Geometry and Symmetry in Quantum Field Theory



Scientific activities of the various Research Units

Quantum spacetimes
Quantum group symmetries
Non-commutative geometry
Algebraic and Topological QFT
Geometrical methods in statistical physics and complex systems

Experimental evidence of physics beyond the standard model of elementary particles is still lacking, and a large debate has emerged for providing new theoretical scenarios that call into play at a deeper level the role of gravitational physics and spacetime geometry. We are experiencing more and more the predictive power of General Relativity both from the experimental point of view, witnessed by the results of the LIGO-VIRGO and of the Event Horizon Telescope, as well as from the theoretical  side with a rich landscape of quantum gravity models that bear relevance to fundamental physics. A predictive power that  keeps on highlighting  the dual role that gravity plays in governing the universe at both the largest and the smallest  scales.

Einstein’s General Relativity, Yang-Mills and String theory embody the time-honoured interplay between the advancement of physical knowledge and the application of increasingly sophisticated mathematical methods. This fruitful framework is also typical of the research project we wish to address. Its range of topics cover a rich variety of models based on standard (perturbative and non-perturbative) approaches to QFT on flat and curved spacetimes, as well as the analysis of fundamental interactions exploiting non-commutative geometry, quantum group symmetries, and geometric flows.

The mathematical background of the participants ranges from differential, algebraic and complex geometry, geometric analysis, geometric and algebraic topology, Poisson and non-commutative geometry, representation theory of Lie groups, Lie and Hopf algebras and their generalizations, up to functional analysis and PDE theory.

Although the core targets are framed within (classical and) quantum field theories –algebraic and topological QFTs, quantum gravity models, particle phenomenology, relativistic cosmology, statistical field theory- the competences of the participants spread also over topics such as non-equilibrium statistical mechanics, quantum many-body systems, quantum computing, and modeling of complex systems.


Localization in Topological Field Theory: We plan to continue the study of the equivariant BV formalism. In the case of the AKSZ theory we introduced a general procedure for including the equivariant differential and studied the equivariant BV observables. Examples addressed so far include two dimensional topological Yang-Mills, Chern-Simons and Donaldson-Witten theory.

Shifted Poisson stacks: We intend to continue the study of Poisson structures on differentiable stacks and study several related aspects: the modular class for Poisson stacks and the implications of Morita equivalence in the study of generalized Kahler structures.

Generalized Kahler Structure:  We plan to extend our computation of the Kahler potential for a toric generalized structure on CP_2 to more general toric manifolds.

Physics of cold atoms: we are currently studying the effects of PT invariant pseudo Hermitian Hamiltonians in transport phenomena.

Classical phase transitions: recently, the phase transition undergone by a 3D lattice gauge model in the absence of a global symmetry-breaking has been successfully described in a topological framework. We shall pursue the investigation of the deep origin of phase transitions by means of topological concepts and methods applied to the energy level sets in phase space.



Spectral Action approach to the Standard Model. We plan to further study its symmetry, the finite mode (eigenvalue cutoff) regularization and its physical consequences.

Kappa-Minkowski NC spaces: We will study quantization/dequantization maps that act  as a Mellin transform and its inverse, the associated star-product (a variant of the Moyal-Weyl one); in particular, whether the quantizer maps L2 functions into Hilbert-Schmidt operators,  real functions into Hermitean operators. We will also study the curved momentum space of this theory, its symmetries for the variants of the commutation relations (space-, time-, or light-like).

Localizability in NC spaces and quantum reference frames. We will analyze the philosophical consequences of a NC ``pointless'' geometry, within the general interpretation of spacetime for general relativity and quantum gravity. We will speculate on the form of  space(time) symmetry transformations involving changes of quantum reference frames.

Submanifolds in NCG. We wish to investigate in several frameworks (Drinfel’d twist deformation, fuzzy spaces,…) when the notion makes sense, its geometrical meaning, induced metric and curvature, possible physical applications (e.g as boundaries) to QFT or QG on NC spaces.

Fuzzy spaces. We will look for a fuzzy (finite-dimensional matrix) approximation of complex projective and (anti)-De Sitter spaces. We will further  study our (recently introduced) fully O(d+1)-covariant fuzzy spheres SdΛ, construct those of dimensions d>2, build examples of QFT on them (with Λ as a UV cutoff), look for applications to condensed matter physics (e.g.quantum wires or graphene tubes/spheres, with systems of  fermions/bosons on S1Λ, S2Λ)  or quantum gravity (Λ might e.g. set a cutoff to the number of BMS charges associated to a black hole).

NC spaces with Lie algebra noncommutativity. We plan to study differential calculi, to develop gauge theories and to study the corresponding Yang-Mills equations on some types thereof.

Noncommutative gauge theory. We will proceed by a novel strategy, based on the conjecture that all well-defined NC gauge theory should  be consistently constructed bootstrapping some starting commutative gauge theory by a noncommutative deformation, so as to complete some L-infinity algebra. In particular, studying the kappa-Minkowski noncommutativity within this formalism looks very promising.

