Deformation Quantization of Shifted Poisson Structures

Conference Date: 
Monday, April 18, 2016 (All day) to Friday, April 22, 2016 (All day)
Scientific Areas: 
Mathematical Physics

 

Within the past five years derived geometry has become a central tool in the mathematics of quantum field theory. Even more recently, shifted Poisson structures (generalizing those of classical mechanics) and their quantization have found application in both mathematics and quantum fields and strings. This conference will allow for the review of recent advances in derived geometry and applications thereof to various moduli spaces by leading experts. In addition, the conference seeks to facilitate the expansion of these techniques into the realm of supersymmetric gauge theory in dimensions three and four.

  • Dima Arinkin, University of Wisconsin
  • Oren Ben-Bassat, University of Haifa
  • Christopher Brav, National Research University Higher School of Economics
  • Damien Calaque, IMAG, University of Montpellier 2
  • Andrei Caldararu, University of Wisconsin
  • David Gepner, Purdue University
  • Julien Grivaux, Aix-Marseille Université
  • Rune Haugseng, Max Planck Institute of Mathematics
  • Benjamin Hennion, Max Planck Institute of Mathematics
  • Dominic Joyce, Oxford University
  • Mauro Porta, Institut de Mathematiques Jussieu 
  • Nick Rozenblyum, University of Chicago
  • Pavel Safronov, Oxford University
  • Theodore Spaide, University of Vienna
  • David Treumann, Boston College
  • Michel Vaquie, Universite Paul Sabatier 
  • Dima Arinkin, University of Wisconsin
  • Oren Ben-Bassat, University of Haifa
  • Christopher Brav, National Research University Higher School of Economics
  • Alexander Braverman, Perimeter Institute & University of Toronto
  • Damien Calaque, IMAG, University of Montpellier 2
  • Andrei Caldararu, University of Wisconsin
  • Kevin Costello, Perimeter Institute
  • Chris Dodd, Perimeter Institute
  • Chris Elliott, Northwestern University
  • David Gepner, Purdue University
  • Ryan Grady, Perimeter Institute
  • Julien Grivaux, Aix-Marseille Université
  • Rune Haugseng, Max Planck Institute of Mathematics
  • Benjamin Hennion, Max Planck Institute of Mathematics
  • Dominic Joyce, Oxford University
  • Mykola Matviichuk, University of Toronto
  • Tony Pantev, University of Pennsylvania
  • Mauro Porta, Institut de Mathematiques Jussieu 
  • Nick Rozenblyum, University of Chicago
  • Pavel Safronov, Oxford University
  • Theodore Spaide, University of Vienna
  • David Treumann, Boston College
  • Michel Vaquie, Universite Paul Sabatier 
  • Philsang Yoo, Northwestern University

Monday, April 18, 2016

Time

Event

Location

9:00 – 9:30am

Registration

Reception

9:30 – 9:35am

Welcome and Opening Remarks

Alice Room

9:35 – 10:30am

Michel Vaquie, University Paul Sabatier
Formal derived stack and Formal localization

Alice Room

10:30 – 11:00am

Coffee Break

Bistro – 1st Floor

11:00 – 12:00pm

Mauro Porta, Institut de Mathematiques Jussieu
An overview of derived analytic geometry

Alice Room

12:00 – 2:00pm

Lunch

Bistro – 2nd Floor

2:00 – 3:00pm

Dominic Joyce, Oxford University
Categorification of shifted symplectic geometry
using perverse sheaves

Alice Room

3:00 – 3:30pm

Coffee Break

Bistro – 1st Floor

3:30pm – 4:30pm

Tony Pantev, University of Pennsylvania
Shifted structures and quantization

Alice Room

 

Tuesday, April 19, 2016

Time

Event

Location

9:30 – 10:30am

Andrei Caldararu, University of Wisconsin
What is the Todd class of an orbifold?

