arXiv subjects

Harold C. Steinacker

Publications and source records attributed to Harold C. Steinacker.

At least 19 recordsLinked to original sources

arXiv2026

Quantum spacetime and gravity from the IKKT matrix model: an invitation

This is a concise and compact introduction to a framework for spacetime, gravity, and fundamental physics based on the IKKT matrix model, focusing on conceptual and structural aspects. The approach rests on nontrivial vacua or matrix backgrounds, which serve as 3+1-dimensional noncommutative spacetime. We discuss a specific class of backgrounds describing expanding homogeneous and isotropic FLRW spacetime, which provide a solution of the one-loop effective action with finitely many dof per volume. Perturbations of the background give rise to fields propagating on spacetime governed by a weakly-coupled gauge theory, as well as a nonstandard description of (quantum) gravity leading to some IR modifications. Maximal supersymmetry of the model is essential to ensure effective locality.

hep-th↗

arXiv2026

Quantum spacetime and quantum fluctuations in the IKKT model at weak coupling

This paper aims to clarify conceptual aspects of emergent structure in IKKT-type matrix models. Even without any adjustable parameters in the action, non-trivial matrix vacua do acquire a meaningful coupling constant, as well as two distinct uncertainty scales: a) the scale of noncommutativity of the matrix background, and b) the scale of quantum fluctuations of the matrices under the path integral. These scales are estimated for two prototypes of matrix backgrounds, known as Moyal-Weyl quantum plane and covariant quantum spacetime. Their relative importance separates two regimes: 1) the semi-classical regime interpreted in terms of semi-classical noncommutative geometry, and 2) the deep quantum regime usually interpreted in terms of holography. The quantum fluctuations are shown to be negligible in the weak coupling regime. This justifies previous work on the emergent 3+1-dimensional semi-classical geometry and (quantum) gravity in suitable vacua.

hep-th↗

arXiv2026

Modified gravity at large scales on quantum spacetime in the IKKT model

The gravitational dynamics of 3+1 dimensional covariant quantum spacetime in the IKKT or IIB matrix model is studied at one loop, combining the Yang-Mills-type matrix action with the induced Einstein-Hilbert action. This combined action leads to interesting modifications of the gravitational dynamics at long distances, governed by modified Einstein equations including an extra geometrical tensor interpreted as ''mirage matter''. In particular we find extra non-Ricci flat geometric modes with a non-standard dispersion relation, with features reminiscent of dark matter.

hep-th↗

arXiv2025

Spatially flat cosmological quantum spacetimes

We recently described a cosmological quantum spacetime of vanishing spatial curvature, which can be considered as background for the IKKT matrix model, assuming that the resulting gauge theory couples weakly. Building on this example, we construct a large class of spatially flat cosmological quantum spacetimes. We also elaborate on various details of their algebraic and semi-classical structure as well as the higher spin modes present in these models. In particular, we introduce the notion of approximate diffeomorphisms on the cosmological quantum spacetimes that stem from gauge transformations of the underlying matrix model, and investigate how different gauges are related in the semi-classical regime by approximate diffeomorphisms. Finally, we briefly outline how the described quantum spacetimes could be incorporated into the full IKKT model.

hep-th↗

arXiv2025

Dynamical Covariant Quantum Spacetime with Fuzzy Extra Dimensions in the IKKT model

We consider general $k=-1$ FLRW covariant quantum spacetimes $\mathcal{M}^{3,1} \times \mathcal{K}$ with fuzzy extra dimensions $\mathcal{K}$ as classical solutions of the IKKT matrix model. The coupled equations of motion are recast in terms of conservation laws, which allow to determine the evolution of spacetime in a transparent way. We show that $\mathcal{K}$ is stabilized as a classical solution in the presence of a large $R$ charge, corresponding to internal angular momentum. This provides a mechanism to maintain a large hierarchy between UV and IR scales. We also argue that the evolution of spacetime is determined by a balance between classical and quantum effects, leading to a cosmic scale factor $a(t) \sim t$ and constant dilaton at late times. On such a background, the undeformed IKKT model leads to a higher-spin gauge theory including gravity.

hep-th↗

arXiv2025

General Relativity in IIB Matrix Model

The matrix models are non-perturbative formulations of string theory, from which many believe that spacetime arises. The matrix fluctuations around the spacetime thus created should represent both matter and gravitational fields. In this paper, we discuss how the gravitational field emerges from the IIB matrix model. In particular, we consider how diffeomorphism invariance arises and how unitarity is guaranteed in this theory. Specifically, we consider matrices as bilocal fields and discuss how the Lorentz-invariant vacuum and low-energy excitations around it can be expressed. We then discuss how the conditions for the theory to be unitary can be written in terms of bilocal fields. We argue that in the low-energy limit, the bilocal fields are reduced to local fields consisting of a finite number of massless fields and an infinite number of massive fields, satisfying unitarity.

