1,085 research outputs found
Odd structures are odd
By an odd structure we mean an algebraic structure in the category of graded
vector spaces whose structure operations have odd degrees. Particularly
important are odd modular operads which appear as Feynman transforms of modular
operads and, as such, describe some structures of string field theory.
We will explain how odd structures are affected by the choice of the monoidal
structure of the underlying category. We will then present two `natural' and
`canonical' constructions of an odd modular endomorphism operad leading to
different results, only one being correct. This contradicts the generally
accepted belief that the systematic use of the Koszul sign convention leads to
correct signs.Comment: Minor revision and a reference added. Accepted for publication in
Advances in Applied Clifford Algebra
Centers and homotopy centers in enriched monoidal categories
We consider a theory of centers and homotopy centers of monoids in monoidal
categories which themselves are enriched in duoidal categories. Duoidal
categories (introduced by Aguillar and Mahajan under the name 2-monoidal
categories) are categories with two monoidal structures which are related by
some, not necessary invertible, coherence morphisms. Centers of monoids in this
sense include many examples which are not `classical.' In particular, the
2-category of categories is an example of a center in our sense. Examples of
homotopy center (analogue of the classical Hochschild complex) include the
Gray-category Gray of 2-categories, 2-functors and pseudonatural
transformations and Tamarkin's homotopy 2-category of dg-categories,
dg-functors and coherent dg-transformations.Comment: 52 page
Wheeled PROPs, graph complexes and the master equation
We introduce and study wheeled PROPs, an extension of the theory of PROPs
which can treat traces and, in particular, solutions to the master equations
which involve divergence operators. We construct a dg free wheeled PROP whose
representations are in one-to-one correspondence with formal germs of
SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky
quantization. We also construct minimal wheeled resolutions of classical
operads Com and Ass as rather non-obvious extensions of Com_infty and
Ass_infty, involving, e.g., a mysterious mixture of associahedra with
cyclohedra. Finally, we apply the above results to a computation of cohomology
of a directed version of Kontsevich's complex of ribbon graphs.Comment: LaTeX2e, 63 pages; Theorem 4.2.5 on bar-cobar construction is
strengthene
A Generalization of Connes-Kreimer Hopf Algebra
``Bonsai'' Hopf algebras, introduced here, are generalizations of
Connes-Kreimer Hopf algebras, which are motivated by Feynman diagrams and
renormalization. We show that we can find operad structure on the set of
bonsais. We introduce a new differential on these bonsai Hopf algebras, which
is inspired by the tree differential. The cohomologies of these are computed
here, and the relationship of this differential with the appending operation
of Connes-Kreimer Hopf algebras is investigated
Crossed interval groups and operations on the Hochschild cohomology
We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades
Higher Poincare Lemma and Integrability
We prove the non-abelian Poincare lemma in higher gauge theory in two
different ways. The first method uses a result by Jacobowitz which states
solvability conditions for differential equations of a certain type. The second
method extends a proof by Voronov and yields the explicit gauge parameters
connecting a flat local connective structure to the trivial one. Finally, we
show how higher flatness appears as a necessary integrability condition of a
linear system which featured in recently developed twistor descriptions of
higher gauge theories.Comment: 1+21 pages, presentation streamlined, section on integrability for
higher linear systems significantly improved, published versio
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