Gauge Theory and Langlands Duality

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is the celebrated Shimura–Taniyama–Weil conjecture (which implies Fermat’s Last Theorem), proved by A. Wiles and others, which says that information about Galois representations associated to elliptic curves over Q is encoded in the Fourier expansion of certain modular forms on the upper-half plane. One of the most fascinating and mysterious aspects of the Langlands Program is the appearance of the Langlands dual group. Given a reductive algebraic group G, one constructs its Langlands dual G by applying an involution to its root data. Under the Langlands correspondence, automorphic representations of the group G correspond to Galois representations with values in G. Surprisingly, the Langlands dual group also appears in Quantum Physics in what looks like an entirely different context; namely, the electro-magnetic duality. Looking at the Maxwell equations describing the classical electromagnetism, one quickly notices that they are invariant under the exchange of the electric and magnetic fields. It is natural to ask whether this duality exists at the quantum level. In quantum theory there is an important parameter, the electric charge e. Physicists have speculated that there is an electro-magnetic duality in the quantum theory under which e ←→ 1/e. Under this duality the electrically charged particle should be exchanged with a magnetically charged particle, called magnetic monopole, first theorized by P. Dirac (so far, it has not been discovered experimentally). In modern terms, Maxwell theory is an example of 4D gauge theory (or Yang–Mills theory) which is defined, classically, on the space of connections on various Gc-bundles