81 research outputs found
Shifted Symplectic Structures
This is the first of a series of papers about \emph{quantization} in the
context of \emph{derived algebraic geometry}. In this first part, we introduce
the notion of \emph{-shifted symplectic structures}, a generalization of the
notion of symplectic structures on smooth varieties and schemes, meaningful in
the setting of derived Artin n-stacks. We prove that classifying stacks of
reductive groups, as well as the derived stack of perfect complexes, carry
canonical 2-shifted symplectic structures. Our main existence theorem states
that for any derived Artin stack equipped with an -shifted symplectic
structure, the derived mapping stack is equipped with a
canonical -shifted symplectic structure as soon a satisfies a
Calabi-Yau condition in dimension . These two results imply the existence of
many examples of derived moduli stacks equipped with -shifted symplectic
structures, such as the derived moduli of perfect complexes on Calabi-Yau
varieties, or the derived moduli stack of perfect complexes of local systems on
a compact and oriented topological manifold. We also show that Lagrangian
intersections carry canonical (-1)-shifted symplectic structures.Comment: 52 pages. To appear in Publ. Math. IHE
Chern character, loop spaces and derived algebraic geometry.
International audienceIn this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere
Geometric Phantom Categories
In this paper we give a construction of phantom categories, i.e. admissible
triangulated subcategories in bounded derived categories of coherent sheaves on
smooth projective varieties that have trivial Hochschild homology and trivial
Grothendieck group. We also prove that these phantom categories are phantoms in
a stronger sense, namely, they have trivial K-motives and, hence, all their
higher K-groups are trivial too.Comment: LaTeX, 18 page
The homotopy theory of dg-categories and derived Morita theory
The main purpose of this work is the study of the homotopy theory of
dg-categories up to quasi-equivalences. Our main result provides a natural
description of the mapping spaces between two dg-categories and in
terms of the nerve of a certain category of -bimodules. We also prove
that the homotopy category is cartesian closed (i.e. possesses
internal Hom's relative to the tensor product). We use these two results in
order to prove a derived version of Morita theory, describing the morphisms
between dg-categories of modules over two dg-categories and as the
dg-category of -bi-modules. Finally, we give three applications of our
results. The first one expresses Hochschild cohomology as endomorphisms of the
identity functor, as well as higher homotopy groups of the \emph{classifying
space of dg-categories} (i.e. the nerve of the category of dg-categories and
quasi-equivalences between them). The second application is the existence of a
good theory of localization for dg-categories, defined in terms of a natural
universal property. Our last application states that the dg-category of
(continuous) morphisms between the dg-categories of quasi-coherent (resp.
perfect) complexes on two schemes (resp. smooth and proper schemes) is
quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect)
on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm.
8.15 is new. Minor corrections. Final version, to appear in Inventione
Lectures on mathematical aspects of (twisted) supersymmetric gauge theories
Supersymmetric gauge theories have played a central role in applications of
quantum field theory to mathematics. Topologically twisted supersymmetric gauge
theories often admit a rigorous mathematical description: for example, the
Donaldson invariants of a 4-manifold can be interpreted as the correlation
functions of a topologically twisted N=2 gauge theory. The aim of these
lectures is to describe a mathematical formulation of partially-twisted
supersymmetric gauge theories (in perturbation theory). These partially twisted
theories are intermediate in complexity between the physical theory and the
topologically twisted theories. Moreover, we will sketch how the operators of
such a theory form a two complex dimensional analog of a vertex algebra.
Finally, we will consider a deformation of the N=1 theory and discuss its
relation to the Yangian, as explained in arXiv:1308.0370 and arXiv:1303.2632.Comment: Notes from a lecture series by the first author at the Les Houches
Winter School on Mathematical Physics in 2012. To appear in the proceedings
of this conference. Related to papers arXiv:1308.0370, arXiv:1303.2632, and
arXiv:1111.423
Principal infinity-bundles - General theory
The theory of principal bundles makes sense in any infinity-topos, such as
that of topological, of smooth, or of otherwise geometric
infinity-groupoids/infinity-stacks, and more generally in slices of these. It
provides a natural geometric model for structured higher nonabelian cohomology
and controls general fiber bundles in terms of associated bundles. For suitable
choices of structure infinity-group G these G-principal infinity-bundles
reproduce the theories of ordinary principal bundles, of bundle
gerbes/principal 2-bundles and of bundle 2-gerbes and generalize these to their
further higher and equivariant analogs. The induced associated infinity-bundles
subsume the notions of gerbes and higher gerbes in the literature.
We discuss here this general theory of principal infinity-bundles, intimately
related to the axioms of Giraud, Toen-Vezzosi, Rezk and Lurie that characterize
infinity-toposes. We show a natural equivalence between principal
infinity-bundles and intrinsic nonabelian cocycles, implying the classification
of principal infinity-bundles by nonabelian sheaf hyper-cohomology. We observe
that the theory of geometric fiber infinity-bundles associated to principal
infinity-bundles subsumes a theory of infinity-gerbes and of twisted
infinity-bundles, with twists deriving from local coefficient infinity-bundles,
which we define, relate to extensions of principal infinity-bundles and show to
be classified by a corresponding notion of twisted cohomology, identified with
the cohomology of a corresponding slice infinity-topos.
In a companion article [NSSb] we discuss explicit presentations of this
theory in categories of simplicial (pre)sheaves by hyper-Cech cohomology and by
simplicial weakly-principal bundles; and in [NSSc] we discuss various examples
and applications of the theory.Comment: 46 pages, published versio
Equivalences between GIT quotients of Landau-Ginzburg B-models
We define the category of B-branes in a (not necessarily affine)
Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition
is a direct generalization of the category of perfect complexes. We then
consider pairs of Landau-Ginzburg B-models that arise as different GIT
quotients of a vector space by a one-dimensional torus, and show that for each
such pair the two categories of B-branes are quasi-equivalent. In fact we
produce a whole set of quasi-equivalences indexed by the integers, and show
that the resulting auto-equivalences are all spherical twists.Comment: v3: Added two references. Final version, to appear in Comm. Math.
Phy
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