Publications
Lie Theory, Cartan Geometry, Quantum Groups, Integrable Systems, Vision
The results established by people involved in CaLISTA Project will give rise to publications, which will be listed in this page.
The results established by people involved in CaLISTA Project will give rise to publications, which will be listed in this page.
Publications labelled "COST publications" regard authors from at least two different COST countries.
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WG3
The Geometry of Quantum Computing
https://arxiv.org/pdf/2312.14807
E. Ercolessi, R. Fioresi, T. Weber
In this expository paper we present a brief introduction to the ge-
ometrical modeling of some quantum computing problems. After a
brief introduction to establish the terminology, we focus on quantum
information geometry and ZX-calculus, establishing a connection be-
tween quantum computing questions and quantum groups, i.e. Hopf
algebras.
WG2
One loop effective actions in Kerr-(A)dS Black Holes
https://arxiv.org/abs/2405.13830
Paolo Arnaudo, Giulio Bonelli, Alessandro Tanzini
We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ODEs with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasi-normal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulae for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss exact properties of the thermal version of the BH effective actions. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulae can be given for higher spin matter and radiation fields in more general gravitational backgrounds.
WG2 and WG3
Rita Fioresi, Maria A. Lledo, Junaid Razzaq
Quantum Chiral Superfields
We study the supersymmetric extension of the conformal symmetry and ordinary super spacetime over the complex field. The different types of superspaces can all be seen Grassmannians or flag supermanifolds (that is, sets of subspaces of a given space). We introduce non commutative coordinates in (super) spacetime in such a way that the symmetries are preserved. In order to do so, the symmetry (super) group become also non commutative space or 'quantum group'.
DOI: 10.1088/1742-6596/2531/1/012015
COST publication, PR2
WG1
Johnson Allen Kessy, Dennis The
On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class
A well-known classical result due to Sophus Lie is the classification of scalar ODEs with maximal and submaximal symmetry. Proving analogous results for vector ODEs in fully generality is not feasible using Lie-theoretic techniques. Instead, Cartan geometric and representation-theoretic tools are used to completely resolve the problem for vector ODEs in the so-called C-class.
Rita Fioresi, Bin Shu
Basic quasi-reductive root data and supergroups
arxiv: https://arxiv.org/pdf/2303.18065.pdf
WG2
We investigate pairs (G,Y ), where G is a reductive algebraic group and Y a purely-odd G-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup G, that is, Gevis isomorphic to G, and the quotient G/Gevis G-equivariantly isomorphic to Y . We prove that, if Y satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. We call the corresponding supergroups basic quasi-reductive. We then classify connected quasi-reductive algebraic supergroups of monodromy type provided that: (i) the root system does not contain 0; (ii) g := Lie(G) admits a non-degenerate even symmetric bilinear form.
WG1
Nathan Couchet, Robert Yuncken
A groupoid approach to the Wodzicki residue
arxiv: https://arxiv.org/pdf/2303.15787.pdf
https://doi.org/10.1016/j.jfa.2023.110268
Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis [33] and Guillemin [19]. In this article we give a definition of the Wodzicki residue, using the language of r-fibered distributions from [24], [30], in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition [25] for contact and codimension 1 foliation Heisenberg manifolds.
WG1 and WG2
Vladislav G. Kupriyanov, Alexey A. Sharapov, Richard J. Szabo
Symplectic Groupoids and Poisson Electrodynamics
arxiv: https://arxiv.org/pdf/2308.07406.pdf
We develop a geometric approach to Poisson electrodynamics, that is, the semi-classical limit of noncommutative U(1) gauge theory. Our framework is based on an integrating symplectic groupoid for the underlying Poisson brackets, which we interpret as the classical phase space of a point particle on noncommutative spacetime. In this picture gauge fields arise as bisections of the symplectic groupoid while gauge transformations are parameterized by Lagrangian bisections. We provide a geometric construction of a gauge invariant action functional which minimally couples a dynamical charged particle to a background electromagnetic field. Our constructions are elucidated by several explicit examples, demonstrating the appearances of curved and even compact momentum spaces, the interplay between gauge transformations and spacetime diffeomorphisms, as well as emergent gravity phenomena.
