Yoshiyuki Watabiki

The spectral dimension of 2D quantum gravity

We show that the spectral dimension ds of two-dimensional quantum gravity coupled to Gaussian fields is two for all values of the central charge c ≤ 1. The same arguments provide a simple proof of the known result ds = 4/3 for branched polymers.

A string field theory based on causal dynamical triangulations

We formulate the string field theory in zero-dimensional target space corresponding to the two-dimensional quantum gravity theory defined through Causal Dynamical Triangulations. This third quantization of the quantum gravity theory allows us in principle to calculate the transition amplitudes of processes in which the topology of space changes in time, and to include non-trivial topologies of space-time. We formulate the corresponding Dyson-Schwinger equations and illustrate how they can be solved iteratively.

A new continuum limit of matrix models

We define a new scaling limit of matrix models which can be related to the method of causal dynamical triangulations (CDT) used when investigating two-dimensional quantum gravity. Surprisingly, the new scaling limit of the matrix models is also a matrix model, thus explaining why the recently developed CDT continuum string field theory [J. Ambjorn, R. Loll, Y. Watabiki, W. Westra, S. Zohren, arXiv: 0802.0719] has a matrix-model representation [J. Ambjorn, R. Loll, Y. Watabiki, W. Westra, S. Zohren, arXiv: 0804.0252].

2d CDT is 2d Hořava–Lifshitz quantum gravity

Abstract Causal Dynamical Triangulations (CDT) is a lattice theory where aspects of quantum gravity can be studied. Two-dimensional CDT can be solved analytically and the continuum (quantum) Hamiltonian obtained. In this Letter we show that this continuum Hamiltonian is the one obtained by quantizing two-dimensional projectable Hořava–Lifshitz gravity.

Nonperturbative effect in the β functions and the equations of motion for the string

Abstract We study systematically how to calculate the β functions nonperturbatively in the sigma model which describes the string propagation in arbitrary background fields. In particular, we calculate explicity the β functions for a tachyon and massless background fields up to the three-string tree interactions and also the four-tachyon tree interactions.

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 dH = 3.58 ± 0.04. This result supports the conjecture dH = −2α1/α−1, where αn is the gravitational dressing exponent of a spinless primary field of conformal weight (n + 1, n + 1), and it disfavours the alternative prediction dH = 2/|γ|. On the other hand, 〈ln〉 ∼ r2n for n > 1 with good accuracy, i.e. the boundary length l has an anomalous dimension relative to the area of the surface.

EFFECTIVE SUPERPOTENTIALS FOR SO/Sp WITH FLAVOR FROM MATRIX MODELS

We study matrix models related to SO/Sp gauge theories with flavors. We give the effective superpotentials for gauge theories with arbitrary tree level superpotential up to first instanton level. For quartic tree level superpotential, we obtained exact one-cut solution. We also derive Seiberg–Witten curve for these gauge theories from matrix model argument.

Quantum geometry of topological gravity

Abstract We study a c = −2 conformal field theory coupled to two-dimensional quantum gravity by means of dynamical triangulations. We define the geodesic distance r on the triangulated surface with N triangles, and show that dim[ r d H ] = dim[ N ], where the fractal dimension d H = 3.58 ± 0.04. This result lends support to the conjecture d H = −2α 1 α −1 , where α − n is the gravitational dressling exponent of a spin-less primary field of conformal weight ( n + 1, n + 1), and it disfavors the alternative prediction d H = −2 γ str . On the other hand, we find dim[ l ] = dim[ r 2 ] with good accuracy, where l is the length of one of the boundaries of a circle with (geodesic) radius r , i.e. the length l has an anomalous dimension relative to the area of the surface. It is further shown that the spectral dimension d s = 1.980±0.014 for the ensemble of (triangulated) manifolds used. The results are derived using finite size scaling and a very efficient recursive sampling technique known previously to work well for c = −2.

Quantum geometry and diffusion

We study the diffusion equation in two-dimensional quantum gravity, and show that the spectral dimension is two despite the fact that the intrinsic Hausdorff dimension of the ensemble of two-dimensional geometries is very different from two. We determine the scaling properties of the quantum gravity averaged diffusion kernel.

Noncritical string field theory for two-D quantum gravity coupled to (p, q) conformal fields

We propose a noncritical string field theory for 2d quantum gravity coupled to (p, q) conformal fields. The Hamiltonian is described by the generators of the Wp algebra, and the Schwinger–Dyson equation is equivalent to a vacuum condition imposed on the generators of Wp algebra.