220 research outputs found
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
Formality of the chain operad of framed little disks
We extend Tamarkin's formality of the little disk operad to the framed little
disk operad.Comment: 5 page
One-dimensional Chern-Simons theory
We study a one-dimensional toy version of the Chern-Simons theory. We
construct its simplicial version which comprises features of a low-energy
effective gauge theory and of a topological quantum field theory in the sense
of Atiyah.Comment: 37 page
Formality theorems for Hochschild chains in the Lie algebroid setting
In this paper we prove Lie algebroid versions of Tsygan's formality
conjecture for Hochschild chains both in the smooth and holomorphic settings.
In the holomorphic setting our result implies a version of Tsygan's formality
conjecture for Hochschild chains of the structure sheaf of any complex manifold
and in the smooth setting this result allows us to describe quantum traces for
an arbitrary Poisson Lie algebroid. The proofs are based on the use of
Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]],
Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and
Fedosov's resolutions of the natural analogues of Hochschild (co)chain
complexes associated with a Lie algebroid.Comment: 40 pages, no figure
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
Hypercommutative operad as a homotopy quotient of BV
We give an explicit formula for a quasi-isomorphism between the operads
Hycomm (the homology of the moduli space of stable genus 0 curves) and
BV/ (the homotopy quotient of Batalin-Vilkovisky operad by the
BV-operator). In other words we derive an equivalence of Hycomm-algebras and
BV-algebras enhanced with a homotopy that trivializes the BV-operator.
These formulas are given in terms of the Givental graphs, and are proved in
two different ways. One proof uses the Givental group action, and the other
proof goes through a chain of explicit formulas on resolutions of Hycomm and
BV. The second approach gives, in particular, a homological explanation of the
Givental group action on Hycomm-algebras.Comment: minor corrections added, to appear in Comm.Math.Phy
The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations
We study decidability and complexity questions related to a continuous
analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of
a linear recurrent sequence. In particular, we show that the continuous version
of the nonnegativity problem is NP-hard in general and we show that the
presence of a zero is decidable for several subcases, including instances of
depth two or less, although the decidability in general is left open. The
problems may also be stated as reachability problems related to real zeros of
exponential polynomials or solutions to initial value problems of linear
differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur
Interpolating Coherent States for Heisenberg-Weyl and Single-Photon SU(1,1) Algebras
New quantal states which interpolate between the coherent states of the
Heisenberg_Weyl and SU(1,1) algebras are introduced. The interpolating states
are obtained as the coherent states of a closed and symmetric algebra which
interpolates between the two algebras. The overcompleteness of the
interpolating coherent states is established. Differential operator
representations in suitable spaces of entire functions are given for the
generators of the algebra. A nonsymmetric set of operators to realize the
Heisenberg-Weyl algebra is provided and the relevant coherent states are
studied.Comment: 13 pages nd 5 ps figure
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