 0$ and $k \neq \infty$ this is the sheaf of differential operators acting on $\mathcal{L}^{\otimes k}$;
for $\lambda \neq 0$ and $k = \infty$ this is the pushforward $p_*(\mathcal{O}_{Loc_G})$ of the sheaf of $\mathcal{O}$-modules along the canonical $P : Loc_G \to Bun_G$ that sends a local system to its underlying bundle;
for $\lambda = 0$ and $k$ arbitrary this is $p_*(\mathcal{O}_{T^* Bun_G^\circ})$.
In the case where $G$ is abelian, the quantum correspondence is given by a Fourier-Mukai transform and has been constructed in (Polishuk-Rothenstein)
quantum geometric Langlands correspondence
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