The divine sympathy: an essay on kenotic Christology

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Cohomology of skew-holomorphic Lie algebroids

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.Comment: 16 pages. v2: Final version to be published in Theor. Math. Phys. (incorporates only very minor changes

Modular classes of Poisson-Nijenhuis Lie algebroids

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.Comment: To appear in Letters in Mathematical Physic

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.Comment: 28 pages. v2: Some results strengthened and references added. v3: Minor corrections, section numbering changed to match published version. v4: Sign errors in Proposition 6.8(d) corrected. This version incorporates an erratum to the published versio

Integral Grothendieck-Riemann-Roch theorem

We show that, in characteristic zero, the obvious integral version of the Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck's strategy: we use resolution of singularities and the weak factorization theorem for birational maps.Comment: 24 page

On localization in holomorphic equivariant cohomology

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.Comment: 16 pages. Completely rewritten, new title. v3: Minor changes in the exposition. v4: final version to appear in Centr. Eur. J. Mat

Weak splittings of quotients of Drinfeld and Heisenberg doubles

We investigate the fine structure of the simplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson-Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties such as the open Richardson varieties.Comment: 20 pages, AMS Late

Continuous non-perturbative regularization of QED

We regularize in a continuous manner the path integral of QED by construction of a non-local version of its action by means of a regularized form of Dirac's $\delta$ functions. Since the action and the measure are both invariant under the gauge group, this regularization scheme is intrinsically non-perturbative. Despite the fact that the non-local action converges formally to the local one as the cutoff goes to infinity, the regularized theory keeps trace of the non-locality through the appearance of a quadratic divergence in the transverse part of the polarization operator. This term which is uniquely defined by the choice of the cutoff functions can be removed by a redefinition of the regularized action. We notice that as for chiral fermions on the lattice, there is an obstruction to construct a continuous and non ambiguous regularization in four dimensions. With the help of the regularized equations of motion, we calculate the one particle irreducible functions which are known to be divergent by naive power counting at the one loop order.Comment: 23 pages, LaTeX, 5 Encapsulated Postscript figures. Improved and revised version, to appear in Phys. Rev.

A comparison of precipitation and filtration-based SARS-CoV-2 recovery methods and the influence of temperature, turbidity, and surfactant load in urban wastewater

Wastewater-based epidemiology (WBE) has become a complimentary surveillance tool during the SARS-CoV-2 pandemic. Viral concentration methods from wastewater are still being optimised and compared, whilst viral recovery under different wastewater characteristics and storage temperatures remains poorly understood. Using urban wastewater samples, we tested three viral concentration methods; polyethylene glycol precipitation (PEG), ammonium sulphate precipitation (AS), and CP select™ InnovaPrep® (IP) ultrafiltration. We found no major difference in SARS-CoV-2 and faecal indicator virus (crAssphage) recovery from wastewater samples (n = 46) using these methods, PEG slightly (albeit non-significantly), outperformed AS and IP for SARS-CoV-2 detection, as a higher genome copies per litre (gc/l) was recorded for a larger proportion of samples. Next generation sequencing of 8 paired samples revealed non-significant differences in the quality of data between AS and IP, though IP data quality was slightly better and less variable. A controlled experiment assessed the impact of wastewater suspended solids (turbidity; 0–400 NTU), surfactant load (0–200 mg/l), and storage temperature (5–20 °C) on viral recovery using the AS and IP methods. SARS-CoV-2 recoveries were >20% with AS and  0.05), whilst surfactant and storage temperature combined were significant negative correlates (p < 0.001 and p < 0.05, respectively). In conclusion, our results show that choice of methodology had small effect on viral recovery of SARS-CoV-2 and crAssphage in wastewater samples within this study. In contrast, sample turbidity, storage temperature, and surfactant load did affect viral recovery, highlighting the need for careful consideration of the viral concentration methodology used when working with wastewater samples

Ultraviolet Complete Electroweak Model Without a Higgs Particle

An electroweak model with running coupling constants described by an energy dependent entire function is utraviolet complete and avoids unitarity violations for energies above 1 TeV. The action contains no physical scalar fields and no Higgs particle and the physical electroweak model fields are local and satisfy microcausality. The $W$ and $Z$ masses are compatible with a symmetry breaking $SU(2)_L\times U(1)_Y \rightarrow U(1)_{\rm em}$, which retains a massless photon. The vertex couplings possess an energy scale $\Lambda_W > 1$ TeV predicting scattering amplitudes that can be tested at the LHC.Comment: 19 pages, no figures, LaTex file. Equation and text corrected. Reference added. Results remain the same. Final version published in European Physics Journal Plus, 126 (2011