1,620 research outputs found
Morita classes in the homology of automorphism groups of free groups
Using Kontsevich's identification of the homology of the Lie algebra l_infty
with the cohomology of Out(F_r), Morita defined a sequence of 4k-dimensional
classes mu_k in the unstable rational homology of Out(F_{2k+2}). He showed by a
computer calculation that the first of these is non-trivial, so coincides with
the unique non-trivial rational homology class for Out(F_4). Using the
"forested graph complex" introduced in [Algebr. Geom. Topol. 3 (2003)
1167--1224], we reinterpret and generalize Morita's cycles, obtaining an
unstable cycle for every connected odd-valent graph. (Morita has independently
found similar generalizations of these cycles.) The description of Morita's
original cycles becomes quite simple in this interpretation, and we are able to
show that the second Morita cycle also gives a nontrivial homology class.
Finally, we view things from the point of view of a different chain complex,
one which is associated to Bestvina and Feighn's bordification of outer space.
We construct cycles which appear to be the same as the Morita cycles
constructed in the first part of the paper. In this setting, a further
generalization becomes apparent, giving cycles for objects more general than
odd-valent graphs. Some of these cycles lie in the stable range. We also
observe that these cycles lift to cycles for Aut(F_r).Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper40.abs.htm
Deformation quantization of Poisson manifolds, I
I prove that every finite-dimensional Poisson manifold X admits a canonical
deformation quantization. Informally, it means that the set of equivalence
classes of associative algebras close to the algebra of functions on X is in
one-to-one correspondence with the set of equivalence classes of Poisson
structures on X modulo diffeomorphisms. In fact, a more general statement is
proven ("Formality conjecture"), relating the Lie superalgebra of polyvector
fields on X and the Hochschild complex of the algebra of functions on X.
Coefficients in explicit formulas for the deformed product can be interpreted
as correlators in a topological open string theory, although I do not use
explicitly the language of functional integrals. One of corollaries is a
justification of the orbit method in the representation theory.Comment: plain TeX and epsf.tex, 46 pages, 24 figure
Topological conformal field theories and gauge theories
This paper gives a construction, using heat kernels, of differential forms on
the moduli space of metrised ribbon graphs, or equivalently on the moduli space
of Riemann surfaces with boundary. The construction depends on a manifold with
a bundle of Frobenius algebras, satisfying various conditions. These forms
satisfy gluing conditions which mean they form an open topological conformal
field theory, i.e. a kind of open string theory.
If the integral of these forms converged, it would yield the purely quantum
part of the partition function of a Chern-Simons type gauge theory. Yang-Mills
theory on a four manifold arises as one of these Chern-Simons type gauge
theories.Comment: A few more typos correcte
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