Algebraic Analysis and Geometry

23-24 September 2013
Dipartimento di Matematica
via Trieste, 63 - Padova
room 2BC60

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Schedule

(Last update: Monday September 23, 2013)

Abstracts

Ugo Bruzzo Nonabelian Lie Algebroid Extensions

Abstract: We classify nonabelian extensions of Lie algebroids in the holomorphic or algebraic category, and introduce and study a spectral sequence that one can attach to any such extension and generalizes the Hochschild-Serre spectral sequence associated to an ideal in a Lie algebra. We compute the differentials of the spectral sequence up to $d_2$.

Alessandro Chiodo Applications of quantum theories for Landau-Ginzburg models

Abstract: In mirror symmetry, Landau-Ginzburg models are singularities of the form $f\colon X \to B$ arising as mirror partners of Fano manifolds. The symmetry relates two local systems. The first local system is classical in algebraic geometry and encodes the cohomology of the fibers of $f\colon X \to B$. The second local system embodies quantum periods of the given manifold and arises from Gromov-Witten theory. The isomorphism has been proven for several Fano manifolds but it remains wide open in several cases and generalizations. Fan-Jarvis-Ruan and Polishchuk-Vaintrob set up analogue quantum theories also for Landau-Ginzburg models and allow us to cast mirror symmetry in a global, more flexible, framework where some well known obstacles in Gromov-Witten theory, such as non-convexity or non-concavity, become more approachable. As an example I will describe recent work by Jérémy Guéré proving a mirror theorem relating Landau-Ginzburg models without concavity.

Barbara Fantechi On the quantization of the Behrend function

Ezra Getzler Derived stacks

Abstract: I define a functor that takes differential graded categories (or L-infinity algebras) to derived stacks.

Chris Rogers $L_{\infty}$-algebras and geometric prequantization

Abstract: I will describe a "homotopical analog" of the prequantization procedure developed by Kostant, Kirillov, and Souriau in which symplectic geometry is used to produce central extensions of Lie algebras. Analogously, our construction geometrically produces $L_{\infty}$-extensions using higher-degree closed differential forms. Such a form canonically gives an $L_{\infty}$-cocycle whose homotopy fiber acts as the $L_{\infty}$ analog of the Poisson algebra. When the form represents an integral cohomology class, this $L_{\infty}$-algebra is homotopy equivalent to a DGLA corresponding to the infinitesimal autoequivalences of a higher gerbe, in analogy with the prequantization of the Poisson algebra as vector fields on a principal circle bundle. Applications of this procedure include constructing Heisenberg-like $L_{\infty}$-algebras such as the "string Lie 2-algebra".
This is joint work with Domenico Fiorenza and Urs Schreiber.

Vladimir Rubtsov Poisson structure on representation spaces and double brackets

Francesco Sala Instanton counting on ALE spaces by means of framed sheaves

Abstract: In the present talk I describe some relations between moduli spaces of framed sheaves on 2-dimensional toric orbifolds and gauge theories on ALE spaces of type $A_n$. After a brief introduction about instantons and gauge theories on an ALE space $X$ of type $A_n$, I explain a conjectured relation between instantons on $X$ and framed sheaves on a "stacky compactification" of $X$. In the second part of the talk, the geometry of the moduli spaces of framed sheaves and the "instanton counting" over them is introduced. The last part of the talk focuses on the case of framed sheaves on rank one. In particular I describe a relation (the Alday-Gaiotto-Tachikawa conjecture) between the equivariant cohomology of the moduli spaces of framed sheaves and basic representations of certain infinite-dimensional Lie algebras.

