CDT open-closed surface field theory of a 3D tensor-matrix model

TL;DR

This paper develops a 3D tensor-matrix model for open-closed surface CDT, introducing novel IK-type interactions and analyzing the algebraic structure of surface generators in the Fokker-Planck framework.

Contribution

It constructs a 3D CDT open-closed surface field theory with unique IK-type interactions and examines the algebraic closure of surface generators.

Findings

The model includes IK-type interactions as quantum corrections.

Generators close in their algebra for surfaces with minimal boundary loops.

The 3D model differs from 2D in the algebraic structure of generators.

Abstract

We construct a tensor-matrix model which describes 3-dimensional (3D) Causal Dynamical Triangulation (CDT) of open-closed surface. Though the usual splitting interaction of a surface is not derived from the stochastic quantization procedure, it provides another interaction of IK-type, which becomes the sole quantum correction. Through the double scaling limit, it realizes CDT open-closed surface field theory including the IK-type interactions in the same condition with the closed surface CDT model. Further, we investigate the commutation relations of the generators, for surfaces with the lowest numbers of boundary loops, in the Fokker-Planck (FP) Hamiltonian. The generators seem to close in their commutation relations in our 3D model, differently from the 2D model, where the algebraic structure was not exact when commutators contain the generator of the open string edge.