## SFB-Seminar on Supersymmetries

Next seminars:

Speaker Date Room Title
Dominik Ostermayr (Universität zu Köln) Monday, 13.07.2015,
14:15 - 16:00
NA 4/64 Automorphism Supergroups of Supermanifolds
Abstract.

A classical theorem states that the group of automorphisms of a manifold M preserving a G-structure of finite type is a Lie group. We show how to generalize this statement to the category of cs manifolds and give some examples, some of which being generalizations of classical notions, others being particular to the super case. Notably, we have to introduce a new notion of supermanifolds which we call mixed supermanifolds.

## Archive

Speaker Date Room Title
Tilmann Wurzbacher (RUB) Wednesday, 04.03.2015,
10:15 - 12:00
NA 4/24 On orbits of Lie supergroups
Abstract.

We study actions of Lie supergroups, in particular, the hitherto elusive notion of orbits through odd (or more general) points'. Following categorical principles, we derive a conceptual framework for their treatment and therein prove general existence theorems for the isotropy supergroups and orbits at general points. In this setting, we show that the coadjoint orbits always admit a (relative) supersymplectic structure of Kirillov-Kostant- Souriau type. Applying a generalisation of Kirillov's orbit method to the parity changed variants of the Heisenberg group, the Clifford supergroup and the odd Heisenberg supergroup, we obtain universal' families of representations parametrised by a supermanifold.

Speaker Date Room Title
Ricardo Kennedy (Universität zu Köln) Friday, 14.11.2014,
10:15 - 11:45
NA 4/64 Homotopy theory of topological insulators
Speaker Date Room Title
Hermann Schulz-Baldes (FAU Erlangen-Nürnberg) Thursday, 26.06.2014,
13:00 - 14:30
NA 4/64 Topological insulators and index theory
Abstract.

According to a widely accepted terminology, a topological insulator is a (independent) Fermion system which has surface modes that are not exposed to Anderson localization. This stability results from topological constraints given by non-trivial invariants like non-commutative Chern numbers and higher winding numbers, but sometimes also more subtle $Z_2$ invariants associated to adequate Fredholm operators with symmetries. Prime examples are quantum Hall systems, but the talk also considers chiral and BdG systems as well as time-reversal symmetric systems with $Z_2$ invariants.

Speaker Date Room Title
David Kyed (Odense) Friday, 25.04.2014,
10:15
NA 4/64 L2-Betti numbers for locally compact groups
Abstract.

In a recent joint work with Petersen and Vaes, we prove that the L2-Betti numbers of a locally compact group coincide, up to a natural scaling constant, with the L2-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of it. As a consequence, we obtain that the L2-Betti numbers of any lattice equal the ones for the ambient group scaled with the covolume of the lattice in question. Moreover, several vanishing results are derived from this, including the vanishing of the reduced L2-cohomology for amenable locally compact groups. In my talk I will elaborate on these results and survey the theory of L2-Betti numbers.

Speaker Date Room Title
Michel Egeileh (American Univ. Beirut/Lebanon) Friday, 10.01.2014,
10:15
NA 4/64 Smoothness of maps and of spaces in infinite dimensions
Abstract.

In his thesis, Adrien Douady introduced the notion of (infinite-dimensional) Banach analytic spaces using the language of "functored spaces". On the other hand, Pierre Mazet defined a notion of analytic spaces in the context of locally convex spaces. We discuss the second approach and show, by a theorem of Douady, that the structure of infinite-dimensional analytic space is very weak, allowing for pathological phenomena. In particular, there exists maps defined on analytic spaces which are "weakly analytic" but not analytic. Finally, we present a new result showing that, for an appropriate category of locally convex spaces (including Fréchet spaces), smoothness of maps is equivalent to their "scalarwise smoothness" and to their "weak smoothness". This allows the definition of a large class of infinite-dimensional manifolds via locally-ringed spaces.

Speaker Date Room Title
Mauro Spera (Brescia) Friday, 20.09.2013,
10:15
NA 4/64 On geometric and physical aspects of some unitary Riemann surface braid group representations
Abstract.

In this talk we discuss the simplest unitary Riemann surface braid group representations, which are constructed geometrically via representations of the Weyl-Heisenberg group- by means of stable bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail with the help of noncommutative geometric tools, and, time permitting, an application of Fourier-Mukai techniques is given as well.

Speaker Date Room Title
Christian Becker (Universität Potsdam) Tuesday, 16.07.2013,
10:15 h
NA 2/24 Introduction to differential cohomology
Abstract.

