Deformation Spaces (eBook)

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.

Preface 5
Contents 6
On the Hochschild and Harrison (co)homology of C8-algebras and applications to string topology 7
1. Hochschild (co)homology of an A8-algebra with values in a bimodule 10
2. C8-algebras, C8-bimodules, Harrison (co)homology 16
3. .-operations and Hodge decomposition 23
4. An exact sequence `a laJacobi-Zariski 37
5. Applications to string topology 47
6. Concluding remarks 55
References 55
What is the Jacobian of a Riemann Surface with Boundary? 58
1. Introduction 58
2. Open abelian varieties 61
3. Gluing and SPCMC structure 66
4. The Jacobian of a worldsheet with boundary 71
5. The lattice conformal .eld theory on the SPCMC of open abelian varieties 76
References 78
Pure weight perfect Modules on divisorial schemes 80
1. Introduction 80
2. Preliminary 81
3. Weight on pseudo-coherent Modules 87
4. Proof of the main theorem 88
5. Applications 92
References 93
Higher localized analytic indices and strict deformation quantization 95
1. Introduction 95
2. Index theory for Lie groupoids 98
3. Index theory and strict deformation quantization 104
4. Higher localized indices 109
References 114
An algebraic proof of Bogomolov-Tian-Todorov theorem 116
Introduction 116
1. Review of DGLAs and L8-algebras 118
2. The Thom-Whitney complex 121
3. Semicosimplicial differential graded Lie algebras and mapping cones 124
4. Semicosimplicial Cartan homotopies 127
5. Semicosimplicial Lie algebras and deformations of smooth varieties 129
6. Proof of the main theorem 131
References 134
Quantizing deformation theory 137
1. Introduction 137
2. Linear algebra 139
References 143
L8-interpretation of a classication of deformations of Poisson structures in dimension three 144
1. Introduction 144
2. Preliminaries: L8-algebras and Poisson algebras 148
3. Choice in a transfer of L8-algebra structure 154
4. Deformations of Poisson structures via L8-algebras 164
References 173

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.