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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.