13 research outputs found
Non-perturbative 3d Lorentzian Quantum Gravity
We have recently introduced a discrete model of Lorentzian quantum gravity, given as a regularized non-perturbative state sum over simplicial Lorentzian space-times, each possessing a unique Wick rotation to Euclidean signature. We investigate here the phase structure of the Wick-rotated path integral in three dimensions with the aid of computer simulations. After fine-tuning the cosmological constant to its critical value, we find a whole range of the gravitational coupling constant for which the functional integral is dominated by non-degenerate three-dimensional space-times. We therefore have a situation in which a well-defined ground state of extended geometry is generated dynamically from a non-perturbative state sum of fluctuating geometries. Remarkably, its macroscopic scaling properties resemble those of a semi-classical spherical universe. Measurements so far indicate that defines an overall scale in this extended phase, without affecting the physics of the continuum limit. These findings provide further evidence that discrete {it Lorentzian} gravity is a promising candidate for a non-trivial theory of quantum gravit
Relativistic center-vortex dynamics of a confining area law
We offer a physicists' proof that center-vortex theory requires the area in
the Wilson-loop area law to involve an extremal area. Area-law dynamics is
determined by integrating over Wilson loops only, not over surface fluctuations
for a fixed loop. Fluctuations leading to to perimeter-law corrections come
from loop fluctuations as well as integration over finite -thickness
center-vortex collective coordinates. In d=3 (or d=2+1) we exploit a contour
form of the extremal area in isothermal which is similar to d=2 (or d=1+1) QCD
in many respects, except that there are both quartic and quadratic terms in the
action. One major result is that at large angular momentum \ell in d=3+1 the
center-vortex extremal-area picture yields a linear Regge trajectory with Regge
slope--string tension product \alpha'(0)K_F of 1/(2\pi), which is the canonical
Veneziano/string value. In a curious effect traceable to retardation, the quark
kinetic terms in the action vanish relative to area-law terms in the large-\ell
limit, in which light-quark masses \sim K_F^{1/2} are negligible. This
corresponds to string-theoretic expectations, even though we emphasize that the
extremal-area law is not a string theory quantum-mechanically. We show how some
quantum trajectory fluctuations as well as non-leading classical terms for
finite mass yield corrections scaling with \ell^{-1/2}. We compare to old
semiclassical calculations of relativistic q\bar{q} bound states at large \ell,
which also yield asymptotically-linear Regge trajectories, finding agreement
with a naive string picture (classically, not quantum-mechanically) and
disagreement with an effective-propagator model. We show that contour forms of
the area law can be expressed in terms of Abelian gauge potentials, and relate
this to old work of Comtet.Comment: 20 pages RevTeX4 with 3 .eps figure
Nexus solitons in the center vortex picture of QCD
It is very plausible that confinement in QCD comes from linking of Wilson
loops to finite-thickness vortices with magnetic fluxes corresponding to the
center of the gauge group. The vortices are solitons of a gauge-invariant QCD
action representing the generation of gluon mass. There are a number of other
solitonic states of this action. We discuss here what we call nexus solitons,
in which for gauge group SU(N), up to N vortices meet a a center, or nexus,
provided that the total flux of the vortices adds to zero (mod N). There are
fundamentally two kinds of nexuses: Quasi-Abelian, which can be described as
composites of Abelian imbedded monopoles, whose Dirac strings are cancelled by
the flux condition; and fully non-Abelian, resembling a deformed sphaleron.
