nLab
quantum geometric Langlands correspondence

Idea

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 $𝒪$-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 ${\mathrm{Bun}}_G^{\circ }$ for the connected component of the moduli stack of $G$-bundles, there is a line bundle denoted ${ℒ}^{\otimes k}$ on ${\mathrm{Bun}}_G^{\circ }$ and dually a line bundle ${\widehat{ℒ}}^{\otimes k}$ on ${\mathrm{Bun}}_{{}^{L}G}^{\circ }$ and these serve to defined sheaves ${𝒟}^{(k,\lambda )}$ of twisted differential operators on ${\mathrm{Bun}}_G^{\circ }$ and dually sheaves ${\widehat{𝒢}}^{(k,\lambda )}$ of twisted differential operators on ${\mathrm{Bun}}_{{}^{L}G}^{\circ }$, where the parameters lie in

$$(k,\lambda )\in {\mathrm{ℂ\mathbb{P}}}^1\times ℂ.$$(k, \lambda) \in \mathbb{CP}^1 \times \mathbb{C} \,.

Writing then ${𝒟}^{k,\lambda }\mathrm{Mod}({\mathrm{Bun}}_G)$ and ${\widehat{𝒟}}^{k,\lambda }\mathrm{Mod}({\mathrm{Bun}}_{{}^{L}G})$, respectively, for the derived categories of modules over these sheaves, the conjectured statement is:

Quantum geometric Langlands correspondence

There is an equivalence

$${𝒟}^{k,\lambda }\mathrm{Mod}({\mathrm{Bun}}_G)\simeq {\widehat{𝒟}}^{\frac{1}{k},\lambda }\mathrm{Mod}({\mathrm{Bun}}_{{}^{L}G}).$$\mathcal{D}^{k,\lambda} Mod(Bun_{G}) \simeq \hat \mathcal{D}^{\frac{1}{k},\lambda} Mod(Bun_{{}^L G}) \,.

(In fact $k$ and $\frac{1}{k}$ here should further be shifted by the dual Coxeter number? of $G$ and ${}^{L}G$, respectively.)

Examples

Limiting cases

The twisted sheaves of differential operators ${𝒟}^{k,\lambda }$ have the following limits:

• For $\lambda \ne 0$ and $k\ne \mathrm{\infty }$ this is the sheaf of differential operators acting on ${ℒ}^{\otimes k}$;

• for $\lambda \ne 0$ and $k=\mathrm{\infty }$ this is the pushforward $p_{*}({𝒪}_{{\mathrm{Loc}}_G})$ of the sheaf of $𝒪$-modules along the canonical $P:{\mathrm{Loc}}_G\to {\mathrm{Bun}}_G$ that sends a local system to its underlying bundle;

• for $\lambda =0$ and $k$ arbitrary this is $p_{*}({𝒪}_{T^{*}{\mathrm{Bun}}_G^{\circ }})$.

Abelian case

In the case where $G$ is abelian, the quantum correspondence is given by a Fourier-Mukai transform and has been constructed in (Polishuk-Rothenstein)

Related concepts

• S-duality

  • electro-magnetic duality

  • geometric Langlands correspondence

  • quantum geometric Langlands correspondence

References

The statement of the quantum geometric Langlands correspondence is surveyed on page 70-71 of

• Edward Frenkel, Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172).

The construction of the correspondence in the abelian case, where it is given by a Fourier-Mukai transform, is given in

• A. Polishchuk and M. Rothstein, Fourier transform for D-algebras , DukeMath. J. 109 (2001) 123–146.

An interpretation of the quantum Langlands correspondence in terms of the B-model is given in

• Anton Kapustin, A Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections (arXiv:0811.3264)