It is well known that topological sigma-models in two dimensions constitute a path-integral approach to the study of holomorphic maps from a Riemann surface SIGMA to an almost complex manifold K, the most interesting case being that were K is a Kahler manifold. We show that, in the same way, topological sigma-models in four dimensions introduce a path-integral approach to the study of triholomorphic maps q: M --> N between a four-dimensional riemannian manifold M and an almost quaternionic manifold N. The most interesting cases are those where M, N are hyper-Kahler or quaternionic Kahler. BRST-cohomology translates into intersection theory in the moduli-space of this new class of instantonic maps, that are named hyperinstantons by us. The definition of triholomorphicity 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 structure 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 topological sigma-model as a twisted version of the N = 2 globally supersymmetric sigma-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyze the coupling of the topological sigma-model to topological gravity. The classification of triholomorphic maps and the analysis of their moduli-space is a new and fully open mathematical problem that we believe deserves the attention of both mathematicians and physicists.
TOPOLOGICAL SIGMA-MODELS IN 4 DIMENSIONS AND TRIHOLOMORPHIC MAPS
ANSELMI, DAMIANO;
1994-01-01
Abstract
It is well known that topological sigma-models in two dimensions constitute a path-integral approach to the study of holomorphic maps from a Riemann surface SIGMA to an almost complex manifold K, the most interesting case being that were K is a Kahler manifold. We show that, in the same way, topological sigma-models in four dimensions introduce a path-integral approach to the study of triholomorphic maps q: M --> N between a four-dimensional riemannian manifold M and an almost quaternionic manifold N. The most interesting cases are those where M, N are hyper-Kahler or quaternionic Kahler. BRST-cohomology translates into intersection theory in the moduli-space of this new class of instantonic maps, that are named hyperinstantons by us. The definition of triholomorphicity 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 structure 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 topological sigma-model as a twisted version of the N = 2 globally supersymmetric sigma-model. We discuss various examples of hyperinstantons, in particular on the torus and the K3 surface. We also analyze the coupling of the topological sigma-model to topological gravity. The classification of triholomorphic maps and the analysis of their moduli-space is a new and fully open mathematical problem that we believe deserves the attention of both mathematicians and physicists.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
