G. Amelino-Camelia	Curved momentum spaces and quantum-gravity phenomenology I illustrate recent progress in the understanding of theories with Planck-scale-curved momentum space, focusing on cases where momentum space has de Sitter geometry. Of particular interest is to compare the relativistic properties of the case with Levi-Civita connection on momentum space and the case with the so-called kappa-Poincaré connection on momentum space. I also show that the geometry of momentum space provides a valuable tool for analyzing possible phenomenological implications, including dual redshift and dual gravity lensing, and observe that Planck-scale sensitivity for such experimental studies is within reach.
M. Arzano	Purity is not eternal at the Planck scale Curved momentum space realized in terms of a Lie group manifold provides the main prototype for modelling relativistic symmetry deformation at the Planck scale. Under a generalization of the ordinary Fourier transform a field theory on momentum group manifold exhibits a "dual" representation in terms of a non-commutative field theory. The appeal of these models is that they naturally emerge in the description of a gravitating point particle in three dimensional gravity and in four dimensions they can lead to testable predictions, mainly based on their deformed energy-momentum dispersion relations. In this talk I will briefly review the basic notions of group momentum space in terms of non-local symmetry generators and how they emerge in the context of three-dimensional gravity. After introducing the kappa-Poincaré algebra as a four dimensional analogue of such model, I will discuss how the non-trivial integration measure on momentum space and deformed dispersion relation are related to the phenomenon of running dimensions and they can realize a scale-invariant spectrum of quantum fluctuations in the early universe. Finally I will introduce a new physical interpretation of the deformed action of time translation generators on operators which characterizes these models. I will show how such action leads to a Lindblad-like evolution equation for density matrices when expanded at leading order in the Planckian deformation parameter. This evolution equation allows pure states to evolve into mixed states. This observation has potential applications for the black hole information paradox and can open new phenomenological windows for Planck scale physics.
P. Aschieri	Deformation quantization of principal bundles Drinfeld twisting procedure leads to noncommutative (*-product) deformations of the algebra of functions of a usual manifold. First I briefly recall how this procedure can be extended to deform the differential and Riemannian geometry of the manifold, including diffeomorphisms and connections not necessarily equivariant under the Hopf algebra of the twist. Then I present how similar techniques allow to canonically deform principal G-bundles (and in general how Hopf-Galois extensions are canonically deformed to new Hopf-Galois extensions). Both the structure group and the base manifold can be rendered noncommutative, thus some examples studied in the literature are recovered by applying this general theory.
E. J.Beggs	Semiquantisation Functor and Poisson-Riemannian Geometry I will begin with some remarks on noncommutative geometry in Physics. Next I will consider the first order deformation of a Riemannian manifold, including the vector bundles, differential calculus and metric. One example will be the Schwarzschild solution, which illustrates that not all the properties of the classical case can be simply carried into the quantum case. The other example is quantising the Kahler manifold, complex projective space. This case is much simpler, and here the complex geometry is also preserved. I will end with some comments on the connection between noncommutative complex geometry and noncommutative algebraic geometry.
J. Buchbinder	Towards Harmonic Superfield Formulation of N=4, USp(4) Supersymmetric Yang-Mills Theory We develop a superfield formulation of N=4 supersymmetric Yang-Mills theory with the rigid central charge in USp(4) harmonic superspace. The component formulation of this theory was given by Sohnius, Stelle and West in 1980, but its superfield formulation is not constructed so far. We construct the superfield action, corresponding to this model, and show that it reproduces the known component.
G. Fiore	On covariance of QFT on Moyal spaces under finite deformed Poincare On covariance of QFT on Moyal spaces under finite deformed Poincaré transformations, and quantum reference frames Abstract. We point out why "finite" space-time transformations on a quantum group covariant noncommutative space are to be described using the coaction of the (deformed) Hopf algebra H of functions on the group G, rather than the action of the Hopf algebra H' deformation of the UEA of Lie(G), dual to H. The noncommutativity of H, i.e. of the parameters of a change of reference frames, hints at the quantum nature of the latter. Using H we then propose, at least at the formal level, a formulation of "twisted" Poincaré covariance of quantum field theory on Groenewold-Moyal-Weyl noncommutative spaces, as developed in previous works, and explore its consequences. A distinction between active and passive transformations is also proposed.
K. Fredenhagen	A generally covariant approach to perturbative quantum gravity The perturbative approach to quantum gravity is reconsidered, from the point of view of generally covariant local quantum field theory. Following older ideas of Nakanishi and De Witt, it is shown that a consistent perturbative treatment is possible. Crucial is the tretament of coordinates as dynamical quantum fields. In view of non-renormalizability the theory has to be interpreted as an effective theory.
