5,307 research outputs found
On membrane interactions and a three-dimensional analog of Riemann surfaces
Membranes in M-theory are expected to interact via splitting and joining
processes. We study these effects in the pp-wave matrix model, in which they
are associated with transitions between states in sectors built on vacua with
different numbers of membranes. Transition amplitudes between such states
receive contributions from BPS instanton configurations interpolating between
the different vacua. Various properties of the moduli space of BPS instantons
are known, but there are very few known examples of explicit solutions. We
present a new approach to the construction of instanton solutions interpolating
between states containing arbitrary numbers of membranes, based on a continuum
approximation valid for matrices of large size. The proposed scheme uses
functions on a two-dimensional space to approximate matrices and it relies on
the same ideas behind the matrix regularisation of membrane degrees of freedom
in M-theory. We show that the BPS instanton equations have a continuum
counterpart which can be mapped to the three-dimensional Laplace equation
through a sequence of changes of variables. A description of configurations
corresponding to membrane splitting/joining processes can be given in terms of
solutions to the Laplace equation in a three-dimensional analog of a Riemann
surface, consisting of multiple copies of R^3 connected via a generalisation of
branch cuts. We discuss various general features of our proposal and we also
present explicit analytic solutions.Comment: 64 pages, 17 figures. V2: An appendix, a figure and references added;
various minor changes and improvement
Correlation functions of boundary field theory from bulk Green's functions and phases in the boundary theory
In the context of the bulk-boundary correspondence we study the correlation
functions arising on a boundary for different types of boundary conditions. The
most general condition is the mixed one interpolating between the Neumann and
Dirichlet conditions. We obtain the general expressions for the correlators on
a boundary in terms of Green's function in the bulk for the Dirichlet, Neumann
and mixed boundary conditions and establish the relations between the
correlation functions. As an instructive example we explicitly obtain the
boundary correlators corresponding to the mixed condition on a plane boundary
of a domain in flat space . The phases of the boundary theory
with correlators of the Neumann and Dirichlet types are determined. The
boundary correlation functions on sphere are calculated for the Dirichlet
and Neumann conditions in two important cases: when sphere is a boundary of a
domain in flat space and when it is a boundary at infinity of Anti-De
Sitter space . For massless in the bulk theory the Neumann
correlator on the boundary of AdS space is shown to have universal logarithmic
behavior in all AdS spaces. In the massive case it is found to be finite at the
coinciding points. We argue that the Neumann correlator may have a dual
two-dimensional description. The structure of the correlators obtained, their
conformal nature and some recurrent relations are analyzed. We identify the
Dirichlet and Neumann phases living on the boundary of AdS space and discuss
their evolution when the location of the boundary changes from infinity to the
center of the AdS space.Comment: 32 pages, latex, no figure
Solving Open String Field Theory with Special Projectors
Schnabl recently found an analytic expression for the string field tachyon
condensate using a gauge condition adapted to the conformal frame of the sliver
projector. We propose that this construction is more general. The sliver is an
example of a special projector, a projector such that the Virasoro operator
\L_0 and its BPZ adjoint \L*_0 obey the algebra [\L_0, \L*_0] = s (\L_0 +
\L*_0), with s a positive real constant. All special projectors provide abelian
subalgebras of string fields, closed under both the *-product and the action of
\L_0. This structure guarantees exact solvability of a ghost number zero string
field equation. We recast this infinite recursive set of equations as an
ordinary differential equation that is easily solved. The classification of
special projectors is reduced to a version of the Riemann-Hilbert problem, with
piecewise constant data on the boundary of a disk.Comment: 64 pages, 6 figure
Loop Gas Model for Open Strings
The open string with one-dimensional target space is formulated in terms of
an SOS, or loop gas, model on a random surface. We solve an integral equation
for the loop amplitude with Dirichlet and Neumann boundary conditions imposed
on different pieces of its boundary. The result is used to calculate the mean
values of order and disorder operators, to construct the string propagator and
find its spectrum of excitations. The latter is not sensible neither to the
string tension \L nor to the mass of the ``quarks'' at the ends of the
string. As in the case of closed strings, the SOS formulation allows to
construct a Feynman diagram technique for the string interaction amplitudes
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
A Polyakov formula for sectors
We consider finite area convex Euclidean circular sectors. We prove a
variational Polyakov formula which shows how the zeta-regularized determinant
of the Laplacian varies with respect to the opening angle. Varying the angle
corresponds to a conformal deformation in the direction of a conformal factor
with a logarithmic singularity at the origin. We compute explicitly all the
contributions to this formula coming from the different parts of the sector. In
the process, we obtain an explicit expression for the heat kernel on an
infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute
the zeta-regularized determinant of rectangular domains of unit area and prove
that it is uniquely maximized by the square.Comment: 51 pages, 2 figures. Major modification of Lemma 4, it was revised
and corrected. Other small misprints were corrected. Accepted for publication
in The Journal of Geometric Analysi
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