nLab Hecke correspondence

Contents

Contents

Idea

A Hecke correspondence is a certain correspondence between a moduli stacks of bundles. The integral transform induced by a Hecke correspondence is called a Hecke transform. The objects preserves by such a transform up to a tensor multiple are called Hecke eigensheaves.

These are central objects of interest in geometric Langlands duality.

For modular curves

The reference for this is section 1.4 of Calegari13.

Let X 0(p)X_{0}(p) be the moduli space of pairs (E,E)(E,E') where EE and EE' are pp-isogenous elliptic curves. This has maps π 1\pi_{1} and π 2\pi_{2} to the moduli space of elliptic curves XX, sending (E,E)(E,E') to EE and EE' respectively:

X 0(p) π 1 π 2 X X \array{ && X_{0}(p) \\ & {}_{\mathllap{\pi_{1}}}\swarrow && \searrow_{\mathrlap{\pi_{2}}} \\ X && && X }

We can think of the Hecke correspondence as the multivalued function π 2*π 1 *\pi_{2 *}\circ\pi_{1}^{*}. Although this is a multivalued function, upon linearizing (by, say, passing to modular forms i.e. global sections of the sheaf ω k\omega^{k}) we get a legitimate function

pT p:H 0(X(Γ),ω k)H 0(X(Γ),ω k)pT_{p}:H^{0}(X(\Gamma),\omega^{k})\to H^{0}(X(\Gamma),\omega^{k})

where T pT_{p} is the pp-th Hecke operator.

Alternatively, we can embed X 0(p)X_{0}(p) inside X×XX\times X as the graph of the above-mentioned multivalued function. In this way it is more properly seen as a correspondence.

For moduli stacks of bundles

A reference for this is section 3.7 of Frenkel05. A discussion can also be found in Lafforgue18.

Hecke correspondences for moduli stacks of bundles follow roughly the same idea but instead of XX we have Bun GBun_{G} and instead of X 0X_{0} we have the Hecke stack HckHck, which parametrizes pairs of bundles 1\mathcal{E}_{1}, 2\mathcal{E}_{2} together with a modification which is an isomorphism between 1\mathcal{E}_{1} and 2\mathcal{E}_{2} away from some points:

Hck π 1 π 2 Bun G Bun G \array{ && Hck \\ & {}_{\mathllap{\pi_{1}}}\swarrow && \searrow_{\mathrlap{\pi_{2}}} \\ Bun_{G} && && Bun_{G} }

Properties

Relation to S-duality in super Yang-Mills theory

geometric Langlands correspondenceS-duality in N=4 D=4 super Yang-Mills theory
Hecke transformation't Hooft operator
local system/flat connectionelectric eigenbrane (eigenbrane of Wilson operator)
Hecke eigensheafmagnetic eigenbrane (eigenbrane of 't Hooft operator )

(Kapustin-Witten 06)

References

Last revised on July 4, 2022 at 18:52:01. See the history of this page for a list of all contributions to it.