Motivated by the construction of the Chern-Simons invariants of closed 3- manifolds, Cheeger and Simons in the late 1970’s introduced the notion of differential characters. From a topological point of view, the ring of differen- tial characters is a refinement of integral cohomology by differential forms. Differential characters provide a particular model for what is now called differential cohomology. From a geometric point of view, differential char- acters are higher degree analogues of (isomorphism classes of) hermitean line bundles with connection. They are used in mathematical physics to describe higher degree analogues of gauge fields and field strengths for the gauge group U(1). Elementary concepts available for complex line bun- dles with connection, like holonomy along closed paths, parallel transport, (nonvanishing) sections and their covariant derivatives, have counterparts for higher degree differential cohomology classes. In the talk I will consider differential cohomology from a topological and a geometric perspective. I will focus on differential characters as a model for differential cohomology. I will also explain the similarities and differences between differential cohomology and singular cohomology.

Speaker Date Room Title
Christian Becker (Universität Potsdam) Tuesday, 16.07.2013,
16:15 h
NA 1/58 Differential cohomology, stratifolds and the Cheeger-Chern-Simons construction
Abstract.

Differential extensions of cohomology theories have appeared in several con- texts in topology, geometry and mathematical physics. For example, com- plex line bundles with connection are classified by the differential cohomol- ogy group of degree 2. This defines a U(1) gauge theory wich classically models electromagnetic potentials and fields. Complex line bundles are also the starting point of geometric quantization. Higher differential cohomology groups classify gerbes and higher form fields used in String theory. In the same way as line bundles allow to compute holonomy along closed curves, higher differential cohomology classes have holonomies along higher dimensional closed oriented manifolds. More generally, we may re- place smooth manifolds by stratifolds. This allows a clear geometric ap- proach to basic constructions for differential cohomology like (uniqueness of) the ring structure and fiber integration. In the talk I will introduce differential cohomology, mainly by the spe- cific model of differential characters. I will explain the identification of low degree differential cohomology groups with well-known objects from geometry, topology and mathematical physics, including the relation to Chern-Simons forms and Chern-Weil theory. Finally I will explain the use of stratifolds to geometrically construct the ring structure, fiber integration and transgression to loop space.

Am 28.06.2013 findet das "SFB-Seminar on symmetries" nicht statt, aufgrund des am gleichen Tage in Wuppertal stattfindenden Seminares "Complex algebraic geometry and complex analysis" (gemeinsam mit der Universität Wuppertal).

Am 21.06.2013 findet das "SFB-Seminar on symmetries" nicht statt, aufgrund eins SFB TR 12 Workshops in Bad Honnef (20.06.2013 bis 23.06.2013).

Speaker Date Room Title
Wolfgang Palzer (Universität Köln) Friday, 14.06.2013,
10:15 h
NA 4/64 The Fourier Transformation on Symmetric Superspaces – The Rank One Case
Abstract.

In order to have a useful Fourier transform on symmetric superspaces, we need to understand the asymptotic behaviour of certain spherical functions, which are eigenfunctions of the Laplacian. In studying their asymptotic behaviour, we are led to the so-called Harish-Chandra c-function - a necessary ingredient to define the inverse Fourier transform. The proof of the inversion formula involves coordinate transformations resulting in boundary terms, which have to be dealt with appropriately.

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Speaker Date Room Title
Zain Shaikh (Universität Köln) Friday, 17.05.2013,
10:15 h
NA 4/64 Superbosonization and the Laplace transform of conical superfunctions
Abstract.

The superbosonisation formula of Littelmann-Sommers-Zirnbauer relates a flat Berezin integral over a linear superspace (LHS) to an integral over a symmetric superspace Omega (RHS). The LHS can be interpreted as averages of observables in condensed matter physics, and the importance of the formula stems from the fact that the RHS is well-suited to calculate asymptotics of such averages. Alexander Alldridge and I have given a short conceptual proof of this formula by developing a theory of Laplace transforms on the relevant superspaces. The Laplace transform of the LHS is straightforward, whereas the RHS is more intricate. The real work is to define and calculate the Laplace transform of conical superfunctions over Omega. This leads to a function that is analogous to the Gindikin Gamma-function in the purely even case, which will have consequences for analytic representations of the Lie supergroups in the picture. In this talk, I will discuss this proof, define the conical superfunctions, give some indication of how to calculate their Laplace transform, and (if time permits) talk about the representation theory. Am 26.04.2013 fand das "SFB-Seminar on symmetries" nicht statt, aufgrund des am gleichen Tage in Bochum stattfindenden Seminares "Complex algebraic geometry and complex analysis" (gemeinsam mit der Universität Wuppertal).

