Schedule for: 23w5081 - Gravity, Noncommutative Geometry, Cosmology

When studying quantum geometry models, it is worth asking how the basic types of space-time, such as the universe and black holes, look according to them. How, for example, quantum effects modify the known Schwarzschild spacetime. Some light is shed on this matter by applying the Oppenheimer-Snyder method of space-time construction.

We study the quantum dynamics of the Lemaître-Tolman-Bondi space-times using a polymer quantization prescription based on loop quantum cosmology that incorporates fundamental discreteness. By solving an effective equation derived from this quantization, we find solutions for a variety of asymptotically flat collapsing dust profiles. We find that the black hole singularity is resolved by a non-singular bounce, and the event horizon is replaced by transient apparent horizons.

A noncommutative extension of general relativity based on a twist-deformed spacetime naturally leads to an extended gravitational sector that includes two independent tetrad fields. Such extra gravitational degrees of freedom survive in the commutative limit of the theory, and thus may lead to potentially interesting deviations from general relativity on large scales. I will review the formulation of such a theory and discuss its connections with bimetric gravity, as well as its implications for the dynamics of cosmological models.

Both teleparallel and bigravity can be motivated by their actions being more reminiscent to the standard model of particle physics, compared to the one of Einstein and Hilbert. Furthermore, GR can be formulated as a teleparallel theory without the need for a York-Gibbons-Hawking boundary term. Bigravity on the other hand has the potential of introducing partially massless fields consistently (without introducing ghosts), which is a promising direction towards formulating a healthy renormalizable theory of gravity. However, explicit example is yet to be constructed. In this talk it will be shown explicit examples of teleparallel bigravity theories and a subclass will be ruled out using perturbation theory. Finally, future directions to investigate if healthy theories with partially massless fields will be indicated.

This talk will introduce the general conditions on covariance in gravitational theories from a canonical perspective. Evaluations of these conditions lead to viable candidate theories of modified gravity, including new ones that are not of higher curvature form. They can be understood as consistent effective space-time descriptions of fundamental quantum gravity theories and help to evaluate physical implications.

The interactions of the gravitational metric with other spin-2 fields are strongly constrained by the absence of ghost instabilities. We will motivate the study of such theories and then describe the ghost free theories of two interacting spin-2 fields (bimetric theories) and their known extensions to multiple spin-2 fields (multimetric theories). We will also comment on the geometrical implications of such theories.

Spectral methods applied beyond the framework of almost-commutative geometries allow for deriving an intriguing class of models resembling bimetric gravity theories. I will briefly discuss the main ideas, necessary tools, and the consequences for several cosmological scenarios. When applied to two-dimensional situations, we obtain models interpretable as describing interacting strings in the Higgs background. This talk is based on joint work with A. Sitarz.

By using the theory of quantum principal bundles and quantum principal connections, in this talk we are going to present the electromagnetic theory with magnetic monopoles in the Moyal-Weyl spacetime. This model closely follows the classical formulation of electromagnetic theory by considering the trivial principal U(1)--bundle on the Minkowski space-time, the second Bianchi identity to get two of the Maxwell equations and the Yang-Mills functional to get the other two equations. It is worth remarking that even when the topology of the bundle remains trivial, this formulation presents magnetic monopoles.

Spectral triples have emerged as the preferred means to encode geometric and physical information over noncommutative spaces. An interesting object is then the space of all spectral triples whose induced Connes' metric is topologically well behaved --- so called metric spectral triples. We present a distance on the space of metric spectral triple, the spectral propinquity, which evolved from the Gromov-Hausdorff distance, though it encodes additional spectral information, as illustrated for instance by the fact that distance zero between two metric spectral triples is null if, and only if, the two spectral triples are unitarily equivalent. We will explain with some examples how such a metric can be used to discuss approximations of classical and noncommutative spaces by finite dimensional "matrix models", as used sometimes in mathematical physics. We also will show that in some appropriate sense, the spectrum of the Dirac operators of spectral triples, as well as the bounded continuous functional calculus for these operators, and even action functionals associated with spectral triples, are continuous with respect to this new distance.

We define weakly isolated horizon as a quasilocal generalization of event horizon, in purely geometrical terms. We then perform a canonical analysis of general relativity in first order formalism in self-dual variables, with a special attention in the treatment and interpretation of boundary terms.

We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to a finite resolution. In our approach the traditional role played by C*-algebras is taken over by so-called operator systems. We consider C*-envelopes and introduce a propagation number for operator systems, which we show to be an invariant under stable equivalence and use it to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle, and of metric spaces up to finite resolution.

