The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.Comment: 21 pages, 15 figure

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra

We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Furusho, stating that the pentagon equation for grt_1-elements implies the hexagon equation

GENERATION OF BLOOD THROMBIN IN PATIENTS WITH CANCER OF PANCREATOBILIARY ZONE COMPLICATED WITH OBSTRUCTIVE JAUNDICE

Obstructive jaundice is one of the most frequent complications of cancer of pancreatobiliary zone that starts the developments of hemorrhagic complications. This group of patients has decrease of coagulation potential that is expressed in changes of process of thrombin generation even before the operative treatment. The article presents the results of the research of indices of process of thrombin generation by Calibrated. Automated. Thrombogram. method in 57 patients with cancer of pancreatobiliary zone complicated with obstructive jaundice before the operative treatment

Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism

In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality $L_\infty$-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality $L_\infty$-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of {\bf Kontsevich eye}. In particular we clarify the r\^ole played by the {\bf I-cube} introduced in \cite{CR1}. Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular we prove that they naturally inherit the structure of an $A_\infty$-algebra with an $A_\infty$-(bi)module.Comment: The first and second Section on $B_\infty$-algebras and modules have been completely re-written, with new results; partial revision of Section 3; the proofs in Section 4 and 5 have been re-formulated in a more general context; we added Section 8 on globalisatio

Formality theorems for Hochschild complexes and their applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.Comment: Submitted to proceedings of Poisson 200

Symmetry, Gravity and Noncommutativity

We review some aspects of the implementation of spacetime symmetries in noncommutative field theories, emphasizing their origin in string theory and how they may be used to construct theories of gravitation. The geometry of canonical noncommutative gauge transformations is analysed in detail and it is shown how noncommutative Yang-Mills theory can be related to a gravity theory. The construction of twisted spacetime symmetries and their role in constructing a noncommutative extension of general relativity is described. We also analyse certain generic features of noncommutative gauge theories on D-branes in curved spaces, treating several explicit examples of superstring backgrounds.Comment: 52 pages; Invited review article to be published in Classical and Quantum Gravity; v2: references adde

Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains

Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.Comment: 27 page