ar X iv : h ep - t h / 91 10 04 2 v 1 1 5 O ct 1 99 1 KÄHLER - CHERN - SIMONS THEORY

Kähler-Chern-Simons theory describes antiself-dual gauge fields on a four-dimensional Kähler manifold. The phase space is the space of gauge potentials , the symplectic reduction of which by the constraints of antiself-duality leads to the moduli space of antiself-dual instantons. We outline the theory highlighting symmetries, their canonical realization and some properties of the quantum wave functions. The relationship to integrable systems via dimensional reduction is briefly discussed. In this talk, I shall describe some recent work done in collaboration with Jeremy Schiff on what we refer to as Kähler-Chern-Simons (KCS) theory. 1 The theory basically provides an action description of antiself-dual gauge fields, i.e. instantons on four-dimensional Kähler manifolds. The motivation for seeking such a theory is essentially twofold. There is considerable evidence that antiself-dual gauge fields may be considered as a 'master' integrable system. 2 For example, we can consider R 4 as a Kähler manifold, pairing up the standard coordinates into complex ones as z = x 2 + ix 1 , w = x 4 + ix 3. The conditions of antiself-duality are then given by F zw = F ¯ z ¯ w = F z ¯ z + F w ¯ w = 0 (1) where F ab denotes the (ab) component of the field strength, which is as usual valued in the algebra of a Lie group G. Dimensional reduction of these equations, such as the requirement of the fields being independent of, say ¯ w, gives a large class of known two-dimensional integrable theories such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, Boussinesq and other equations. 3,4 It is likely that all inte-grable theories can be considered as special cases of the antiself-dual gauge theory. Now, integrable theories themselves are of interest because they may describe certain types of perturbations of conformal field theories and also because the Poisson bracket structures associated with integrable theories are related to the Virasoro and W N algebras. 5 These algebras in turn can be the chiral algebras of conformal field theories. Independently, the study of gravity theories associated to these algebras also seem to lead to four-dimensional Kähler manifolds. 6 A unified description of integrable theories in terms of antiself-dual gauge fields, especially in a Lagrangian framework with associated Hamiltonian and Poisson bracket structures, can thus be useful in understanding these theories and algebras.