Using the noncommutative residue we define certain functionals of differential forms, the densities of which yield such tensors as: metric, Einstein, and torsion. We generalise these concepts in non-commutative geometry and show e.g. that for the conformally rescaled noncommutative two-torus the Einstein and the torsion functionals vanish. Also Hodge-de Rham, Einstein-Yang-Mills, and quantum $SU(2)$ group geometries are torsion-free, while the almost commutative $M\times Z_2$ has torsion if the two sheets are nontrivialy coupled.

As a general principle, the Reflection positivity (RF) correspondence has proved useful in mathematics and in many neighboring areas. In its original form, RP successfully combines powerful tools from analysis, from geometry, from representation theory to questions in quantum physics. For example, due to work by many authors, the RF-correspondence has served to link abelian (commutative) properties of Gaussian processes/fields in the Euclidean setting, to the context of non-commutativity in the study of quantum fields. And by now, RP has further become a powerful tool in non-commutative harmonic analysis, and in the theory of unitary representations of Lie groups.

Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. This lays down on operator ideal techniques that are used in both fields. In this talk we shall present semiclassical Weyl’s laws for Schrödinger operators on noncommutative manifolds (i.e., spectral triples). This shows that well known semiclassical Weyl’s laws in the commutative setting ultimately holds in a purely noncommutative setting. This extends and simplifies previous work of McDonald-Sukochev-Zanin. In particular, this allows us to get semiclassical Weyl’s laws on noncommutative tori of any dimension $n\geq 2$, which were only accessible in dimension $n\geq 3$ by the MSZ approach. There are numerous other examples as well. The approach relies on spectral asymptotics for some weak Schatten class operators. As a further application of these asymptotics we obtain far reaching extensions of Connes’ integration formulas for noncommutative manifolds. (For Riemannian closed manifolds, Connes’ integration shows that Connes’ NC integral recaptures the Riemannian measure.)

We consider the problem of Levi-Civita connections of arbitrary metrics on noncommutative spaces. After reviewing the triangular quantum group and the associated quantum (homogeneous) algebras cases we study the case of arbitrary Hopf algebras. In the context of Woronowicz bicovariant differential calculi we show how connections on one forms or vector fields extend to the braided symmetric tensor product of forms or vector fields. This allows to define a metric compatibility condition between an arbitrary connection (not necessarily a bimodule connection) and an arbitrary braided symmetric metric.The metric compatibility condition and the torsion free condition are solved for metrics conformal to central and equivariant metrics provided the braiding given by the differential calculus is diagonalizable (and an associated map invertible). Thus, existence and uniqueness of the Levi-Civita connection on a quantum group for this class of metrics that are neither central nor equivariant is proven. This includes the $SU_q(2)$ example.

Quantum gravity requires a quantum spacetime, which in turn requires quantum symmetries. This was the parallel development of noncommutative geometry and Hopf algebras/quantum groups. I will describe this with the examples of some Lie-algebra type noncommutative geometry, and add a further element: the need to have quantum observers in the theory.

We shall review the IH formalism and how the degrees of freedom at the horizon come about. We focus on possible non-commutativity connections at the horizon.

In a recent paper I have clarified the structure of standard quantum theory by placing (quantum) events in the central role. A brief exposition will be given on how that works and its relation to quantization in general. This then leads me to make some guesses about how GR might be quantized.

Motivated by the usefulness of graded geometry in physics, we introduce a graded geometric framework suitable for describing tensor fields of mixed symmetry. We test this framework in a variety of physical contexts such as (a) the construction of universal kinetic, mass and interaction terms in linear and nonlinear gauge theories, (b) the universal description of various single- and multi-field dualities and their higher Buscher rules, (c) the interpretation of generalised global symmetries as generalised isometries on graded manifolds and (d) the identification of a topological theta term in gravity and its physical consequences.

We investigate the notion of subsystem in the framework of spectral triple as a generalised notion of noncommutative submanifold. In the case of manifolds, we have found conditions on Dirac operators which turn embedded submanifolds into isometric submanifolds. We then suggest the definition of spectral subtriple based on the notion of submanifold algebras and the existing notion of Riemannian, isometric, and totally geodesic subtriples. Some example of subtriples are given in both commutative and some noncommutative cases, for example, an almost commutative spectral triple (describes physics of standard model). In addition, the links between these different notions of subtriples are investigated.

It is always interesting to find connections between NCG and other central areas of mathematics. Recent work gradually unravels a deep connections between NCG and random matrix theory. In this talk I shall explain some of the techniques we have employed so far. In some cases one can apply the Coulomb gas method to find the empirical spectral distribution and rigorously prove existence of phase transition. More recently we applied the newly developed bootstrap technique to find the moments of certain multi-trace and multi-matrix random matrix models suggested by noncommutative geometry. Using bootstrapping we are able to find the relationships between the coupling constants of these models and their second moments. Using the Schwinger-Dyson equations, all other moments can be expressed in terms of the coupling constant and the second moment. Explicit relations for higher moments are obtained. (Based on joint works with Hamed Hessam and Nathan Pagliaroli).