https://doi.org/10.1007/JHEP03(2024)039
WG1
Keegan J. Flood, Mauro Mantegazza, Henrik Winther
Symbols in Noncommutative Geometry
In this paper we prove that the classical Lie bracket of vector fields can be generalized to the noncommutative setting by antisymmetrizing (in a suitable noncommutative sense) their compositions. To accomplish this, we study linear differential operators on modules over a unital associative algebra equipped with an exterior algebra. We provide necessary and sufficient conditions for jet modules to be representing objects for differential operators, which is, a priori, a key desideratum for any theory of noncommutative differential geometry. Further, we construct natural symbol maps, which play a crucial r\^ole in the construction of the aforementioned Lie bracket.
arxiv: https://arxiv.org/pdf/2308.00835.pdf
DOI:10.48550/arXiv.2308.00835
COST Publication, PR2
WG1 and WG4
R. Fioresi, A. Marraffa, J. Petkovic
A new perspective on border completion in visual cortex as bicycle rear wheel geodesics paths via sub-Riemannian Hamiltonian formalism
We present a review of known models and a new simple mathematical modelling for border completion in the visual cortex V1 highlighting the striking analogies with bicycle rear wheel motions in the plane.
arxiv: https://arxiv.org/pdf/2304.00084.pdf
https://doi.org/10.1016/j.difgeo.2024.102125
COST Publication PR2
ON RELATIVE TRACTOR BUNDLES
https://arxiv.org/pdf/2405.13614
ANDREAS CAP, ZHANGWEN GUO, AND MICHAL WASILEWICZ
This article contributes to the relative BGG-machinery for para-
bolic geometries. Starting from a relative tractor bundle, this machinery con-
structs a sequence of differential operators that are naturally associated to the
geometry in question. In many situations of interest, it is known that this se-
quence provides a resolution of a sheaf that can locally be realized as a pullback
from a local leaf space of a foliation that is naturally available in this situation.
An explicit description of the latter sheaf was only available under much more
restrictive assumptions.
WG1
WG1
Indranil Biswas, Benjamin McKay
Locally homogeneous holomorphic geometric structures on Moishezon manifolds
For any smooth projective variety with holomorphic locally homogeneous structure modelled on a homogeneous algebraic variety, we determine all the subvarieties of it which develop to the model.
arxiv: https://arxiv.org/abs/2302.13649
https://doi.org/10.3842/SIGMA.2024.030
WG1
Benjamin McKay
An introduction to Cartan geometries
We explain what Cartan geometries are, aiming at an audience of graduate students familiar with manifolds, Lie groups and differential forms.
arxiv: https://arxiv.org/pdf/2302.14457.pdf
10.48550/arXiv.2302.14457
R. Fioresi, F. Zanchetta
Deep Learning and Geometric Deep Learning: an introduction for mathematicians and physicists
arxiv: https://arxiv.org/pdf/2305.05601.pdf
WG4
In this expository paper we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successfull algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph Neural Networks. We go over the key ingredients for these algorithms: the score and loss function and we explain the main steps for the training of a model. We do not aim to give a complete and exhaustive treatment, but we isolate few concepts to give a fast introduction to the subject. We provide some appendices to complement our treatment discussing Kullback-Leibler divergence, regression, Multi-layer Perceptrons and the Universal Approximation Theorem.
WG1 and WG2
Laurenţiu Bubuianu, Douglas Singleton, Sergiu. I. Vacaru
Nonassociative black holes in R-flux deformed phase spaces and relativistic models of G. Perelman thermodynamics
This paper explores new classes of black hole (BH) solutions in nonassociative and noncommutative gravity, focusing on features that generalize to higher dimensions. The theories we study are modelled on (co) tangent Lorentz bundles with a star product structure determined by R-flux deformations in string theory. For the nonassociative vacuum Einstein equations we consider both real and complex effective sources. In order to analyze the nonassociative vacuum Einstein equations we develop the anholonomic frame and connection deformation methods, which allows one to decoupled and solve these equations. The metric coefficients can depend on both space-time coordinates and energy-momentum. By imposing conditions on the integration functions and effective sources we find physically important, exact solutions: (1) 6-d Tangherlini BHs, which are star product and R-flux distorted to 8-d black ellipsoids (BEs) and BHs; (2) nonassocitative space-time and co-fiber space double BH and/or BE configurations generalizing Schwarzschild-de Sitter metrics. We also investigate the concept of Bekenstein-Hawking entropy and find it applicable only for very special classes of nonassociative BHs with conventional horizons and (anti) de Sitter configurations. Finally, we show how analogs of the relativistic Perelman W-entropy and related geometric thermodynamic variables can be defined and computed for general classes of off-diagonal solutions with nonassociative R-flux deformations.
