Abbaspour / Marcolli / Tradler Deformation Spaces

Perspectives on algebro-geometric moduli
2010
ISBN: 978-3-8348-9680-3
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 173 Seiten, eBook

Reihe: Aspects of Mathematics

The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics.
This volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras. They originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn.


Dr. Hossein Abbaspour, Department of Mathematics, Université de Nantes, France.
Prof. Dr. Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, California, USA.
Dr. Thomas Tradler, Department of Mathematics, New York City College of Technology (CUNY), New York, USA.

Zielgruppe

Research

Autoren/Hrsg.

Abbaspour, Hossein

Marcolli, Matilde

Tradler, Thomas

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;On the Hochschild and Harrison (co)homology of C8-algebras and applications to string topology;7
3.1;1. Hochschild (co)homology of an A8-algebra with values in a bimodule;10
3.2;2. C8-algebras, C8-bimodules, Harrison (co)homology;16
3.3;3. .-operations and Hodge decomposition;23
3.4;4. An exact sequence `a laJacobi-Zariski;37
3.5;5. Applications to string topology;47
3.6;6. Concluding remarks;55
3.7;References;55
4;What is the Jacobian of a Riemann Surface with Boundary?;58
4.1;1. Introduction;58
4.2;2. Open abelian varieties;61
4.3;3. Gluing and SPCMC structure;66
4.4;4. The Jacobian of a worldsheet with boundary;71
4.5;5. The lattice conformal .eld theory on the SPCMC of open abelian varieties;76
4.6;References;78
5;Pure weight perfect Modules on divisorial schemes;80
5.1;1. Introduction;80
5.2;2. Preliminary;81
5.3;3. Weight on pseudo-coherent Modules;87
5.4;4. Proof of the main theorem;88
5.5;5. Applications;92
5.6;References;93
6;Higher localized analytic indices and strict deformation quantization;95
6.1;1. Introduction;95
6.2;2. Index theory for Lie groupoids;98
6.3;3. Index theory and strict deformation quantization;104
6.4;4. Higher localized indices;109
6.5;References;114
7;An algebraic proof of Bogomolov-Tian-Todorov theorem;116
7.1;Introduction;116
7.2;1. Review of DGLAs and L8-algebras;118
7.3;2. The Thom-Whitney complex;121
7.4;3. Semicosimplicial differential graded Lie algebras and mapping cones;124
7.5;4. Semicosimplicial Cartan homotopies;127
7.6;5. Semicosimplicial Lie algebras and deformations of smooth varieties;129
7.7;6. Proof of the main theorem;131
7.8;References;134
8;Quantizing deformation theory;137
8.1;1. Introduction;137
8.2;2. Linear algebra;139
8.3;References;143
9;L8-interpretation of a classication of deformations of Poisson structures in dimension three;144
9.1;1. Introduction;144
9.2;2. Preliminaries: L8-algebras and Poisson algebras;148
9.3;3. Choice in a transfer of L8-algebra structure;154
9.4;4. Deformations of Poisson structures via L8-algebras;164
9.5;References;173

On the Hochschild and Harrison (co)homology of C ?-algebras and applications to string topology.- What is the Jacobian of a Riemann Surface with Boundary?.- Pure weight perfect Modules on divisorial schemes.- Higher localized analytic indices and strict deformation quantization.- An algebraic proof of Bogomolov-Tian-Todorov theorem.- Quantizing deformation theory.- L ?-interpretation of a classification of deformations of Poisson structures in dimension three.

Leseproben

What is the Jacobian of a Riemann Surface with Boundary? (S. 52-53)

Thomas M. Fiore and Igor Kriz Abstract.

We de?ne the Jacobian of a Riemann surface with analytically parametrized boundary components. These Jacobians belong to a moduli space of “open abelian varieties” which satis?es gluing axioms similar to those of Riemann surfaces, and therefore allows a notion of “conformal ?eld theory” to be de?ned on this space. We further prove that chiral conformal ?eld theories corresponding to even lattices factor through this moduli space of open abelian varieties.

1. Introduction

The main purpose of the present note is to generalize the notion of the Jacobian of a Riemann surface to Riemann surfaces with real-analytically parametrized boundary (or, in other words, conformal ?eld theory worldsheets). The Jacobian of a closed surface is an abelian variety. What structure of “open abelian variety” captures the relevant data in the “Jacobian” of a CFT worldsheet? If we considered Riemann surfaces with punctures instead of parametrized boundary components, the right answer could be easily phrased in terms of mixed Hodge structures.

But in worldsheets, we see more structure, and some of it is in?nite-dimensional. For example, even to a disk with analytically parametrized boundary, one naturally assigns an in?nite-dimensional symplectic form and a restricted maximal isotropic space (cf. [7]). Any structure we propose should certainly include such data. Additionally, in worldsheets, boundary components can have inbound or outbound orientation, and an inbound and outbound boundary component can be glued to produce another worldsheet. So another test of having the right notion of “open abelian variety” is that it should enjoy a similar gluing structure.

We should point out that it is actually a remarkably strong requirement that a structure such as a (closed) abelian variety could somehow be “glued together” from “genus 0” data similar to the situation we described above for a disk. One quickly convinces oneself that naive approaches based on modelling somehow the 1-forms on a Riemann surface, together with mixed Hodge-type integral structure data, fail to produce the required gluing. In fact, in some sense, the desired structure must be “pure” rather than “mixed”.

Note that there is no way of “gluing” a pure Hodge structure out of a mixed Hodge structure which does not already contain it: in the case of a closed Riemann surface with punctures, the mixed Hodge structure on its ?rst cohomogy contains the pure Hodge structure of the original closed surface, so no gluing is involved. Clearly, the situation is di?erent when we are gluing a non-zero genus surface from a genus 0 surface with parametrized boundary.

There is, however, a yet stronger test. When L is an even lattice (together with a Z/2-valued bilinear form b satisfying a suitable condition), one has a notion of conformal ?eld theory associated with L ([9, 4]). It could be argued that the de?nition only uses additive data, so the lattice conformal theories should “factor through open abelian varieties”. In some sense, if one considers the conjectured space of open abelian varieties to be the “Jacobian” of the moduli space of worldsheets (with all its structure), then one could interpret this as a sort of “Abelian Langlands correspondence” for that space.

Über Autor(innen)

Dr. Hossein Abbaspour, Department of Mathematics, Université de Nantes, France. Prof. Dr. Matilde Marcolli, Department of Mathematics, California Institute of Technology, Pasadena, California, USA. Dr. Thomas Tradler, Department of Mathematics, New York City College of Technology (CUNY), New York, USA.