Transformation Groups and Mathematical Physics 7
Paderborn, April 15/16, 2011
Supported by the DFG Priority Project 1388 Representation Theory
Program:
Speakers:
A. Alldridge (Köln)
G. Ben Simon (Zürich)
K. Rejzner (Hamburg)
M. Kalus (Bochum)
Th. Nikolaus (Hamburg)
Ch. Sachse (Hamburg)
G. Tuynman (Lille)
S. Wagner (Erlangen)
J. Winkelmann (Bochum)
M. Zirnbauer (Köln)
Schedule: ll talks will take place in room A3.301 on the main campus of Paderborn University
Friday, April 15
13:00 - 13:50 K. Reijzner: Batalin-Vilkovisky formalism and General Relativity
14:00 - 14:50 A. Alldridge: A Paley-Wiener theorem on the super-hyperbolic plane
Coffee Break
15:30 - 16:20 G. Ben Simon: Lie semigroups and Quasi-morphisms
16:30 - 17:20 Th. Nikolaus: String structures and 2-bundles
17:30 - 18:20 M. Kalus: On radial parts of super differential operators
19:00 Conference Dinner
Saturday, April 16
09:00 - 09:50 M. Zirnbauer: Weyl symmetry of an orbital integral transform for symmetric superspaces
Coffee Break
10:30 -11:20 Ch. Sachse: Diffeomorphism supergroups
11:30 - 12:20 J. Winkelmann: Entire Curves and Rational Points: Lifting along fiber bundles
Lunch Break (Sandwiches)
13:00 - 13:50 S. Wagner: A geometric approach to noncommutative principal torus-bundles
14:00 - 14:50 G. Tuynman: Super Heisenberg orbits: a case study
Logistics:
Possible train connections:
April 15:
Bremen-Paderborn: 8:44 - 11:39
Erlangen-Paderborn: 7:09 - 11:15
Hamburg-Paderborn: 8:24 - 11:46
Köln-Paderborn: 9:28 - 12:09
April 16:
Paderborn-Bremen: 16:16 - 19:14
Paderborn-Erlangen: 15:49 - 20:04
Paderborn-Hamburg: 16:15 - 19:35
Paderborn-Köln: 15:51 - 18:32
Accomodation:
Campus Lounge, Hotel zur Mühle, Galerie-Hotel Abdinghof, Hotel Kaup
Registration: Please send your date of arrival and departure to B. Borchert
If you request financial support, please contact J. Hilgert
Previous workshops in this series:
TGMP 4: Bremen, March 24/25, 2007
TGMP 5: Hamburg, December 5/6, 2009
TGMP 6: Bremen, May 28/29, 2010
Abstracts:
A. Alldridge: Motivated by applications to disordered metals, Zirnbauer defined in 1991 a spherical Fourier transform for the super-hyperbolic plane $SO^+(2,1|2)/SO(2|2)$. We prove a Paley-Wiener theorem, characterising the Fourier image of compactly supported superfunctions, in this context and discuss this in the framework of harmonic analysis on more general Riemannian symmetric superspaces.
G. Ben Simon: A finite dimensional connected simple Lie group carries an invariant continuous partial order (a Lie semigroups of the groups) if and only if it carries a non trivial homogeneous quasi-morphism. This remarkable fact has been noticed only recently. The "proof" to this fact can be obtained by looking at a classification lists obtained by the Lie semigroups schools of most notably Vinberg and Olshanskii on the one hand and of Hofmann, Hilgert and Neeb on the other hand. However, this classification proof does not provide any explanations to this fact. It is the purpose of this talk to explain why. The link between these two structures has found implications, which is under study, to bounded cohomology of Lie groups and Teichmuller theory-on which I will say a few words if time will permit. It is based on a joint work with T.Hartnick.
K. Fredenhagen/K. Rejzner: The treatment of gauge symmetries by the Batalin-Vilkovisky formalism is applied to General Relativity (GR). A conceptual difficulty is the nonexistence of local observables in GR which, on a fixed spacetime, would render the Batalin-Vilokovisky cohomology trivial.
It turns out, however, that the framework can be extended to the locally covariant fields (in the sense of natural transformations beween suitable functors) and leads there to a nontrivial cohomology which can be identified with the Poisson algebra of observables of classical gravity.
M. Kalus: In a certain geometric framework it is possible to define the restriction of super differential operators to embedded subsupermanifolds. In such a setting the Laplacian is analyzed and its radial part is computed. This is used along with a complex analytic version of Kostant's Lie-Hopf superalgebra construction to obtain Helgason's local root-torus restriction formula for type I Lie supergroups.
Th. Nikolaus: String structures are important for the well-definedness of the Feynman amplitudes in supersymmetric sigma models. They are usually defined using topological models of the string group. In order to discuss geometric string structures we have to find a smooth model of the string group and string structures. We discuss an approach using smooth 2-groups and 2-bundles. Furthermore we compare our approach with the one of Stolz-Teichner and with the one of Waldorf.
Ch. Sachse: Automorphisms of supermanifolds naturally only form groups but a way to construct proper diffeomorphism supergroups has in principle been known for a while. We review the construction and give some explicit details. As usual, these groups are often infinite-dimensional which poses additional technical problems in the super case. We will outline a convenient way to handle infinite-dimensional supermanifolds and give examples of such infinite-dimensional diffeomorphism supergroups. Time permitting we will discuss some applications of these groups in supersymmetric sigma models.
G. Tuynman: The philosophy of the orbit method is that there is a bijection between a class of coadjoint orbits of a given Lie group and the set of all irreducible unitary representaions of that group. In order to understand what happens for Lie supergroups, we take a particular Heisenberg-like Lie supergroup of dimension 8. For this group we decompose the regular representation into irreducible components using the Berezin-Fourier transform. We also compute the representations associated to coadjoint orbits. It turns out that if we restrict attention to coadjoint orbits with an even symplectic form, not all irreducible components in the regular representation are recovered. Adding the coadjoint orbits with an odd symplectic form recovers some of the missing components, but not all. It is only when we consider also coadjoint orbits with a non-homogeneous symplectic form that we recover all irreducible components appearing in the regular representation.
S. Wagner: The noncommutative geometry of principal bundles is not really well understood so far. Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An important handicap of this approach is the ignorance of any topological and geometrical aspects. In this talk we present a new, geometrically oriented approach to the noncommutative geometry of principal bundles based on dynamical systems and irreducible representations of the corresponding transformation group. Indeed, given a dynamical system (A,G,a), consisting of a commutative locally convex algebra A, a locally compact group G and a group homomorphism a:G-->Aut(A), which induces a continuous action of G on A, we provide conditions including irreducible representations of G which ensure that the corresponding action µ:OA x G --> OA, (d,g) |--> d o a(g) of G on the spectrum OA of A is free. From this ``new characterization of free actions" we obtain a reasonable definition of trivial noncommutative principal Tn-bundles. The step from the trivial to the non-trivial case is then carried out by introducing a localization concept for algebras and saying that an algebra is a \emph{non-trivial noncommutative principal Tn-bundle if ``localization" around characters of the fixalgebra of CA are trivial noncommutative principal Tn-bundles. Finally, we discuss some examples.
M. Zirnbauer: Recent numerical and experimental studies on disordered quantum Hamiltonian systems have revealed a surprising symmetry property of the distribution function of the local density of states. I will argue that the observed symmetry is related to the Weyl group invariance of an orbital integral transform for the Iwasawa decomposition of a noncompact symmetric space. More precisely, it results from an adaptation of the Iwasawa decomposition to the case of Riemannian symmetric superspaces.