18 research outputs found
Poisson bivectors and Poisson brackets on affine derived stacks
Let Spec(A) be an affine derived stack. We give two proofs of the existence
of a canonical map from the moduli space of shifted Poisson structures (in the
sense of Pantev-To\"en-Vaqui\'e-Vezzosi, see http://arxiv.org/abs/1111.3209 )
on Spec(A) to the moduli space of homotopy (shifted) Poisson algebra structures
on A. The first makes use of a more general description of the Poisson operad
and of its cofibrant models, while the second in more computational and
involves an explicit resolution of the Poisson operad.Comment: 23 pages. Revised argument in section 3, results unchange
The derived moduli stack of shifted symplectic structures
We introduce and study the derived moduli stack of
-shifted symplectic structures on a given derived stack , as introduced
by [PTVV] (IHES Vol. 117, 2013). In particular, under reasonable assumptions on
, we prove that carries a canonical shifted quadratic
form. This generalizes a classical result of Fricke and Habermann, which was
established in the -setting, to the broader context of derived
algebraic geometry, thus proving a conjecture stated by Vezzosi.Comment: 17 page
Formality criteria for algebras over operads
We study some formality criteria for differential graded algebras over
differential graded operads. This unifies and generalizes other known
approaches like the ones by Manetti and Kaledin. In particular, we construct
general operadic Kaledin classes and show that they provide obstructions to
formality. Moreover, we show that an algebra A is formal if and only if its
operadic cohomology spectral sequence degenerates at .Comment: 17 page
A flag version of Beilinson-Drinfeld Grassmannian for surfaces
In this paper we define and study a generalization of the Belinson-Drinfeld
Grassmannian to the case where the curve is replaced by a smooth projective
surface , and the trivialization data are given with respect to a flag of
closed subschemes. In order to do this, we first establish some general formal
gluing results for moduli of almost perfect complexes, perfect complexes and
torsors. We then construct a simplicial object of flags of closed subschemes of
a smooth projective surface , naturally associated to the operation of
taking union of flags. We prove that this simplicial object has the -Segal
property. For an affine complex algebraic group , we finally define a flag
analog of the Beilinson-Drinfeld Grassmannian of -bundles
on the surface , and show that most of the properties of the
Beilinson-Drinfeld Grassmannian for curves can be extended to our flag
generalization. In particular, we prove a factorization formula, the existence
of a canonical flat connection, we construct actions of the loop group and of
the positive loop group on , and define a fusion product on
sheaves on .Comment: 78 page
Shifted Coisotropic Correspondences
We define (iterated) coisotropic correspondences between derived Poisson
stacks, and construct symmetric monoidal higher categories of derived Poisson
stacks where the -morphisms are given by -fold coisotropic
correspondences. Assuming an expected equivalence of different models of higher
Morita categories, we prove that all derived Poisson stacks are fully
dualizable, and so determine framed extended TQFTs by the Cobordism Hypothesis.
Along the way we also prove that the higher Morita category of -algebras
with respect to coproducts is equivalent to the higher category of iterated
cospans.Comment: 51 pages, v2: accepted versio
Local opers with two singularities: the case of
We study local opers with two singularities for the case of the Lie algebra
sl(2), and discuss their connection with a two-variables extension of the
affine Lie algebra. We prove an analogue of the Feigin-Frenkel theorem
describing the centre at the critical level, and an analogue of a result by
Frenkel and Gaitsgory that characterises the endomorphism rings of Weyl modules
in terms of functions on the space of opers.Comment: 56 pages. Comments are very welcome
The semi-infinite cohomology of Weyl modules with two singular points
In their study of spherical representations of an affine Lie algebra at the
critical level and of unramified opers, Frenkel and Gaitsgory introduced what
they called the Weyl module corresponding to a dominant
weight . This object plays an important role in the theory. In
arXiv:2012.01858, we introduced a possible analogue
of the Weyl module in the setting of opers with
two singular points, and in the case of sl(2) we proved that it has the
"correct" endomorphism ring. In this paper, we compute the semi-infinite
cohomology of and we show that it does not share
some of the properties of the semi-infinite cohomology of the Weyl module of
Frenkel and Gaitsgory. For this reason, we introduce a new module
which, in the case of sl(2), enjoys all
the expected properties of a Weyl module.Comment: 23 page
Derived coisotropic structures II: stacks and quantization
We extend results about -shifted coisotropic structures from part I of
this work to the setting of derived Artin stacks. We show that an intersection
of coisotropic morphisms carries a Poisson structure of shift one less. We also
compare non-degenerate shifted coisotropic structures and shifted Lagrangian
structures and show that there is a natural equivalence between the two spaces
in agreement with the classical result. Finally, we define quantizations of
-shifted coisotropic structures and show that they exist for .Comment: 45 pages. Contains the second half of arXiv:1608.01482v1 with new
material adde
Derived coisotropic structures I: affine case
We define and study coisotropic structures on morphisms of commutative dg
algebras in the context of shifted Poisson geometry, i.e. -algebras.
Roughly speaking, a coisotropic morphism is given by a -algebra acting
on a -algebra. One of our main results is an identification of the space
of such coisotropic structures with the space of Maurer--Cartan elements in a
certain dg Lie algebra of relative polyvector fields. To achieve this goal, we
construct a cofibrant replacement of the operad controlling coisotropic
morphisms by analogy with the Swiss-cheese operad which can be of independent
interest. Finally, we show that morphisms of shifted Poisson algebras are
identified with coisotropic structures on their graph.Comment: 49 pages. v2: many proofs rewritten and the paper is split into two
part