Higher-spin gravity and torsion on quantized space-time in matrix models

Abstract

A geometric formalism is developed which allows to describe the non-linear regime of higher-spin gravity emerging on a cosmological quantum space-time in the IKKT matrix model. The vacuum solutions are Ricci-flat up to an effective vacuum energy-momentum tensor quadratic in the torsion, which arises from a Weitzenböck-type higher spin connection. Torsion is expected to be significant only at cosmic scales and around very massive objects, and could behave like dark matter. A non-linear equation for the torsion tensor is found, which encodes the Yang-Mills equations of the matrix model. The metric and torsion transform covariantly under a higher-spin generalization of volume-preserving diffeomorphisms, which arises from the gauge invariance of the matrix model. known as UV/IR mixing [2][3][4]. They cancel only in the maximally supersymmetric IKKT matrix model [5], which should thus have the best chance to describe physics, leading to an unexpected link with string theory. However, it is not obvious how to obtain gravity from Yang-Mills-type matrix models such as the IKKT model. There are intriguing hints such as non-local gauge transformations and Ricci-flat propagating metric fluctuations [6][7][8], but the presence of an anti-symmetric tensor θ µν in space-time leads to a dangerous breaking of Lorentz invariance. This problem can be overcome by considering a higher-spin generalization, where θ µν is replaced by a twisted bundle of such tensors over space-time. This is realized in a simple solution of the IKKT model with a mass term interpreted as cosmological FLRW space-time M, based on the doubleton representations of so(4, 2) [9, 10], cf. [11]. It leads to a higher spin gauge theory which is invariant under a higher spin generalization of volume-preserving diffeomorphisms, ghost-free at the linearized level, and includes spin 2 gravitons. A linearized Schwarzschild-like solution was also found [12]. The theory has intriguing structural similarities with Vasilievs higherspin gravity [13,14], but also crucial differences 2 : it is defined through an action, both IR and UV scale parameters are present, and there are 5 propagating metric modes which could be interpreted as "would-be massive" gravitons. In the present paper, we study that higher-spin theory at the non-linear level. This is not straightforward because the model is of Yang-Mills type, there is no Einstein-Hilbert action, and everything is based on Poisson brackets (or commutators). The gauge symmetry corresponds to generalized diffeomorphisms rather than local Lorentz transformations 3 , hence it is not some reformulation of GR in the spirit of MacDowell-Mansouri [23]. A more appropriate approach can be found in a paper by Langmann and Szabo (LS) [24], who pointed out that a dimensional reduction of a gauge theory in 8-dimensional phase space with suitable constraints can be interpreted in terms of 4-dimensional teleparallel gravity through torsion. Although their setup does not provide a complete theory, a similar strategy provides a geometric unders...

Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution

“…In particular, there is no manifest local Lorentz (gauge) invariance acting on the frame. That is not a problem per se, and invariance under volume-preserving diffeos is manifest [1]. The vacuum solutions…”

“…}, which arises from the basic matrices Zα of the matrix model via the Poisson structure in the semi-classical limit. The semi-classical equations of the matrix model were recast in [1] as non-linear geometric equations, in terms of the torsion of the Weitzenböck connection associated to this frame. Covariance under volume-preserving diffeomorphisms is manifest in this formalism, which originates from the gauge invariance of the matrix model.…”

“…At the linearized level, the resulting higher-spin gauge theory was elaborated in [14,21]. A suitable formalism to understand the gravity sector at the non-linear level was developed in [1], based on the Weitzenböck connection associated to the frame…”

“…At the reference point x = (x 0 , 0, 0, 0), these relations imply t 0 = 0 and t i t i =R −2 cosh 2 (η). While x ∈ M (3,1) is invariant under its local SO(3) stabilizer, the t µ transform as vectors under this SO(3), leading to the structure of a twisted or equivariant bundle. Accordingly, the space of functions…”

“…The basic geometric structure is a frame, which arises from the Poisson structure on an underlying S 2 bundle over spacetime. The equations of motion for the associated Weitzenböck torsion obtained in [1] are rewritten in the form of Yang-Mills-type equations for the frame. An effective action is found which reproduces these equations of motion, which contains an Einstein-Hilbert term coupled to a dilaton, an axion and a Maxwell-type term for the dynamical frame.…”