arxiv: https://arxiv.org/pdf/2207.05157.pdf
https://doi.org/10.1007/JHEP05(2023)057
COST Publication, PR2
WG1
Ivan Dimitrov, Rita Fioresi
Generalized root systems
arxiv: https://arxiv.org/pdf/2308.06852.pdf
We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs for short). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras.
The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.
WG3
P. Aschieri, R. Fioresi, E. Latini, T. Weber
Differential Calculi on Quantum Principal Bundles over Projective Bases
arxiv: https://arxiv.org/pdf/2110.03481.pdf
We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of covariant bimodules together with a morphism of sheaves -- the differential -- such that the Leibniz rule and surjectivity hold locally. The main class of examples is given by covariant calculi over quantum flag manifolds, which we provide via an explicit Ore extension construction. In a second step we introduce principal covariant calculi by requiring a local compatibility of the calculi on the total sheaf, base sheaf and the structure Hopf algebra in terms of exact sequences. In this case Hopf--Galois extensions of algebras lift to Hopf--Galois extensions of exterior algebras with compatible differentials. In particular, the examples of principal (covariant) calculi on the quantum principal bundles q(SL2(ℂ)) and q(GL2(ℂ)) over the projective space P1(ℂ) are discussed in detail.
https://doi.org/10.1007/s00220-024-05007-5
WG3
Milagrosa Aldana, María A. Lledó
The fuzzy bit
In this paper we revise the idea of appliying fuzzy sets to quantum theory. It is established that, appropriately using a particular Universe of discourse, Quantum Mechanics can be expressed as a theory of fuzzy sets, according to a result by Pycakz for quantum logics in general.
DOI: https://doi.org/10.3390/sym15122103
on Simmetry vol. 15 year 2023
WG3
Reduction of Quantum Principal Bundles over non affine bases
Rita Fioresi, Emanuele Latini, Chiara Pagani
https://arxiv.org/html/2403.06830v1
In this paper we develop the theory of reduction of quantum principal bundles over projective bases. We show how the sheaf theoretic approach can be effectively applied to certain relevant examples as the Klein model for the projective spaces; in particular we study in the algebraic setting the reduction of the principal bundle GL(n)→GL(n)/P=Pn−1(ℂ) to the Levi subgroup G0 inside the maximal parabolic subgroup P of GL(n). We characterize reductions in the sheaf theoretic setting.
WG3
QUANTISED sl2-DIFFERENTIAL ALGEBRAS
https://arxiv.org/pdf/2403.08521
ANDREY KRUTOV AND PAVLE PANDZIC
WG1
Geometric quantization and unitary highest weight Harish-Chandra supermodules
Meng-Kiat Chuah, Rita Fioresi
https://arxiv.org/abs/2405.16251
Geometric quantization transforms a symplectic manifold with Lie group action to a unitary representation. In this article, we extend geometric quantization to the super setting. We consider real forms of contragredient Lie supergroups with compact Cartan subgroups, and study their actions on some pseudo-Kähler supermanifolds. We construct their unitary representations in terms of sections of some line bundles. These unitary representations contain highest weight Harish-Chandra supermodules, whose occurrences depend on the image of the moment map. As a result, we construct a Gelfand model of highest weight Harish-Chandra supermodules. We also perform symplectic reduction, and show that quantization commutes with reduction.
