Character Expansion Methods for Matrix Models of Dually Weighted Graphs

We consider generalized one-matrix models in which external fields allow control over the coordination numbers on both the original and dual lattices. We rederive in a simple fashion a character expansion formula for these models originally due to Itzykson and Di Francesco, and then demonstrate how to take the large N limit of this expansion. The relationship to the usual matrix model resolvent is elucidated. Our methods give as a by-product an extremely simple derivation of the Migdal integral equation describing the large $N$ limit of the Itzykson-Zuber formula. We illustrate and check our methods by analyzing a number of models solvable by traditional means. We then proceed to solve a new model: a sum over planar graphs possessing even coordination numbers on both the original and the dual lattice. We conclude by formulating equations for the case of arbitrary sets of even, self-dual coupling constants. This opens the way for studying the deep problem of phase transitions from random to flat lattices.Comment: 22 pages, harvmac.tex, pictex.tex. All diagrams written directly into the text in Pictex commands. (Two minor math typos corrected. Acknowledgements added.

On the connection between Quantum Mechanics and the geometry of two-dimensional strings

On the basis of an area-preserving symmetry in the phase space of a one-dimensional matrix model - believed to describe two-dimensional string theory in a black-hole background which also allows for space-time foam - we give a geometric interpretation of the fact that two-dimensional stringy black holes are consistent with conventional quantum mechanics due to the infinite gauged `W-hair' property that characterises them.Comment: 19 page

The Origin of Space-Time as WW Symmetry Breaking in String Theory

Physics in the neighbourhood of a space-time metric singularity is described by a world-sheet topological gauge field theory which can be represented as a twisted $N=2$ superconformal Wess-Zumino model with a $W_{1+\infty} \otimes W_{1+\infty}$ bosonic symmetry. The measurable $W$-hair associated with the singularity is associated with Wilson loop integrals around gauge defects. The breaking of $W_{1+\infty}$ $\otimes$ $W_{1+\infty}$ $\rightarrow$ $W_{1+\infty}$ is associated with expectation values for open Wilson lines that make the metric non-singular away from the singularity. This symmetry breaking is accompanied by massless discrete `tachyon' states that appear as leg poles in $S$-matrix elements. The triviality of the $S$-matrix in the high-energy limit of the $c=1$ string model, after renormalisation by the leg pole factors, is due to the restoration of double $W$-symmetry at the singularity.Comment: 13 page

Quantum Mechanics and Black Holes in Four-Dimensional String Theory

In previous papers we have shown how strings in a two-dimensional target space reconcile quantum mechanics with general relativity, thanks to an infinite set of conserved quantum numbers, ``W-hair'', associated with topological soliton-like states. In this paper we extend these arguments to four dimensions, by considering explicitly the case of string black holes with radial symmetry. The key infinite-dimensional W-symmetry is associated with the $\frac{SU(1,1)}{U(1)}$ coset structure of the dilaton-graviton sector that is a model-independent feature of spherically symmetric four-dimensional strings. Arguments are also given that the enormous number of string {\it discrete (topological)} states account for the maintenance of quantum coherence during the (non-thermal) stringy evaporation process, as well as quenching the large Hawking-Bekenstein entropy associated with the black hole. Defining the latter as the measure of the loss of information for an observer at infinity, who - ignoring the higher string quantum numbers - keeps track only of the classical mass,angular momentum and charge of the black hole, one recovers the familiar a quadratic dependence on the black-hole mass by simple counting arguments on the asymptotic density of string states in a linear-dilaton background.Comment: 18 page

String Theory Modifies Quantum Mechanics

We argue that the light particles in string theory obey an effective quantum mechanics modified by the inclusion of a quantum-gravitational friction term, induced by unavoidable couplings to unobserved massive string states in the space-time foam. This term is related to the $W$-symmetries that couple light particles to massive solitonic string states in black hole backgrounds, and has a formal similarity to simple models of environmental quantum friction. It increases apparent entropy, and may induce the wave functions of macroscopic systems to collapse.Comment: 18 pages, CERN-6595/9

