Langlands Duality and G2 Spectral Curves

Abstract

We first demonstrate how duality for the fibres of the so-called Hitchin fibration works for the Langlands dual groups Sp(2m) and SO(2m + 1). We then show that duality for G 2 is implemented by an involution on the base space which takes one fibre to its dual. A formula for the natural cubic form is given and shown to be invariant under the involution.

Electric-magnetic duality and the geometric Langlands program

“…For the exceptional Lie group G 2 , this duality has been established recently by Hitchin [81]. The question has also been analyzed by Donagi and Pantev for any semi-simple Lie group using an abstract approach to spectral covers [82].…”

Lectures on Higgs moduli and abelianisation

“…Such an approach can also be adapted to work for the exceptional group G 2 , see [Hit07]. For a complete treatment, then, one has to study the spectral correspondence for pairs consisting of a group G and a representation G → GL n , see [Don95].…”

“…The calculation then is completed as in §6.5 of [Hit07], but using the expression from Theorem 9.1 for the cubic.…”

Higgs bundles and exceptional isogenies

“…The induced map M Spin(2n,C) →M SO(2n,C) is a finite map. For any n, the moduli space M SO(2n,C) has two components corresponding to the two possible values for the second Stiefel-Whitney class of an SO(2n, C)-principal bundle (see for example [16]). In contrast, the underlying holomorphic bundles for the Higgs bundles in M SL(2,C)×SL(2,C) and M SL(4,C) have just one topological type, and the moduli spaces are connected.…”