Analytic solutions are available for the quasi-Abelian case, and we discuss
variational estimates of the action of the fully non-Abelian nexus solitons in
SU(2). The non-Abelian nexuses carry Chern-Simons number (or topological charge
in four dimensions). Their presence does not change the fundamentals of
confinement in the center-vortex picture, but they may lead to a modified
picture of the QCD vacuum.Comment: LateX, 24 pages, 2 .eps figure
Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality
We exploit the properties of the three-dimensional hyperbolic space to
discuss a simplicial setting for open/closed string duality based on (random)
Regge triangulations decorated with null twistorial fields. We explicitly show
that the twistorial N-points function, describing Dirichlet correlations over
the moduli space of open N-bordered genus g surfaces, is naturally mapped into
the Witten-Kontsevich intersection theory over the moduli space of N-pointed
closed Riemann surfaces of the same genus. We also discuss various aspects of
the geometrical setting which connects this model to PSL(2,C) Chern-Simons
theory.Comment: 35 pages, references added, slightly revised introductio
Center Vortices, Nexuses, and Fractional Topological Charge
It has been remarked in several previous works that the combination of center
vortices and nexuses (a nexus is a monopole-like soliton whose world line
mediates certain allowed changes of field strengths on vortex surfaces) carry
topological charge quantized in units of 1/N for gauge group SU(N). These
fractional charges arise from the interpretation of the standard topological
charge integral as a sum of (integral) intersection numbers weighted by certain
(fractional) traces. We show that without nexuses the sum of intersection
numbers gives vanishing topological charge (since vortex surfaces are closed
and compact). With nexuses living as world lines on vortices, the contributions
to the total intersection number are weighted by different trace factors, and
yield a picture of the total topological charge as a linking of a closed nexus
world line with a vortex surface; this linking gives rise to a non-vanishing
but integral topological charge. This reflects the standard 2\pi periodicity of
the theta angle. We argue that the Witten-Veneziano relation, naively violating
2\pi periodicity, scales properly with N at large N without requiring 2\pi N
periodicity. This reflects the underlying composition of localized fractional
topological charge, which are in general widely separated. Some simple models
are given of this behavior. Nexuses lead to non-standard vortex surfaces for
all SU(N) and to surfaces which are not manifolds for N>2. We generalize
previously-introduced nexuses to all SU(N) in terms of a set of fundamental
nexuses, which can be distorted into a configuration resembling the 't
Hooft-Polyakov monopole with no strings. The existence of localized but
widely-separated fractional topological charges, adding to integers only on
long distance scales, has implications for chiral symmetry breakdown.Comment: 15 pages, revtex, 6 .eps figure
The modular geometry of Random Regge Triangulations
We show that the introduction of triangulations with variable connectivity
and fluctuating egde-lengths (Random Regge Triangulations) allows for a
relatively simple and direct analyisis of the modular properties of 2
dimensional simplicial quantum gravity. In particular, we discuss in detail an
explicit bijection between the space of possible random Regge triangulations
(of given genus g and with N vertices) and a suitable decorated version of the
(compactified) moduli space of genus g Riemann surfaces with N punctures. Such
an analysis allows us to associate a Weil-Petersson metric with the set of
random Regge triangulations and prove that the corresponding volume provides
the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio
The quantum space-time of c=-2 gravity
We study the fractal structure of space-time of two-dimensional quantum
gravity coupled to c=-2 conformal matter by means of computer simulations. We
find that the intrinsic Hausdorff dimension d_H = 3.58 +/- 0.04. This result
supports the conjecture d_H = -2 \alpha_1/\alpha_{-1}, where \alpha_n is the
gravitational dressing exponent of a spinless primary field of conformal weight
(n+1,n+1), and it disfavours the alternative prediction d_H = 2/|\gamma|. On
the other hand ~ r^{2n} for n>1 with good accuracy, i.e. the boundary
length l has an anomalous dimension relative to the area of the surface.Comment: 46 pages, 16 figures, 32 eps files, using psfig.sty and epsf.st
Triangulated surfaces and polyhedral structures
In this chapter we introduce the foundational material that will be used in our
analysis of triangulated surfaces and of their quantum geometry. We start by
recalling the relevant definitions from Piecewise–Linear (PL) geometry, (for which
we refer freely to [20, 21]). After these introductory remarks we specialize to the
case of Euclidean polyhedral surfaces whose geometrical and physical properties
will be the subject of the first part of the book. The focus here is on results which
are either new or not readily accessible in the standard repertoire. In particular we
discuss from an original perspective the structure of the space of all polyhedral
surfaces of a given genus and their stable degenerations. This is a rather delicate
point which appears in many guises in quantum gravity [6], and string theory, and
which is related to the role that Riemann moduli space plays in these theories.
Not surprisingly, the Witten–Kontsevich model [10] lurks in the background of our
analysis, and some of the notions we introduce may well serve for illustrating, from
a more elementary point of view, the often deceptive and very technical definitions
that characterize this subject. In such a framework, and in the whole landscaping
of the space of polyhedral surfaces an important role is played by the conical
singularities associated with the Euclidean triangulation of a surface.We provide, in
the final part of the chapter, a detailed analysis of the geometry of these singularities