Gravity theory on Poisson manifold with R‐flux

A novel gravity theory based on Poisson Generalized Geometry is investigated. A gravity theory on a Poisson manifold equipped with a Riemannian metric is constructed from a contravariant version of the Levi‐Civita connection, which is based on the Lie algebroid of a Poisson manifold. Then, we show that in Poisson Generalized Geometry the R‐fluxes are consistently coupled with such a gravity. An R‐flux appears as a torsion of the corresponding connection in a similar way as an H‐flux which appears as a torsion of the connection formulated in the standard Generalized Geometry. We give an analogue of the Einstein‐Hilbert action coupled with an R‐flux, and show that it is invariant under both β‐diffeomorphisms and β‐gauge transformations.     

Electromagnetism,Local Covariance,the Aharonov–Bohm Effect and Gauss’ Law

We quantise the massless vector potential A of electromagnetism in the presence of a classical electromagnetic (background) current, j, in a generally covariant way on arbitrary globally hyperbolic spacetimes M. By carefully following general principles and procedures we clarify a number of topological issues. First we combine the interpretation of A as a connection on a principal U(1)-bundle with the perspective of general covariance to deduce a physical gauge equivalence relation, which is intimately related to the Aharonov–Bohm effect. By Peierls’ method we subsequently find a Poisson bracket on the space of local, affine observables of the theory. This Poisson bracket is in general degenerate, leading to a quantum theory with non-local behaviour. We show that this non-local behaviour can be fully explained in terms of Gauss’ law. Thus our analysis establishes a relationship, via the Poisson bracket, between the Aharonov–Bohm effect and Gauss’ law – a relationship which seems to have gone unnoticed so far. Furthermore, we find a formula for the space of electric monopole charges in terms of the topology of the underlying spacetime. Because it costs little extra effort, we emphasise the cohomological perspective and derive our results for general p-form fields A (p <  dim(M)), modulo exact fields, for the Lagrangian density ${\mathcal{L} = \frac{1}{2} dA\wedge*dA+ A\wedge*j}$ . In conclusion we note that the theory is not locally covariant, in the sense of Brunetti–Fredenhagen–Verch. It is not possible to obtain such a theory by dividing out the centre of the algebras, nor is it physically desirable to do so. Instead we argue that electromagnetism forces us to weaken the axioms of the framework of local covariance, because the failure of locality is physically well-understood and should be accommodated.     

0-extended supergravity and Chern-Simons theories

We give generalizations of extended Poincaré supergravity with arbitrarily many supersymmetries in the absence of central charges in three dimensions by gauging its intrinsic global SO(N) symmetry. We call these ₀ (Aleph-null) supergravity theories. We further couple a non-Abelian supersymmetric Chern-Simons theory and an Abelian topological BF theory to ₀ supergravity. Our result overcomes the previous difficulty for supersymmetrization of Chern-Simons theories beyond N = 4. This feature is peculiar to the Chern-Simons and BF theories including supergravity in three dimensions. We also show that dimensional reduction schemes for four-dimensional theories such as N = 1 self-dual supersymmetric Yang-Mills theory or N = 1 supergravity theory that can generate ₀ globally and locally supersymmetric theories in three dimensions. As an interesting application, we present ₀ supergravity Liouville theory in two dimensions after appropriate dimensional reduction from three dimensions.     

Effective Field Theory for Fermi Systems in a Large N Expansion

A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where N is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.     

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a generalization, known as p-form electrodynamics, in which (p−1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.     

Universal Star Products

One defines the notion of universal deformation quantization: given any manifold M, any Poisson structure Λ on M and any torsionfree linear connection ? on M, a universal deformation quantization associates to this data a star product on (M, Λ) given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor Λ, the curvature tensor R and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poissoncohomology.     

Letter: Ultra-High Energy Cosmic Rays and Symmetries of Spacetime

High energy cosmic rays allow probing phenomena that are inaccessible to accelerators. Observation of cosmic rays, presumably protons, with energies beyond 4 × 10¹⁹ eV, the so-called Greisen-Zatsepin-Kuzmin (GZK) cut-off, give origin to two puzzles: How do particles accelerate to such energies ? Are their sources within 50–100 Mpc from Earth, or Lorentz invariance is actually a broken symmetry ? We suggest an astrophysical test to verify the latter alternative and explore a possible connection with an alternative theory of gravity that exhibits preferred-frame effects.     