hep-th↗

arXiv2025

Quantum Geometry of Data

We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by learned Hermitian matrices, and data points are mapped to states in Hilbert space. The quantum geometry description endows the dataset with rich geometric and topological structure - including intrinsic dimension, quantum metric, and Berry curvature - derived directly from the data. QCML captures global properties of data, while avoiding the curse of dimensionality inherent in local methods. We illustrate this on a number of synthetic and real-world examples. Quantum geometric representation of QCML could advance our understanding of cognitive phenomena within the framework of quantum cognition.

cs.LG↗

arXiv2025

One-loop effective action of the IKKT model for cosmological backgrounds

We study cosmological solutions of the IKKT model with $k=-1$ FLWR geometry, taking into account one-loop corrections. A previously discussed covariant quantum spacetime is found to be stabilized through one-loop effects at early times, without adding a mass term to the model. At late times, this background is modified and approaches a solution of the classical model where $a(t) \sim const$, but the dilaton decreases in time. This suggests that a more complete treatment of the system is required in the late-time regime.

hep-th↗

arXiv2025

Minimal covariant quantum space-time

We discuss minimal covariant quantum space-time ${\cal M}^{1,3}_0$, which is defined through the minimal doubleton representation of $\mathfrak{so}(4,2)$. An elementary definition in terms of generators and relations is given. This space is shown to admit a semi-classical interpretation as quantized twistor space ${\mathbb C} P^{1,2}$, viewed as a quantized $S^2$-bundle over a 3+1-dimensional $k=-1$ FLRW space-time. In particular we find an over-complete set of (quasi-) coherent states, with a large hierarchy between the uncertainty scale and the geometric curvature scale. This provides an interesting background for the IKKT model, leading to a $\mathfrak{hs}$-extended gravitational gauge theory, which is free of ghosts due to the constraints on phase space arising from the doubleton representation.

hep-th↗

arXiv2024

Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model

We discuss a $(3{+}1)$-dimensional covariant quantum space-time describing a FLRW cosmology with Big Bounce, obtained by a projection of the fuzzy hyperboloid $H^4_n$. This provides a background solution of the IKKT matrix model with mass term. We characterize the bosonic fluctuation spectrum, which consists of a tower of higher-spin modes. The modes are organized in terms of an underlying $SO(4,2)$ structure group, which is broken to the $SO(3,1)$ isometry of the background. The resulting higher-spin gauge theory includes all degrees of freedom required for gravity, and should be well suited for quantization. All modes propagate with the same speed of light, even though local boost invariance is not manifest. The propagating metric perturbation modes comprise those of a massless graviton, as well as a scalar mode. Gauge invariance allows to obtain the analog of the linearized Einstein-Hilbert action, which is expected to be induced upon quantization.

hep-th↗

arXiv2024

$\mathfrak{hs}$-extended gravity from the IKKT matrix model

We elaborate further on the one-loop effective action of the IKKT model on 3 + 1 dimensional covariant quantum spacetime in the presence of fuzzy extra dimensions. In particular, we describe the one-loop effective action in terms of a remarkable $SO(1, 9)$ character, which allows to evaluate the pertinent traces over the internal modes explicitly. This also allows to estimate the higher-order contributions (in the internal flux $\mathcal{F}_{\mathtt{IJ}}$) to the one-loop effective action in a systematic way. We show that all higher-order contributions are generally suppressed and UV finite, which justifies the previous treatment of the induced gravitational action. We also obtain explicit expressions for the effective Newton constant, and determine the dynamics of the Kaluza-Klein scale $Δ_{\mathcal{K}}$ of the fuzzy extra dimensions $\mathcal{K}$.

hep-th↗

arXiv2024

Interactions in the IKKT matrix model on covariant quantum spacetime

We study the interactions of the higher-spin gauge theory arising from the IKKT matrix model on a covariant quantum FLRW quantum space-time $\mathcal{M}^{1,3}_{\mathtt{J}}$, denoted as HS-IKKT. In particular, we elaborate some of the vertices and observe that they are not manifestly Lorentz invariant in the unitary formulation. We argue that Lorentz invariance of HS-IKKT can be recovered since Lorentz transformations are part of the gauge invariance in the covariant formulation. This statement is verified for some vertices with the lowest number of derivatives. The lowest-derivative sector of this theory is expected to be governed by an ``almost''-Lorentz-invariant Yang-Mills theory coupled to emergent gravity.

