arXiv research

The paper realizes Lie superalgebras G(3) and F(4) as symmetries of supergeometries.

Extends Tanaka theory to supergeometry for upper bounds on supersymmetry.

The paper realizes 6 supergeometries for the Lie superalgebra D(2,1;a).

Expands Euler-Poincare characteristic to supergeometry.

Introduces basics of supergeometry for PhD students.

We reformulate superalgebra and supergeometry in completely categorical terms by a consequent use of the functor of points. The increased abstraction of this approach is rewarded by a number of great advantages. First, we show that one can extend supergeometry completely naturally to infinite-dimensional contexts. Seco…

Extends transversality to supergeometry, proving stability and genericity.

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n \geq 2$, do appear quite frequently. The corresponding sign rules are determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The present paper is the first of a series on $\mathbb{Z}_2^n$-Supergeometry. The new theory exhibits challenging…

The concept of $\Zn$-supermanifold has been recently proposed as a natural generalization of classical ($\Zs$-graded) supergeometry, allowing for more complicated commutativity constraints. Here we continue the study of $\Zn$-supergeometry by developing the foundations of differential calculus on $\Zn$-supermanifolds.

Explains basics of supersymmetry and supergeometry.

We continue the development of $\mathbb{Z}^n_2$-supergeometry, a natural generalization of classical ($\mathbb{Z}_2$-graded) supergeometry, by proving the Frobenius theorem for integrable distributions on differentiable $\mathbb{Z}^n_2$-supermanifolds. Both the local and global versions of the theorem are addressed.

New algebraic structures on manifolds generalize supergeometry concepts.

Generalizes Lie supergroups to Lie superheaps.

New equivalences found between graded supermanifolds and vector bundles.

Derives hyperbolic laws of cosines and sines with fermionic corrections.

This is the author's Master's thesis written under the supervision of Dr. Gregor Weingart at the National Autonomous University of Mexico. The purpose of this study is to rewrite differential supergeometry in terms of classical differential geometry. This rewriting from "first principles" has two main motivations: 1 av…

This is an expository paper about the geometry of the torsion constraints in the superspace formulation of supergravity theories. It was prepared for the 2001 Park City Research Program in Supergeometry.

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmüller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.

Study classifies super vector bundles and proves universality.

A supermanifold M is canonically associated to any pseudo Riemannian spin manifold (M_0,g_0). Extending the metric g_0 to a field g of bilinear forms g(p) on T_p M, p\in M_0, the pseudo Riemannian supergeometry of (M,g) is formulated as G-structure on M, where G is a supergroup with even part G_0\cong Spin(k,l); (k,l) …

These notes are based on a series of lectures given by the first author at the school of `Poisson 2010', held at IMPA, Rio de Janeiro. They contain an exposition of the theory of super- and graded manifolds, cohomological vector fields, graded symplectic structures, reduction and the AKSZ-formalism.

Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of…

The paper examines Riemannian structures on $\mathbb{Z}_2^n$-manifolds.

Motivated by a paper of Zirnbauer, we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g. geodesics, isometry groups and inv…

A new generalization of Grassmannians, called ν-grassmannians, and a canonical super vector bundle over this new space, say Γ, are introduced. Then, constructing a Gauss supermap of a super vector bundle, the universal property of Γ is discussed. Finally, we generalize one of the main theorems of homotopy classificatio…

A new generalization of Grassmannians to supergeometry, different from the well known supergrassmannian, is introduced. These are constructed by gluing a finite number of copies of a ν\- domain, i.e. a superdomain with an odd involution, say ν\, on their structure sheaf considered as a sheaf of C^\infty_{R^m}-modules.

Some cohomology elements, called $ν$ classes, as a supergeneralization of universal Chern classes, are introduced for canonical super line bundles over $ν$ projective spaces, a novel supergeometric generalization of projective spaces. It is shown that these classes may be described by analytic representatives of elemen…

We investigate (pseudo)differential forms in the framework of supergeometry. Definitions, basic properties and Cartan calculus (DeRham differential, Lie derivative, inner product, Hodge operator) are presented; the symplectic supermechanics (even and odd) is formulated; and the question of quantization is discussed. In…

A new generalization of Grassmannians in supergeometry, called $ν-$Grassmannians, are constructed by gluing $ν-$domains. By a $ν-$domain, we mean a superdomain with an odd involution say $ν$ on its structure sheaf, as morphism of modules. Then we show that $ν-$Grassmannians are homogeneous superspaces. In addition, in …

The harmonic action functional allows a natural generalisation to semi-Riemannian supergeometry, referred to as superharmonic action, which resembles the supersymmetric sigma models studied in high energy physics. We show that Killing vector fields are infinitesimal supersymmetries of the superharmonic action and prove…

This paper concerns constructing topological sigma models governing maps from semirigid super Riemann surfaces to general target supermanifolds. We define both the A model and B model in this general setup by defining suitable BRST operators and physical observables. Using supersymmetric localization, we express correl…

Develops derived differential geometry for supermanifolds.

A new concept of Loday algebroid (and its pure algebraic version - Loday pseudoalgebra) is proposed and discussed in comparison with other similar structures present in the literature. The structure of a Loday pseudoalgebra and its natural reduction to a Lie pseudoalgebra is studied. Further, Loday algebroids are inter…

In this thesis we investigate a new formalism for supergeometry which focuses on the categorical properties of the theory. This approach is our main tool in the subsequent investigation of a global analytic approach to the construction of super Teichmueller spaces. These results should be of importance to various other…

The aim of this paper is to present a short introduction to supergeometry on pure odd supermanifolds. (Pseudo)differential forms, Cartan calculus (DeRham differential, Lie derivative, "inner" product), metric, inner product, Killing's vector fields, Hodge star operator, integral forms, co-differential and connection on…

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n}…

There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions …

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This…

The transformation formula of the Berezin integral holds, in the non-compact case, only up to boundary integrals, which have recently been quantified by Alldridge-Hilgert-Palzer. We establish divergence theorems in semi-Riemannian supergeometry by means of the flow of vector fields and these boundary integrals, and sho…

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional t…

Study on $GL(1|1)$ Higgs bundles over Riemann surfaces.

We discuss of the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M. We discuss the two main approaches to supermanifolds, and …

We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the…

We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…

Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.

Let G be a compact Lie group. Let M be a smooth G-manifold and V --> M be an oriented G-equivariant vector bundle. One defines the spaces of equivariant forms with generalized coefficients on V and M. An equivariant Thom form $θ$ on V is a compactly supported closed equivariant form such that its integral along the fib…

In Physics and in Mathematics $\mathbb{Z}_2^n$-gradings, $n>1$, appear in various fields. The corresponding sign rule is determined by the `scalar product' of the involved $\mathbb{Z}_2^n$-degrees. The $\mathbb{Z}_2^n$-Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinate…

In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a f…