69 research outputs found
Free Lie algebroids and the space of paths
We construct algebraic and algebro-geometric models for the spaces of
unparametrized paths. This is done by considering a path as a holonomy
functional on indeterminate connections.
For a manifold X, we construct a Lie algebroid P which serves as the tangent
space to X (punctual paths) inside the space of all unparametrized paths. It
serves as a natural receptacle of all "covariant derivatives of the curvature"
for all bundles with connections on X.
If X is an algebraic variety, we integrate P to a formal groupoid G which can
be seen as the formal neighborhood of X inside the space of paths. We establish
a relation of G with the stable map spaces of Kontsevich.Comment: 42 pages, revised version, to appear in Selecta Mat
Representation and character theory in 2-categories
We define the character of a group representation in a 2-category C. For
linear C, this notion yields a Hopkins-Kuhn-Ravenel type character theory
defined on pairs of commuting elements of the group. We discuss some examples
and prove a formula for the character of the induced representation.Comment: 34 pages, revised version, to appear in Advances in Mathematic
Supersymmetry and the formal loop space
For any algebraic super-manifold M we define the super-ind-scheme LM of
formal loops and study the transgression map (Radon transform) on differential
forms in this context. Applying this to the super-manifold M=SX, the spectrum
of the de Rham complex of a manifold X, we obtain, in particular, that the
transgression map for X is a quasi-isomorphism between the [2,3)-truncated de
Rham complex of X and the additive part of the [1,2)-truncated de Rham complex
of LX. The proof uses the super-manifold SSX and the action of the Lie
superalgebra sl(1|2) on this manifold. This quasi-isomorphism result provides a
crucial step in the classification of sheaves of chiral differential operators
in terms of geometry of the formal loop space
Microlocal sheaves and quiver varieties
We relate Nakajima Quiver Varieties (or, rather, their multiplicative
version) with moduli spaces of perverse sheaves. More precisely, we consider a
generalization of the concept of perverse sheaves: microlocal sheaves on a
nodal curve X. They are defined as perverse sheaves on normalization of X with
a Fourier transform condition near each node and form an abelian category M(X).
One has a similar triangulated category DM(X) of microlocal complexes. For a
compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all
components of X are rational, M(X) is equivalent to the category of
representations of the multiplicative pre-projective algebra associated to the
intersection graph of X. Quiver varieties in the proper sense are obtained as
moduli spaces of microlocal sheaves with a framing of vanishing cycles at
singular points. The case when components of X have higher genus, leads to
interesting generalizations of preprojective algebras and quiver varieties. We
analyze them from the point of view of pseudo-Hamiltonian reduction and
group-valued moment maps.Comment: 49 page
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