Alice Room

10:30 – 11:00am

Coffee Break

Bistro – 1st Floor

11:00 – 12:00pm

Dima Arinkin, University of Wisconsin
Singular support of categories

Alice Room

12:00 – 2:00pm

Lunch

Bistro – 2nd Floor

2:00 – 3:00pm

Ted Spaide, University of Vienna
Symplectic and Lagrangian structures on mapping stacks

Alice Room

3:00 – 3:30pm

Coffee Break

Bistro – 1st Floor

3:30pm – 4:30pm

David Treumann, Boston College
The Maslov cycle and the J-homomorphism

Alice Room

 

Wednesday, April 20, 2016

Time

Event

Location

9:30 – 10:30am

Christopher Brav, Higher School of Economics (Moscow)
Relative non-commutative Calabi-Yau structures and shifted Lagrangians

Alice Room

10:30 – 11:00am

Coffee Break

Bistro – 1st Floor

11:00 – 12:00pm

Julien Grivaux, Aix-Marseille Université
Towards a general description of derived self-intersections

Alice Room

12:00 – 2:00pm

Lunch

Bistro – 2nd Floor

2:00 – 3:30pm

Colloquium
Damien Calaque, IMAG, University of Montpellier 2
Derived symplectic geometry and classical Chern-Simons theory

Time Room

3:30 – 4:00pm

Coffee Break

Bistro – 1st Floor

5:30pm

Banquet

Bistro – 2nd Floor

 

Thursday, April 21, 2016

Time

Event

Location

9:30 – 10:30am

David Gepner, Purdue University
On the stable homotopy theory of stacks and elliptic cohomology

Alice Room

10:30 – 11:00am

Coffee Break

Bistro – 1st Floor

11:00 – 12:00pm

Rune Haugseng, Max Planck Institute
Free linear BV-quantization as an infinity-functor

Alice Room

12:00 – 2:00pm

Lunch

Bistro – 2nd Floor

2:00 – 3:00pm

Benjamin Hennion, Max Planck Institute
Formal loop spaces

Alice Room

3:00 – 3:15pm

Conference Photo

TBA

3:15 – 4:15pm

Coffee Break

Bistro – 1st Floor

4:15pm

Collaboration

Alice Room

 

Friday, April 22, 2016

Time

Event

Location

9:30 – 10:30am

 Nick Rozenbluym, University of Chicago
AKSZ quantization of shifted Poisson structures

Alice Room

10:30 – 11:00am

Coffee Break

Bistro – 1st Floor

11:00 – 12:00pm

Pavel Safronov, Oxford University
Derived coisotropic structures

Alice Room

12:00 – 2:00pm

Lunch

Bistro – 2nd Floor

2:00 – 3:00pm

Oren Ben-Bassat, University of Hafia
A perspective on derived analytic geometry

Alice Room

3:00 – 3:10pm

Wrap-up and Good-bye

Alice Room

 

Dima Arinkin, University of Wisconsin

Singular support of categories

In many situations, geometric objects on a space have some kind of singular support, which refines the usual support. For instance, for smooth X, the singular support of a D-module (or a perverse sheaf) on X is as a conical subset of the cotangent bundle; similarly, for quasi-smooth X, the singular support of a coherent sheaf on X is a conical subset of the cohomologically shifted cotangent bundle. I would like to describe a higher categorical version of this notion.

Let X be a smooth variety, and let Z be a closed conical isotropic subset of the cotangent bundle of X. I will define a 2-category associated with Z; its objects may be viewed as `categories over X with singular support in Z'. In particular, if Z is the zero section, we simply consider categories over Z in the usual sense. 
 
This talk is based on a joint project with D.Gaitsgory. The project is motivated by the local geometric Langlands correspondence; I plan to sketch the relation with the Langlands correspondence in the talk. 
 
Oren Ben-Bassat, University of Hafia
 
A perspective on derived analytic geometry
 
I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry.
 
Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in various geometrical theories and how it is characterized categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces.  The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans.  In fact, we work with the larger category of  Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of derived analytic geometry (my joint project with Kobi Kremnizer). We compare this approach with standard standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions), the theory of Stein domains and others.  I will formulate derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.
 
This talk involves various joint work with Federico Bambozzi and Kobi Kremnizer.
 
Christopher Brav, Higher School of Economics (Moscow)
 
Relative non-commutative Calabi-Yau structures and shifted Lagrangians
 
We give a definition of relative Calabi-Yau structure on a dg functor f: A --> B, discussing a examples coming from algebraic geometry, homotopy theory, and representation theory. When A=0, this returns the usual definition of Calabi-Yau structure on a smooth dg category B. When A itself is endowed with a Calabi-Yau structure and relative Calabi-Yau structure on f is compatible with the absolute structure on A, then we sketch the construction of a shifted symplectic structure on the derived moduli space M_A of pseudo-perfect A-modules, as well as the construction of a Lagrangian structure on the induced map f* : M_B --> M_A of derived moduli. This is joint work with Tobias Dyckerhoff.
 
