Gauge theory, ramification, and the geometric Langlands program.

*(English)*Zbl 1237.14024
Jerison, David (ed.) et al., Current developments in mathematics, 2006. Somerville, MA: International Press (ISBN 978-1-57146-167-4/hbk). 35-180 (2008).

This long paper is a detailed exposition of the geometric Langlands program with tame ramification from the gauge theoretic point of view. Very roughly, the geometric Langlands correspondence relates flat connections on a Riemann surface \(C\) to \(\mathcal{D}\)-modules over it. The focus of the paper is on a generalization of this correspondence to the case of punctured surfaces \(C\backslash\{p_1,\dots,p_n\}\) with the flat connection having prescribed singularities at the punctures. In the original Langlands program for number theory singularities correspond to ramification. Accordingly, the cases of singularities being simple and higher order poles are known as tame and wild ramification respectively.

Physically, the correspondence can be described in terms of \(\mathrm{GL}\)-twisted \(\mathcal{N}=4\) super Yang-Mills theory that reduces at low energies to a sigma-model given by Hitchin’s equations on \(C\). The gauge group is a compact Lie group often assumed to be simple. Solutions to Hitchin’s equations describe Higgs bundles, i.e. pairs \((E,\varphi)\) with a holomorphic \(G\)-bundle \(E\) and a holomorphic section \(\varphi\) of \(K_C\otimes \mathrm{ad}(E)\), where \(K_C\) is the canonical bundle of \(C\). The ramification is implemented by introducing surface operators analogous to Wilson loop and t’Hooft operators, but supported on \(2\)-submanifolds. Just like t’Hooft operators the surface operators are specified by requiring that in the normal planes to the surface the gauge fields look like solutions to Hitchin’s equations with a prescribed singularity. In its turn, the singularity is specified by a triple of parameters, which turn out to be the Kähler moduli of the hyper-Kähler moduli space \(\mathcal{M}_H(G)\) of the correspondingly ramified Higgs bundles. A highlight of the paper is a detailed proposal for the geometric Langlands correspondence with tame ramification in these terms.

The authors go on to describe the action of the electric-magnetic \(S\)-duality of the super Yang-Mills theory on the ramification parameters, the action of the affine braid group on the cohomology of \(\mathcal{M}_H(G)\), local singularities of \(\mathcal{M}_H(G)\) and other phenomena relevant to the geometric Langlands. The affine braid action is further extended to \(A\)- and \(B\)-branes. These constructions are in close parallel to some structures of the representation theory, like Springer representations of the affine Weyl group and the Kazhdan-Lusztig theory for the affine Hecke algebras. Some aspects of local behavior at the ramification points are captured by studying sigma-models having as targets complex coadjoint orbits endowed with the hyper-Kähler metrics (local models for \(\mathcal{M}_H(G)\)). The authors also consider line operators that can act on branes at ramification points, and describe string-theoretic constructions of the surface operators.

For the entire collection see [Zbl 1149.00018].

Physically, the correspondence can be described in terms of \(\mathrm{GL}\)-twisted \(\mathcal{N}=4\) super Yang-Mills theory that reduces at low energies to a sigma-model given by Hitchin’s equations on \(C\). The gauge group is a compact Lie group often assumed to be simple. Solutions to Hitchin’s equations describe Higgs bundles, i.e. pairs \((E,\varphi)\) with a holomorphic \(G\)-bundle \(E\) and a holomorphic section \(\varphi\) of \(K_C\otimes \mathrm{ad}(E)\), where \(K_C\) is the canonical bundle of \(C\). The ramification is implemented by introducing surface operators analogous to Wilson loop and t’Hooft operators, but supported on \(2\)-submanifolds. Just like t’Hooft operators the surface operators are specified by requiring that in the normal planes to the surface the gauge fields look like solutions to Hitchin’s equations with a prescribed singularity. In its turn, the singularity is specified by a triple of parameters, which turn out to be the Kähler moduli of the hyper-Kähler moduli space \(\mathcal{M}_H(G)\) of the correspondingly ramified Higgs bundles. A highlight of the paper is a detailed proposal for the geometric Langlands correspondence with tame ramification in these terms.

The authors go on to describe the action of the electric-magnetic \(S\)-duality of the super Yang-Mills theory on the ramification parameters, the action of the affine braid group on the cohomology of \(\mathcal{M}_H(G)\), local singularities of \(\mathcal{M}_H(G)\) and other phenomena relevant to the geometric Langlands. The affine braid action is further extended to \(A\)- and \(B\)-branes. These constructions are in close parallel to some structures of the representation theory, like Springer representations of the affine Weyl group and the Kazhdan-Lusztig theory for the affine Hecke algebras. Some aspects of local behavior at the ramification points are captured by studying sigma-models having as targets complex coadjoint orbits endowed with the hyper-Kähler metrics (local models for \(\mathcal{M}_H(G)\)). The authors also consider line operators that can act on branes at ramification points, and describe string-theoretic constructions of the surface operators.

For the entire collection see [Zbl 1149.00018].

Reviewer: Sergiy Koshkin (Houston)

##### MSC:

14D24 | Geometric Langlands program (algebro-geometric aspects) |

53C26 | Hyper-Kähler and quaternionic Kähler geometry, “special” geometry |

81T60 | Supersymmetric field theories in quantum mechanics |