hep-th↗

arXiv2024

The fuzzy 4-hyperboloid $H^4_n$ and higher-spin in Yang-Mills matrix models

We consider the $SO(4,1)$-covariant fuzzy hyperboloid $H^4_n$ as a solution of Yang-Mills matrix models, and study the resulting higher-spin gauge theory. The degrees of freedom can be identified with functions on classical $H^4$ taking values in a higher-spin algebra associated to $\mathfrak{so}(4,1)$. We develop a suitable calculus to classify the higher-spin modes, and show that the tangential modes are stable. The metric fluctuations encode one of the spin 2 modes, however they do not propagate in the classical matrix model. Gravity is argued to arise upon taking into account induced gravity terms. This formalism can be applied to the cosmological FLRW space-time solutions of [1], which arise as projections of $H^4_n$. We establish a one-to-one correspondence between the tangential fluctuations of these spaces.

hep-th↗

arXiv2024

Quantum $\mathfrak{hs}$-Yang-Mills from the IKKT matrix model

We study the one-loop effective action of the higher-spin gauge theory induced by the IKKT matrix model on a $\mathcal{M}^{1,3}\times \mathcal{K}$ background, where $\mathcal{M}^{1,3}$ is an FLRW cosmological spacetime brane and $\mathcal{K}$ are compact fuzzy extra dimensions. In particular, we show that all non-abelian ($\mathfrak{hs}$-valued) gauge fields in this model acquire mass via quantum effects, thus avoiding no-go theorems. This leads to a massive non-abelian quantum $\mathfrak{hs}$-Yang-Mills theory, whose detailed structure depends on $\mathcal{K}$. The stabilization of $\mathcal{K}$ at one loop is understood as a result of the coupling between $\mathcal{K}$ and the $U(1)$-flux bundle on space-time. This flux stabilization induces the KK scale into the $\mathcal{N} = 4$ SYM sector of the model, which break superconformal symmetry.

hep-th↗

arXiv2024

Oxidation, Reduction and Semi-Classical Limit for Quantum Matrix Geometries

Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including noncommutative gauge theory and emergent gravity. Refining the construction in [1], we construct a semi-classical limit through an immersed submanifold of complex projective space based on quasi-coherent states. We observe the phenomenon of oxidation, where the resulting semi-classical space acquires spurious extra dimensions. We propose to remove this artifact by passing to a leaf of a carefully chosen foliation, which allows to extract the geometrical content of the noncommutative spaces. This is demonstrated numerically via multiple examples.

hep-th↗

arXiv2023

Spinorial description for Lorentzian $\mathfrak{hs}$-IKKT

We introduce a novel spinorial description for the higher-spin gauge theory induced by the IKKT matrix model on an FLRW spacetime with Lorentzian signature, called Lorentzian $\mathfrak{hs}$-IKKT theory. The new description is based on Weyl spinors transforming under the space-like isometry subgroup $SL(2,\mathbb{C})$ of the structure group $SO(2, 4) \simeq SU(2,2)$. It allows us to exploit the full power of the spinor formalism in Lorentzian signature, in contrast to a previous formalism based on the compact subgroup $SU(2)_L\times SU(2)_R$ of $SU(2,2)$. Some cubic vertices of the Yang-Mills sector and the corresponding scattering amplitudes are computed. We observe that the $n$-point (for $n \geq 4$) tree-level amplitudes are typically non-trivial on-shell, but exponentially suppressed in the late-time regime. While Lorentz invariance of the higher-spin amplitudes is not manifest, it is expected to be restored by higher-spin gauge invariance.

hep-th↗

arXiv2023

Modified Einstein equations from the 1-loop effective action of the IKKT model

We derive the equations of motion that arise from the one-loop effective action for the geometry of 3+1 dimensional quantum branes in the IKKT matrix model. These equations are cast into the form of generalized Einstein equations, with extra contributions from dilaton and axionic fields, as well as a novel anharmonicity tensor C_{μν} capturing the classical Yang-Mills-type action. The resulting gravity theory approximately reduces to general relativity in some regime, but differs significantly at cosmic scales, leading to an asymptotically flat FLWR cosmological evolution governed by the classical action.

hep-th↗

arXiv2023

On the propagation across the big bounce in an open quantum FLRW cosmology

Recently, solutions of the Ishibashi, Kawai, Kitazawa and Tsuchiya matrix theory have been found, which can be interpreted as 3+1-dimensional quantum geometries describing an effective Friedmann-Lemaître-Robertson-Walker cosmology with a big bounce. In this paper, we examine the propagation of a scalar field in an open Friedmann-Lemaître-Robertson-Walker spacetime arising within this framework. The paper is divided into two parts. In the first one, we perform a classical investigation by resorting to general-relativity tools where we show that both massless and massive non-interacting particles can travel across the big bounce. In the second part, we evaluate the scalar field propagator by means of quantum-field-theory techniques. This analysis reveals that in the late-time regime the scalar propagator resembles the standard Feynman propagator of flat Minkowski space, whereas for early times it gives rise to a well-defined correlation between two points on opposite sheets of the spacetime.

gr-qc↗