35 research outputs found
Goldman-Turaev formality from the Knizhnik-Zamolodchikov connection
For an oriented 2-dimensional manifold of genus with
boundary components the space
carries the Goldman-Turaev Lie bialgebra structure defined in terms of
intersections and self-intersections of curves. Its associated graded (under
the natural filtration) is described by cyclic words in and
carries the structure of a necklace Schedler Lie bialgebra. The isomorphism
between these two structures in genus zero has been established in [G.
Massuyeau, Formal descriptions of Turaev's loop operations] using Kontsevich
integrals and in [A. Alekseev, N. Kawazumi, Y. Kuno and F. Naef, The
Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem]
using solutions of the Kashiwara-Vergne problem.
In this note we give an elementary proof of this isomorphism over
. It uses the Knizhnik-Zamolodchikov connection on
. The proof of the isomorphism for Lie
brackets is a version of the classical result by Hitchin. Surprisingly, it
turns out that a similar proof applies to cobrackets.Comment: 12 pages, 1 figure, section 3 adde
Chern-Simons, Wess-Zumino and other cocycles from Kashiwara-Vergne and associators
Descent equations play an important role in the theory of characteristic
classes and find applications in theoretical physics, e.g. in the Chern-Simons
field theory and in the theory of anomalies. The second Chern class (the first
Pontrjagin class) is defined as where is the
curvature 2-form and is an invariant scalar
product on the corresponding Lie algebra . The descent for
gives rise to an element of
mixed degree. The 3-form part is the Chern-Simons form. The 2-form
part is known as the Wess-Zumino action in physics. The 1-form
component is related to the canonical central extension of the loop
group .
In this paper, we give a new interpretation of the low degree components
and . Our main tool is the universal differential calculus
on free Lie algebras due to Kontsevich. We establish a correspondence between
solutions of the first Kashiwara-Vergne equation in Lie theory and universal
solutions of the descent equation for the second Chern class . In more
detail, we define a 1-cocycle which maps automorphisms of the free Lie
algebra to one forms. A solution of the Kashiwara-Vergne equation is mapped
to . Furthermore, the component is related to the
associator corresponding to . It is surprising that while and
satisfy the highly non-linear twist and pentagon equations, the elements
and solve the linear descent equation
The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera
For a compact oriented surface of genus with boundary
components, the space spanned by free homotopy classes of loops
in carries the structure of a Lie bialgebra. The Lie bracket was
defined by Goldman and it is canonical. The Lie cobracket was defined by Turaev
and it depends on the framing of . The Lie bialgebra has
a natural decreasing filtration such that both the Goldman bracket and the
Turaev cobracket have degree .
In this paper, we address the following Goldman-Turaev formality problem:
construct a Lie bialgebra homomorphism from to its
associated graded such that . In
order to solve it, we define a family of higher genus Kashiwara-Vergne (KV)
problems for an element , where is a free Lie algebra. In the
case of and , the problem for is the classical KV problem from
Lie theory. For , these KV problems are new.
Our main results are as follows. On the one hand, every solution of the KV
problem induces a GT formality map. On the other hand, higher genus KV problems
admit solutions for any and . In fact, the solution reduces to two
important cases: which admits solutions by Alekseev and Torossian
and for which we construct solutions in terms of certain elliptic
associators following Enriquez. By combining these two results, we obtain a
proof of the GT formality for any and .
We also study the set of solutions of higher genus KV problems and introduce
pro-unipotent groups which act on them freely and transitively.
These groups admit graded pro-nilpotent Lie algebras . We show
that the elliptic Lie algebra contains a copy of the
Grothendieck-Teichmuller Lie algebra as well as symplectic derivations
.Comment: 61 pages, 5 figure
Torsion volume forms
We introduce volume forms on mapping stacks in derived algebraic geometry
using a parametrized version of the Reidemeister-Turaev torsion. In the case of
derived loop stacks we describe this volume form in terms of the Todd class. In
the case of mapping stacks from surfaces, we compare it to the symplectic
volume form. As an application of these ideas, we construct canonical
orientation data for cohomological DT invariants of closed oriented
3-manifolds.Comment: 58 page
Higher genus Kashiwara–Vergne problems and the Goldman–Turaev Lie bialgebra
We define a family of Kashiwara-Vergne problems associated
with compact connected oriented 2-manifolds of genus with boundary
components. The problem is the classical Kashiwara-Vergne
problem from Lie theory. We show the existence of solutions of for arbitrary and . The key point is the solution of based on the results by B. Enriquez on elliptic associators. Our
construction is motivated by applications to the formality problem for the
Goldman-Turaev Lie bialgebra . In more detail, we show
that every solution of induces a Lie bialgebra isomorphism
between and its associated graded . For , a similar result was obtained by G.
Massuyeau using the Kontsevich integral.
This paper is a summary of our results. Details and proofs will appear
elsewhere.Comment: 6 pages, 2 figure