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Curved flats in symmetric spaces
In this paper we study maps (curved flats) into symmetric spaces which are
tangent at each point to a flat of the symmetric space. Important examples of
such maps arise from isometric immersions of space forms into space forms via
their Gauss maps. Further examples are found in conformal geometry, e.g. the
curved flats obtained from isothermic surfaces and conformally flat 3-folds in
the 4-sphere. Curved flats admit a 1-parameter family of deformations (spectral
parameter) which enables us to make contact to integrable system theory. In
fact, we give a recipe to construct curved flats (and thus the above mentioned
geometric objects) from a hierarchy of finite dimensional algebraically
completely integrable flows.Comment: 9 pages, latex2e, no figures, also available at
http://www_sfb288.math.tu-berlin.de/preprints.htm
A Geometry for Multidimensional Integrable Systems
A deformed differential calculus is developed based on an associative
star-product. In two dimensions the Hamiltonian vector fields model the algebra
of pseudo-differential operator, as used in the theory of integrable systems.
Thus one obtains a geometric description of the operators. A dual theory is
also possible, based on a deformation of differential forms. This calculus is
applied to a number of multidimensional integrable systems, such as the KP
hierarchy, thus obtaining a geometrical description of these systems. The limit
in which the deformation disappears corresponds to taking the dispersionless
limit in these hierarchies.Comment: LaTeX, 29 pages. To be published in J.Geom.Phy
Axion Monodromy and the Weak Gravity Conjecture
Axions with broken discrete shift symmetry (axion monodromy) have recently
played a central role both in the discussion of inflation and the `relaxion'
approach to the hierarchy problem. We suggest a very minimalist way to
constrain such models by the weak gravity conjecture for domain walls: While
the electric side of the conjecture is always satisfied if the
cosine-oscillations of the axion potential are sufficiently small, the magnetic
side imposes a cutoff, , independent of the height
of these `wiggles'. We compare our approach with the recent related proposal by
Ibanez, Montero, Uranga and Valenzuela. We also discuss the non-trivial
question which version, if any, of the weak gravity conjecture for domain walls
should hold. In particular, we show that string compactifications with branes
of different dimensions wrapped on different cycles lead to a `geometric weak
gravity conjecture' relating volumes of cycles, norms of corresponding forms
and the volume of the compact space. Imposing this `geometric conjecture',
e.g.~on the basis of the more widely accepted weak gravity conjecture for
particles, provides at least some support for the (electric and magnetic)
conjecture for domain walls.Comment: 22 pages, 2 figures; v2: references added, typos corrected; v3:
published version + minor clarifications on self dual flux in Sec. 3.
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