Contents

duality

# Contents

## 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 $\mathcal{O}$-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

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

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

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

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

## Embedding in string theory

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).

## Examples

### Limiting cases

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 $p_*(\mathcal{O}_{T^* Bun_G^\circ})$.

### 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)

## References

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

• 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)

A general quantum geometric Langlands correspondence is produced in

Relation to semi-holomorphic 4d Chern-Simons theory:

• Peter Koroteev, Daniel S. Sage, Anton Zeitlin, $(SL(N),q$)-opers, the $q$-Langlands correspondence, and quantum/classical duality (arXiv:1811.09937)