Covariant Quantum Spaces, Higher Spin, and Gravity

supported by the FWF, P32086

principal investigator: Harold Steinacker

project members: Yuhma Asano (postdoc)
                            Emmanuele Battista (postdoc),
                            Emil Broukal (Master's student)                         
                            Laura Felder (Master's student)

The starting point of this project is a covariant cosmological quantum space-time solution of the IIB or IKKT matrix model as described in   

•  M. Sperling and H. C. Steinacker, ``Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model,''  JHEP 1907, 010 (2019)  [arXiv:1901.03522 [hep-th]].

• H. C. Steinacker, ``On the quantum structure of space-time, gravity, and higher spin in matrix models,'
  Class. Quant. Grav. \textbf{37} (2020) no.11, 113001 [arXiv:1911.03162 [hep-th]]
  (this is an introductory review paper).
  

The mathematical basis is the quantized twistor space, which also underlies fuzzy H⁴. As a solution of the matrix model it defines a 3+1 dimensional covariant quantum space-time describing a FLRW cosmology with Big Bounce. This leads to a higher-spin gauge theory, which includes all degrees of freedom required for gravity, and should be well suited for quantization within the IIB model.
The aim of the project is to develop and elaborate this theory from various points of view, including the resulting gravity theory in the non-linear regime, and more generally the resulting higher spin gauge theory.

As a start, the linearized Schwarzschild solution of the classical model was found in

• H. C. Steinacker, “Scalar modes and the linearized Schwarzschild solution on a quantized FLRW space-time in Yang–Mills matrix models", Class.Quant.Grav. 36 (2019) no.20, 205005 [arXiv:1905.07255]

 and the crucial no-ghost-theorem was established in

• H. C. Steinacker, “Higher-spin kinematics & no ghosts on quantum space-time in Yang-Mills matrix models"  [arXiv:1910.00839]

This means that the theory is free of pathologies, and provides a solid basis for further work.
A useful description of the non-linear regime of the emergent gravity theory was developed in

• H. C. Steinacker, `Higher-spin gravity and torsion on quantized space-time in matrix models,'' JHEP \textbf{04} (2020), 111  [arXiv:2002.02742].
  
  
• S. Fredenhagen and H. C. Steinacker, ``Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution,''   [arXiv:2101.07297].

Unlike in general relativity, it turns out that the geometry is most conveniently described using a Weitzenböck connection, rather than the Levi-Civita connection. The  reason is that the matrix model provides a preferred frame, which naturally defines an associated Weitzenböck connection.

The most general rotationally invariant solutions of the classical matrix model action were found in https://arxiv.org/abs/2112.08204. They reproduce the linearized Schwarzschild solution, but differ from Schwarzschild at the non-linear level (keep in mind that this is just the classical theory; the Einstein-Hilbert action comes into play only at the quantum level, as explained below).
Further details, including the explicit form of the symplectic form on the underlying 6-dimensional bundle space (a.k.a. twistor space) were elaborated in the Master's thesis of Emil Broukal.

Furthermore, in collaboration with Jurai Tekel the useful technique of “string states” was elaborated further in the example of the fuzzy sphere. This was introduced some years ago as a technique to evaluate one-loop integrals in non-commutative field theory and matrix models. In https://arxiv.org/abs/2203.02376, we pushed this technique further and established several novel properties.

Finally, an alternative formulation of the higher-spin theory emerging from the matrix model in terms of spinors was proposed in collaboration with Tung Tran in https://arxiv.org/abs/2203.05436, and developed further in https://arxiv.org/abs/2305.19351. This should be useful to compute scattering amplitudes in the higher-spin gauge theory, as explored in follow-up work with Tung.

We also studied more physical aspects of the underlying cosmological space-time:

•  In https://arxiv.org/abs/2207.01295, we first studied the geodesic motion of particles across the Big Bounce (BB). It turns out that the singularity is sufficiently mild to allow a well-defined propagation across the Big Bounce. Even though this is only a classical analysis, it does suggest that some information can be transmitted through the Big Bounce in the Matrix Model. This could have important implications for early universe cosmology, however a more detailed understanding of the model in the quantum regime is required.
  
  
• The propagator for scalar fields on this background  is computed in https://arxiv.org/abs/2207.01295 using QFT techniques. It resembles the standard Feynman propagator of flat Minkowski space in the late-time regime, whereas for early times it gives rise to a remarkable correlation between two points on opposite side of the BB. Similar results were obtained in a related collaboration with Joanna Karczmarek in 1+1 dimensions in https://arxiv.org/abs/2207.00399.
  
  
• Extending the analysis to fermions, the propagation of fermions on generic, curved branes in IKKT-type matrix models is
  studied in https://arxiv.org/abs/2212.08611. In particular, we provide the relation between the matrix model Dirac operator and the standard Dirac operator, which differ only by an extra coupling to the totally antisymmetric part of the torsion. Using this result, the coupling of spin to the background geometry was studied in detail using the JWKB approximation.
  
  
• The one-loop corrections of the cosmological solutions of the IKKT model was computed in https://arxiv.org/pdf/2310.11126.pdf. The quantum spacetime was found to be stabilized through one-loop effects at early times, but it is modified at late times, going through a period of acceleration. This is qualitatively close to observation without any fine-tuning, irrespective of the detailed matter content of the universe.

Quasi-coherent states and general framework of quantum spaces

A systematic way to extract the semi-classical geometry associated to some given matrix configuration was developed in https://arxiv.org/abs/2009.03400, based on quasi-coherent states. This is a rather general and conceptual work on the general foundation of quantum or matrix geometry, which sheds new light on the underlying mathematical structures
    (this also led to a Master’s thesis by Laura Felder).

One-loop effective action

One of the main results (and goals) of this project was the computation of the one-loop effective action for the geometry, and to clarify its significance for gravity. This was achieved in https://arxiv.org/abs/2110.03936, and elaborated in more detail in https://arxiv.org/abs/2303.08012. These papers establish a new mechanism for 3+1-dimensional gravity arising as a quantum effect on quantum space-time.

More specifically, the 3+1-dimensional Einstein-Hilbert action arises in the 1-loop effective action on noncommutative branes in the IKKT matrix model. The presence of compact fuzzy extra dimensions K as well as maximal supersymmetry of the model is essential. The effective Newton constant is determined by the Kaluza-Klein scale of K. Moreover, the vacuum energy does not act as cosmological constant, since is given in terms of the symplectic volume form rather than the Riemannian volume form. This result suggests that the IKKT model may indeed lead to a viable theory of gravity at the quantum level.

This even allows to derive modified Einstein equations, see https://arxiv.org/pdf/2312.01317

here is a colloqium talk given at DAMPT Cambridge, explaining some of these results.

... and here is a little fun talk on "Leben wir in der Matrix?" given at the pint of science festival in 2022.

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