Pierre Schapira Sheaves on the subanalytic topology and filtered $\mathcal D$-modules

Abstract: As a by product of the theory of ind-sheaves, we have introduced with M. Kashiwara in [KS01] the Grothendieck subanalytic topology on a real analytic manifold $M$ (see also [Pr08]). In this topology, we only consider open subanalytic sets and finite coverings. This allows us to define on a complex manifold the sheaf of holomorphic functions with temperate growth. Recently, with S. Guillermou [GS12] we have refined this topology and obtained the linear subanalytic topology, with same open sets but less coverings. We can then define sheaves of holomorphic functions with a given growth and thus a filtration (in the derived sense, following [Sn99,SSn13]) on the sheaf of temperate holomorphic functions. Using Kashiwara's solution of the Riemann-Hilbert correspondence, we can then endow functorially regular holonomic $\mathcal D$-modules with a filtration. We can also define sheaves with Gevrey growth which appear naturally in the study of irregular $\mathcal D$-modules.
This is a joint work with S. Guillermou.
Bibliography
[GS12] S. Guillermou and P. Schapira, Subanalytic topologies I. Construction of sheaves, arXiv:math.AG:1212.4326
[KS01] M.Kashiwara and P.Schapira, Ind-sheaves, Astérisque Soc. Math. France. 271 (2001).
[Pr08] L.Prelli, Sheaves on subanalytic sites, Rend. Sem. Mat. Univ. Padova, 120 (2008) p.167–216.
[Sn99] J.-P. Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999).
[SSn13] P.Schapira and J.-P. Schneiders, Derived category of filtered objects, arXiv:math.AG:1306.1359

Paolo Stellari Fourier-Mukai functors and dg enhancements

Abstract: When dealing with the categories of perfect complexes on projective varieties and exact functors between them, one immediately encounters some natural questions. Does the category admit a unique dg enhancement? Are all exact functors between these categories of Fourier-Mukai type? We discuss the recent advances in these subjects and propose a generalization of a result of Lunts-Orlov about the uniqueness of dg enhancements. Then we consider the reformulation of the second question in the context of dg categories where the problem has been settled by B. Toën. In this talk we propose a simpler approach not based on the notion of model category.
These are joint (partly in progress) works with A. Canonaco.

Pietro Tortella Contractions of Lie algebroids and the moduli space of flat connections

Abstract: The non abelian Hodge correspondence establishes a (smooth) isomorphism between the moduli spaces of stable flat connections and stable Higgs bundles over a smooth projetive complex variety. Both flat connections and Higgs bundles are representations of different Lie algebroid structures one can put on the tangent bundle (respectively, the canonical and the trivial one). Moreover, let us remark that the trivial Lie algebroid structure of the tangent bundle is a contraction of the canonical one. We construct a moduli space $M_\tau$ parametrizing semistable pairs $(S, D)$, where $S$ is a Lie algebroid structure of the tangent bundle and $D$ is a representation of $S$. This space may be used to study the moduli space of flat connections and of Higgs bundles. For instance, Simpson's lambda connections are naturally included in this space, and many constructions can be generalized to this setting. In particular, there is an action of the space of automorphisms of the tangent bundle over $M_\tau$ that generalize the $C^*$ action on the moduli space of lambda connections, and a quotient of $M_\tau$ by this action provides a (partial) compactification of the moduli space of flat connections.

Amnon Yekutieli Twisted Deformation Quantization of Algebraic Varieties

Abstract: Let $X$ be a smooth algebraic variety over a field of characteristic 0, with structure sheaf ${\cal O}_X$. I will begin by explaining what are associative (resp. Poisson) deformations of ${\cal O}_X$. These deformations form a stack of crossed groupoids on $X$.
Next I will introduce the concept of twisted object of a stack of crossed groupoids. This includes gerbes and stacks of algebroids as particular examples. A twisted deformation (associative or Poisson) is a twisted object of the stack of (associative or Poisson) deformations.
Our main result is the Twisted Quantization Theorem states that there is a canonical bijection between gauge equivalence classes of twisted Poisson deformations and twisted Poisson deformations. If time permits I will state several intermediate results, that lead to the proof of the main result.
There are lecture notes (pdf), a paper on this material arXiv:0905.0488, and also a survey article arxiv:0801.3233.

Marco Zambon Multisymplectic manifolds, moment maps, and conserved quantities

Abstract: Multisymplectic structures are higher generalizations of symplectic structures, where forms of higher degree are considered. We introduce a notion of moment map for such structures, and by relating it to equivariant cohomology we are able to produce many examples. Further, we give some geometric justification for the theory: the existence of such moment maps implies the existence of several conserved quantities.
This is joint work with Yael Fregier, Chris Rogers and Camille Laurent-Gengoux.

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