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quantum geometric Langlands correspondence

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 GG-principal bundles over some complex curve for some reductive group GG, and that of DD-modules with a dual twist on the stack of LG{}^L G-bundles, for LG{}^L G the Langlands dual group.

In a certain limit of the twisting parameter the DD-modules on one side of this correspondence become equivalent to just π’ͺ\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 ∘Bun_G^\circ for the connected component of the moduli stack of GG-bundles, there is a line bundle denoted β„’ βŠ—k\mathcal{L}^{\otimes k} on Bun G ∘Bun_G^\circ and dually a line bundle β„’^ βŠ—k\hat \mathcal{L}^{\otimes k} on Bun LG ∘Bun_{{}^L G}^\circ and these serve to defined sheaves π’Ÿ (k,Ξ»)\mathcal{D}^{(k,\lambda)} of twisted differential operators on Bun G ∘Bun_{G}^\circ and dually sheaves 𝒒^ (k,Ξ»)\hat \mathcal{G}^{(k,\lambda)} of twisted differential operators on Bun LG ∘Bun_{{}^L G}^\circ, where the parameters lie in

(k,Ξ»)βˆˆβ„‚β„™ 1Γ—β„‚. (k, \lambda) \in \mathbb{CP}^1 \times \mathbb{C} \,.

Writing then π’Ÿ k,Ξ»Mod(Bun G)\mathcal{D}^{k,\lambda} Mod(Bun_{G}) and π’Ÿ^ k,Ξ»Mod(Bun LG)\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

π’Ÿ k,Ξ»Mod(Bun G)β‰ƒπ’Ÿ^ 1k,Ξ»Mod(Bun LG). \mathcal{D}^{k,\lambda} Mod(Bun_{G}) \simeq \hat \mathcal{D}^{\frac{1}{k},\lambda} Mod(Bun_{{}^L G}) \,.

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

Examples

Limiting cases

The twisted sheaves of differential operators π’Ÿ k,Ξ»\mathcal{D}^{k,\lambda} have the following limits:

  • For Ξ»β‰ 0\lambda \neq 0 and kβ‰ βˆžk \neq \infty this is the sheaf of differential operators acting on β„’ βŠ—k\mathcal{L}^{\otimes k};

  • for Ξ»β‰ 0\lambda \neq 0 and k=∞k = \infty this is the pushforward p *(π’ͺ Loc G)p_*(\mathcal{O}_{Loc_G}) of the sheaf of π’ͺ\mathcal{O}-modules along the canonical P:Loc Gβ†’Bun GP : Loc_G \to Bun_G that sends a local system to its underlying bundle;

  • for Ξ»=0\lambda = 0 and kk arbitrary this is p *(π’ͺ T *Bun G ∘)p_*(\mathcal{O}_{T^* Bun_G^\circ}).

Abelian case

In the case where GG 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)

Revised on August 20, 2012 14:18:32 by Urs Schreiber (89.204.138.243)