nLab Hecke correspondence

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

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

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

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

where $T_{p}$ is the $p$-th Hecke operator.

Alternatively, we can embed $X_{0}(p)$ inside $X\times X$ 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 $X$ we have $Bun_{G}$ and instead of $X_{0}$ we have the Hecke stack $Hck$, which parametrizes pairs of bundles $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ together with a modification which is an isomorphism between $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ away from some points:

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

Properties

Relation to S-duality in super Yang-Mills theory

geometric Langlands correspondence	S-duality in N=4 D=4 super Yang-Mills theory
Hecke transformation	't Hooft operator
local system/flat connection	electric eigenbrane (eigenbrane of Wilson operator)
Hecke eigensheaf	magnetic eigenbrane (eigenbrane of 't Hooft operator )

(Kapustin-Witten 06)

Related concepts

• modular form

• Hecke algebra

• Hecke eigensheaf

• Hecke category

References

• Edward Frenkel, section 3.7 in Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172)

• Frank Calegari, Congruences between modular forms, 2013 pdf

• Vincent Lafforgue, Shtukas for reductive groups and Langlands correspondence for function fields, arXiv:1803.03791