13,228 research outputs found
Cohomological Yang-Mills Theory in Eight Dimensions
We construct nearly topological Yang-Mills theories on eight dimensional
manifolds with a special holonomy group. These manifolds are the Joyce manifold
with holonomy and the Calabi-Yau manifold with SU(4) holonomy. An
invariant closed four form on the manifold allows us to
define an analogue of the instanton equation, which serves as a topological
gauge fixing condition in BRST formalism. The model on the Joyce manifold is
related to the eight dimensional supersymmetric Yang-Mills theory. Topological
dimensional reduction to four dimensions gives non-abelian Seiberg-Witten
equation.Comment: 9 pages, latex, Talk given at APCTP Winter School on Dualities in
String Theory, (Sokcho, Korea), February 24-28, 199
Special Quantum Field Theories In Eight And Other Dimensions
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.Comment: 36 pages, latex. References have been added together with a not
Fractional analytic index
For a finite rank projective bundle over a compact manifold, so associated to
a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of
differential operators `acting on sections of the projective bundle' in a
formal sense. In particular, any oriented even-dimensional manifold carries a
projective spin Dirac operator in this sense. More generally the corresponding
space of pseudodifferential operators is defined, with supports sufficiently
close to the diagonal, i.e. the identity relation. For such elliptic operators
we define the numerical index in an essentially analytic way, as the trace of
the commutator of the operator and a parametrix and show that this is homotopy
invariant. Using the heat kernel method for the twisted, projective spin Dirac
operator, we show that this index is given by the usual formula, now in terms
of the twisted Chern character of the symbol, which in this case defines an
element of K-theory twisted by w; hence the index is a rational number but in
general it is not an integer.Comment: 23 pages, Latex2e, final version, to appear in JD
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