12,152 research outputs found
On the fields generated by the lengths of closed geodesics in locally symmetric spaces
This paper is the next installment of our analysis of length-commensurable
locally symmetric spaces begun in Publ. math. IHES 109(2009), 113-184. For a
Riemannian manifold , we let be the weak length spectrum of , i.e.
the set of lengths of all closed geodesics in , and let
denote the subfield of generated by . Let now be an
arithmetically defined locally symmetric space associated with a simple
algebraic -group for . Assuming Schanuel's
conjecture from transcendental number theory, we prove (under some minor
technical restrictions) the following dichotomy: either and are
length-commensurable, i.e. ,
or the compositum has infinite transcendence
degree over for at least one or (which means
that the sets and are very different)
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
A finite model of two-dimensional ideal hydrodynamics
A finite-dimensional su() Lie algebra equation is discussed that in the
infinite limit (giving the area preserving diffeomorphism group) tends to
the two-dimensional, inviscid vorticity equation on the torus. The equation is
numerically integrated, for various values of , and the time evolution of an
(interpolated) stream function is compared with that obtained from a simple
mode truncation of the continuum equation. The time averaged vorticity moments
and correlation functions are compared with canonical ensemble averages.Comment: (25 p., 7 figures, not included. MUTP/92/1
Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
In three spacetime dimensions, general relativity drastically simplifies,
becoming a ``topological'' theory with no propagating local degrees of freedom.
Nevertheless, many of the difficult conceptual problems of quantizing gravity
are still present. In this review, I summarize the rather large body of work
that has gone towards quantizing (2+1)-dimensional vacuum gravity in the
setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms,
additions, missing references welcome; v2: minor changes, added reference
Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
In topologically ordered quantum states of matter in 2+1D (space-time
dimensions), the braiding statistics of anyonic quasiparticle excitations is a
fundamental characterizing property which is directly related to global
transformations of the ground-state wavefunctions on a torus (the modular
transformations). On the other hand, there are theoretical descriptions of
various topologically ordered states in 3+1D, which exhibit both point-like and
loop-like excitations, but systematic understanding of the fundamental physical
distinctions between phases, and how these distinctions are connected to
quantum statistics of excitations, is still lacking. One main result of this
work is that the three-dimensional generalization of modular transformations,
when applied to topologically ordered ground states, is directly related to a
certain braiding process of loop-like excitations. This specific braiding
surprisingly involves three loops simultaneously, and can distinguish different
topologically ordered states. Our second main result is the identification of
the three-loop braiding as a process in which the worldsheets of the three
loops have a non-trivial triple linking number, which is a topological
invariant characterizing closed two-dimensional surfaces in four dimensions. In
this work we consider realizations of topological order in 3+1D using
cohomological gauge theory in which the loops have Abelian statistics, and
explicitly demonstrate our results on examples with topological
order
Discrete Minimal Surface Algebras
We consider discrete minimal surface algebras (DMSA) as generalized
noncommutative analogues of minimal surfaces in higher dimensional spheres.
These algebras appear naturally in membrane theory, where sequences of their
representations are used as a regularization. After showing that the defining
relations of the algebra are consistent, and that one can compute a basis of
the enveloping algebra, we give several explicit examples of DMSAs in terms of
subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by
itself). A special class of DMSAs are Yang-Mills algebras. The representation
graph is introduced to study representations of DMSAs of dimension d<=4, and
properties of representations are related to properties of graphs. The
representation graph of a tensor product is (generically) the Cartesian product
of the corresponding graphs. We provide explicit examples of irreducible
representations and, for coinciding eigenvalues, classify all the unitary
representations of the corresponding algebras
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