6 research outputs found
Solutions of the Einstein-Dirac and Seiberg-Witten Monopole Equations
We present unique solutions of the Seiberg-Witten Monopole Equations in which
the U(1) curvature is covariantly constant, the monopole Weyl spinor consists
of a single constant component, and the 4-manifold is a product of two Riemann
surfaces of genuses p_1 and p_2. There are p_1 -1 magnetic vortices on one
surface and p_2 - 1 electric ones on the other, with p_1 + p_2 \geq 2 p_1 =
p_2= 1 being excluded). When p_1 = p_2, the electromagnetic fields are
self-dual and one also has a solution of the coupled euclidean
Einstein-Maxwell-Dirac equations, with the monopole condensate serving as
cosmological constant. The metric is decomposable and the electromagnetic
fields are covariantly constant as in the Bertotti-Robinson solution. The
Einstein metric can also be derived from a K\"{a}hler potential satisfying the
Monge-Amp\`{e}re equations.Comment: 22 pages. Rep. no: FGI-99-
Generalization of Weierstrassian Elliptic Functions to
The Weierstrassian and functions are generalized to
. The and cases have already been used in
gravitational and Yang-Mills instanton solutions which may be interpreted as
explicit realizations of spacetime foam and the monopole condensate,
respectively. The new functions satisfy higher dimensional versions of the
periodicity properties and Legendre's relations obeyed by their familiar
complex counterparts. For , the construction reproduces functions found
earlier by Fueter using quaternionic methods. Integrating over lattice points
along all directions but two, one recovers the original Weierstrassian elliptic
functions.Comment: pp. 9, Late
Fake R^4's, Einstein Spaces and Seiberg-Witten Monopole Equations
We discuss the possible relevance of some recent mathematical results and
techniques on four-manifolds to physics. We first suggest that the existence of
uncountably many R^4's with non-equivalent smooth structures, a mathematical
phenomenon unique to four dimensions, may be responsible for the observed
four-dimensionality of spacetime. We then point out the remarkable fact that
self-dual gauge fields and Weyl spinors can live on a manifold of Euclidean
signature without affecting the metric. As a specific example, we consider
solutions of the Seiberg-Witten Monopole Equations in which the U(1) fields are
covariantly constant, the monopole Weyl spinor has only a single constant
component, and the 4-manifold M_4 is a product of two Riemann surfaces
Sigma_{p_1} and Sigma_{p_2}. There are p_{1}-1(p_{2}-1) magnetic(electric)
vortices on \Sigma_{p_1}(\Sigma_{p_2}), with p_1 + p_2 \geq 2 (p_1=p_2= 1 being
excluded). When the two genuses are equal, the electromagnetic fields are
self-dual and one obtains the Einstein space \Sigma_p x \Sigma_p, the monopole
condensate serving as the cosmological constant.Comment: 9 pages, Talk at the Second Gursey Memorial Conference, June 2000,
Istanbu