Speaker Date Room Title
Natalie Schieferstein (Ruhr-Universität Bochum) Friday, 03.05.2013,
10:15 h
NA 4/64 Elliptische Integrale, der elliptische Sinus und meromorphe Funktionen auf komplex-eindimensionalen Tori (Präsentation BSc-Arbeit).
Speaker Date Room Title
Andreas Osterloh (Universität Duisburg-Essen) Friday, 19.04.2013,
10:15 h
NA 4/64 Invariant theory and entanglement: a connection of the polynomial degree of an invariant and the length of the state
Abstract.

Entanglement is a phenomenon that is invariant with respect to local unitaries, or even more, the local SL transformations. The theory of multipartite entanglement for qubits is therefore well described by the Cayley-Hilbert omega-process and singlet contractions. Here I show how it is correlated to the antilinear expectation value of an SL invariant operator. Consequences of this is a balancedness condition that points towards the length of the state, i.e. the way it is decomposed into product states of some local basis.

Speaker Date Room Title
Benjamin Schwarz (Universität Paderborn) Friday, 12.04.2013,
10:15 h
NA 4/64 Nearly holomorphic sections on Kähler manifolds
Abstract.

We describe the notion of nearly holomorphic functions on Kähler manifolds which was first introduced by G. Shimura. This notion extends to vector bundles and corresponding nearly holomorphic sections. We show that if $X$ is a compact Hermitian symmetric manifold, and $E \to X$ is anirreducible homogeneous Hermitian vector bundle, then the space of nearly holomorphic sections is dense in the space of $L^2$-sections. Finally, we give some representation theoretic applications of this result.

Speaker Date Room Title
Nathan Grieve (Queen's University, Canada) Tuesday, 09.04.2013,
14:15 h
NA 2/24 Index conditions, cup-products and number fields
Abstract.

We will discuss a cohomological condition which we place on a pair of vector bundles on an abelian variety. This condition is motivated on several fronts, notably D. Mumford's index theorem, as well as my own results concerning cup-products. I will explain these results, consider some examples, and show that this condition can be satisfied, in all possible instances, by a large class of abelian varieties. To see why, we will consider this condition as it applies to abelian varieties with real multiplication.

Speaker Date Room Title
Hannah Bergner (Ruhr-Universität Bochum) Tuesday, 12.02.2013,
14:15 h
NA 4/64 Lie superalgebras of vector fields and local actions
Speaker Date Room Title
Björn Villa (Ruhr-Universität Bochum) Friday, 01.02.2013,
12:15 h
NA 4/64 Kählerian structures on coadjoint orbits
Speaker Date Room Title
Florian Hanisch (Universität Potsdam) Friday, 18.01.2013,
10:15 h
NA 4/64 Variational problems on supermanifolds and applications to supergeodesics
Speaker Date Room Title
Christopher Graw (Ruhr-Universität Bochum) Friday, 11.01.2013,
10:15 h
NA 4/64 Real supergroup orbits in complex Supergrassmannians
Speaker Date Room Title
Alexander Kahle (MPIM Bonn) Tuesday, 18.12.2012,
16:15 h
NA 1/58 Supersymmetric field theories and differential K-theory
Speaker Date Room Title
Majdi BEN HALIMA (Sfax, Tunisia) Friday, 07.12.2012,
10:15 h
NA 4/64 On sums of coadjoint orbits
Abstract.

For compact connected Lie groups, a connection between the restriction of irreducible representations and the projection of coadjoint orbits has been discovered by Heckman, and then generalized by Guillemin and Sternberg in the framework of symplectic geometry. This provides in particular a connection between sums of coadjoint orbits of a compact connected Lie group G and tensor products of irreducible representations of G. In the first half of this talk, we explain applications of the latter result to certain matrix spectral problems. In the second half, using elementary representation-theoretic arguments, we prove an extension of the above-mentioned result of Heckman to all irreducible finite-dimensional representations of a compact semisimple Lie group.

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Speaker Date Room Title
Prof. Dr. Alexander Isaev (Canberra/Australia) Tuesday, 27.11.2012,
14:15 h
NA 1/58 Explicit characterization of isolated hypersurface singularities by their moduli algebras
Abstract.

The Mather-Yau theorem states that a germ of a hypersurface in complex space is completely characterized by a certain commutative algebra, the Tjurina algebra. The proof of the Mather-Yau theorem does not provide an explicit procedure for recovering the germ from the algebra, and finding a way of doing it is an interesting open problem. In this talk I will concentrate on the case of homogeneous singularities. One way to make the Mather-Yau theorem explicit in this case is to utilize classical invariant theory. This approach leads to an intriguing conjecture concerning classical invariants, and I will discuss progress on the conjecture to date. The talk will mainly involve elementary concepts from commutative algebra and classical invariant theory and should be suitable for a broad mathematical audience.

