84 research outputs found
Higher Genus Correlators for the Complex Matrix Model
We describe an iterative scheme which allows us to calculate any multi-loop
correlator for the complex matrix model to any genus using only the first in
the chain of loop equations. The method works for a completely general
potential and the results contain no explicit reference to the couplings. The
genus contribution to the --loop correlator depends on a finite number
of parameters, namely at most . We find the generating functional
explicitly up to genus three. We show as well that the model is equivalent to
an external field problem for the complex matrix model with a logarithmic
potential.Comment: 17 page
Matrix Model Calculations beyond the Spherical Limit
We propose an improved iterative scheme for calculating higher genus
contributions to the multi-loop (or multi-point) correlators and the partition
function of the hermitian one matrix model. We present explicit results up to
genus two. We develop a version which gives directly the result in the double
scaling limit and present explicit results up to genus four. Using the latter
version we prove that the hermitian and the complex matrix model are equivalent
in the double scaling limit and that in this limit they are both equivalent to
the Kontsevich model. We discuss how our results away from the double scaling
limit are related to the structure of moduli space.Comment: 44 page
4D Quantum Gravity Coupled to Matter
We investigate the phase structure of four-dimensional quantum gravity
coupled to Ising spins or Gaussian scalar fields by means of numerical
simulations.
The quantum gravity part is modelled by the summation over random simplicial
manifolds, and the matter fields are located in the center of the 4-simplices,
which constitute the building blocks of the manifolds. We find that the
coupling between spin and geometry is weak away from the critical point of the
Ising model. At the critical point there is clear coupling, which qualitatively
agrees with that of gaussian fields coupled to gravity. In the case of pure
gravity a transition between a phase with highly connected geometry and a phase
with very ``dilute'' geometry has been observed earlier. The nature of this
transition seems unaltered when matter fields are included.
It was the hope that continuum physics could be extracted at the transition
between the two types of geometries. The coupling to matter fields, at least in
the form discussed in this paper, seems not to improve the scaling of the
curvature at the transition point.Comment: 15 pages, 9 figures (available as PS-files by request). Late
The Two-point Function of c=-2 Matter Coupled to 2D Quantum Gravity
We construct a reparametrization invariant two-point function for c=-2 conformal matter coupled to two-dimensional quantum gravity. From the two-point function we extract the critical indices \nu and \eta. The results support the quantum gravity version of Fisher's scaling relation. Our approach is based on the transfer matrix formalism and exploits the formulation of the c=-2 string as an O(n) model on a random lattice
Integrable 2D Lorentzian Gravity and Random Walks
We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-to-one correspondence between Lorentzian triangulations and directed Random Walks. This gives a simple explanation why the Lorentzian triangulations have fractal dimension 2 and why the curvature model lies in the universality class of pure Lorentzian gravity. We also study integrable generalizations of the curvature model with arbitrary polygonal tiles. All of them are found to lie in the same universality class
Thermodynamics of the quantum Landau-Lifshitz model
We present thermodynamics of the quantum su(1,1) Landau-Lifshitz model,
following our earlier exposition [J. Math. Phys. 50, 103518 (2009)] of the
quantum integrability of the theory, which is based on construction of
self-adjoint extensions, leading to a regularized quantum Hamiltonian for an
arbitrary n-particle sector. Starting from general discontinuity properties of
the functions used to construct the self-adjoint extensions, we derive the
thermodynamic Bethe Ansatz equations. We show that due to non-symmetric and
singular kernel, the self-consistency implies that only negative chemical
potential values are allowed, which leads to the conclusion that, unlike its
su(2) counterpart, the su(1,1) LL theory at T=0 has no instabilities.Comment: 10 page
Generalized Penner models to all genera
We give a complete description of the genus expansion of the one-cut solution
to the generalized Penner model. The solution is presented in a form which
allows us in a very straightforward manner to localize critical points and to
investigate the scaling behaviour of the model in the vicinity of these points.
We carry out an analysis of the critical behaviour to all genera addressing all
types of multi-critical points. In certain regions of the coupling constant
space the model must be defined via analytical continuation. We show in detail
how this works for the Penner model. Using analytical continuation it is
possible to reach the fermionic 1-matrix model. We show that the critical
points of the fermionic 1-matrix model can be indexed by an integer, , as it
was the case for the ordinary hermitian 1-matrix model. Furthermore the 'th
multi-critical fermionic model has to all genera the same value of
as the 'th multi-critical hermitian model. However, the
coefficients of the topological expansion need not be the same in the two
cases. We show explicitly how it is possible with a fermionic matrix model to
reach a multi-critical point for which the topological expansion has
alternating signs, but otherwise coincides with the usual Painlev\'{e}
expansion.Comment: 27 pages, PostScrip
New Penrose Limits and AdS/CFT
We find a new Penrose limit of AdS_5 x S^5 giving the maximally
supersymmetric pp-wave background with two explicit space-like isometries. This
is an important missing piece in studying the AdS/CFT correspondence in certain
subsectors. In particular whereas the Penrose limit giving one space-like
isometry is useful for the SU(2) sector of N=4 SYM, this new Penrose limit is
instead useful for studying the SU(2|3) and SU(1,2|3) sectors. In addition to
the new Penrose limit of AdS_5 x S^5 we also find a new Penrose limit of AdS_4
x CP^3.Comment: 30 page
The Morphology of N=6 Chern-Simons Theory
We tabulate various properties of the language of N=6 Chern-Simons Theory, in
the sense of Polyakov. Specifically we enumerate and compute character formulas
for all syllables of up to four letters, i.e. all irreducible representations
of OSp(6|4) built from up to four fundamental fields of the ABJM theory. We
also present all tensor product decompositions for up to four singletons and
list the (cyclically invariant) four-letter words, which correspond to
single-trace operators of length four. As an application of these results we
use the two-loop dilatation operator to compute the leading correction to the
Hagedorn temperature of the weakly-coupled planar ABJM theory on R \times S^2.Comment: 41 pages, 1 figure; v2: minor correction
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