Robust Limits on Lorentz Violation from Gamma-Ray Bursts

We constrain the possibility of a non-trivial refractive index in free space corresponding to an energy-dependent velocity of light: c(E) \simeq c_0 (1 - E/M), where M is a mass scale that might represent effect of quantum-gravitational space-time foam, using the arrival times of sharp features observed in the intensities of radiation with different energies from a large sample of gamma-ray bursters (GRBs) with known redshifts. We use wavelet techniques to identify genuine features, which we confirm in simulations with artificial added noise. Using the weighted averages of the time-lags calculated using correlated features in all the GRB light curves, we find a systematic tendency for more energetic photons to arrive earlier. However, there is a very strong correlation between the parameters characterizing an intrinsic time-lag at the source and a distance-dependent propagation effect. Moreover, the significance of the earlier arrival times is less evident for a subsample of more robust spectral structures. Allowing for intrinsic stochastic time-lags in these features, we establish a statistically robust lower limit: M > 0.9x10^{16} GeV on the scale of violation of Lorentz invariance.Comment: 18 pages, 4 eps figure

Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy

We present three holographic constructions of fractional quantum Hall effect (FQHE) via string theory. The first model studies edge states in FQHE using supersymmetric domain walls in N=6 Chern-Simons theory. We show that D4-branes wrapped on CP^1 or D8-branes wrapped on CP^3 create edge states that shift the rank or the level of the gauge group, respectively. These holographic edge states correctly reproduce the Hall conductivity. The second model presents a holographic dual to the pure U(N)_k (Yang-Mills-)Chern-Simons theory based on a D3-D7 system. Its holography is equivalent to the level-rank duality, which enables us to compute the Hall conductivity and the topological entanglement entropy. The third model introduces the first string theory embedding of hierarchical FQHEs, using IIA string on C^2/Z_n.Comment: 36 pages, 6 figures; v2: with an improved derivation of Hall conductivity in section 3.2, typo corrections, and additional references; v3: explanations and comments adde

On the Evaporation of Black Holes in String Theory

We show that, in string theory, the quantum evaporation and decay of black holes in two-dimensional target space is related to imaginary parts in higher-genus string amplitudes. These arise from the regularisation of modular infinities due to the sum over world-sheet configurations, that are known to express the instabilities of massive string states in general, and are not thermal in character. The absence of such imaginary parts in the matrix model limit confirms that the latter constitutes the final stage of the evaporation process, at least in perturbation theory. Our arguments appear to be quite generic, related only to the summation over world-sheet surfaces, and hence should also apply to higher-dimensional target spaces.Comment: 17 page

Notes on D-branes in 2D Type 0 String Theory

In this paper we construct complete macroscopic operators in two dimensional type 0 string theory. They represent D-branes localized in the time direction. We give another equivalent description of them as deformed Fermi surfaces. We also discuss a continuous array of such D-branes and show that it can be described by a matrix model with a deformed potential. For appropriate values of parameters, we find that it has an additional new sector hidden inside its strongly coupled region.Comment: harvmac, 18 pages, 2 figures, references adde

Lowest-Landau-level theory of the quantum Hall effect: the Fermi-liquid-like state

A theory for a Fermi-liquid-like state in a system of charged bosons at filling factor one is developed, working in the lowest Landau level. The approach is based on a representation of the problem as fermions with a system of constraints, introduced by Pasquier and Haldane (unpublished). This makes the system a gauge theory with gauge algebra W_infty. The low-energy theory is analyzed based on Hartree-Fock and a corresponding conserving approximation. This is shown to be equivalent to introducing a gauge field, which at long wavelengths gives an infinite-coupling U(1) gauge theory, without a Chern-Simons term. The system is compressible, and the Fermi-liquid properties are similar, but not identical, to those in the previous U(1) Chern-Simons fermion theory. The fermions in the theory are effectively neutral but carry a dipole moment. The density-density response, longitudinal conductivity, and the current density are considered explicitly.Comment: 32 pages, revtex multicol