# From Poisson to String Geometry

## Erlangen, September 11 – 14 2012

Photograph by Johanna Kulzer

## Scientific Objective

Both, applications in mathematical physics and the study of Poisson geometry have lead to the consideration of higher categorical geometrical structures, for example higher generalizations of bundles. A profound source for such structures has been the study of sigma models, in particular in its application to the quantization of Poisson manifolds. We plan to bring together people working on Poisson geometry, on higher categorical structures and mathematical physicists. A direction that we want to emphasize for future research are higher categorical structures occurring in the description of geometric string structures.

## Speakers

Confirmed speakers include:

• Christian Becker, Universität Potsdam
• Lawrence Breen, Université Paris 13
• Ulrich Bunke, Universität Regensburg
• Henrique Bursztyn, IMPA, Rio de Janeiro
• Marius Crainic, Utrecht University
• Giovanni Felder, ETH Zürich
• Ezra Getzler, Northwestern University
• Gerd Laures, Ruhr-Universität Bochum
• Pavel Mnev, ETH Zürich
• Jouko Mickelsson, University of Helsinki
• Dmitry Roytenberg, University of Nijmegen / Utrecht University
• Urs Schreiber, Utrecht University
• Danny Stevenson, University of Glasgow
• Christian Voigt, University of Glasgow
• Tilmann Wurzbacher, Ruhr-Universität Bochum

## Programme

Time Tuesday Sept. 11 Wednesday Sept 12 Thursday Sept 13 Friday Sept 14
9:30 Registration     Schreiber
10:00   Getzler Bursztyn Breen
10:30 Coffee Break
11:00 Coffee break Coffee break Coffee break Zambon
11:30 Felder Voigt Mnev
12:00 Lunch break
12:30 Lunch break Lunch break Lunch break
13:30 Stevenson
14:00
14:30 Bunke Mickelsson Becker Coffee Break
15:00 Wurzbacher
15:30 Coffee Break Coffee Break Coffee Break
16:00 Crainic Roytenberg Laures
16:30
17:00 Welcome
18:30
19:00 Conference Dinner
19:30

## Lunch Breaks

Lunch will be provided at the university restaurant in the same building. There will be a choice of different dishes, including meat or fish, vegetarian options and a salad bar.

## Welcome

On Tuesday, September 11, there will be an informal welcome with beer and pretzels directly after the last talk. It will take place in the conference building.

## Conference Dinner

The conference dinner will take place Thursday, September 13, 7:00 pm at the restaurant Schwarzer Bär, Innere Brucker Straße 19, 91054 Erlangen. The restaurant is in Erlangen city centre in walking distance from all hotels (Map).

