An $OSp$ extension of Canonical Tensor Model

TL;DR

This paper extends the canonical tensor model to include fermionic degrees of freedom by developing a super-extension with an extended symmetry group, maintaining the algebraic structure but introducing challenges like negative norm states.

Contribution

It formulates a super-extension of the canonical tensor model with an extended symmetry group and first-class constraint algebra, enabling inclusion of fermions in tensor models of quantum gravity.

Findings

Super-extension maintains algebraic structure of bosonic case

Negative norm states arise in quantization, indicating need for further work

Results suggest parallels with bosonic tensor models, including wave functions and dual systems

Abstract

Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have a similar algebraic structure as the constraints of the ADM formalism of general relativity. In this paper, we formulate a super-extension of CTM as an attempt to incorporate fermionic degrees of freedom. The kinematical symmetry group is extended from $O(N)$ to $OSp(N,\tilde N)$, and the constraints are constructed so that they form a first-class constraint super-Poisson algebra. This is a straightforward super-extension, and the constraints and their algebraic structure are formally unchanged from the purely bosonic case, except for the additional signs associated to the order of the fermionic indices and dynamical variables. However, this extension of CTM leads to the existence of negative norm states in the quantized case, and requires some future improvements as quantum gravity with fermions. On the other hand, since this is a straightforward super-extension, various results obtained so far for the purely bosonic case are expected to have parallels also in the super-extended case, such as the exact physical wave functions and the connection to the dual statistical systems, i.e. randomly connected tensor networks.