Deformation theory of holomorphic vector bundles, extended conformal symmetry and extensions of 2D gravity

Developing the ideas of Stora and coworkers, a formulation of two-dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle E over a closed surface endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group Aut(E) of E and the extended Weyl group Weyl(E) of E and acts on the holomorphic and Hermitian structures. The extended Weyl anomaly can be shifted into a chirally split automorphism anomaly by adding to the action a local counterterm, as in ordinary conformal field theory. The dependence on the scale of the metric on the fibre of E is encoded in the Donaldson action, a vector bundle generalization of the Liouville action. The Weyl and automorphism anomaly split into two contributions corresponding, respectively, to the determinant and projectivization of E. The determinant part induces an effective ordinary Weyl or diffeomorphism anomaly and the induced central charge can be computed. As an application, it is shown that to any embedding t of into a simple Lie algebra and any faithful representation R of one can naturally associate a flat vector bundle DS(t,R) on . It is further shown that there is a deformation of the holomorphic structure of such a bundle whose parameter fields are generalized Beltrami differentials of the type appearing in light-cone W geometry, and that the projective part of the automorphism anomaly reduces to the standard W anomaly in the large central charge limit. A connection between the Donaldson action and Toda field theory is also observed.