## Talks

• Christian Becker, Universität Potsdam: Cheeger-Chern-Simons Theory and Geometric String StructuresI will explain the notion of differential characters with sections along a smooth map and their covariant derivatives. The Cheeger-Simons characters have canonical sections with covariant derivative the Chern-Simons form.This yields a notion of geometric String structures. I will further explain fiber integration for differential characters. This leads to higher dimensional transgressions with properties analogous to topological quantum field theories in the sense of Atiyah.(Slides)
• Lawrence Breen, Université Paris 13: Functorial homology and geometryI will discuss certain functorial aspects of the homology and cohomology of groups and Eilenberg-Mac Lane spaces, and related geometric constructions.
• Ulrich Bunke, Universität Regensburg: Differential cohomology – the spectrum aspect
• Henrique Bursztyn, IMPA, Rio de Janeiro: Multiplicative structures on Lie groupoids Symplectic groupoids are central objects in Poisson geometry which naturally arise in the theory of (Poisson) sigma models. A simplectic groupoid is a Lie groupoid equipped with a symplectic form which is suitably compatible with the groupoid structure, in the sense that it is „multiplicative“. The talk will discuss more general multiplicative geometric structures on Lie groupoids and describe their associated infinitesimal geometries, extending the correspondence between Poisson structures and symplectic groupoids.
• Marius Crainic, Utrecht University: Multiplicative forms and Spencer operatorsI will discuss multiplicative forms with coefficients and explain an integrability theorem in this context, inspired from Cartan’s theory of Lie pseudogroups and literature that followed it (the geometry of PDE and exterior differential systems). This is based on joint work with Maria Amelia Salazar and Ivan Struchiner.
• Giovanni Felder, ETH Zürich: The classical master equationWe formalize the construction of Batalin and Vilkovisky of a solution of the master equation associated with a polynomial in n variables (or a regular function on a nonsingular affine variety). We show existence and uniqueness up to „stable equivalence“ and discuss the associated BRST cohomology (joint work with David Kazhdan).
• Ezra Getzler Northwestern University: The Poisson Lie n-algebra in classical field theory I will discuss the role of L-infinity algebras in classical field theory, especially in the presence of supersymmetry.
• Gerd Laures, Ruhr-Universität Bochum: On characteristic classes in TMF
• Pavel Mnev, ETH Zürich: TBA
• Jouko Mickelsson, University of Helsinki: 3-cocycles, QFT anomalies, and gerbal representationsThe purpose of this contribution is to point out connections between recent ideas about the gerbes and gerbal representations (as higher categorical extension of representation theory) and the old discussion in quantum field theory on commutator anomalies, gauge group extensions, and 3-cocycles. The unifying concept is the classical obstruction theory for group extensions as explained in the monograph S. MacLane: Homology. (Slides)
• Dmitry Roytenberg, University of Nijmegen / Utrecht University: Courant algebroids and Poisson brackets in (1+1)-dimensional field theory The structure of a Courant algebroid is intimately related to both the string geometry of a manifold and the Poisson geometry of its loop space. The latter link is more or less straightforward, the former –much more intriguing.
• Urs Schreiber, Utrecht University: Higher quantomorphism groups on n-plectic higher stacks n-Plectic geometry is an interpretation of themultisymplectic description of n-dimensional field theory in terms ofhigher algebra/higher geometry. Chris L. Rogers has proposed a definition of Poisson L-infinity algebras over n-plectic manifolds. In the talk I give a simple definition of quantomorphism n-groups over n-plectic cohesive infinity-stacks. Then I discuss that they integrate these L-infinity algebras in the case that the infinity-stack is just a smooth manifold, and hence generalize them to the case that it is not. I end by indicating how for n=2 and n=3 the construction subsumes the higher gauge coupling behaviour of the open type II string and the open membrane.
• Danny Stevenson, University of Glasgow: Classifying theory for parametrized simplicial groups I will describe a classifying theory for parametrized simplicial groups, i.e. simplicial group objects in a suitable category of spaces over a fixed base space, in terms of a certain universal cocycle.
• Christian Voigt, University of Glasgow: The string group and vertex algebras I shall discuss a tentative realization of the string 2-group using unitary vertex superalgebras and unitary full field algebras. This includes a categorified analogue of the Clifford algebra and its spinor representation, and I will explain carefully the analogy to the situation for the spin group.(work in progress)
• Tilmann Wurzbacher, Ruhr-Universität Bochum: Dirac operators on loop spaces ?! We recall the Sigma-model derivation of the classical indextheorem on a Spin manifold to motivate the quest for a Dirac operator on a loop space and its hypothetical relation to elliptic genera and cohmology via its rotation equivariant index. We then review work of several people (including the speaker) on the construction of Dirac operators on loop spaces and the ensuing problems in Spin geometry of loop spaces/String geometry of manifolds.
• Marco Zambon, Universidad Autonoma de Madrid, ICMAT: Homotopy moment maps The notion of moment map is central in symplectic geometry, where thefunctions on the symplectic manifold (the „observables“) form a Lie algebra. We extend this notion to higher differential forms, defining ahomotopy moment map to be an $L_{\infty}$-algebra morphism into the observables. We give a cohomological interpretation (which provides a natural notion of equivalence), show that certain equivariant cocycles induce homotopy moment maps, and discuss obstructions. This is joint work in progress with Chris L. Rogers (Göttingen) and Yael Fregier (MIT).

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## Travel and Location

The workshop will take place in Erlangen, which is a university town in the south-east of Germany in the region of Franconia. It is easy to reach by a combination of air and train travel. The conference venue is the Department of Mathematics, University of Erlangen-Nürnberg, Cauerstraße 11, 91058 Erlangen. (Map, Campus Map ) The talks will take place in the lecture theatre Hörsaal 13 on the first floor.

## Support

This workshop is an activity of the Research Network String Geometry and the Emerging Fields Project Quantum Geometry and funded by the University of Erlangen-Nuremberg via its Emerging Fields Initiative.