# Noncommutative Geometry
and Physics.

Quantum Spacetimes.

## November 29th - December 1st, 2021

MINI-WORKSHOP

(Horizon Europe Week)

A mini-workshop to review the state of the art of research on NCG and Quantum Spacetime organized at Jagiellonian University, Krakow.

**Organizer:**Andrzej Sitarz (UJ)

**Supported by:** Jagiellonian University and NCN project UMO-2020/37/B/ST1/01540

**Talks (click to see slides)**

29.11, 11:00 José Mourão

Infinite dimensional ambiguity in quantization and Quantum Gravity
Despite huge progresses in several fronts it is embarrassing that a fully consistent Quantum Theory of Gravity has not yet been found.
The difficulties motivate us to look back at the foundations of Quantum Physics and, in particular, to the nonuniqueness of quantization
of a physical system with phase space M. Quantization usually uses explicitly or implicitly a (possibly local) choice of subalgebra of "preferred observables".
Different choices typically lead to inequivalent quantizations predicting different spectra for the same observables.
Geometric quantization provides us with a mathematically convenient way of parametrizing infinite dimensional families of nonequivalent choices
of maximal local algebras of complex valued observables in evolution (or "polarizations") by reducing this problem
to the study of an extension of the moduli space of Kahler structures on M, Kah(M). This (infinite dimensional) moduli space
has (in every cohomology class) a beautiful metric, uncovered by Mabuchi and further studied by Donaldson and Semmes,
giving Kah(M) the structure of a symmetric space, Kah(M) = G_C/G, G being the group of Hamiltonian symplectomorphisms.
The geodesics in the space Kah(M) of quantizations of M are then naturally associated with imaginary time
symplectomorphisms of M and geometric quantization further provides liftings of such geodesics to the
quantum bundle in the form of coeherent state transforms, allowing us to compare different
quantizations.
We will comment on applications of this formalism to the fractional quantum Hall effect and
to Quantum Gravity in Ashtekar variables.

29.11, 11:55 Jerzy Lewandowski

29.11, 12:50 Joakim Arnlind

Quantized Minimal Surfaces
I will discuss and motivate different ways of approaching solutions to a particular set of double commutator equations that arise in the quantization of minimal surfaces, related to both string and membrane theories.

29.11, 18:00 Giacomo Rosati (zoom)

Testing quantum spacetime with gamma ray burst neutrinos and photons
I will review some recent results on tests of signatures of Planck-scale effects in the propagation of ultra-high energy astrophysical particles based on the possibility of in-vacuo dispersion suggested in some quantum gravity/quantum spacetime scenarios. My talk will be based mostly on these two papers, "Phenomenology of curvature-induced quantum-gravity effects" Phys.Lett.B 820 (2021) 136595,
"In-vacuo-dispersion features for GRB neutrinos and photons" Nature Astron. 1 (2017) 0139,
and references therein.

30.11, 9:00 Richard Szabo

30.11, 9:55 Larissa Jonke

Hopf algebra underlying L_infinity algebra
We show that a curved $L_\infty$-algebra can be considered as a graded Hopf algebra with compatible codifferential. We use this observation to construct the recently proposed braided $L_\infty$-algebra using Drinfel'd twist of the underlying Hopf algebra.

30.11, 10:50 Patrizia Vitale

30.11, 11:45 Denjoe O’Connor

Multi-matrix Trace relations and Hagedorn Transitions
For a system of D N×N matrices I will shown that trace
relations dominate the entropy for traces of length greater than N^2/4.
I will describe the implications for Hagedorn transitions in matrix
models.

30.11, 12:40 Wojciech Szymanski

30.11, 14:45 George Zoupanos

30.11, 15:45 Walter van Suijlekom (zoom)

Cyclic cocycles in the spectral action and one-loop corrections
We show that the spectral action, when perturbed by a gauge potential, can be written as a series of Chern-Simons actions and Yang-Mills actions of all orders. In the odd orders, generalized Chern-Simons forms are integrated against an odd $(b,B)$-cocycle, whereas, in the even orders, powers of the curvature are integrated against $(b,B)$-cocycles that are Hochschild cocycles as well. In both cases, the Hochschild cochains are derived from the Taylor series expansion of the spectral action $\tr(f(D+V))$ in powers of $V=\pi_D(A)$, but unlike the Taylor expansion we expand in increasing order of the forms in $A$. We then analyze the perturbative quantization of the spectral action in noncommutative geometry and establish its one-loop renormalizability as a gauge theory. We show that the one-loop counterterms are of the same Chern-Simons-Yang-Mills form so that they can be safely subtracted from the spectral action. A crucial role will be played by the appropriate Ward identities, allowing for a fully spectral formulation of the quantum theory at one loop.

30.11, 16:45 Harold Steinacker (zoom)

Gravity as a Quantum Effect on Quantum Space-Time
The 3+1-dimensional Einstein-Hilbert action is obtained from the 1-loop effective action
on noncommutative branes in the IIB or IKKT matrix model.
The presence of compact fuzzy extra dimensions $\cK$ as well as maximal supersymmetry of the model are essential.
The E-H action can be interpreted as interaction of $\cK$ with the
space-time brane via IIB supergravity, and
the effective Newton constant is determined by the Kaluza-Klein scale of $\cK$.
%, which is quasi-local due to the maximal supersymmetry of the model.
The bare matrix model defines a pre-gravity action with 2 derivatives less
than the induced E-H action, governing the cosmological regime.
The perturbative physics is confined to the space-time brane,
which for covariant quantum space-times includes
all dof of gravity, as well as a tower of higher-spin modes.
The vacuum energy is given in terms of the symplectic volume form, and hence does not
gravitate.

30.11, 17:45 Peter Schupp

1.12, 9:00 Paolo Aschieri / Thomas Weber

Levi-Civita connection in noncommutative Riemannnian geometry: the symmetric and the quantum group cases

1.12, 10:00 Arkadiusz Bochniak