WG4
Spontaneous Emergence of Robustness to Light Variation in CNNs With a Precortically Inspired Module
J. Petkovic, R. Fioresi
The analogies between the mammalian primary visual cortex and the structure of CNNs used for image classification tasks suggest that the introduction of an additional preliminary convolutional module inspired by the mathematical modeling of the precortical neuronal circuits can improve robustness with respect to global light intensity and contrast variations in the input images. We validate this hypothesis using the popular databases MNIST, FashionMNIST, and SVHN for these variations once an extra module is added.
https://doi.org/10.1162/neco_a_01691
COST Publication, PR2
WG4
Geometric deep learning for enhanced quantitative analysis of microstructures in X-ray computed tomography data
Research
M. Lapenna, A. Tsamos, F. Faglioni, R. Fioresi, F. Zanchetta & G. Bruno
Quantitative microstructural analysis of XCT 3D images is key for quality assurance of materials and components. In this paper we implement a Graph Convolutional Neural Network (GCNN) architecture to segment a complex Al-Si Metal Matrix composite XCT volume (3D image). We train the model on a synthetic dataset and we assess its performance on both synthetic and experimental, manually-labeled, datasets. Our simple GCNN shows a comparable performance, measured via the Dice score, to more standard machine learning methods, but uses a greatly reduced number of parameters (less than 1/10 of parameters), features low training time, and needs little hardware resources. Our GCNN thus achieves a cost-effective reliable segmentation.
https://doi.org/10.1007/s42452-024-05985-0
COST Publication PR2
WG4
Synthetic Data Generation for Automatic Segmentation of X-ray Computed Tomography Reconstructions of Complex Microstructures
by Athanasios Tsamos,Sergei Evsevleev, Rita Fioresi, Francesco Faglioni and Giovanni Bruno
The greatest challenge when using deep convolutional neural networks (DCNNs) for automatic segmentation of microstructural X-ray computed tomography (XCT) data is the acquisition of sufficient and relevant data to train the working network. Traditionally, these have been attained by manually annotating a few slices for 2D DCNNs. However, complex multiphase microstructures would presumably be better segmented with 3D networks. However, manual segmentation labeling for 3D problems is prohibitive. In this work, we introduce a method for generating synthetic XCT data for a challenging six-phase Al–Si alloy composite reinforced with ceramic fibers and particles. Moreover, we propose certain data augmentations (brightness, contrast, noise, and blur), a special in-house designed deep convolutional neural network (Triple UNet), and a multi-view forwarding strategy to promote generalized learning from synthetic data and therefore achieve successful segmentations. We obtain an overall Dice score of 0.77. Lastly, we prove the detrimental effects of artifacts in the XCT data on achieving accurate segmentations when synthetic data are employed for training the DCNNs. The methods presented in this work are applicable to other materials and imaging techniques as well. Successful segmentation coupled with neural networks trained with synthetic data will accelerate scientific output.
https://doi.org/10.3390/jimaging9020022
COST Publication PR2
WG1: Cartan Geometry and Representation theory
P. Truini, A. Marrani, M.Rios, W. de Graaf
Exceptional Periodicity and Magic Star Algebras. Generalized Roots, Gradings, Hermitian Vinberg T-Algebras and their Derivations
The introduction of Magic Star Algebras, a class of non-Lie, finite-dimensional algebras, generalizing, in an infinite yet periodic way, the exceptional Lie algebras, and the account for the appearance of Hermitian Vinberg cubic T-algebras, which generalize the Jordan cubic simple algebras, especially the celebrated Albert algebra
https://doi.org/10.1016/j.exmath.2024.125621
WG1: Cartan Geometry and Representation theory
Andreas Cap, Kaibo Hu
Bounded Poincaré operators for twisted and BGG complexes
Connects ideas from parabolic geometries to applied mathematics, in particular to the theory of elasticity. We construct bounded Poincare operators for BGG complexes of Sobolev forms coming from projective and conformal differential geometry on bounded Lipschitz domains in R^n. This also leads to complexes of polynomial forms which are an important ingredient for the construction of finite element spaces.
