5,907 research outputs found
The augmented marking complex of a surface
We build an augmentation of the Masur-Minsky marking complex by
Groves-Manning combinatorial horoballs to obtain a graph we call the augmented
marking complex, . Adapting work of Masur-Minsky, we prove
that is quasiisometric to Teichm\"uller space with the
Teichm\"uller metric. A similar construction was independently discovered by
Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy
machinery to to build flexible families of uniform
quasigeodesics in Teichm\"uller space. As an application, we give a new proof
of Rafi's distance formula for the Teichm\"uller metric.Comment: 30 pages; significantly rewritten to strengthen main construction
Shadows of the Planck Scale: The Changing Face of Compactification Geometry
By studying the effects of the shape moduli associated with toroidal
compactifications, we demonstrate that Planck-sized extra dimensions can cast
significant ``shadows'' over low-energy physics. These shadows can greatly
distort our perceptions of the compactification geometry associated with large
extra dimensions, and place a fundamental limit on our ability to probe the
geometry of compactification simply by measuring Kaluza-Klein states. We also
discuss the interpretation of compactification radii and hierarchies in the
context of geometries with non-trivial shape moduli. One of the main results of
this paper is that compactification geometry is effectively renormalized as a
function of energy scale, with ``renormalization group equations'' describing
the ``flow'' of geometric parameters such as compactification radii and shape
angles as functions of energy.Comment: 7 pages, LaTeX, 2 figure
A model for emergence of space and time
We study string field theory (third quantization) of the two-dimensional
model of quantum geometry called generalized CDT ("causal dynamical
triangulations"). Like in standard non-critical string theory the so-called
string field Hamiltonian of generalized CDT can be associated with W-algebra
generators through the string mode expansion. This allows us to define an
"absolute" vacuum. "Physical" vacua appear as coherent states created by vertex
operators acting on the absolute vacuum. Each coherent state corresponds to
specific values of the coupling constants of generalized CDT. The cosmological
"time" only exists relatively to a given "physical" vacuum and comes into
existence before space, which is created because the "physical" vacuum is
unstable. Thus each CDT "universe" is created as a "Big Bang" from the absolute
vacuum, its time evolution is governed by the CDT string field Hamiltonian with
given coupling constants, and one can imagine interactions between CDT
universes with different coupling constants ("fourth quantization"
Integrable hierarchies and the mirror model of local CP1
We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light
of its realization as a two-component reduction of the two-dimensional Toda
hierarchy, and establish new results on its connection to the Gromov-Witten
theory of local CP1. We first of all elaborate on the relation to the Toeplitz
lattice and obtain a neat description of the Lax formulation of the AL system.
We then study the dispersionless limit and rephrase it in terms of a conformal
semisimple Frobenius manifold with non-constant unit, whose properties we
thoroughly analyze. We build on this connection along two main strands. First
of all, we exhibit a manifestly local bi-Hamiltonian structure of the
Ablowitz-Ladik system in the zero-dispersion limit. Secondarily, we make
precise the relation between this canonical Frobenius structure and the one
that underlies the Gromov-Witten theory of the resolved conifold in the
equivariantly Calabi-Yau case; a key role is played by Dubrovin's notion of
"almost duality" of Frobenius manifolds. As a consequence, we obtain a
derivation of genus zero mirror symmetry for local CP1 in terms of a dual
logarithmic Landau-Ginzburg model.Comment: 27 pages, 1 figur
The local Gromov-Witten theory of CP^1 and integrable hierarchies
In this paper we begin the study of the relationship between the local
Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line
and the theory of integrable hierarchies. We first of all construct explicitly,
in a large number of cases, the Hamiltonian dispersionless hierarchies that
govern the full descendent genus zero theory. Our main tool is the application
of Dubrovin's formalism, based on associativity equations, to the known results
on the genus zero theory from local mirror symmetry and localization. The
hierarchies we find are apparently new, with the exception of the resolved
conifold O(-1) + O(-1) -> P1 in the equivariantly Calabi-Yau case. For this
example the relevant dispersionless system turns out to be related to the
long-wave limit of the Ablowitz-Ladik lattice. This identification provides us
with a complete procedure to reconstruct the dispersive hierarchy which should
conjecturally be related to the higher genus theory of the resolved conifold.
We give a complete proof of this conjecture for genus g<=1; our methods are
based on establishing, analogously to the case of KdV, a "quasi-triviality"
property for the Ablowitz-Ladik hierarchy at the leading order of the
dispersive expansion. We furthermore provide compelling evidence in favour of
the resolved conifold/Ablowitz-Ladik correspondence at higher genus by testing
it successfully in the primary sector for g=2.Comment: 30 pages; v2: an issue involving constant maps contributions is
pointed out in Sec. 3.3-3.4 and is now taken into account in the proofs of
Thm 1.3-1.4, whose statements are unchanged. Several typos, formulae,
notational inconsistencies have been fixed. v3: typos fixed, minor textual
changes, version to appear on Comm. Math. Phy
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