3,435 research outputs found
On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture
For a finite-dimensional Lie algebra over a field , we deduce from the compatibility between cup products
Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an
alternative way of re-writing Kontsevich product on by means of the Alekseev--Torossian flat connection (Alekseev
and Torossian, 2010). We deduce a similar formula directly from the
Kashiwara--Vergne conjecture (Kashiwara and Vergne, 1978).Comment: 8 pages, 1 figure; notation changed; corrected many other misprints
recently noticed; comments are very welcome
The explicit equivalence between the standard and the logarithmic star product for Lie algebras
The purpose of this short note is to establish an explicit equivalence
between the two star products and on the symmetric
algebra of a finite-dimensional Lie algebra over a field of characteristic 0 associated with
the standard angular propagator and the logarithmic one: the differential
operator of infinite order with constant coefficients realizing the equivalence
is related to the incarnation of the Grothendieck-Teichm\"uller group
considered by Kontsevich.Comment: 2 figures; corrected and completed the formulation of Theorem 3.7.
Comments are very welcome
Deformation quantization with generators and relations
In this paper we prove a conjecture of B. Shoikhet which claims that two
quantization procedures arising from Fourier dual constructions actually
coincide.Comment: 9 pages; 4 figures; many typos have been corrected; the introduction
has been considerably extended and a more detailed exposition of the Koszul
theory behind the main idea has been added; the proof of Proposition 2.4
(iii) has been also extended; Subsection 3.2 has been enlarged, and a more
detailed exposition of how the Duflo element arises has been adde
Loop observables for BF theories in any dimension and the cohomology of knots
A generalization of Wilson loop observables for BF theories in any dimension
is introduced in the Batalin-Vilkovisky framework. The expectation values of
these observables are cohomology classes of the space of imbeddings of a
circle. One of the resulting theories discussed in the paper has only trivalent
interactions and, irrespective of the actual dimension, looks like a
3-dimensional Chern-Simons theory.Comment: 13 page
A gerbe for the elliptic gamma function
The identities for elliptic gamma functions discovered by A. Varchenko and
one of us are generalized to an infinite set of identities for elliptic gamma
functions associated to pairs of planes in 3-dimensional space. The language of
stacks and gerbes gives a natural framework for a systematic description of
these identities and their domain of validity. A triptic curve is the quotient
of the complex plane by a subgroup of rank three (it is a stack). Our
identities can be summarized by saying that elliptic gamma functions form a
meromorphic section of a hermitian holomorphic abelian gerbe over the universal
oriented triptic curve.Comment: 54 page
Hochschild cohomology for Lie algebroids
We define the Hochschild (co)homology of a ringed space relative to a locally
free Lie algebroid. Our definitions mimic those of Swan and Caldararu for an
algebraic variety. We show that our (co)homology groups can be computed using
suitable standard complexes.
Our formulae depend on certain natural structures on jetbundles over Lie
algebroids. In an appendix we explain this by showing that such jetbundles are
formal groupoids which serve as the formal exponentiation of the Lie algebroid.Comment: The authors were informed that the fact that jetbundles are formal
groupoids is already contained in arXiv:0904.4736 (with a somewhat different
proof
- …