Quantum categories: quantization of the category of linear spaces

We generalize the notion of bialgebras or Hopf algebras and on this basis we define quantum categories. The quantization of the category of linear (super)spaces is constructed, with the examples. We establish a criterion for the classical value of the dimension of the polynomial function algebra on the full quantum subcategory defined by the Sudbery type commutational relations for quantum vector spaces (the ``Poincare -- Birkhoff -- Witt property''). The criterion is the equality of the ``quantum constants'' c_\alpha=c_\beta^{\pm 1} for quantum spaces in consideration. We also establish links with the Yang -- Baxter equation and the Yang -- Baxter structures of quantum linear spaces. The role of categories as a generalization of groups and the related topic of a ``category programme'' are discussed in the introduction

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent \Delta operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the "big bracket" of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.Comment: 42 pages, LaTeX. v2: Added remarks in Section 5. v3: Added further explanation. v4: Minor adjustments. v5: Section 5 completely rewritten to include covariant construction. v6: Minor adjustments. v7: Added references and explanation to Section

On the AKSZ formulation of the Rozansky-Witten theory and beyond

Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the Rozansky-Witten model, which can be defined for any complex manifold with a closed (2,0)-form. We also construct the holomorphic version of Rozansky-Witten theory defined over Calabi-Yau 3-fold.Comment: 12 page

The symplectic origin of conformal and Minkowski superspaces

Supermanifolds provide a very natural ground to understand and handle supersymmetry from a geometric point of view; supersymmetry in $d=3,4,6$ and $10$ dimensions is also deeply related to the normed division algebras. In this paper we want to show the link between the conformal group and certain types of symplectic transformations over division algebras. Inspired by this observation we then propose a new\,realization of the real form of the 4 dimensional conformal and Minkowski superspaces we obtain, respectively, as a Lagrangian supermanifold over the twistor superspace $\mathbb{C}^{4|1}$ and a big cell inside it. The beauty of this approach is that it naturally generalizes to the 6 dimensional case (and possibly also to the 10 dimensional one) thus providing an elegant and uniform characterization of the conformal superspaces.Comment: 15 pages, references added, minor change

Are Nonrenormalizable Gauge Theories Renormalizable?

We raise the issue whether gauge theories, that are not renormalizable in the usual power-counting sense, are nevertheless renormalizable in the modern sense that all divergences can be cancelled by renormalization of the infinite number of terms in the bare action. We find that a theory is renormalizable in this sense if the {\em a priori} constraints that we impose on the form of the bare action correspond to the cohomology of the BRST transformations generated by the action. Recent cohomology theorems of Barnich, Brandt, and Henneaux are used to show that conventionally nonrenormalizable theories of Yang-Mills fields (such as quantum chromodynamics with heavy quarks integrated out) and/or gravitation are renormalizable in the modern sense.Comment: 28 pages, no figures. This is a plain LaTeX file. No special macros are needed. Numerous changes are made in the previous version, including added reference