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Speaker Date Room Title
Roman Avdeev (Moscow) Friday, 23.11.2012,
10:15 h
NA 4/64 Spherical homogeneous spaces and harmonic analysis on them (II)
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Speaker Date Room Title
Roman Avdeev (Moscow) Friday, 16.11.2012,
10:15 h
NA 4/64 Spherical homogeneous spaces and harmonic analysis on them (I)
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Speaker Date Room Title
Prof. Dr. Bruce Gilligan (Regina/Canada) Friday, 09.11.2012,
10:15 h
NA 4/64 Holomorphic reductions of pseudoconvex homogeneous manifolds
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Speaker Date Room Title
Stéphane Garnier (Ruhr-Universität) Friday, 02.11.2012,
10:15 h
NA 4/64 Vector fields and metrics on supermanifolds
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Speaker Date Room Title
Stéphane Garnier (Ruhr-Universität) Friday, 19.10.2012,
14:15 h
NA 4/64 Generalized cohomology theories and spectra (Intro)
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Speaker Date Room Title
Aprameyan Parthasarathy (Universität Marburg) Friday, 29.06.2012,
10:15 h
NA 4/64 A fixed point formula on the Oshima compactification of a Riemannian symmetric space
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Speaker Date Room Title
Ricardo Kennedy Friday, 18.05.2012,
10:15 h
NA 4/64 Homotopy classification of topological insulators II
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Speaker Date Room Title
Hannah Bergner Friday, 11.05.2012,
10:15 h
NA 4/64 Palais' Konstruktion lokaler Lie Gruppenwirkungen
Abstract.

In dem Vortag werde ich die in "Global formulation of the Lie theory of transformation groups" von Palais beschriebene Konstruktion erkären, welche zeigt, dass zu jeder endlich-dimensionalen Lie Unteralgebra der Vektorfelder auf einer Mannigfaltigkeit eine lokale Lie Gruppenwirkung existiert, deren infinitesimale Wirkung genau der Wirkung der Lie Unteralgebra auf der Mannigfaltigkeit entspricht.

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Speaker Date Room Title
Arturo Prat-Waldron Friday, 04.05.2012,
10:15 h
NA 4/64 On the Construction of the Dirac Operator of a Spin Manifold via Geometric Quantization
Abstract.

In this talk we revisit the geometric quantization of a certain symplectic supermanifold obtained from considering a one-dimensional supersymmetric sigma model on a Riemannian manifold. We study the relation between the metaplectic correction in the super-setting and spin structures and we construct the heat kernel of the Dirac operator of a spin manifold by quantizing an appropriate Hamiltonian function using a supersymmetric version of the Blattner-Kostant-Stenberg kernel method. We explain how this point of view fits into the Stolz-Teichner program of studying cohomology theories via super-symmetric field theories.

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Speaker Date Room Title
Ricardo Kennedy Thursday, 03.05.2012,
10:15 h
NA 4/64 Homotopy classification of topological insulators
Abstract.

After their theoretical prediction, several examples of topological insulators have been realised experimentally. Despite this success, the question of the very definition of a 'topological insulator' is not settled. In this talk, I will motivate a mathematical definition of a topological insulator as a homotopy equivalence class of maps from a sphere or a torus into a complex Grassmannian, subject to contraints due to symmetries. Furthermore, I will present some methods of determining these equivalence classes in special cases.

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Speaker Date Room Title
Dr. Christian Miebach Friday, 27.04.2012,
10:15 h
NA 4/64 Holomorphe Reduktion von komplex-homogenen Mannigfaltigkeiten

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Speaker Date Room Title
Wolfgang Palzer (University of Cologne) Friday, 13.04.2012,
10:15 h
NA 4/64 The Stokes theorem on supermanifolds
Wolfgang Palzer (University of Cologne) Friday, 13.04.2012,
14:15 h
NA 4/64 Discussion with Wolfgang Palzer
Abstract.

We investigate integration over non-compact supermanifolds. Since the Berezin integral on superdomains is not invariant under changes of coordinates, a global definition of this integral is only possible after fixing a retraction. Once this is done and a reasonable analog of differential forms are defined — integral forms in the case of supermanifolds — we will deduce a super version of Stokes' Theorem. We conclude by investigating the behavior of ‘boundary terms’, which appear under the aforementioned change of coordinates/retractions.

