21,437 research outputs found
Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; additionally, it can be proven that sequences of
approximately Einstein manifolds and sequences of approximately Einstein
conifolds both converge to Einstein conifolds. Consequently, generalized
Euclidean functional integrals based on these conifold histories yield
semiclassical amplitudes for sequences of both manifold and conifold histories
that approach a stationary point of the Einstein action. Therefore sums over
conifold histories provide a useful and self-consistent starting point for
further study of topological effects in quantum gravity. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.
Geometric Cobordism Categories
In this paper we study cobordism categories consisting of manifolds which are
endowed with geometric structure. Examples of such geometric structures include
symplectic structures, flat connections on principal bundles, and complex
structures along with a holomorphic map to a target complex manifold. A general
notion of "geometric structure" is defined using sheaf theoretic constructions.
Our main theorem is the identification of the homotopy type of such cobordism
categories in terms of certain Thom spectra. This extends work of
Galatius-Madsen-Tillmann-Weiss who identify the homotopy type of cobordism
categories of manifolds with fiberwise structures on their tangent bundles.
Interpretations of the main theorem are discussed which have relevance to
topological field theories, moduli spaces of geometric structures, and
h-principles. Applications of the main theorem to various examples of interest
in geometry, particularly holomorphic curves, are elaborated upon.Comment: 82 pages. Second version with remarks on higher category approaches
and various minor correction
The homotopy type of the topological cobordism category
We define a cobordism category of topological manifolds and prove that if its classifying space is weakly equivalent to , where is the Thom spectrum of the inverse of the
canonical bundle over . We also give versions with tangential
structures and boundary. The proof uses smoothing theory and excision in the
tangential structure to reduce the statement to the computation of the homotopy
type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.Comment: 61 pages, 9 figures. Minor correction
Generalized Sums over Histories for Quantum Gravity II. Simplicial Conifolds
This paper examines the issues involved with concretely implementing a sum
over conifolds in the formulation of Euclidean sums over histories for gravity.
The first step in precisely formulating any sum over topological spaces is that
one must have an algorithmically implementable method of generating a list of
all spaces in the set to be summed over. This requirement causes well known
problems in the formulation of sums over manifolds in four or more dimensions;
there is no algorithmic method of determining whether or not a topological
space is an n-manifold in five or more dimensions and the issue of whether or
not such an algorithm exists is open in four. However, as this paper shows,
conifolds are algorithmically decidable in four dimensions. Thus the set of
4-conifolds provides a starting point for a concrete implementation of
Euclidean sums over histories in four dimensions. Explicit algorithms for
summing over various sets of 4-conifolds are presented in the context of Regge
calculus. Postscript figures available via anonymous ftp at
black-hole.physics.ubc.ca (137.82.43.40) in file gen2.ps.Comment: 82pp., plain TeX, To appear in Nucl. Phys. B,FF-92-
Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds
We study Lie group structures on groups of the form C^\infty(M,K)}, where M
is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie
group. First we prove that there is at most one Lie group structure with Lie
algebra C^\infty(M,k) for which the evaluation map is smooth. We then prove the
existence of such a structure if the universal cover of K is diffeomorphic to a
locally convex space and if the image of the left logarithmic derivative in
\Omega^1(M,k) is a smooth submanifold, the latter being the case in particular
if M is one-dimensional. We also obtain analogs of these results for the group
O(M,K) of holomorphic maps on a complex manifold with values in a complex Lie
group. We show that there exists a natural Lie group structure on O(M,K) if K
is Banach and M is a non-compact complex curve with finitely generated
fundamental group.Comment: 39 page
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