Damien Calaque, IMAG, University of Montpellier 2
 
Derived symplectic geometry and classical Chern-Simons theory
 
In this talk we will review various point-of-views on classical Chern-Simons theory and moduli of flat connections. We will explain how derived symplectic geomletry (after Pantev-Toën-Vaquié-Vezzosi) somehow reconciles all of these. If time permits, we will discuss a bit the quantization problem.
 
Andrei Caldararu, University of Wisconsin
 
What is the Todd class of an orbifold?
 
The Todd class enters algebraic geometry in two places, in the Hirzebruch-Riemann-Roch formula and in the correction of the HKR isomorphism needed to make the Hochschild cohomology isomorphic to polyvector field cohomology (Kontsevich’s claim, proved by Calaque and van den Bergh). In the case of orbifolds the Riemann-Roch formula is known, but not the analogue of Kontsevich’s result. However, we can try to use the former as a guide towards a conjectural formulation for the latter. 
 
The problem with this approach is that in the case of an orbifold it is not obvious what the Todd class actually is. This happens because the Riemann-Roch formula mixes the Todd class with the Chern character and it is difficult to separate one from the other. In my talk I shall discuss what the study of loop groups of orbifolds predicts the correct Todd class to be, and then I shall explain how the orbifold Riemann-Roch formula can be rewritten to make this prediction consistent.
 
David Gepner, Purdue University
 
On the stable homotopy theory of stacks and elliptic cohomology
 
In this talk, we'll discuss what it means to be a cohomology theory for topological stacks, using a notion of local symmetric monoidal inversion of objects in families. While the general setup is abstract, it specializes to many cases of interest, including Schwede's global spectra. We will then go on to discuss various examples with particular emphasis on elliptic cohomology. It turns out that TMF sees more objects as dualizable (or even invertible) than one might naively expect.
 
Julien Grivaux, Aix-Marseille Université
 
Towards a general description of derived self-intersections
 
Thanks to a result of Arinkin and Cāldāru, the derived self-intersection of a closed smooth subscheme of an ambiant scheme (over a field of characteristic zero) is a formal object if and only if the conormal bundle of the  subscheme extends to a locally free sheaf at the first order. In this talk, we will explain a program as well as new  results in order to describe these derived self-intersections in the non-formal case.
 
Rune Haugseng, Max Planck Institute
 
Free linear BV-quantization as an infinity-functor
 
I will describe a functorial construction of the free BV-quantization of chain complexes equipped with antisymmetric forms of degree 1 in the context of infinity-categories. This is joint work with Owen Gwilliam.
 
Benjamin Hennion, Max Planck Institute
 
Formal loop spaces
 
Formal loop spaces are algebraic analogs to smooth loops. They were introduced and studied extensively in the 2000' by Kapranov and Vasserot for their link to chiral algebras.
In this talk, we will introduced higher dimensional analogs of K. and V. formal loop spaces. We will show how derived methods allow such a definition. We will then study their tangent complexes: even though formal loop spaces are "of infinite dimension", their tangent has enough structure so that we can speak of symplectic forms on them.
 
Dominic Joyce, Oxford University
 
Categorification of shifted symplectic geometry using perverse sheaves
 
Let (X,w) be a -1-shifted symplectic derived scheme or stack over C in the sense of Pantev-Toen-Vaquie-Vezzosi with an "orientation" (square root of det L_X). We explain how to construct a perverse sheaf P on the classical truncation X=t_0(X), over a base ring A. The hypercohomology H*(P) is regarded as a categorification of X. 
 
Now suppose i : L --> X is a Lagrangian in (X,w) in the sense of PTVV, with a "relative orientation". We outline a programme (work in progress) to construct a natural morphism
 
 \mu : A_L[vdim L] --> i^!(P)
 
of constructible complexes on L=t_0(L). If i is proper this is equivalent to a  hypercohomology  in H^{-vdim L}(P). These natural morphisms / hypercohomology classes \mu satisfy various identities under products, composition of Lagrangian correspondences, etc.
 
This programme will have interesting applications. In particular:
 
(a) Take (X,w) to be the derived moduli stack of coherent sheaves on a Calabi-Yau 3-fold Y, so that the orientation is essentially "orientation data" in the sense of Kontsevich-Soibelman 2008. Then we regard H*(P) as being the Cohomological Hall Algebra of Y (cf Kontsevich and Soibelman 2010 for quivers). Consider 
 
i :  Exact --> (X,w) x (X,-w) x (X,w)  
 
the moduli stack of exact sequences of coherent sheaves on Y, with projections to first, second and third factors. This is a Lagrangian in -1-shifted symplectic. Suppose we have a relative orientation. Then the hypercohomology element \mu associated to Exact should give the COHA multiplication on H*(P), and identities on \mu should imply associativity of multiplication. 
 