Random sets,q-distributions and quantum groups

We have shown that the non-extensivity of classical set theory is related to unitary quantum groups. Using this non-extensivity property, we define a q-distribution, a binomial q-distribution and a Poisson q-distribution.     

Kontsevich Deformation Quantization and Flat Connections

In Torossian (J Lie Theory 12(2):597–616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ω _n on the compactified configuration spaces [`(C)]_n,0{\overline{C}_{n,0}} of n points on the upper half-plane. Connections ω _n take values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω _n is flat.     

Reply to Comment on “Towards a large deviation theory for strongly correlated systems”

The computational study commented by Touchette opens the door to a desirable generalization of standard large deviation theory for special, though ubiquitous, correlations. We focus on three inter-related aspects: (i) numerical results strongly suggest that the standard exponential probability law is asymptotically replaced by a power-law dominant term; (ii) a subdominant term appears to reinforce the thermodynamically extensive entropic nature of q-generalized rate function; (iii) the correlations we discussed, correspond to Q-Gaussian distributions, differing from Lévy?s, except in the case of Cauchy–Lorentz distributions. Touchette has agreeably discussed point (i), but, unfortunately, points (ii) and (iii) escaped to his analysis. Claiming the absence of connection with q-exponentials is unjustified.     

The entropic boundary law in BF theory

We compute the entropy of a closed bounded region of space for pure 3d Riemannian gravity formulated as a topological BF theory for the gauge group SU(2)$\mathrm{SU}(2)$ and show its holographic behavior. More precisely, we consider a fixed graph embedded in space and study the flat connection spin network state without and with particle-like topological defects. We regularize and compute exactly the entanglement for a bipartite splitting of the graph and show it scales at leading order with the number of vertices on the boundary (or equivalently with the number of loops crossing the boundary). More generally these results apply to BF theory with any compact gauge group in any space–time dimension.     

On Quantizable Odd Lie Bialgebras

Motivated by the obstruction to the deformation quantization of Poisson structures in infinite dimensions, we introduce the notion of a quantizable odd Lie bialgebra. The main result of the paper is a construction of the highly non-trivial minimal resolution of the properad governing such Lie bialgebras, and its link with the theory of so-called quantizable Poisson structures.     

Semiclassical Limits of Ore Extensions and a Poisson Generalized Weyl Algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A₁, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A₁ is Poisson simple and established that the Poisson endomorphisms of A₁ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.     

EPR spectroscopy of poorly characterized systems: A historical and current view

Modeling of the CW-EPR spectra due to transition metal ions in non-crystalline systems grew in sophistication from the early 1960’s until fairly recently. A number of important effects have been introduced into the simulation of CW spectra over the past decade or so. These include allowance for local strain effects, the so-called “g-strain” and correlated “g-A strain”, the latter especially important in multifrequency spectroscopy. Limits have been reached in what is reasonable by way of guesses regarding the underlying lineshape function and further information can only be sought via ENDOR or ESEEM experiments in favourable cases. A clear understanding of the simplest way to describe field swept EPR via a master equation derived in the frequency domain will be reviewed in which asymmetric field swept lines sometimes occur naturally as a consequence of the theory. The paper will provide a review of the history of computer simulations and various statistical approaches to correlation effects covering examples as diverse as proteins and glasses. It is hoped that sufficient progress will have been achieved with a new Bruker ESP 380 pulsed spectrometer that some new results can also be reported.     

Spectral Duality Between Heisenberg Chain and Gaudin Model

In our recent paper we described relationships between integrable systems inspired by the AGT conjecture. On the gauge theory side an integrable spin chain naturally emerges while on the conformal field theory side one obtains some special reduced Gaudin model. Two types of integrable systems were shown to be related by the spectral duality. In this paper we extend the spectral duality to the case of higher spin chains. It is proved that the N-site GL _k Heisenberg chain is dual to the special reduced k + 2-points gl _N Gaudin model. Moreover, we construct an explicit Poisson map between the models at the classical level by performing the Dirac reduction procedure and applying the AHH duality transformation.     

Theoretical treatment of volume localized NMR spectroscopy in weakly coupled spin systems

A pulse sequence is proposed for obtaining volume selective echoes with both chemical shift and homonuclear scalar coupling refocused. Theory, in terms of product operator formalism, is presented and the full response of the spin system is computed. The theoretical spatial dependence of the power signal given by spins localized in different planes (x=0,y=0, respectivez=0) is presented.