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.
Highlights
Tensor models [1,2,3] were introduced with the hope to analytically describe simplicial quantum gravity for arbitrary dimensions1 by extending the matrix models which successfully describe the two-dimensional simplicial quantum gravity [6]
We have made an attempt to include fermionic degrees of freedom in CTM, which initially was purely bosonic in nature
We have introduced such degrees of freedom by allowing the dynamical rank-three tensors to be either Grassmann even or odd in accordance with the Grassmann natures associated to the indices
Summary
Tensor models [1,2,3] were introduced with the hope to analytically describe simplicial quantum gravity for arbitrary dimensions by extending the matrix models which successfully describe the two-dimensional simplicial quantum gravity [6]. The large N analyses of the colored tensor models have shown that the generated simplicial manifolds are dominated by branched polymers [14, 26, 27]. In Causal Dynamical Triangulation, which is a Lorentzian model of simplicial quantum gravity, it has been shown that de Sitter-like space-times, similar to our actual universe, are generated [32]. This success can be contrasted with the unsuccessful situation in Dynamical Triangulation, which is the original Euclidean model. This success would indicate the importance of a time-like direction in quantum gravity, and would raise the possibility of improving the tensor models above, which basically dealt with
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