Tensor models and 3-ary algebras

Sasakura, Naoki

doi:10.1063/1.3654028

Cited by 23 publications

(49 citation statements)

References 60 publications

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“…Therefore, in general a fuzzy space is non-local and can be of any dimension. In the canonical tensor model, in order to define the physics controlling the dynamics, two external conditions are imposed [50]. These are the reality conditions given by…”

Section: Canonical Tensor Modelmentioning

confidence: 99%

Physical states in the canonical tensor model from the perspective of random tensor networks

Narain

Sasakura

Sato

2015

J. High Energ. Phys.

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Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for N = 2, 3, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general N . Then, by generalizing this form, we also obtain various solutions for general N . Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased N . This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in N = 3, and comment on an extension of Airy function related to the solutions.

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Section: Canonical Tensor Modelmentioning

confidence: 99%

Physical states in the canonical tensor model from the perspective of random tensor networks

Narain

Sasakura

Sato

2015

J. High Energ. Phys.

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“…In the present application of this paper, we take the following assumptions for the choice of N : N is a homogeneous function of M, and is covariant under the orthogonal transformation (8). These properties make the flow consistent with the gauge invariance (8) and (9). Then the simplest choice is…”

Section: The Hamiltonian Vector Field Of the Canonical Tensor Modelmentioning

confidence: 99%

“…The setup we consider in this paper is a rank-three tensor model [2,3,4] in the canonical formalism [5,6,7,8], dubbed canonical tensor model for short. It is supposed to be a theory of dynamical fuzzy spaces [9], and its dynamical variables are a canonical conjugate pair of tensors with three indices of a certain cardinality * N. The model is formulated as a totally constrained system with a number of first-class constraints including Hamiltonian ones. Remarkably, the Hamiltonian constraints can be uniquely fixed by the algebraic consistency of the constraints and some other physically reasonable assumptions.…”

Section: Introductionmentioning

confidence: 99%

Ising model on random networks and the canonical tensor model

Sasakura

Sato

2014

Progress of Theoretical and Experimental Physics

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We introduce a statistical system on random networks of trivalent vertices for the purpose of studying the canonical tensor model, which is a rank-three tensor model in the canonical formalism. The partition function of the statistical system has a concise expression in terms of integrals, and has the same symmetries as the kinematical ones of the canonical tensor model. We consider the simplest nontrivial case of the statistical system corresponding to the Ising model on random networks, and find that its phase diagram agrees with what is implied by regrading the Hamiltonian vector field of the canonical tensor model with N = 2 as a renormalization group flow. Along the way, we obtain an explicit exact expression of the free energy of the Ising model on random networks in the thermodynamic limit by the Laplace method. This paper provides a new example connecting a model of quantum gravity and a random statistical system. * sasakura@yukawa.kyoto-u.ac.jp † Yuki.Sato@wits.ac.za * Namely, the indices can take N values, say 1, 2, . . . , N .

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“…This would explain the presence of c R even for S 1 , which has no curvatures, in the following sense. As derived in [23], the right-hand side of the second equation in (45) actually contains the terms like β µµ β νν (∇ µ β)(∇ ν β)/β 2 and β µν β µ ν (∇ µ ∇ ν β)/β, which can potentially contribute to c R . We have ignored these terms because of the homogeneity of the spaces, but the discreteness locally violates this assumption in short distances.…”

Section: Correspondence To a General Relativistic Systemmentioning

confidence: 97%

“…Real symmetric three-way tensors may be used to describe spaces through the algebra of functions acting on these spaces [43,44,45,46,47]. In this section we describe a systematic method to construct such tensors from their corresponding algebra.…”

Section: Real Symmetric Three-way Tensors Corresponding To Fuzzy Spacesmentioning

confidence: 99%

Canonical tensor model through data analysis: Dimensions, topologies, and geometries

2018

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The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (or CP etc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties are extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two-and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.

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Tensor models and 3-ary algebras

Cited by 23 publications

References 60 publications

Physical states in the canonical tensor model from the perspective of random tensor networks

Physical states in the canonical tensor model from the perspective of random tensor networks

Ising model on random networks and the canonical tensor model

Canonical tensor model through data analysis: Dimensions, topologies, and geometries

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