(b) Let (S,w) be a classical symplectic C-scheme, or complex symplectic manifold, of dimension 2n, and L --> S, M --> S be algebraic / complex Lagrangians (or derived Lagrangians in the PTVV sense), proper over S. Suppose we are given "orientations" on L,M, i.e. square roots of the canonical bundles K_L,K_M. Then the derived intersection X = L x_S M is -1-shifted symplectic and oriented, so we get a perverse sheaf P on X. We regard the shifted hypercohomology H^{*-n}(P) as being a version of the "Lagrangian Floer cohomology" HF*(L,M), and the morphisms L --> M in a "Fukaya category" of (S,w).
 
If L,M,N are oriented Lagrangians in (S,w), then the triple intersection L x_S M x_S N is Lagrangian in the triple product (L x_S M) x (M x_S N) x (N x_S L). The associated hypercohomology element should correspond to the product HF*(L,M) x HF*(M,N) --> HF*(L,N) which is composition of morphisms in the "Fukaya category". Using these techniques we intend to define "Fukaya categories" of algebraic symplectic / complex symplectic manifolds, with many nice properties.
 
Different parts of this programme are joint work with subsets of Lino Amorim, Oren Ben-Bassat, Chris Brav, Vittoria Bussi, Delphine Dupont, Pavel Safronov, and Balazs Szendroi.
 
Tony Pantev, University of Pennsylvania
 
Shifted structures and quantization
 
I will discuss the comparison of shifted Poisson and symplectic geometry and applications to the shifted quantization of moduli spaces.
 
Mauro Porta, Institut de Mathematiques Jussieu
 
An overview of derived analytic geometry
 
After the pioneering work of J. Lurie in [DAG-IX], the possibility of a derived version of analytic geometry drew the attention of several mathematicians. The goal of this talk is to provide an overview of the state of art of derived analytic geometry, addressing both the complex and the non-archimedean setting.
 
After providing a series of motivations for derived analytic geometry, I will survey the main results obtained in my PhD thesis: derived versions of GAGA theorems, the existence of the analytic cotangent complex and an analytic version of Lurie's representability theorem. If time will permit, I will conclude the talk by discussing the possible future directions.
 
Parts of the results I will talk about have been obtained in collaboration with T. Y. Yu.
 
Nick Rozenbluym, University of Chicago
 
AKSZ quantization of shifted Poisson structures
 
One of the key constructions in the PTVV theory of shifted symplectic structures is the construction, via transgression, of a shifted symplectic structure on the derived mapping stack from an oriented manifold to a shifted symplectic stack vastly generalizing the AKSZ construction (which was formulated in the context of super manifolds). I will explain local-to-global approach to this construction, which also generalizes the construction to shifted Poisson structures and shows that the AKSZ/PTVV construction is compatible with quantization in a strong sense. One pleasant consequence is that every deformation quantization problem reduces to a version of BV-quantization. Time permitting, I will describe several geometric applications of the theory.
 
Pavel Safronov, Oxford University
 
Derived coisotropic structures
 
I will define coisotropic structures in the setting of shifted Poisson geometry in two ways and show their equivalence. The interplay between the definitions allows one to produce nontrivial statements. I will also describe some examples of coisotropic structures. This is a report on joint work with V. Melani.
 
Theodore Spaide, University of Vienna
 
Symplectic and Lagrangian structures on mapping stacks
 
An important result in shifted symplectic geometry is the existence of shifted symplectic forms on mapping spaces with symplectic target and oriented source.  I provide several examples of more complicated situations where stacks of maps shifted symplectic structures, or maps between them have Lagrangian structures. These include spaces of framed maps, pushforwards of perfect complexes, and perfect complexes on open varieties.
 
David Treumann, Boston College
 
The Maslov cycle and the J-homomorphism
 
Let L be an exact Lagrangian submanifold of a cotangent bundle T^* M.  If a topological obstruction vanishes, a local system of R-modules on L determines a constructible sheaf of R-modules on M -- this is the Nadler-Zaslow construction.  I will discuss a variant of this construction that avoids Floer theory, and that allows R to be a ring spectrum.  The talk is based on joint work with Xin Jin.
 