K-theory of quantum spaces described by noncommutative C*-algebras. For its computation we plan to develop and use the theory of Waldhausen categories, in alternative to attempted NC generalizations of  homotopy  and model categories, and to apply it to quantum spaces obtained by "gluing" simpler spaces (multipullback). A related project is to study Yetter-Drinfeld cohomology of Hopf algebroids, which are a NC generalization of (functions on) groupoids.

QFT on spaces with boundaries: We plan on one hand to develop the general theory studying the eta invariant (which is closely related to the boundary Chern-Simons terms and the parity anomaly), on the other to look for applications of this QFT framework to real physical systems, such as Dirac materials and  Weyl semi-metals.

Double and Generalised Geometries: our research will proceed in relation with invariance of string and field theories under T-duality. Both DG and GG provide the geometry of  Double Field Theory, which in turn furnishes generalisations of Einsten-Hilbert action of GR. Poisson-Lie T-duality of sigma models and its formulation in terms of Drinfel'd doubles will be further analyzed.

Quantum information geometry. We plan to use NCG techniques (C*-algebras, etc.) to study vector fields on infinite dimensional Banach manifolds.

Deformed Canonical (Anti)Commutation Relations. We will study their connections with: i) non-unitary dynamics via pseudo-hermitian operators, with possible alternative definitions of probability transitions; ii) exactly solvable quantum Hamiltonians; iii) quons, truncated bosons, bicoherent and bisqueezed states. Motivated by the study of some dissipative quantum systems we will also explore extensions to a distributional setting of pseudo-bosonic ladder operators.


Stochastic quantization and regularity structures: As first step we intend to provide an alternative and algorithmically less demanding approach to the existence of solutions of elliptic and parabolic stochastic PDEs. Moreover, we shall study the long time behaviour of these solutions in order to obtain an Euclidean quantum field theory.

Hadamard states on globally hyperbolic spacetimes with timelike boundaries: This line will be further improved to address arbitrary boundary conditions yielding a well-defined mixed boundary/Cauchy problem.

Unruh-de Witt detectors and Robin boundary conditions: We shall extend existing approaches   so to better characterize the dependence of the detector response from the boundary data.

Non-linear sigma model,  Renormalization Group and Ricci flow: The recent rigorous result  on the geometric flow (RG-2 flow), associated with the 2-loop  perturbative RG flow for the non-linear sigma model, has opened the way to the characterization of functionals which extend Perelman entropy to higher order and to address their physical applications.

Averaging procedures and cosmological backreaction: We plan in particular to  address the “cosmographic” approach: the general strategy exploits a subtle connection between the spacetime geometry of the past light-cone and the scale-averaging on observational data.

Geometrical and topological methods in complex systems and data science: The simplicial category -used for discretized TFQT and quantum gravity models- provides an ideal arena to address geometrical and topological aspects of large data sets and complex systems. In particular we intend to carry out an analysis of the geometrical quantities built from discrete curvatures (such as Regge’s, and Forman’s) and discuss their computational complexity.

Theory and phenomenology of amorphous solids: The cellular model introduced in the last few years has been shown to described very well the observed phenomenology  of real glasses.  From the theoretical point of view, the program is to develop in full a theoretical description of the equilibrium thermodynamic of glass melting, by resorting in particular to a description in terms of topological excitations that destroy the cellular jammed solid.

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Neutrino mixing in QFT and General Covariance: By considering role of neutrino mixing in the context of the decay of accelerated protons, we have found that the only setting compatible with a generally covariant formulation of QFT is the one based on flavor states.  We plan to improve the study of  neutrino oscillations in the Unruh radiation as well as the phenomenological implications of the condensate structure of the vacuum for flavor fields, in particular in connection with a possibile dynamical generation of fermion mixing.

GUP effects on QFT in flat and curved spacetimes: The possibility to test experimentally our predictions will be further discussed, in particular GUP (generalized uncertainty principle) effects on the Hawking and Unruh temperatures as well as corrections at the horizon scale of a black hole induced by GUP. Phenomenological applications of this have been considered  in the astrophysical context and will be further investigated.

‘t Hooft deterministic quantization and gauge theories: We will continue  on the study of 't Hooft quantization scheme, in particular gauge theories will be considered. 

Entanglement for relativistic fermions: We have further investigated the effects of  Lorentz boosts on the quantum correlations carried by a pair of massive spin-1/2 particles, by means of a formalism based on Dirac bispinors. The results will be applied to scattering processes, e.g. in QED.

Large deviations in statistical mechanics systems: We have studied  large deviations in statistical mechanics systems. Numerical study will be further carried out for interacting systems, like active matter systems, which have applications also in biology.

Axion-photon mixing in QFT: We have analyzed axion--photon mixing in QFT and have shown that the condensate structure of the vacuum for mixed fields could provide an indirect proof of the existence of such mixing. We will further proceed along these lines of research.


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Next meeting

April, 20-21 2022

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