Special Quantum Field Theories¶in Eight and Other Dimensions

Abstract

We build nearly topological quantum field theories in various dimensions. We give special attention to the case of 8 dimensions for which we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU (4) and Spin(7). The choice of SU (4) gives a quantum field theory for a Calabi-Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M -and F -theories. Relations to the eight dimensional supersymmetric Yang-Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg-Witten equations. The latter are thus related to pure Yang-Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang-Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions. *

Topological Gravity versus Supergravity on Manifolds with Special Holonomy

“…This is analogous to what happens for the topological Yang-Mills theory in eight dimensions, for which the manifold must have Spin (7) or SU(4) holonomy [3].…”

“…A promising case is in eight dimensions, for which [3] suggests that one can use the octonionic gravitational instanton equation [17] as a generalization of Eq. (6),…”

“…This has been shown to be true in two dimensions [6], and Anselmi and Frè have given good arguments that the four-dimensional N = 2 supergravity can be twisted in a topological gravity [7]. There is moreover a quite general argument that makes us confident that one can reach 1 By dimensional reduction to four dimensions and a suitable gauge-fixing in the Cartan algebra allowed by topological invariance, this theory gives the abelian monopole theory of Seiberg and Witten [5], which describes the infra-red behavior of the non-abelian Super Yang-Mills theory with extended supersymmetry [3].…”

Yang-Mills instantons and dyons on homogeneous G 2-manifolds

“…Interpreted as a static configuration on S 1 × G/H, the energy of the sphaleron is (6.5) and e.g. for the case of κ = −3 in (6.3) we obtain 6) where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively [29].…”

The octonionic membrane

“…The octonionic structure constants have appeared before in many different physical contexts [14,7,15,8,9,16,18,17,19,20,21,25]. In particular, octonionic string soliton [18] and octonionic membrane soliton [20,21] solutions of the heterotic string have been found which make use of the SO(7) octonionic instanton [7,8,9] in eight dimensions and the G 2 octonionic instanton [20,21] in seven dimensions respectively.…”