TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

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A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY

International Journal of Modern Physics A ◽

10.1142/s0217751x10050433 ◽

2010 ◽

Vol 25 (23) ◽

pp. 4475-4492 ◽

Cited By ~ 5

Author(s):

NAOKI SASAKURA

Keyword(s):

General Relativity ◽

Models Of Quantum Gravity ◽

Quantum Gravity ◽

Matrix Models ◽

Dynamical Variable ◽

Renormalization Procedure ◽

Emergent Phenomena ◽

Tensor Models ◽

Theories Of Gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

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Detecting deformed commutators with exceptional points in optomechanical sensors

New Journal of Physics ◽

10.1088/1367-2630/ac3ff7 ◽

2021 ◽

Author(s):

Dianzhen Cui ◽

Tao Li ◽

Jianning Li ◽

Xuexi Yi

Keyword(s):

Models Of Quantum Gravity ◽

Quantum Gravity ◽

The Other ◽

Exceptional Point ◽

Weak Effect ◽

Third Order ◽

Exceptional Points ◽

The One ◽

Gravity Perturbation ◽

Canonical Position

Abstract Models of quantum gravity imply a modification of the canonical position-momentum commutation relations. In this manuscript, working with a binary mechanical system, we examine the effect of quantum gravity on the exceptional points of the system. On the one side, we find that the exceedingly weak effect of quantum gravity can be sensed via pushing the system towards a second-order exceptional point, where the spectra of the non-Hermitian system exhibits non-analytic and even discontinuous behavior. On the other side, the gravity perturbation will affect the sensitivity of the system to deposition mass. In order to further enhance the sensitivity of the system to quantum gravity, we extend the system to the other one which has a third-order exceptional point. Our work provides a feasible way to use exceptional points as a new tool to explore the effect of quantum gravity.

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Renormalization Group Approach to the Continuum Limit of Matrix Models of Quantum Gravity With Preferred Foliation

Frontiers in Physics ◽

10.3389/fphy.2021.531766 ◽

2021 ◽

Vol 9 ◽

Author(s):

Alicia Castro ◽

Tim Andreas Koslowski

Keyword(s):

Renormalization Group ◽

Quantum Gravity ◽

Matrix Models ◽

Fixed Points ◽

Renormalization Group Equation ◽

Two Dimensions ◽

Gravity Models ◽

Group Equation ◽

Tensor Models ◽

The Matrix

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

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Gauged double field theory as an L∞ algebra

Journal of High Energy Physics ◽

10.1007/jhep06(2021)058 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Eric Lescano ◽

Martín Mayo

Keyword(s):

Field Theory ◽

Algebraic Structure ◽

Classical Field ◽

Field Theories ◽

Lie Derivative ◽

Linear Perturbation ◽

Double Field Theory ◽

Generalized Frame ◽

Symmetry Transformations ◽

Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

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A COMPACT BOSON COUPLED TO TWO-DIMENSIONAL GRAVITY

International Journal of Modern Physics A ◽

10.1142/s0217751x91001337 ◽

1991 ◽

Vol 06 (15) ◽

pp. 2743-2754 ◽

Cited By ~ 22

Author(s):

NORISUKE SAKAI ◽

YOSHIAKI TANII

Keyword(s):

Field Theory ◽

Partition Function ◽

Quantum Gravity ◽

Matrix Models ◽

Riemann Surfaces ◽

Partition Functions ◽

Two Dimensional ◽

Dimensional Gravity ◽

The Continuum ◽

Continuum Field

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

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Matrix models, quantum gravity and the spectral dimension

10.22323/1.020.0257 ◽

2005 ◽

Author(s):

Fermin Viniegra ◽

Michael J. Peardon ◽

Sinead Ryan

Keyword(s):

Quantum Gravity ◽

Matrix Models ◽

Spectral Dimension

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Quantum gravity as a multitrace matrix model

International Journal of Modern Physics A ◽

10.1142/s0217751x17501809 ◽

2017 ◽

Vol 32 (31) ◽

pp. 1750180

Author(s):

Badis Ydri ◽

Cherine Soudani ◽

Ahlam Rouag

Keyword(s):

Quantum Gravity ◽

Matrix Models ◽

Large Class ◽

Matrix Model ◽

Two Dimensions ◽

Two Dimensional ◽

Topology Change ◽

New Model ◽

And Topology ◽

Emergent Geometry

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of two-dimensional quantum gravity which works away from two dimensions and captures a large class of spaces admitting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change.

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From random matrices to random tensors

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0011 ◽

2021 ◽

pp. 166-177

Author(s):

Adrian Tanasa

Keyword(s):

Matrix Models ◽

Random Matrices ◽

Mathematical Physics ◽

Natural Generalization ◽

Random Surface ◽

The Third ◽

Planar Surfaces ◽

Tensor Models ◽

Feynman Graphs ◽

The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

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