GEOMETRIC HAMILTON–JACOBI FIELD THEORY

Geometric Class Field Theory I

1.1. The Artin map. Let’s start off by reviewing the classical origins of class field theory. The motivating problem is basically to “describe” in some meaningful way the abelian extensions of a number field F , or what is essentially the same: the abelianized Galois group Gal(K/K)ab (since its finite quotients are the Galois groups of finite abelian extensions of K). If we’re going to describe...

Geometric Class Field Theory Iii

Geometric Structures in Field Theory

This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper reviews, compares and constrasts the various generalizations in order to bring some unity to the field of study. The generalizations seem to fall into two categorie...

Geometric Class Field Theory Ii

Homcts(π1,ét(C),Q × ` ) ∼= Hom(Pic0(k),Q` ) + (Ẑ Frobc −−−→ Q` ). where c ∈ C(Fq) is a fixed rational point. Our proof will proceed by upgrading this equality to an equivalence of geometric objects. First, we’ll interpretHomcts(π1,ét(C),Q × ` ) in terms of rank one `-adic local systems on C. Similarly, we’ll interpret the datum of Hom(Pic0(k),Q` ) + (Ẑ Frobc −−−→ Q` ) as a “character sheaf” on ...

Geometric Construction of Modular Functors from Conformal Field Theory

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [48] and [51] by a certain fractional power of the abelian theory first considered in [32] and further studied in [2].