Експрес-оцінка репродуктивної здатності псів в умовах розплідника

In recent decades doctors of veterinary medicine have devoted particular attention to dogs and, in particular, to the questions of physiology and pathology of their reproduction. The result is the publication of a significant number of manuals and monographs devoted to the actual issues of diagnosis and treatment of small animal diseases. Little remains the problems of andrological pathology of dogs, while the requirements for quality and number of offspring in high-value breeds of dogs and, consequently, reproductive health dogs. An important direction in solving these issues is the holding of andrological dispensary, which allows to diagnose the form of impotence, to identify the causes, to develop medical and preventive measures. However, in the conditions of large nurseries to study a significant number of dogs in limited time, there is a need for an accelerated, but at the same time, objective assessment of reproductive capacity of males. Proceeding from this, the aim of the work was to validate the method developed by us for express evaluation of the reproductive capacity of dogs in the conditions of a particular nursery.We have developed an express assessment of the reproductive capacity of dogs, which involves conducting a general clinical trial of males, a survey of the genitourinary system, determining the manifestation of sexual reflexes and the withdrawal of spermograms. The obtained results of research were distributed on a scale: «G» (good) – pathological changes were not found, all indicators of homeostasis and spermograms within the norm. The pedigree can be used without restrictions; «S» (satisfactory) – slight deviations from the norm, which have a minor effect on the body and partly reduce the reproductive potential of the dog. The herb is allowed to reproduce with a restriction on the careful control of its reproductive use; «U» (unsatisfactory) – a pathological state of the body, hypotension or impotence. Breeding is not allowed.According to the results of the research, it was determined that express assessment of the reproductive capacity of the dogs makes it possible to carry out a quick and fairly objective assessment of the reproductive potential of a large number of dogs for a limited period, to identify diseased animals, to develop and implement the necessary preventive or curative measures.It was established that from the total number of evaluated dogs of the «Zvezdnoye schastye» nursery, 12 dogs (or 76%) were rated «good» and recommended for breeding, 4 ‒ were «satisfactorily» (18%) and could be allowed to be restrained with careful control their reproductive use, 1 ‒ «unsatisfactory» (6%) with the recommendation to carry out the treatment of concomitant pathologies and pass a repeat test in a month. Розроблено експрес-оцінку репродуктивної здатності псів, яка передбачає проведення загального клінічного дослідження самців, обстеження стану сечостатевої системи, визначення прояву статевих рефлексів та виведення спермограми. Одержані результати розподіляються за шкалою: «Д» (добре) – патологічних змін не виявлено, усі показники гомеостазу і спермограми в межах норми. Плідника можна використовувати без обмежень; «З» (задовільно) – незначні відхилення від норми, що мають несуттєвий вплив на організм і частково знижують репродуктивний потенціал пса. Самца допускають до відтворення з обмеженням за ретельного контролю його репродуктивного використання; «Н» (незадовільно) – патологічний стан організму, гіпо- чи імпотенція. До розведення не допускається.За результатами досліджень визначено, що експрес-оцінка репродуктивної здатності псів дозволяє провести швидку та досить об’єктивну оцінку репродуктивного потенціалу великої кількості псів за обмежений термін, виявити неплідних тварин, розробити та здійснити необхідні превентивні чи лікувальні заходи.Встановлено, що із загальної кількості оцінених псів розплідника «Zvezdnoye schastye» оцінку «добре» отримали 12 псів (або 76%) і рекомендовані до розведення, 4 – «задовільно» (18%) і можуть бути допущеними до відтворення з обмеженням за ретельного контролю їх репродуктивного використання, 1 – «незадовільно» (6%) з рекомендацією провести лікування супутніх патологій та пройти повторний тест через місяць.

Courant-Dorfman algebras and their cohomology

We introduce a new type of algebra, the Courant-Dorfman algebra. These are to Courant algebroids what Lie-Rinehart algebras are to Lie algebroids, or Poisson algebras to Poisson manifolds. We work with arbitrary rings and modules, without any regularity, finiteness or non-degeneracy assumptions. To each Courant-Dorfman algebra (\R,\E) we associate a differential graded algebra \C(\E,\R) in a functorial way by means of explicit formulas. We describe two canonical filtrations on \C(\E,\R), and derive an analogue of the Cartan relations for derivations of \C(\E,\R); we classify central extensions of \E in terms of H^2(\E,\R) and study the canonical cocycle \Theta\in\C^3(\E,\R) whose class $[\Theta]$ obstructs re-scalings of the Courant-Dorfman structure. In the nondegenerate case, we also explicitly describe the Poisson bracket on \C(\E,\R); for Courant-Dorfman algebras associated to Courant algebroids over finite-dimensional smooth manifolds, we prove that the Poisson dg algebra \C(\E,\R) is isomorphic to the one constructed in \cite{Roy4-GrSymp} using graded manifolds.Comment: Corrected formulas for the brackets in Examples 2.27, 2.28 and 2.29. The corrections do not affect the exposition in any wa

Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory

We study an SU(3) gauge theory with Nf=8 degenerate flavors of light fermions in the fundamental representation. Using the domain wall fermion formulation, we investigate the light hadron spectrum, chiral condensate and electroweak S parameter. We consider a range of light fermion masses on two lattice volumes at a single gauge coupling chosen so that IR scales approximately match those from our previous studies of the two- and six-flavor systems. Our results for the Nf=8 spectrum suggest spontaneous chiral symmetry breaking, though fits to the fermion mass dependence of spectral quantities do not strongly disfavor the hypothesis of mass-deformed infrared conformality. Compared to Nf=2 we observe a significant enhancement of the chiral condensate relative to the symmetry breaking scale F, similar to the situation for Nf=6. The reduction of the S parameter, related to parity doubling in the vector and axial-vector channels, is also comparable to our six-flavor results