After reviewing some key hints and puzzles from the early universe, I will introduce recent joint work with Neil Turok suggesting a rigid and predictive new approach to addressing them. Our universe seems to be dominated by radiation at early times, and positive vacuum energy at late times. Taking the symmetry and analyticity properties of such a spacetime seriously leads to a new formula for the gravitational entropy of our universe, and a picture in which the Big Bang may be regarded as a kind of mirror. I will explain how this line of thought suggests new explanations for a number of observed properties of the universe, including: its homogeneity, isotropy and flatness; the arrow of time (i.e. the fact that entropy increases *away* from the bang); several properties of the primordial perturbations; the nature of dark matter (which, in this picture, is a right-handed neutrino, radiated from the early universe like Hawking radiation from a black hole); the origin of the primordial perturbations; and even the existence of three generations of standard model fermions. I will mention some observational predictions that will be tested in the coming decade, some open issues, and a question for which I hope NCG might help resolve.

We explore a matrix realization of a three-dimensional quantum space. It has an onion-like structure composed of concentric fuzzy spheres of increasing radius. The angular part of the Laplace operator is inherited from that of the fuzzy sphere. The radial part is constructed using operators that relate matrices of various sizes using the matrix harmonic expansion. As an example of this approach, we produce a numerical simulation of a scalar field theory, the heat transfer, study the hydrogen atom, and consider some analytical aspects of the scalar field theory on this space. We also compare this construction to related previous realizations a three-dimensional quantum space.

Following steps analogous to classical Kaluza-Klein theory, we solve for the quantum Riemannian geometry on $C^\infty(M) \otimes M_2(\mathbb{C})$ in terms of classical Riemannian geometry on a smooth manifold M, a finite quantum geometry on the algebra $M_2(\mathbb{C})$ of 2×2 matrices, and a quantum metric cross term. Fixing a standard form of quantum metric on $M_2(\mathbb{C})$, we show that this cross term data amounts in the simplest case to a 1-form $A_\mu$ on M, which we regard as like a gauge-fixed background field. We show in this case that a real scalar field on the product algebra with its noncommutative Laplacian decomposes on M into two real neutral fields and one complex charged field minimally coupled to $A_\mu$. We show further that the quantum Ricci scalar on the product decomposes into a classical Ricci scalar on M, the Ricci scalar on $M_2(\mathbb{C})$, the Maxwell action ∣∣F∣∣^2 of A and a higher order ∣∣A.F∣∣^2 term. Another solution of the QRG on the product has A = 0 and a dynamical real scalar field φ on M which imparts mass-splitting to some of the components of a scalar field on the product. Joint work with C. Liu arXiv:2303.06239 (gr-qc)

"The talk will discuss the rotation of the contour of functional integration in quantum gravity from Lorentzian geometries to Euclidean geometries. It uses the usual framework of an action for fields on a manifold, but under the assumption that this is a low-energy approximation only and that there is a high-energy cut-off on the modes of all fields. The contour rotation is used to explain the relation between the Lorentzian and Euclidean spectral triple formulations of gravity and the standard model of particle physics, explaining some features of the Euclidean models. It is hoped that these formulas will provide exact mathematical results when applied to theories that are fully non-commutative, though this is beyond the scope of this talk."

We review the applications of twisted spectral triples to the Standard Model. The initial motivation was to generate a scalar field, required to stabilise the electroweak vacuum and fit the Higgs mass, while respecting the first-order condition. Ultimately, it turns out that the truest interest of the twist lies in a new—and unexpected—field of 1-forms, which generates a torsion in the spin connection and is related to the transition from Euclidean to Lorentzian signature.

After a short review of main spectral functions and relations between them I will concentrate on their boundary parts. I will describe the cases when a boundary part of spectral function of a Dirac operator can be expressed through a spectral function of a Dirac operator on the boundary. I will also formulate some conditions when these boundary contributions may be related to boundary modes.

We discuss connections on higher structures such as Lie and Courant algebroids and explore the role of their basic curvature tensor and of the Atiyah cocycle in topological sigma models and higher gauge theories.

For noncommutative principal bundles which are equivariant for a triangular Hopf algebra we study an associated Atiyah sequence of braided infinite dimensional Lie algebras and corresponding splittings (a way to define connections). Elements of the sequence act on the (affine) space of connections as gauge transformations. From the sequence we derive a Chern-Weil homomorphism and braided Chern-Simons terms. We present explicit examples (including Levi-Civita connections) over noncommutative spheres.

