Abstracts


R. Fioresi (Università di Bologna)
"Introduction to complex analytic and algebraic super-geometry"

I will give definitions and first properties. At the end, I will briefly give as examples the SUSY curves and some results I obtained with S. Kwok.


F. Gavarini (Università Tor Vergata)
"Algebraic Supergroups"

Basing upon prof. Fioresi's talk on an "Introduction to complex analytic and algebraic super-geometry", I shall swiftly introduce the basics of the theory of "supergroups", freely switching from the real differential or complex analytic framework to the algebraic geometrical one. I shall then pass to the infinitesimal point of view, considering the notions of "Lie superalgebra" and of "Lie superalgebra *tangent to a supergroup*". For these, I shall touch upon the converse step of "integrating" a Lie superalgebra to a supergroup, in the sense of a super-version of Lie's Third Theorem: besides discussing the problem in general, I shall present a concrete, constructive solution available for *simple* Lie superalgebras. Time permitting I shall quickly present the alternative approach to supergroups in terms of "super Harish-Chandra pairs": these are somewhat simpler objects, whose category is the target of a natural functor from supergroups. The key fact is that one can construct a quasi-inverse to the latter, hence the two categories are equivalent. One possible quasi-inverse functor, originally devised by Koszul, has been recently extended by Fioresi and other authors to a widest context. Another, totally different quasi-inverse has been introduced by myself.


S. Kwok (Università di Bologna)
"Pi-projective space, SUSY curves ans super theta functions"

In supergeometry, one finds that many super analogues of classical projective manifolds are not superprojective (e.g. super Grassmannians). As a remedy for this, Manin proposed a new type of projective superspace, the \Pi-projective superspaces. We give a new construction of this superspace, interpreting it in terms of the super skew field D, and the associated theory of \Pi-invertible sheaves. We discuss briefly the connection with Levin's embedding of SUSY-1 elliptic curves into (products of) \Pi-projective superspaces.

G. Codogni (Università Roma Tre)
"Moduli of SUSY curves"

SUSY curves are the generalization of Riemann surfaces. Following Witten and Donagi, I will show that the moduli space of SUSY curves is not projected. According to them, this means that the moduli space has "its life on its own". Then, I will introduce periods of SUSY curves. Where the period matrix makes sense, it provides a map towards the classical moduli space of curves endowed with a highly non-reduced structure.




Bibliography

General Introductions to Super-Geometry



Super-Groups (those with a * are the key ones)

* [1] C. Carmeli, L. Caston, Lauren, R. Fioresi, "Mathematical foundations of supersymmetry", EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zrich, 2011, in particular, Chapters 7 & 11
* [2] R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014, pp. 1-77
[3] R. Fioresi, F. Gavarini, "On the construction of Chevalley Supergroups", in: S. Ferrara, R. Fioresi, V. S. Varadarajan (eds.), Supersymmetry in Mathematics and Physics, UCLA Los Angeles, USA 2010 Lecture Notes in Mathematics, Volume 2027, 2011, Springer & Verlag, Berlin-Heidelberg-New York, pp. 101-123
[4] R. Fioresi, F. Gavarini, "Algebraic supergroups with classical Lie superalgebras", Journal of Lie Group Theory 23 (2013), no. 1, 143-158
[5] F. Gavarini, "Chevalley Supergroups of type D(2,1;a)", Proceedings of the Edinburgh Mathematical Society (to appear), 20 pages - see also arXiv:1006.0464 (2010)
* [6] F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (to appear), 92 pages - see also arXiv:1109.0626 (2011)
* [7] F. Gavarini, "Global splittings and super Harish-Chandra pairs for affine supergroups", Transactions of the American Mathematical Society (to appear) - see also arxiv:1308.0462 (2013)

SUSY curves