9,964 research outputs found
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
Self-improvement of the Bakry-\'Emery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
We prove that the linear heat flow in a RCD(K,\infty) metric measure space
(X,d,m) satisfies a contraction property with respect to every
L^p-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a
precise estimate for the optimal W_\infty-coupling between two fundamental
solutions in terms of the distance of the initial points.
The result is a consequence of the equivalence between the RCD(K,\infty)
lower Ricci bound and the corresponding Bakry-\'Emery condition for the
canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension
to the non-smooth metric measure setting of the Bakry's argument, that allows
to improve the commutation estimates between the Markov semigroup and the
Carr\'e du Champ associated to the Dirichlet form. This extension is based on a
new a priori estimate and a capacitary argument for regular and tight Dirichlet
forms that are of independent interest.Comment: (v2) Minor corrections. A discussion of quasi-regular Dirichlet forms
has been added (Section 2.3) to cover the case of a sigma-finite reference
measure. The proof of the quasi regularity of the Cheeger energy has been
added (Thm. 4.1
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