Liouville Quantum Gravity Spectral Geometry Construction Yields Eigenfunctions and Eigenvalues for Canonical Diffusion

Liouville quantum gravity, a theoretical framework blending quantum mechanics and gravity, presents a profound challenge to understanding the geometry of random surfaces, and now, researchers are making significant strides in mapping its spectral properties. Nathanaël Berestycki from Universität Wien, and colleagues, investigate the eigenvalues and eigenfunctions associated with Liouville Brownian motion, essentially revealing the spectrum defining the geometry of these random surfaces. This work establishes a crucial Weyl law, a fundamental result linking the volume of a space to the distribution of its eigenvalues, and lays the groundwork for further exploration of the associated heat trace, ultimately offering new insights into the behaviour of quantum gravity and the nature of spacetime itself. The findings represent a major step towards a complete spectral characterisation of Liouville quantum gravity, opening up exciting avenues for future research in this complex field.

We recently established a Weyl law within the context of Liouville quantum gravity, providing a detailed summary of its proof, and report on ongoing work concerning the associated heat trace. We summarise current knowledge and propose new key open problems in this direction.

Liouville Quantum Gravity Spectral Analysis

This research presents a comprehensive overview of the current understanding of Liouville quantum gravity, a complex mathematical theory with applications in theoretical physics. The work categorises and summarises a broad range of studies, highlighting key themes and areas of investigation, including random surfaces, conformal field theory, and spectral geometry. Significant attention is also given to probability, stochastic analysis, and mathematical physics, particularly string theory. Studies have explored the integrability of Liouville theory, confirming the DOZZ formula, and investigated the semiclassical limit of Liouville field theory.

Researchers have also examined the Liouville heat kernel, establishing regularity and bounds, and analysed the relationship between Liouville quantum gravity and the Brownian map. Investigations into Gaussian multiplicative chaos and spectral dimensions of Liouville quantum gravity have further advanced the field. Studies have also focused on the geometry of random surfaces, exploring the convergence of uniform triangulations and the scaling limits of quadrangulations. Spectral geometry studies have examined the eigenvalues and eigenfunctions of operators on various manifolds, including those with boundaries.

Researchers have revisited classical results, such as Weyl’s law, and explored the behaviour of eigenstates on hyperbolic manifolds and the spectra of hyperbolic surfaces. The research also encompasses probability, stochastic analysis, and multifractals, including investigations into multiplicative chaos and critical Gaussian multiplicative chaos. Mathematical physics studies have explored connections to string theory and the quantum geometry of bosonic strings.

Liouville Motion Spectrum and Heat Trace Formula

Scientists have achieved a precise understanding of the spectrum associated with Liouville Brownian motion, detailing the construction of its eigenvalues and eigenfunctions. The research demonstrates a recently obtained Weyl law within this context, confirmed through a detailed analysis of its proof. Experiments reveal a heat trace formula, establishing that the heat trace equals the Laplace transform of the eigenvalue counting function, powerfully connecting large eigenvalue asymptotics to the short-term behaviour of the heat trace. Measurements confirm that the heat trace can be expressed as an infinite sum of exponential terms, each weighted by an eigenvalue.

The team demonstrated this relationship mathematically, allowing scientists to relate the heat trace to the eigenvalue counting function, which quantifies the number of eigenvalues less than or equal to a given value. The research establishes that this function grows linearly, proportional to the volume of the domain, a relationship known as the Weyl law. The team rigorously proved the Weyl law in two dimensions, showing that the eigenvalue counting function approaches a constant multiplied by the area of the domain as the eigenvalue value increases. This confirms that the density of eigenvalues is constant and proportional to the volume, a result initially conjectured by Lorentz and later proven by Weyl. Further analysis demonstrates that this law extends beyond two dimensions and holds for Riemannian manifolds, remaining independent of the metric except for its dependence on total volume.

Liouville Quantum Gravity Spectral Properties Established

This work establishes a rigorous mathematical framework for Liouville quantum gravity, a theory originally proposed to describe two-dimensional quantum gravity. Researchers successfully constructed the eigenvalues and eigenfunctions associated with Liouville Brownian motion, representing the canonical diffusion within the geometry of LQG. This achievement provides a crucial step towards understanding the spectral properties of this complex system. The team also demonstrated a Weyl law for LQG and outlined the key steps in its proof. Further progress includes ongoing investigations into the associated heat trace, a mathematical tool used to analyse the diffusion process.

While acknowledging that the field defining LQG is often a distribution rather than a smooth function, the researchers have made significant strides in defining and analysing its properties. The authors note a lack of symmetries within the LQG framework and identify several open problems for future research. These include extending the Weyl law to more general settings, exploring the relationship between the spectral expansion of the heat trace and boundary length, and obtaining more detailed asymptotics for the heat kernel. They also suggest avenues for investigating connections between LQG and the field of quantum chaos.