10.1016/j.matpur.2023.09.008
https://arxiv.org/abs/2304.07185
COST Publication PR2
WG1: Cartan Geometry and Representation theory
Dennis The
On uniqueness of submaximally symmetric parabolic geometries
Among the (regular, normal) parabolic geometries of type (G, P), there is a locally unique maximally symmetric structure and it has the symmetry dimension dim(G). The symmetry gap problem concerns the determination of the next realizable (submaximal) symmetry dimension. When G is a complex or split-real simple Lie group of rank at least three or when (G, P) = (G2, P2), we establish a local uniqueness result for submaximally symmetric structures of type (G, P).
https://arxiv.org/abs/2107.10500
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Andrea Rivezzi
On the universal Drinfeld-Yetter algebra
We give a combinatorial description of the universal Drinfeld-Yetter algebra by means of two original objects, namely the sets of Drinfeld-Yetter looms and Drinfeld-Yetter mosaics.
https://arxiv.org/abs/2404.16786
WG1: Cartan Geometry and Representation theory, WG2: Integrable Systems and Supersymmetry
Andrea Santi, Dennis The
Exceptionally simple super-PDE for F(4)
For the largest exceptional simple Lie superalgebra F(4), having dimension (24|16), we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second- and third-order, respectively.
https://arxiv.org/abs/2207.04531
https://doi.org/10.1142/S0219199723500530
COST Publication PR2
WG1: Cartan Geometry and Representation theory
Andreas Cap, Jan Slovak
Bundles of Weyl structures and invariant calculus for parabolic geometries
This is a rather technical article discussing operators naturally associated to certain geometric structures.
https://arxiv.org/abs/2210.16652
DOI:10.48550/arXiv.2210.16652
COST Publication, PR2
WG1: Cartan Geometry and Representation theory, WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Rita Fioresi & Robert Yuncken
Quantized semisimple Lie groups Introduction on the structure and representation theory of quantized semisimple Lie groups,
based on lectures given at the CaLISTA summer school in Prague in September 2023.
https://arxiv.org/abs/2403.17180
10.48550/arXiv.2403.17180
COST Publication, PR2
WG1: Cartan Geometry and Representation theory, WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Robert Yuncken
Algebraic isomorphisms of quantized homogeneous spaces
We prove that the q-deformations of compact semisimple Lie groups, or their homogeneous spaces, are all distinct as noncommutative algebraic varieties for 0<q≤1.
https://arxiv.org/abs/2409.06139
WG1: Cartan Geometry and Representation theory
Andreas Cap, A. Rod Gover
A Boundary-Local Mass Cocycle and the Mass of Asymptotically Hyperbolic Manifolds
This article relates the notion of mass of asymptotically hyperbolic manifolds that is used in general relativity to conformal geometry, tractor calculus and the theory of parabolic geometries.
https://arxiv.org/abs/2108.01373
WG1: Cartan Geometry and Representation theory
Andreas Cap, Thomas Mettler
Induced almost para-Kähler Einstein metrics on cotangent bundles
This article relates almost Kaehler Einstein metrics constructed using the geometric theory of Weyl structures for parabolic geometries to the classical Patterson-Walker metrics studied in general relativity
https://arxiv.org/abs/2301.03217
https://doi.org/10.1093/qmath/haae047
COST Publication PR2
WG1: Cartan Geometry and Representation theory, WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Andreas Cap, Christoph Harrach, Pierre Julg
Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1)
This article connects ideas from the theory of BGG sequences to representation theory of the group SU(n+1,1) with the aim to study of instances of the Baum-Connes conjecture which has close relations to non-commutative geometry
https://arxiv.org/abs/2402.08262
COST Publication PREPRINT (not in PR2)
WG1: Cartan Geometry and Representation theory
Andreas Cap, Keegan J. Flood, Thomas Mettler
Flat extensions of principal connections and the Chern-Simons 3-form
This is a foundational study of a general version of Chern-Simons 3-forms and the resulting invariants which in special cases are heavily used in theoretical physics.
https://arxiv.org/abs/2409.12811
COST Publication PREPRINT (not in PR2)
WG1: Cartan Geometry and Representation theory
A. S. Cattaneo, L. Menger, M. Schiavina
Gravity with torsion as deformed BF theory
We study a family of (possibly non topological) deformations of BF theory for the Lie algebra obtained by quadratic extension of so(3,1) by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin--Vilkovisky formalism. In particular, both Palatini--Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin--Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields.
