Supported by:

University of Luxembourg

Mathematics Research Unit

The scientific programme will start on the morning of the **December 14th, 2017** and conclude **December 15th, 2017** in the afternoon. The talks will take place in Maison du Savoir, room 2.400, which is listed as building 1 on the Campus map. The schedule is as follows.

*Thursday, December 14, 2017*

10:00 - 11:00 | Hovhannes Khudaverdian | Thick morphisms and higher Koszul brackets |
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11:00 - 11:30 | Coffee | |

11:30 - 12:30 | Janusz Grabowski | Splitting theorem for colored supermanifolds |

12:30 - 15:00 | Lunch | |

15:00 - 16:00 | Rita Fioresi | Admissible systems and Harish-Chandra representations for semisimple real Lie supergroups |

16:00 - 16:30 | Coffee | |

16:30 - 17:30 | Norbert Poncin | Higher supergeometry revisited |

18:30 | Workshop dinner | Restaurant Beeftro à Esch-belval |

*Friday, December 15, 2017*

10:00 - 11:00 | Steven Duplij | Developing new supermanifolds by revitalizing old ideas |
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11:00 - 11:30 | Coffee | |

11:30 - 12:30 | Zoran Škoda | Symmetry objects for geometries based on monoidal categories |

12:30 - 15:00 | Lunch | |

15:00 - 16:00 | Florian Hanish | Berezin integration on super loop space and applications to index theory |

16:00 - 16:30 | Coffee | |

16:30 - 17:30 | Andrew Bruce | On a ℤ_{n} ^{2} - graded version of Minkowski superspace |

**On a ℤ _{n} ^{2} - graded version of Minkowski superspace**

We show one possible way of extending the notion of Minkowski superspace to the setting of ℤ_{n} ^{2} - geometry. The resulting ℤ_{n} ^{2} - manifold formally resembles N-extended superspace (with central charges) but there are subtle diffences due to the exotic nature of the grading employed.

**Developing new supermanifolds by revitalizing old ideas**

First, we apply the regularity concept for supermanifolds and obtain so called obstructed (or regular) supermanifolds (semi-supermanifolds), generizing von Neumann regularity to n-regularity. In the patch definition, the transition functions become noninvertible in some sense and satisfy special cocycle conditions described by semi-commutative diagrams. The obstruction, which measures the distinction from the standard supermanifold, is proportional to the difference of some idempotent self-maps from identical morphisms. The same procedure of regularization is applied for categories and functors. Second, we introduce another noninvertible analog of superconformal transformations which arise from the alternative possiblity of supermatrix reduction in the simplest one-grade case. They are always noninvertible, having nilpotent Berezinian and twist parity of the tangent space together with the so called mixed cocycle condition. For N=2 the role of superjacobian is played by permanent, also the remarkable formula connected Berezinian, permanent and determinant is obtained. Third, we consider some aspects of ternary supersymmetry. Fourth, we introduce the ring of polyadic integer numbers, which leads to unusual polyadic fields. The ways of generalization for the above constructions to the multigraded supersymmetry is outlined.

**Admissible systems and Harish-Chandra representations
for semisimple real Lie supergroups**

Real forms of classical Lie superalgebras are in one to one correspondence with Cartan automorphisms and Cartan decompositions. We introduce the notion of admissible root system with respect to a given Cartan automorphism and we show that the condition of admissibility is equivalent to the existence of infinite dimensional holomorphic representations of the real Lie supergroups underlying the given real form. We briefly discuss generalization to Lie superalgebras infinite dimensional and to Kostant root systems.

**Splitting theorem for colored supermanifolds**

The classical Batchelor-Gawedzki theorem says that any smooth supermanifold is (non-canonocally) diffeomorphic to the `superization' ΠE of a vector bundle E. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of colored supermanifolds (ℤ_{n} ^{2} - supermanifolds) with its local model made of formal power series with ℤ_{n} ^{2} - gradation and ℤ_{n} ^{2} - commutation rules. We show that any smooth colored supermanifold is (non-canonocally) diffeomorphic to the `superization' ΠE of an n-fold vector bundle E. The latter can be chosen split.

**Berezin integration on super loop space and applications to index theory**

The super loop space associated to an ordinary (spin) manifold is the space of ``maps'' from the super circle S^{1|1} to M. We will briefly describe its infinite-dimensional supermanifold structure and show that the resulting algebra functions is closely related to the algebra of differential forms on ordinary loop space. In particular, it is possible to relate the action of S^{1|1} on superfunctions to the equivariant differential on forms. Based on the Wiener integral, it is possible to define an analogue of Berezin integration. Similar to the finite-dimensional case, it turns out that the odd part of this integral may be expressed by an infinite-dimensional Pfaffian, the latter being defined using zeta-regularization. If time allows, we will eventually describe how these concepts can be used to obtain ``path integral-type'' proofs of index theorems by integrating Bismut-Chern characters. The argument indicates that supersymmetry enters these calculations in a crucial way.

This talk is based on joint work (partly in progress) with Matthias Ludewig.

**Thick morphisms and higher Koszul brackets**

It is a classical result in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. One of the manifestations of this structure is the ``Koszul bracket'' of differential forms. There is a natural homomorphism from the resulting differential Lie superalgebra into the superalgebra of multivector fields with respect to the canonical Schouten bracket and the Lichnerowicz differential.

In the talk, we shall present a homotopy analog of the above results. When an ordinary Poisson structure is replaced by a homotopy one, instead of a single Koszul bracket there arises an infinite sequence of ``higher Koszul brackets'' introducing an L_{∞}-algebra structure in the space of differential forms (
Khudaverdian-Voronov, 2008). We shall show how to use thick morphisms of supermanifolds to construct a non-linear transformation, which is an L_{∞}-morphism, from this L_{∞}-algebra of differential forms to the Lie superalgebra of multivector fields with the canonical Schouten bracket. (A thick morphism of manifolds is a new notion recently introduced, see e.g. http://arxiv.org/abs/1411.6720v5.)

Based on joint work with Th. Voronov.

**Higher supergeometry revisited**

The aim of the talk is to present a generalization of superalgebra and supergeometry to ℤ_{n} ^{2} - gradings, ℤ_{n} ^{2} = ℤ_{2} x … x ℤ_{2}, n>1. The corresponding sign rule is not given by the product of the parities, but by the scalar product of the involved ℤ_{n} ^{2} - degrees. This ℤ_{n} ^{2} - supergeometry exhibits interesting differences with classical supergeometry, provides a sharpened viewpoint, and has better categorical properties. Further, it is closely related to Clifford calculus: Clifford algebras have numerous applications in Physics, but the use of ℤ_{n} ^{2} - gradings has never been investigated. In particular, the ℤ_{n} ^{2}-Berezinian determinant and the corresponding integration theory will be discussed.

**Symmetry objects for geometries based on monoidal categories**

As the basic organizing principle of supersymmetry, as well
as of its ℤ_{n} ^{2} refinement, one often emphasizes that all algebraic
constructions live in an appropriate symmetric monoidal category. In
the context of monoidal categories one studies the symmetry objects
like supergroups and their anyonic and braided analogues. Using
actions of these symmetry objects, one comes to important examples of
geometries, the superhomogeneous and some quantum homogeneous spaces.
After presenting some generalities on symmetries in the context of monoidal categories, I will focus on attempts at combining the infinitesimal symmetries or
differential operators with the coordinate superalgebra of the space
into the analogues of groupoids as novel symmetries in this context.
Main examples comprise super-Hopf algebroids. I will also explain
their relevance in the deformation quantization.

Programme and Abstracts in pdf format