A. Goerlich	Geometry of the Universe in CDT Causal Dynamical Triangulations (CDT) introduce a regularization of quantum gravity. In this framework, a quantum universe with the global shape of a Euclidean de Sitter spacetime emerges as dynamically generated background geometry. We investigate the microscopic and macroscopic properties of the geometry of this universe, using geodesic shell decompositions of spacetime. We focus on evidence of fractality and global anisotropy.
T. R. Govindarajan	Noncommutative geometry and manifolds with boundary - Fuzzy disc revisited Quantum Physics on manifolds with boundary exhibits novel features. These arise due to a variety of boundary conditions that will satisfy the requirement of self adjoint Hamiltonian. We study Laplace Beltrami operator on such manifolds and exhibit `edge states'. When fuzzy geometry with `boundary' is considered whose commutative limit leads to such manifolds one gets only Dirichlet conditions. We revisit the fuzzy geometry with boundary to enlarge the set of boundary conditions. We point out new features in quantum field theories on these fuzzy geometries.
D. Gurevich	Noncommutative Geometry on enveloping algebra and its applications to Physics The central problem of Noncommutative Geometry is constructing differential calculus on a given noncommutative algebra. Some known approaches to this problem will be mentioned in my talk. Also, I shall exhibit a new approach to constructing such a calculus on the enveloping algebras of Lie algebras gl(n) and their super-analogs. This approach is based on a new form of the Leibniz rule. As a result, the corresponding differential algebra can be treated as a quantization (deformation) of its commutative counterpart, namely, the differential algebra on the symmetric algebra of a given Lie algebra gl(n). The role of braided algebras (i.e., those related to the corresponding quantum groups) in constructing this calculus will be exhibited. Applications to quantization of some dynamical models by means of so-called "quantum spherical coordinates" will be also exhibited.
E. Ivanov	Deforming Supersymmetric Quantum Mechanics We present new models of N=4 supersymmetric mechanics based on the worldline superspace realizations of the superalgebra su(2|1) and its central extension, following recent works with S. Sidorov (1307.7690, 1312.6821 [hep-th]).
J. Lewandowski	The issues and advances in LQG
S. Majid	Emergence of Riemannian geometry out of the Leibniz rule for quantum spacetime We overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula for the standard Levi-Civita connection, a new point of view of the cosmological constant as a very small mass for the graviton of around 10^{-33}ev, and a weakening of metric-compatibility in the presence of torsion. The same mechanism also provides a new construction for quantum bimodule connections on quantum spacetimes and a new approach to the quantum Ricci tensor.
A. Nowicki	On relation between deformed Heisenberg algebra and finite dimensional Lie algebras The relation between nonlinear algebras and linear ones is discussed. For one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one. We consider two cases: (1) the linear algebra realized as three dimensional Lie algebra iso(1,1) or iso(2), and (2) the linear algebra reali
D. Oriti	Non-commutative tools in loop quantum gravity, spin foam models and group field theory We present recent results on the so-called non-commutative Fourier transform for quantum systems with Lie groups as configuration space and on the associated algebra representation. We highlight its dependence on quantisation maps, and we show the non-commutative plane waves corresponding to a few of them (in particular, the Duflo map). Next, we review the application of these tools to: 1) the definition of a non-commutative flux representation for loop quantum gravity, spin foam models and group field theories; 2) the study of the asymptotic semi-classical limit of spin foam models; 3) the construction of models for 4d quantum gravity.
B. Schroers	Deformed Poincare symmetry, Fourier transformations and wave equations in 2+1 dimensions Lorentz-covariant wave equations like the Proca, Dirac or Klein-Gordon equation can be obtained from the Mackey-Wigner formulation of irreducible representations of the Poincaré group via Fourier transformation. In this talk I will discuss how this picture changes when one deforms the Poincaré symmetry to the quantum double of the Lorentz group in 2+1 dimensions. This deformation arises in 2+1 dimensional quantum gravity, but the focus on this talk will be on various possible notions of Fourier transforms in this context and on the resulting wave equations in non-commutative spacetimes.
V. N. Tolstoy	Relativistic supersymmetries with double geometry
F. Toppan	Snyder non-commutativity from Jordanian twist I review the framework of the "Unfolded Quantization" in application to Drinfeld twistdeformations of non-relativistic quantum mechanical systems. Several examples of abelian twist (leading to constant non-commutativity) will be discussed, as well as the case of the Jordanian twist, leading to a Snyder-type non commutativity for the space coordinates. In the Jordan-Snyder case the deformed Hamiltonian is pseudo-hermitian. The non-additive effects (encoded in the coproduct) of deformed multiparticle Hamiltonians are computed and discussed.
S. Woronowicz	Quantum symmetries and description of composed systems