[Paper Review] Fuzzy spaces from tensor models, cyclicity condition, and n-ary algebras

Naoki Sasakura|arXiv (Cornell University)|Mar 28, 2012

Noncommutative and Quantum Gravity Theories24 references3 citations

TL;DR

This paper proposes that rank-three tensor models describe dynamical fuzzy spaces through a generalized Hermiticity condition that enforces a cyclic property in the function algebra of fuzzy spaces. Despite nonassociativity, the cyclic structure enables quantum-like mechanics, diffeomorphism-like symmetries via n-ary transformations, and systematic truncation for compactifications and coarse-graining, enabling physical applications in quantum gravity and field theory on nonassociative geometries.

ABSTRACT

The rank-three tensor models, which have a rank-three tensor as their only dynamical variable, may be interpreted as models of dynamical fuzzy spaces. In this interpretation, the generalized Hermiticity condition on the rank-three tensor leads to a cyclic property of the algebra of functions on fuzzy spaces. The fuzzy spaces with the cyclic property are shown to have various physically interesting characteristics. (i) Although the function algebras of the kind are nonassociative in general, various properties analogous to quantum mechanics hold on the fuzzy spaces. (ii) The symmetry of the rank-three tensor models can be shown to be represented systematically by n-ary transformations on the fuzzy spaces. The transformations contain, for instance, diffeomorphism on fuzzy spaces. (iii) There exists a systematic procedure of truncating the function algebras of the kind, and it can be used to consider subspaces, compactifications, lattice theories, and coarsegraining procedures of fuzzy spaces in physical applications.

Motivation & Objective

• To establish a correspondence between rank-three tensor models and fuzzy spaces with cyclic function algebras via a generalized Hermiticity condition.
• To demonstrate that nonassociative fuzzy spaces with cyclic structure support quantum-mechanical analogs despite lacking associativity.
• To show that the orthogonal symmetry of tensor models manifests as n-ary transformations on fuzzy spaces, including diffeomorphism-like symmetries.
• To develop a systematic truncation procedure for fuzzy space algebras to model subspaces, compactifications, lattice theories, and coarse-graining.

Proposed method

• Map the rank-three tensor $M_{abc}$ to the structure constants $f_{ab}{}^c$ and metric $g_{ab}$ of a fuzzy space via $M_{abc} = f_{ab}{}^{c'} g_{c'c}$, with $g_{ab} = ar{ ho}_{ab}$ for gauge fixing.
• Impose the generalized Hermiticity condition $M_{abc} = M_{bca} = M_{cab} = M_{bac}^* = M_{acb}^* = M_{cba}^*$, which translates to the cyclic property $\langle \phi_a \phi_b | \phi_c \rangle = \langle \phi_a | \phi_b \phi_c \rangle = \langle \phi_b | \phi_c \phi_a \rangle$.
• Define states $|s\rangle = s_a |\phi_a\rangle$ and operators $\mathcal{O} = v_a \phi_a$, with matrix elements computed via the cyclic algebra, ensuring consistency in inner products.
• Construct scalar field actions on fuzzy spaces by substituting the algebra and metric into a general action form, yielding effective field theories.
• Use damping factors $D(j)$ in spherical harmonic algebras to model fuzzy spheres, with $D(j) \propto I_{j+1/2}(-\beta)$ derived from Gaussian-like damping on the sphere.
• Perform low-momentum or low-$j$ expansions to show that the scalar field action reproduces standard kinetic terms (e.g., Laplacian on a sphere) in the continuum limit.

Experimental results

Research questions

• RQ1 How does the generalized Hermiticity condition on a rank-three tensor lead to a cyclic property in the function algebra of fuzzy spaces?
• RQ2 To what extent do nonassociative fuzzy spaces with cyclic structure still support quantum-mechanical analogs such as consistent inner products and operator traces?
• RQ3 Can the orthogonal symmetry of rank-three tensor models be systematically realized as n-ary transformations on fuzzy spaces, including diffeomorphism-like maps?
• RQ4 What is the systematic procedure for truncating function algebras of fuzzy spaces, and how can it be used to model compactifications, subspaces, and coarse-graining?
• RQ5 Does the scalar field action on a fuzzy sphere derived from the tensor model reproduce the standard Laplacian on a sphere in the low-energy limit?

Key findings

• The generalized Hermiticity condition on the rank-three tensor induces a cyclic property in the function algebra: $\langle \phi_a \phi_b | \phi_c \rangle = \langle \phi_a | \phi_b \phi_c \rangle = \langle \phi_b | \phi_c \phi_a \rangle$, which underpins the physical consistency of the fuzzy space.
• Despite nonassociativity, the fuzzy space supports quantum-like mechanics: states, operators, and traces are consistently defined, and the trace of an operator is independent of the order of multiplication due to cyclicity.
• The orthogonal symmetry $O(N)$ of the tensor model is realized as $n$-ary transformations on the fuzzy space, including diffeomorphism-like maps, providing a systematic symmetry structure.
• A systematic truncation procedure exists for the function algebra, enabling the construction of subspaces, compactifications, lattice theories, and coarse-graining procedures.
• On a fuzzy two-sphere with damping factor $D(j) \propto I_{j+1/2}(-\beta)$, the scalar field action reduces to $S_{sphere} = (c_0 + c_1 j(j+1) + \cdots) \psi_{(j,m)}^* \psi_{(j,m)}$ at low $j$, reproducing the standard Laplacian on a sphere.
• In the low-momentum limit, the scalar field action on a fuzzy flat space yields a kinetic term $\propto 3\alpha c_0 p^2$, matching the standard scalar field action on flat space.