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
- L. Caston, R. Fioresi, Mathematical Foundations of Supersymmetry, preprint, 2007, available on R. Fioresi homepage.
- Section 2 of R. Donagi and E. Witten, Super Moduli Space is not projected, ArXiv
- Y. Manin, Gauge Field Theory and Complex Geometry, Springer
- Witten, Notes on SuperManifolds and Itegration, ArXiv
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
- M. J. Bergvelt and J. M. Rabin, Supercurves, their Jacobians, and super KP equations, Duke Mathematical Journal, 1999
- R.Fioresi and S. Kwok, On SUSY curves, ArXiv
- Y. Manin, Topics in Non Commutative Geometry, M. B. Porter Lectures
- R. Donagi and E. Witten, Super Moduli Space is not projected, ArXiv
- E. Witten, Notes on super Riemann surfaces and their moduli, ArXiv