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Speaker Date Room Title
Prof. Dr. Arkadij Onishchik (Yaroslavl/Russia) Wednesday, 22.02.2012,
14:00 h
NA 4/24 Non-Abelian Cohomology and Supermanifolds
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Speaker Date Room Title
Dr. Valdemar Tsanov (Bochum) Friday, 10.02.2012,
10:15 h
NA 4/64 Branching of representations, semistability and Schubert calculus
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Speaker Date Room Title
Miriam Kümmel (Bochum) Friday, 03.02.2012,
10:15 h
NA 4/64 Anwendungen des Spektralsatz fuer selbstadjungierte Operatoren in der Darstellungstheorie
Abstract.

Wir wiederholen den Spektralsatz für unbeschränkte selbstadjungierte Operatoren in drei Versionen und wenden ihn auf grundlegende Probleme der Darstellungstheorie an (Lemma von Schur bzw. Satz von Stone). Es handelt sich um einen Vortrag im Niveau für fortgeschrittene Studenten/junge Postdoktoranden und NICHT um einen Vortrag über rezente Forschung!

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Speaker Date Room Title
Dr. Valdemar Tsanov (Bochum) Monday, 30.01.2012,
12:00 h
NA 1/58 The Belkale-Kumar product on cohomology of flag manifolds
Abstract.

Let X be a complex flag manifold. Belkale and Kumar have de fined a new product on the cohomology space H*(X,C), which can be obtained as a deformation of the usual cup product. Another way to de fined the new product is by interpreting H*(X,C) as a suitable Lie algebra cohomology space, and taking the natural product there. An insight into the geometric meaning of the new product is gained through the notion of Levi-movability.
The Belkale-Kumar product is a useful technical tool in the study of semistability and branching rules for representations.

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Speaker Date Room Title
Prof. Dr. Gerald Schwarz (Brandeis University) Friday, 27.01.2012,
10:15 h
NA 4/64 Real geometric invariant theory and double coset spaces
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Speaker Date Room Title
Dr. Florin Dumitrescu (Hamburg) Friday, 20.01.2012,
10:15 h
NA 4/64 Superconnections and supergeometry
Abstract.

We use the language of supermanifolds to interpret geometrically superconnections (a la Quillen) as parallel transport along superpaths. This generalizes the usual parallel transport interpretation of connections.

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Speaker Date Room Title
Guillaume Tomasini Wednesday, 18.01.2012,
15:00 h
NA 4/64 The unitary dual of the universal cover of SU(1,1) and tensor products
Abstract.

In this talk, we shall derive by algebraic methods the classification of all simple unitary representation of the universal cover of SU(1,1), recovering a result first due to Pukanszky. This approach does (almost) not require any knowledge from representation theory. In a second part of the talk, we shall use this algebraic situation to understand the (discrete spectrum of the) decomposition of tensor products of two simple unitary representations of the universal cover of SU(1,1). The talk is based on a joint work with Bent Orsted.

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Speaker Date Room Title
Dr. Mathieu Molitor Monday, 19.12.2011,
12:15 h
NA 1/58 Information geometry, Kähler geometry and quantum mechanics
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Speaker Date Room Title
Prof. Dr. Ivan Dimitrov Friday, 16.12.2011,
10:15 h
NA 4/64 Lagrangian subalgebras of classical Lie superalgebras
Abstract.

I will first explain the theorems describing the Lagrangian subalgebras of complex simple Lie algebras (due to Belavin and Drinfeld) and of complex reductive Lie algebras (due to Delorme). I will then define the classical Lie superalgebras and will describe their Lagrangian subalgebras. This is a joint work with Milen Yakimov, LSU.

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Speaker Date Room Title
Dr. Will Kirwin Monday, 28.11.2011,
12:00 h
NA 1/64 Unitarity in quanization commutes with reduction I
Dr. Will Kirwin Monday, 05.12.2011,
12:00 h
NA 1/58 Unitarity in quanization commutes with reduction II
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Speaker Date Room Title
Prof. Dr. Frédéric Campana Friday, 25.11.2011,
10:15 h
NA 4/64 An orbifold version of Miyaoka's generic semi-positivity theorem
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Speaker Date Room Title
Christoph Wockel Friday, 28.10.2011,
10:15 h
NA 4/64 The diffeomorphism supergroup of a finite-dimensional supermanifold
Abstract.

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism su- pergroup of a finite-dimensional supermanifold. The reduced part of this supergroup is the automorphism group of this supermanifold, which can be endowed with the structure of a Frechet-Lie group in case the supermanifold is compact. We will expain this con- struction and how it can be used to endow the dif- feomorphism supergroup with the structure of a Lie supergroup.