Quantum geometric Langlands correspondence in positive characteristic: The GLN case

Travkin, Roman

doi:10.1215/00127094-3449780

Cited by 8 publications

(14 citation statements)

References 17 publications

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“…Then is naturally a -torsor. As explained in [Ari08] and [Tra11, §3.1], the correspondence induces an equivalence of 2-categories between extensions of by and -torsors.…”

Section: Picard Stackmentioning

confidence: 99%

“…By definition the dual of is An element belongs to if and only if the composition is equal to the identity. Equivalently, gives a splitting of the exact sequence (A.7.2) and according to Definition A.5.4 it is a multiplicative splitting of . The following theorem follows immediately from Theorem A.4.7. ([Ari08], [<cite class="target" data-doi="10.1215/00127094-3449780">Tra11</cite>, §3.2]). The Fourier–Mukai functor restricts to an equivalence There is an action of on by tensoring and an action of on by convolution. Those two actions are compatible with the above equivalence in the following sense: there is a canonical isomorphism for and .…”

Section: Picard Stackmentioning

confidence: 99%

“…The case has been considered by Bezrukavnikov and Braverman in [BB07] (see [Gro12, Tra11] for various extensions). The main observation is that the geometric Langlands duality in characteristic formulated in the above form can be thought as a twisted version of its classical limit.…”

Section: Introductionmentioning

confidence: 99%

“…In Appendix B we recall the basic theory of -modules over varieties and stacks in positive characteristic, following [BMR08, BB07, OV07, Tra11].…”

Section: Introductionmentioning

confidence: 99%

“…Then T := P × Z S {1} is naturally a P 1 -torsor. As explained in [Ar] and [Trav,§3.1], the correspondence P → T induces an equivalence of 2-categories between extensions of Z S by P 1 and P 1 -torsors. Now, let P and P 1 be two Picard stacks and let G be a P 1 -torsor over P. Let m : P × P → P, e : T → P be the multiplication morphism and the unit morphism respectively, and let σ : P ×P → P ×P be the flip map σ(x, y) = (y, x).…”

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confidence: 99%

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Geometric Langlands in prime characteristic

Chen¹,

Zhu²

2017

Compositio Math.

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Abstract. Let G be a semisimple algebraic group over an algebraically closed field k, whose characteristic is positive and does not divide the order of the Weyl group of G, and letG be its Langlands dual group over k. Let C be a smooth projective curve over k of genus at least two. Denote by BunG the moduli stack of Gbundles on C and LocSysG the moduli stack ofG-local systems on C. Let DBun G be the sheaf of crystalline differential operators on BunG. In this paper we construct an equivalence between the bounded derived category

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“…Then is naturally a -torsor. As explained in [Ari08] and [Tra11, §3.1], the correspondence induces an equivalence of 2-categories between extensions of by and -torsors.…”

Section: Picard Stackmentioning

confidence: 99%

Section: Picard Stackmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Appendix B we recall the basic theory of -modules over varieties and stacks in positive characteristic, following [BMR08, BB07, OV07, Tra11].…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Geometric Langlands in prime characteristic

Chen¹,

Zhu²

2017

Compositio Math.

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Quantum parameters of the geometric Langlands theory

Zhao

2023

Sel. Math. New Ser.

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Fix a smooth, complete algebraic curve X over an algebraically closed field k of characteristic zero. To a reductive group G over k, we associate an algebraic stack $${\text {Par}}_G$$ Par G of quantum parameters for the geometric Langlands theory. Then we construct a family of (quasi-)twistings parametrized by $${\text {Par}}_G$$ Par G , whose module categories give rise to twisted $${\mathcal {D}}$$ D -modules on $${\text {Bun}}_G$$ Bun G as well as quasi-coherent sheaves on the DG stack $${\text {LocSys}}_G$$ LocSys G .

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Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

2020

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We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SLn and P GLn. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for d coprime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffman in the coprime case.Contents arXiv:1707.06417v2 [math.AG] 8 Jun 2018Proof. We begin the proof by fixing notation. The relative Jacobian of the trivial family of curves X × S A / A will be denoted by J. The relative Jacobian of the universal spectral curve Y / A will be denoted by J. Similarly we denote by J 1 and J 1 the relative moduli spaces of degree 1 line bundles.Henceforth we restrict every A-scheme to the open subset A good . To avoid awkward notation we will omit the corresponding superscript.The relative norm map induces a morphism of abelian A-schemes J Nm / / J. Similarly, pullback of line bundles yields π * : J / / J. We claim that these two morphisms are dual to each other with respect to the canonical isomorphism J ∨ J induced by the Poincaré bundle (and similarly for J). To see this we observe that we have a commutative diagram (the horizontal arrows represent the Abel-Jacobi map)to which we can apply the contravariant Pic 0 (?/ A) functor to obtain the commutative diagram of abelian schemes Furthermore, the top row of the first diagram is the fibre of the vertical arrows and hence the top row of the second diagram is the corresponding cofibre. By explicitly computing the fibres in the second diagram

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Quantum geometric Langlands correspondence in positive characteristic: The GLN case

Cited by 8 publications

References 17 publications

Geometric Langlands in prime characteristic

Geometric Langlands in prime characteristic

Quantum parameters of the geometric Langlands theory

Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

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