TOPOLOGICAL SIGMA-MODELS IN 4 DIMENSIONS AND TRIHOLOMORPHIC MAPS

It is well known that topological sigma-models in two dimensions const itute a path-integral approach to the study of holomorphic maps from a Riemann surface SIGMA to an almost complex manifold K, the most inter esting case being that were K is a Kahler manifold. We show that, in t he same way, topological sigma-models in four dimensions introduce a p ath-integral approach to the study of triholomorphic maps q: M --> N b etween a four-dimensional riemannian manifold M and an almost quaterni onic manifold N. The most interesting cases are those where M, N are h yper-Kahler or quaternionic Kahler. BRST-cohomology translates into in tersection theory in the moduli-space of this new class of instantonic maps, that are named hyperinstantons by us. The definition of triholo morphicity that we propose is expressed by the equation q - J(u) . q . j(u) = 0, where {j(u), u = 1, 2, 31 is an almost quaternionic structu re on M and {J(u), u = 1, 2, 3) is an almost quaternionic structure on N. This is a generalization of the Cauchy-Fueter equations. For M, N hyper-Kahler, this generalization naturally arises by obtaining the to pological sigma-model as a twisted version of the N = 2 globally super symmetric sigma-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyze the co upling of the topological sigma-model to topological gravity. The clas sification of triholomorphic maps and the analysis of their moduli-spa ce is a new and fully open mathematical problem that we believe deserv es the attention of both mathematicians and physicists.