Communications in Mathematical Physics

, Volume 194, Issue 1, pp 149-175

Special Quantum Field Theories¶in Eight and Other Dimensions

  • Laurent BaulieuAffiliated withLPTHE, Universitées Paris VI – Paris VII, URA 280 CNRS, 4 place Jussieu, F-75252 Paris Cedex 05, France. E-mail:
  • , Hiroaki KannoAffiliated withDepartment of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima 739, Japan. E-mail:
  • , I. M. SingerAffiliated withDepartment of Mathematics, MIT, Cambridge, MA 02139, USA. E-mail:

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We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight 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.