Fuzzy geometries are finite matrix models provided with a geometrical structure. I will review some examples and novel features of some or these spaces, discuss how equivariant vector bundles over such are described and explain how such geometries can arise as emergent structures from quantized membranes.

In this talk we will introduce the notion of quantum geometric realisation of spectral triples. Given this axiomatic formalism, a quantum metric and a quantum Levi-Civita Connection we will show models of this on two noncommutative unital algebras: the Noncommutative Torus, C_\theta [T^2], and the Complex 2x2 Matrices, M_2(C). We will see that for the noncommutative torus, we obtain an even standard spectral triple, but now uniquely determined by full geometric realisability. Morover, for M_2(C), we are forced to the flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. This is joint work with Prof. Shahn Majid in arXiv2208.07821

Given linear operators $A_0, ..., A_n$, their projective spectrum is the set of tuples $(z_0, ..., z_n)$ in the complex projective space $P^n$ such that $z_0A_0+...+z_nA_n$ is not invertible. This talk reviews a recent application of projective spectrum to group representation theory and complex dynamics. Roughly speaking, a group representation $(\pi, \mathcal{H})$ is said to be self-similar if it has an intrinsic lifting to the direct sum $\mathcal{H}^d$ for some $d$. Through the consideration of projective spectrum, this lifting gives rise to a rational map F on the projective space. The Julia set of F has been studied in some recent papers. Remarkably, in the case of the infinite dihedral group, the Julia set turns out to coincide with the projective spectrum. This provides a rare nontrivial example in which a multivariable Julia set can be explicitly described.

Making use of its smooth structure only, out of a connected oriented smooth 4-manifold a von Neumann algebra is constructed. It is geometric in the sense that is generated by local operators and as a special four dimensional phenomenon it contains all algebraic (i.e., formal or coming from a metric) curvature tensors of the underlying 4-manifold. The von Neumann algebra itself is a hyperfinite factor of II_1-type hence is unique up to abstract isomorphisms of von Neumann algebras. Over a fixed 4-manifold this universal von Neumann algebra admits a particular representation on a Hilbert space such that its unitary equivalence class is preserved by orientation-preserving diffeomorphisms consequently the Murray--von Neumann coupling constant of this representation is well-defined and gives rise to a new and computable real-valued smooth 4-manifold invariant. Its link with Jones' subfactor theory is noticed as well as computations in the simply connected closed case are carried out. Application to the cosmological constant problem is also discussed. Namely, the aforementioned mathematical construction allows to reformulate the classical vacuum Einstein equation with cosmological constant over a 4-manifold as an operator equation over its tracial universal von Neumann algebra such that the trace of a solution is naturally identified with the cosmological constant. This framework permits to use the observed magnitude of the cosmological constant to estimate by topological means the number of primordial black holes about the Planck era. This number turns out to be negligible which is in agreement with known density estimates based on the Press--Schechter mechanism.

In this talk will review a recent connection established between toy models of Quantum Gravity originating from Noncommutative Geometry and 2D Liouville Quantum Gravity. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned some probability distribution. We refer to such models as Dirac ensembles. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a well-defined noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. Using well-established rigorous techniques of Random Matrix Theory, we derive the critical exponents and the asymptotic expansion of partition functions of various Dirac ensembles which match that of minimal models from Liouville conformal field theory coupled with gravity.

he 3+1-dimensional Einstein-Hilbert action is obtained from the 1-loop effective action on noncommutative branes in the IKKT matrix model. The presence of compact fuzzy extra dimensions as well as maximal supersymmetry of the model are essential. The classical matrix model defines a pre-gravity action with 2 derivatives less than the induced E-H action, governing the cosmological regime. The vacuum energy does not act as cosmological constant, but helps to stabilize the background.

We review the status and current prospects of the "Modular Algebraic Quantum Gravity" proposal (arXiv:1007.4094v1).

We discuss braided (quantum) field theories and their BV (algebraic) quantization. We use the homological (L-infinity algebra) description and the homological perturbation theory. Examples of scalar field theories and U(1) gauge theory are presented and their renormalizability properties are analyzed. The obtained results are compared with the results already present in the literature.

Heisenberg uncertainty principle suggests that one cannot simultaneously measure with precision, both position and momentum of particles. The extension of this idea suggests that space becomes fuzzy as one approaches the early universe. That is to say that all the components of position cannot be simultaneously measured with precision. Such a space is called non-commutative space. This talk probes the quantum mechanics of hydrogen atom on such a fuzzy space. Further corrections to the Hydrogen atomic energy spectrum due to non-commutative space have been evaluated. This asymmetry in spatial direction could lead to violation of the conservation of angular momentum, for instance, in weak interactions the direction of motion of the particle manifests a preference for the direction of its spin.