Naoki Sasakura

TENSOR MODEL FOR GRAVITY AND ORIENTABILITY OF MANIFOLD

We investigate the relation between rank-three tensor models and the dynamical triangulation model of three-dimensional quantum gravity, and discuss the orientability of the manifold and the corresponding tensor models. We generalize the orientable tensor models to arbitrary dimensions, which include the two-dimensional Hermitian matrix model as a special case.

Open membranes in a constant C field background and noncommutative boundary strings

We investigate the dynamics of open membrane boundaries in a constant C-field background. We follow the analysis for open strings in a B-field background, and take some approximations. We find that open membrane boundaries do show noncommutativity in this case by explicit calculations. Membrane boundaries are one-dimensional strings, so we face a new type of noncommutativity, that is, non-commutative strings.

Discrete and Continuum Approaches to Three-Dimensional Quantum Gravity

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k^2, where q=e^(2πi/(k+2)).

Space-time uncertainty relation and Lorentz invariance

We discuss a Lorentz covariant space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period δt is larger than the Planck time lP. At δt lP, this uncertainty relation takes roughly the form δtδx lP2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the representation space. In four-dimensional space-time, the Jacobi identity should be violated in the algebraic representation of the uncertainty relation.

Tensor models and 3-ary algebras

Tensor models are the generalization of matrix models, and are studied as models of quantum gravity in general dimensions. In this paper, I discuss the algebraic structure in the fuzzy space interpretation of the tensor models which have a tensor with three indices as its only dynamical variable. The algebraic structure is studied mainly from the perspective of 3-ary algebras. It is shown that the tensor models have algebraic expressions, and that their symmetries are represented by 3-ary algebras. It is also shown that the 3-ary algebras of coordinates, which appear in the nonassociative fuzzy flat spacetimes corresponding to a certain class of configurations with Gaussian functions in the tensor models, form Lie triple systems, and the associated Lie algebras are shown to agree with those of the Snyders noncommutative spacetimes. The Poincare transformations of the coordinates on the fuzzy flat spacetimes are shown to be generated by 3-ary algebras.

A de-Sitter thick domain wall solution by elliptic functions

We obtain and study an analytical solution of a de-Sitter thick domain wall in five-dimensional Einstein gravity interacting with a scalar field. The scalar field potential is axion-like, V() = a + b cos((2/3)1/2) with constants a,b satisfying −3b < 5a < 3b, and the solution is expressed in terms of elliptic functions.

CANONICAL TENSOR MODELS WITH LOCAL TIME

It is an intriguing question how local time can be introduced in the emergent picture of space–time. In this paper, this problem is discussed in the context of tensor models. To consistently incorporate local time into tensor models, a rank-three tensor model with first class constraints in Hamilton formalism is presented. In the limit of usual continuous spaces, the algebra of constraints reproduces that of general relativity in Hamilton formalism. While the momentum constraints can be realized rather easily by the symmetry of the tensor models, the form of the Hamiltonian constraints is strongly limited by the condition of the closure of the whole constraint algebra. Thus the Hamiltonian constraints have been determined on the assumption that they are local and at most cubic in canonical variables. The form of the Hamiltonian constraints has similarity with the Hamiltonian in the c < 1 string field theory, but it seems impossible to realize such a constraint algebras in the framework of vector or matrix models. Instead these models are rather useful as matter theories coupled with the tensor model. In this sense, a three-index tensor is the minimum-rank dynamical variable necessary to describe gravity in terms of tensor models.

Braided Quantum Field Theories and Their Symmetries

Braided quantum field theories, proposed by Oeckl, can provide a framework for quantum field theories that possess Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We study Hopf algebra symmetries and such relations in the context of braided quantum field theories. We give the four algebraic conditions among Hopf algebra symmetries and braided quantum field theories that are required for the relations to hold. As concrete examples, we apply our

AN INVARIANT APPROACH TO DYNAMICAL FUZZY SPACES WITH A THREE-INDEX VARIABLE

A dynamical fuzzy space might be described in terms of a dynamical threeindex variable Cab c , which determines the algebraic relations fafb = Cab c fc of the functions fa on a fuzzy space. A fuzzy analogue of the general coordinate transformation would be given by the general linear transformation on fa. The solutions to the invariant equations of motion of Cab c can be generally constructed from the invariant tensors of Lie groups. Euclidean models the actions of which are bounded from below are introduced. Lie group symmetric solutions to a class of Euclidean model are obtained. The analysis of the fluctuations around the SO(3) symmetric solution shows that the solution can be regarded as a fuzzy S 2 /Z2.

TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.