https://arxiv.org/abs/2310.01877
DOI 10.1088/1361-6382/ad5135
COST Publication, PR2
WG1: Cartan Geometry and Representation theory
A. S. Cattaneo, F. Fila-Robattino, V. Huang, M. Tecchiolli
Gravity Coupled with Scalar, SU(n), and Spinor Fields on Manifolds with Null-Boundary
In this paper, we present a theory for gravity coupled with scalar, SU(n) and spinor fields on manifolds with null-boundary. We perform the symplectic reduction of the space of boundary fields and give the constraints of the theory in terms of local functionals of boundary vielbein and connection. For the three different couplings, the analysis of the constraint algebra shows that the set of constraints does not form a first class system.
https://arxiv.org/abs/2401.09337
COST Publication, not in PR2
WG1: Cartan Geometry and Representation theory
A. S. Cattaneo, N. Moshayedi
Equivariant BV-BFV Formalism
The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example.
https://arxiv.org/abs/2410.00045
WG1: Cartan Geometry and Representation theory
Athanasios Chatzistavrakidis, Noriaki Ikeda, Larisa Jonke
Geometric BV for twisted Courant sigma models and the BRST power finesse
A target space covariant and geometric formulation of the Batalin-Vilkovisky action for twisted Courant sigma models, whose geometric structure relies on a pre-Courant algebroid and they modify 3D courant sigma models by a 4-form Wess-Zumino term
https://arxiv.org/abs/2401.00425
https://doi.org/10.1007/JHEP07(2024)115
COST Publication PR2
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Stein Meereboer
Symmetries for spherical functions of type $\chi$ for quantum symmetric pairs
In this article certain symmetries of functions related to so called quantum symmetric pairs are shown. Quantum symmetric pairs are a quantum analog of symmetric pairs. The symmetries for classical symmetric pairs are well known, the results extend the symmetries to the quantum case using modern tools.
https://arxiv.org/pdf/2405.15401
WG1: Cartan Geometry and Representation theory
Athanasios Chatzistavrakidis, Toni Kodžoman, Zoran Škoda
Brane mechanics and gapped Lie n-algebroids
The higher structure of a gapped almost Lie n-algebroid is introduced, together with higher notions of connections, torsion and basic curvature tensors associated to higher Koszul multibrackets and their relation to higher dimensional Hamiltonian mechanics in the Batalin-Viskovisky formulation is established.
https://arxiv.org/abs/2404.14126
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Xiao Han and Peter Schauenburg
Hopf Galois extensions of Hopf algebroids
The paper generalise the theory of Hopf Galois extension to the Hopf algebroids case.
https://arxiv.org/abs/2406.11058
10.48550/arXiv.2406.11058
COST Publication, PR2
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Paolo Aschieri and Thomas Weber
Metric compatibility and Levi-Civita Connections on Quantum Groups
Arbitrary connections on a generic Hopf algebra H are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an H-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on SLq(2).
arxiv.org/abs/2209.05453v2
WG3
Emanuele Latini, Antonio Del Donno and Thomas Weber
On the Durdevic approach to quantum principal bundles
We revisit and extend the Durdevic theory of complete calculi on quantum principal bundles. In this setting one naturally obtains a graded Hopf-Galois extension of the higher order calculus and an intrinsic decomposition of degree 1-forms into horizontal and vertical forms. This proposal is appealing, since it is consistently equipped with a canonical braiding and exactness of the Atiyah sequence is guaranteed. Moreover, we provide examples of complete calculi, including the noncommutative 2-torus, the quantum Hopf fibration and differential calculi on crossed product algebras.
arxiv.org/abs/2404.07944
COST Publication PREPRINT (not in PR2)
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction
The Kostant-Kirillov-Souriau construction is generalized to the case of finite-dimensional Jordan algebras, leading to a new interpretation of well-known geometrical objects like the Fisher-Rao metric tensor, the Bures-Helstrom metric tensor, and the Fubini-Study metric tensor.
https://arxiv.org/abs/2112.09781
https://doi.org/10.3842/SIGMA.2023.078
COST Publication PR2
WG3: Noncommutative Geometry and Quantum Homogeneous Spaces
F. M. Ciaglia, S. Jiang, J.Jost, L. Schwachhöfer
A coadjoint orbit-like construction for Jordan superalgebras
The Kostant-Kirillov-Souriau construction is generalized to the case of finite-dimensional Jordan superalgebras.