Michel Vaquie, University Paul Sabatier
 
Formal derived stack and Formal localization
 
A crucial ingredient in the theory of shifted Poisson structures on general derived Artin stacks is the method of formal localization.
 
Formal localization is interesting in its own right as a new, very power ful tool that will prove useful in order to globalize tricky constructions and  results, whose extension from the local case presents obstructions that only vanish formally locally.
 
 

 

Friday Apr 22, 2016
Speaker(s): 

I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry.
Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in various geometrical theories and how it is characterized

Scientific Areas: 

 

Friday Apr 22, 2016
Speaker(s): 

I will define coisotropic structures in the setting of shifted Poisson geometry in two ways and show their equivalence. The interplay between the definitions allows one to produce nontrivial statements. I will also describe some examples of coisotropic structures. This is a report on joint work with V. Melani.

Scientific Areas: 

 

Friday Apr 22, 2016
Speaker(s): 

One of the key constructions in the PTVV theory of shifted symplectic structures is the construction, via transgression, of a shifted symplectic structure on the derived mapping stack from an oriented manifold to a shifted symplectic stack vastly generalizing the AKSZ construction (which was formulated in the context of super manifolds). I will explain local-to-global approach to this construction, which also generalizes the construction to shifted Poisson structures and shows that the AKSZ/PTVV construction is compatible with quantization in a strong sense.

Scientific Areas: 

 

Thursday Apr 21, 2016
Speaker(s): 

Formal loop spaces are algebraic analogs to smooth loops. They were introduced and studied extensively in the 2000' by Kapranov and Vasserot for their link to chiral algebras.
In this talk, we will introduced higher dimensional analogs of K. and V. formal loop spaces. We will show how derived methods allow such a definition. We will then study their tangent complexes: even though formal loop spaces are "of infinite dimension", their tangent has enough structure so that we can speak of symplectic forms on them.

Scientific Areas: 

 

Thursday Apr 21, 2016
Speaker(s): 

I will describe a functorial construction of the free BV-quantization of chain complexes equipped with antisymmetric forms of degree 1 in the context of infinity-categories. This is joint work with Owen Gwilliam.

Scientific Areas: 
 

 

Thursday Apr 21, 2016
Speaker(s): 

In this talk, we'll discuss what it means to be a cohomology theory for topological stacks, using a notion of local symmetric monoidal inversion of objects in families. While the general setup is abstract, it specializes to many cases of interest, including Schwede's global spectra. We will then go on to discuss various examples with particular emphasis on elliptic cohomology. It turns out that TMF sees more objects as dualizable (or even invertible) than one might naively expect.

Scientific Areas: 

 

Wednesday Apr 20, 2016
Speaker(s): 

In this talk we will review various point-of-views on classical Chern-Simons theory and moduli of flat connections. We will explain how derived symplectic geomletry (after
Pantev-Toën-Vaquié-Vezzosi) somehow reconciles all of these. If time permits, we will discuss a bit the quantization problem.

Scientific Areas: 

 

Wednesday Apr 20, 2016
Speaker(s): 

Thanks to a result of Arinkin and Cāldāru, the derived self-intersection of a closed smooth subscheme of an ambiant scheme (over a field of characteristic zero) is a formal object if and only if the conormal bundle of the subscheme extends to a locally free sheaf at the first order. In this talk, we will explain a program as well as new results in order to describe these derived self-intersections in the non-formal case.

Scientific Areas: 

 

Wednesday Apr 20, 2016
Speaker(s): 

We give a definition of relative Calabi-Yau structure on a dg functor f: A --> B, discussing a examples coming from algebraic geometry, homotopy theory, and representation theory. When A=0, this returns the usual definition of Calabi-Yau structure on a smooth dg category B.

Scientific Areas: 

 

Tuesday Apr 19, 2016
Speaker(s): 

Let L be an exact Lagrangian submanifold of a cotangent bundle T^* M. If a topological obstruction vanishes, a local system of R-modules on L determines a constructible sheaf of R-modules on M -- this is the Nadler-Zaslow construction. I will discuss a variant of this construction that avoids Floer theory, and that allows R to be a ring spectrum. The talk is based on joint work with Xin Jin.

Scientific Areas: 

Pages

Scientific Organizers:

  • Damien Calaque, IMAG, University of Montpellier 2
  • Kevin Costello, Perimeter Institute
  • Ryan Grady, Perimeter Institute
  • Tony Pantev, University of Pennsylvania