Bubbling Calabi-Yau geometry from matrix models

We study bubbling geometry in topological string theory. Specifically, we analyse Chern-Simons theory on both the 3-sphere and lens spaces in the presence of a Wilson loop insertion of an arbitrary representation. For each of these three manifolds we formulate a multi-matrix model whose partition function is the vev of the Wilson loop and compute the spectral curve. This spectral curve is the reduction to two dimensions of the mirror to a Calabi-Yau threefold which is the gravitational dual of the Wilson loop insertion. For lens spaces the dual geometries are new. We comment on a similar matrix model which appears in the context of Wilson loops in AdS/CFT.Comment: 30 pages; v.2 reference added, minor correction

Insertion of Continuous Set-Valued Mappings

An interesting result about the existence of "intermediate" set-valued mappings between pairs of such mappings was obtained by Nepomnyashchii. His construction was for a paracompact domain, and he remarked that his result is similar to Dowker's insertion theorem and may represent a generalisation of this theorem. In the present paper, we characterise the $\tau$-paracompact normal spaces by this set-valued "insertion" property and for $\tau=\omega$, i.e. for countably paracompact normal spaces, we show that it is indeed equivalent to the mentioned Dowker's theorem. Moreover, we obtain a similar result for $\tau$-collectionwise normal spaces and show that for normal spaces, i.e. for $\omega$-collectionwise normal spaces, our result is equivalent to the Kat\v{e}tov-Tong insertion theorem. Several related results are obtained as well

A strengthening of the Katětov-Tong insertion theorem

summary:Normal spaces are characterized in terms of an insertion type theorem, which implies the Katětov-Tong theorem. The proof actually provides a simple necessary and sufficient condition for the insertion of an ordered pair of lower and upper semicontinuous functions between two comparable real-valued functions. As a consequence of the latter, we obtain a characterization of completely normal spaces by real-valued functions

Depths in random recursive metric spaces

As a generalization of random recursive trees and preferential attachment trees, we consider random recursive metric spaces. These spaces are constructed from random blocks, each a metric space equipped with a probability measure, containing a labelled point called a hook, and assigned a weight. Random recursive metric spaces are equipped with a probability measure made up of a weighted sum of the probability measures assigned to its constituent blocks. At each step in the growth of a random recursive metric space, a point called a latch is chosen at random according to the equipped probability measure and a new block is chosen at random and attached to the space by joining together the latch and the hook of the block. We prove a law of large numbers and a central limit theorem for the insertion depth, the distance from the master hook to the latch chosen. A classic argument proves that the insertion depth in random recursive trees is distributed as a sum of independent Bernoulli random variables. We generalize this argument by approximating the insertion depth in random recursive metric spaces with a sum of independent random variables.Comment: 18 page

The Dilaton Theorem and Closed String Backgrounds

The zero-momentum ghost-dilaton is a non-primary BRST physical state present in every bosonic closed string background. It is given by the action of the BRST operator on another state \x, but remains nontrivial in the semirelative BRST cohomology. When local coordinates arise from metrics we show that dilaton and \x insertions compute Riemannian curvature and geodesic curvature respectively. A proper definition of a CFT deformation induced by the dilaton requires surface integrals of the dilaton and line integrals of \x. Surprisingly, the ghost number anomaly makes this a trivial deformation. While dilatons cannot deform conformal theories, they actually deform conformal string backgrounds, showing in a simple context that a string background is not necessarily the same as a CFT. We generalize the earlier proof of quantum background independence of string theory to show that a dilaton shift amounts to a shift of the string coupling in the field-dependent part of the quantum string action. Thus the ``dilaton theorem'', familiar for on-shell string amplitudes, holds off-shell as a consequence of an exact symmetry of the string action.Comment: 51 pages, plain tex with phyzzx, two uuencoded figure