31 research outputs found
Generalised shear coordinates on the moduli spaces of three-dimensional spacetimes
We introduce coordinates on the moduli spaces of maximal globally hyperbolic
constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are
derived from the parametrisation of the moduli spaces by the bundle of measured
geodesic laminations over Teichm\"uller space of S and can be viewed as
analytic continuations of the shear coordinates on Teichm\"uller space. In
terms of these coordinates the gravitational symplectic structure takes a
particularly simple form, which resembles the Weil-Petersson symplectic
structure in shear coordinates, and is closely related to the cotangent bundle
of Teichm\"uller space. We then consider the mapping class group action on the
moduli spaces and show that it preserves the gravitational symplectic
structure. This defines three distinct mapping class group actions on the
cotangent bundle of Teichm\"uller space, corresponding to different values of
the curvature.Comment: 40 pages, 6 figure
Weighted Laplacians, cocycles and recursion relations
Hodge's formula represents the gravitational MHV amplitude as the determinant
of a minor of a certain matrix. When expanded, this determinant becomes a sum
over weighted trees, which is the form of the MHV formula first obtained by
Bern, Dixon, Perelstein, Rozowsky and rediscovered by Nguyen, Spradlin,
Volovich and Wen. The gravity MHV amplitude satisfies the Britto, Cachazo, Feng
and Witten recursion relation. The main building block of the MHV amplitude,
the so-called half-soft function, satisfies a different, Berends-Giele-type
recursion relation. We show that all these facts are illustrations to a more
general story.
We consider a weighted Laplacian for a complete graph of n vertices. The
matrix tree theorem states that its diagonal minor determinants are all equal
and given by a sum over spanning trees. We show that, for any choice of a
cocycle on the graph, the minor determinants satisfy a Berends-Giele as well as
Britto-Cachazo-Feng-Witten type recursion relation. Our proofs are purely
combinatorial.Comment: 12 pages, some figures embedded in the tex
Pure connection formalism for gravity: Recursion relations
In the gauge-theoretic formulation of gravity the cubic vertex becomes simple
enough for some graviton scattering amplitudes to be computed using
Berends-Giele-type recursion relations. We present such a computation for the
current with all same helicity on-shell gravitons. Once the recursion relation
is set up and low graviton number cases are worked out, a natural guess for the
solution in terms of a sum over trees presents itself readily. The solution can
also be described either in terms of the half-soft function familiar from the
1998 paper by Bern, Dixon, Perelstein and Rozowsky or as a matrix determinant
similar to one used by Hodges for MHV graviton amplitudes. This solution also
immediate suggests the correct guess for the MHV graviton amplitude formula, as
is contained in the already mentioned 1998 paper. We also obtain the recursion
relation for the off-shell current with all but one same helicity gravitons.Comment: 13 pages, no figure
Pure connection formalism for gravity: Feynman rules and the graviton-graviton scattering
We continue to develop the pure connection formalism for gravity. We derive
the Feynman rules for computing the connection correlation functions, as well
as the prescription for obtaining the Minkowski space graviton scattering
amplitudes from the latter. The present formalism turns out to be simpler than
the metric based one in many aspects. Simplifications result from the fact that
the conformal factor of the metric, a source of complications in the usual
approach, does not propagate in the connection formulation even off-shell. This
simplifies both the linearized theory and the interactions. For comparison, in
our approach the complete off-shell cubic GR interaction contains just 3 terms,
which should be compared to at least a dozen terms in the metric formalism. We
put the technology developed to use and compute the simplest graviton-graviton
scattering amplitudes. For GR we reproduce the well-known result. For our
other, distinct from GR, interacting theories of massless spin 2 particles we
obtain non-zero answers for some parity-violating amplitudes. Thus, in the
convention that all particles are incoming, we find that the 4 minus, as well
as the 3 minus 1 plus amplitudes are zero (as in GR), but the amplitudes with 4
gravitons of positive helicity, as well as the 3 plus 1 minus amplitudes are
different from zero. This serves as a good illustration of the type of parity
violation present in these theories. We find that the parity-violating
amplitudes are important at high energies, and that a general parity-violating
member of our class of theories "likes" one helicity (negative in our
conventions) more than the other in the sense that at high energies it tends to
convert all present gravitons into those of negative helicity.Comment: v2: 57 pages, figures, a missing contribution to the all plus
amplitude added, discussion improve
A 4D gravity theory and G2-holonomy manifolds
Bryant and Salamon gave a construction of metrics of G2 holonomy on the total
space of the bundle of anti-self-dual (ASD) 2-forms over a 4-dimensional
self-dual Einstein manifold. We generalise it by considering the total space of
an SO(3) bundle (with fibers R^3) over a 4-dimensional base, with a connection
on this bundle. We make essentially the same ansatz for the calibrating 3-form,
but use the curvature 2-forms instead of the ASD ones. We show that the
resulting 3-form defines a metric of G2 holonomy if the connection satisfies a
certain second-order PDE. This is exactly the same PDE that arises as the field
equation of a certain 4-dimensional gravity theory formulated as a
diffeomorphism-invariant theory of SO(3) connections. Thus, every solution of
this 4-dimensional gravity theory can be lifted to a G2-holonomy metric. Unlike
all previously known constructions, the theory that we lift to 7 dimensions is
not topological. Thus, our construction should give rise to many new metrics of
G2 holonomy. We describe several examples that are of cohomogeneity one on the
base.Comment: 25 page
Lightlike and ideal tetrahedra
We give a unified description of tetrahedra with lightlike faces in 3d
anti-de Sitter, de Sitter and Minkowski spaces and of their duals in 3d anti-de
Sitter, hyperbolic and half-pipe spaces. We show that both types of tetrahedra
are determined by a generalized cross-ratio with values in a commutative 2d
real algebra that generalizes the complex numbers. Equivalently, tetrahedra
with lightlike faces are determined by a pair of edge lengths and their duals
by a pair of dihedral angles. We prove that the dual tetrahedra are precisely
the generalized ideal tetrahedra introduced by Danciger. Finally, we compute
the volumes of both types of tetrahedra as functions of their edge lengths or
dihedral angles, obtaining generalizations of the Milnor-Lobachevsky volume
formula of ideal hyperbolic tetrahedra.Comment: 39 pages, 5 figures. Revised versio
Symplectic Wick rotations between moduli spaces of 3-manifolds
Given a closed hyperbolic surface , let \cQF denote the space of
quasifuchsian hyperbolic metrics on and \cGH_{-1} the space of
maximal globally hyperbolic anti-de Sitter metrics on . We describe
natural maps between (parts of) \cQF and \cGH_{-1}, called "Wick
rotations", defined in terms of special surfaces (e.g. minimal/maximal
surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at
least smooth and symplectic with respect to the canonical symplectic
structures on both \cQF and \cGH_{-1}. Similar results involving the spaces
of globally hyperbolic de Sitter and Minkowski metrics are also described.
These 3-dimensional results are shown to be equivalent to purely
2-dimensional ones. Namely, consider the double harmonic map
\cH:T^*\cT\to\cTT, sending a conformal structure and a holomorphic
quadratic differential on to the pair of hyperbolic metrics
such that the harmonic maps isotopic to the identity from to
and to have, respectively, Hopf differentials equal to and , and the double earthquake map \cE:\cT\times\cML\to\cTT, sending a
hyperbolic metric and a measured lamination on to the pair
, where and denote the left and right
earthquakes. We describe how such 2-dimensional double maps are related to
3-dimensional Wick rotations and prove that they are also smooth and
symplectic.Comment: 36 pages, 5 figure