Rita Fioresi

Mathematical Foundations of Supersymmetry

We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups.

The Minkowski and conformal superspaces

We define complex Minkowski superspace in four dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this superflag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.

On the Construction of Chevalley Supergroups

We give a description of the construction of Chevalley supergroups, providing some explanatory examples.We avoid the discussion of the A(1, 1), P(3) and Q(n) cases, for which our construction holds, but the exposigetion becomes more complicated. We shall not in general provide complete proofs for our statements, instead we will make an effort to convey the key ideas underlying our construction. A fully detailed account of our work is scheduled to appear in [Fioresi and Gavarini, Chevalley Supergroups, preprint arXiv:0808.0785 Memoirs of the AMS (2008) (to be published).

A DEFORMATION OF THE BIG CELL INSIDE THE GRASSMANNIAN MANIFOLD G(r,n)

In this paper we construct a quantum analogue of the big cell inside the grassmannian manifold. Our deformation comes in tandem with a coaction of the upper parabolic subgroup in SLn(k), giving to the big cell the structure of quantum homogeneous space. At the end we give the De Rham complex of the quantum big cell and we define a ring of differential operators acting on the quantum big cell.

ON ALGEBRAIC SUPERGROUPS AND QUANTUM DEFORMATIONS

We give the definitions of affine algebraic supervariety and affine algebraic supergroup through the functor of points and we relate them to the other definitions present in the literature. We study in detail the algebraic supergroups GL(m|n) and SL(m|n) and give explicitly the Hopf algebra structure of the algebra representing the functors of points. In the end we also give the quantization of GL(m|n) together with its coaction on suitable quantum spaces according to Manins philosophy.

Quantum deformation of the flag variety

We want to define a deformation of the flag variety of SL n (k) the special linear group of k n , where kis an algebraically closed field of characteristic 0. We will construct the quantum flag ring as a subalgebra of k, q , [SL n(k)], the quantum SL n (k) and we then will exhibit it in terms of generators and relations in the case in which k q is replaced by a suitable local ring in k(q). These results are a generalization of those obtained for the quantum grassmannian in [4].

On SUSY curves

We give a summary of some elementary results in the theory of super Rie-mann surfaces (SUSY curves), which are mostly known, but are not readily available in the literature. In particular, we give the classification of all genus 0 SUSY-1 curves and touch on the case of genus 1. We also briefly discuss the related topic of П-projective spaces.

A COMPARISON OF THE FUNCTORS OF POINTS OF SUPERMANIFOLDS

We study the functor of points and the local functor of points (here called the Weil--Berezin functor) for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity theorems.

Symmetry in Mathematics and Physics

Fix a manifold M, and let V be an infinite dimensional Lie algebra of vector fields on M. Assume that V contains a finite dimensional semisimple maximal subalgebra A, the projective or conformal subalgebra. A projective or conformal quantization of a V-module of differential operators on M is a decomposition into irreducible A-modules. We survey recent results on projective quantizations and their applications to cohomology, geometric equivalences and symmetries of differential operator modules, and indecomposable modules.Fix a manifold M , and let V be an infinite dimensional simple Lie subalgebra of the Lie algebra Vec M of vector fields on M . Assume that V contains a finite dimensional simple maximal subalgebra a(V). We define an a(V)-quantization of a V-module of differential operators on M to be a decomposition of the module into irreducible a(V)-modules. In this article we survey some recent results and open problems involving this type of quantization and its applications to cohomology, indecomposable modules, and geometric equivalences and symmetries of differential operator modules. There are several mathematical theories of quantization. Two of the most important are geometric quantization, which hinges on polarization and is linked to the orbit method in the representation theory of Lie groups, and deformation quantization, in which the classical Poisson algebra structure becomes the first order approximation of an associative star product. In its original physical sense, to quantize a system meant to replace the commutative Poisson algebra of functions on the phase space, the classical observables, with a noncommutative algebra of operators on a Hilbert space, the quantum mechanical observables. In the theory of quantization under consideration here, the role of the noncommutative algebra is played by the differential operators and that of the commutative algebra is played by their symbols. We will consider two cases: the case that V is all of Vec M , and the case that M is a contact manifold and V is the Lie algebra Con M of contact vector fields on M . Our approach is algebraic: we assume that M is a Euclidean manifold R and we consider only polynomial vector fields. Thus, writing Di for ∂/∂xi and using the multi-index notation x = x1 1 · · ·xJm m , V ⊆ Vec R := SpanC { xDi : 1 ≤ i ≤ m, J ∈ N }

The projective linear supergroup and the SUSY-preserving automorphisms of P^1|1

The purpose of this paper is to describe the projective linear supergroup, its relation with the automorphisms of the projective superspace and to determine the supergroup of SUSY preserving automorphisms of