27 research outputs found
The Lefschetz-Lunts formula for deformation quantization modules
We adapt to the case of deformation quantization modules a formula of V.
Lunts who calculates the trace of a kernel acting on Hochschild homology.Comment: 19 pages; Mathematische Zeitschrift 201
Non-birational twisted derived equivalences in abelian GLSMs
In this paper we discuss some examples of abelian gauged linear sigma models
realizing twisted derived equivalences between non-birational spaces, and
realizing geometries in novel fashions. Examples of gauged linear sigma models
with non-birational Kahler phases are a relatively new phenomenon. Most of our
examples involve gauged linear sigma models for complete intersections of
quadric hypersurfaces, though we also discuss some more general cases and their
interpretation. We also propose a more general understanding of the
relationship between Kahler phases of gauged linear sigma models, namely that
they are related by (and realize) Kuznetsov's `homological projective duality.'
Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense)
are realized physically in gauged linear sigma models, providing examples of
new types of conformal field theories. Throughout, the physical realization of
stacks plays a key role in interpreting physical structures appearing in GLSMs,
and we find that stacks are implicitly much more common in GLSMs than
previously realized.Comment: 54 pages, LaTeX; v2: typo fixe
From Atiyah Classes to Homotopy Leibniz Algebras
A celebrated theorem of Kapranov states that the Atiyah class of the tangent
bundle of a complex manifold makes into a Lie algebra object in
, the bounded below derived category of coherent sheaves on .
Furthermore Kapranov proved that, for a K\"ahler manifold , the Dolbeault
resolution of is an
algebra. In this paper, we prove that Kapranov's theorem holds in much wider
generality for vector bundles over Lie pairs. Given a Lie pair , i.e. a
Lie algebroid together with a Lie subalgebroid , we define the Atiyah
class of an -module (relative to ) as the obstruction to
the existence of an -compatible -connection on . We prove that the
Atiyah classes and respectively make and
into a Lie algebra and a Lie algebra module in the bounded below
derived category , where is the abelian
category of left -modules and is the universal
enveloping algebra of . Moreover, we produce a homotopy Leibniz algebra and
a homotopy Leibniz module stemming from the Atiyah classes of and ,
and inducing the aforesaid Lie structures in .Comment: 36 page
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
Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
In this paper we prove, with details and in full generality, that the
isomorphism induced on tangent homology by the Shoikhet-Tsygan formality
-quasi-isomorphism for Hochschild chains is compatible with
cap-products. This is a homological analog of the compatibility with
cup-products of the isomorphism induced on tangent cohomology by Kontsevich
formality -quasi-isomorphism for Hochschild cochains.
As in the cohomological situation our proof relies on a homotopy argument
involving a variant of {\bf Kontsevich eye}. In particular we clarify the
r\^ole played by the {\bf I-cube} introduced in \cite{CR1}.
Since we treat here the case of a most possibly general Maurer-Cartan
element, not forced to be a bidifferential operator, then we take this
opportunity to recall the natural algebraic structures on the pair of
Hochschild cochain and chain complexes of an -algebra. In particular
we prove that they naturally inherit the structure of an -algebra
with an -(bi)module.Comment: The first and second Section on -algebras and modules have
been completely re-written, with new results; partial revision of Section 3;
the proofs in Section 4 and 5 have been re-formulated in a more general
context; we added Section 8 on globalisatio