https://arxiv.org/abs/2311.01333
10.48550/arXiv.2311.01333
COST Publication PR2
WG1: Cartan Geometry and Representation theory
Mohammad Enayati, Jean-Pierre Gazeau, Mariano A. del Olmo, Hamed Pejhan
Anti-de Sitterian "massive" elementary systems and their Minkowskian and Newtonian limits
We elaborate the definition and properties of ''massive" elementary systems in the (1+3)-dimensional Anti-de Sitter (AdS4) spacetime, on both classical and quantum levels. We fully exploit the symmetry group Sp(4,R), that is, the two-fold covering of SO0(2,3) (Sp(4,R)∼ SO0(2,3)×Z2), recognized as the relativity/kinematical group of motions in AdS4 spacetime. In particular, we discuss that the group coset Sp(4,R)/S(U(1)x SU(2)), as one of the Cartan classical domains, can be interpreted as a phase space for the set of free motions of a test massive particle on AdS4 spacetime; technically, in order to facilitate the computations, the whole process is carried out in terms of complex quaternions. The (projective) unitary irreducible representations (UIRs) of the Sp(4,R) group, describing the quantum version of such motions, are found in the discrete series of the Sp(4,R) UIRs. We also describe the null-curvature (Poincaré) and non-relativistic (Newton-Hooke) contraction limits of such systems, on both classical and quantum levels. On this basis, we unveil the dual nature of ''massive" elementary systems living in AdS4 spacetime, as each being a combination of a Minkowskian-like massive elementary system with an isotropic harmonic oscillator arising from the AdS4 curvature and viewed as a Newton-Hooke elementary system. This matter-vibration duality will take its whole importance in the quantum regime (in the context of the validity of the equipartition theorem) in view of its possible rôle in the explanation of the current existence of dark matter.
https://arxiv.org/abs/2307.06690
DOI:10.48550/arXiv.2307.06690
COST Publication PR2
WG1: Cartan Geometry and Representation theory
Enrico Celeghini, Manuel Gadella, Mariano A. del Olmo
Gelfand Triplets, Continuous and Discrete Bases and Legendre Polynomials
We consider a basis of square integrable functions on a rectangle, contained in R2, constructed with Legendre polynomials, suitable, for instance, for the analogical description of images on the plane or in other fields of application of the Legendre polynomials in higher dimensions. After extending the Legendre polynomials to any arbitrary interval of the form [a,b], from its original form on [−1,1], we generalize the basis of Legendre polynomials to two dimensions. This is the first step to generalize the basis to n-dimensions. We present some mathematical constructions such as Gelfand triples appropriate on this context. “Smoothness” of functions on space of test functions and some other properties are revisited, as well as te continuity of generators of su(1,1) on this context.
https://arxiv.org/html/2312.17743v2
WG1: Cartan Geometry and Representation theory, WG3: Noncommutative Geometry and Quantum Homogeneous Spaces —
Marija Dimitrijević Ćirić, Biljana Nikolić, Voja Radovanović, Richard J. Szabo, Guillaume Trojani
Braided Scalar Quantum Electrodynamics
New noncommutative deformation of scalar electrodynamics is formulated and its quantization is discussed. We used the formalism of braided L-infinity algebra and the homological perturbation theory. Already at the classical level, this theory is different compared with the "standard" noncommutative electrodynamics. At the quantum level, we showed that there is no UV/IR mixing at one-loop order.
https://arxiv.org/abs/2408.14583
https://doi.org/10.1002/prop.202400190
COST Publication PR2
WG1: Cartan Geometry and Representation theory
A. Chattopadhyay, T. Mandal, A. Marrani
Freudenthal Duality in Conformal Field Theory
The first extension of the non-linear map named "Freudenthal duality" in Conformal Field Theory
https://arxiv.org/abs/2406.09259
DOI:10.48550/arXiv.2406.09259
WG1: Cartan Geometry and Representation theory
A. Marrani, D. Corradetti, F. Zucconi
Physics with non-unital algebras? An invitation to the Okubo algebra
The first appearance of the Okubo algebra (lacking the unity!) in attempts to provide an algebraic model of Quantum Chromodynamics, the quantum field theory describing the strong nuclear fundamental interaction
https://arxiv.org/abs/2309.17435
10.48550/arXiv.2309.17435
COST Publication PR2