Minkowski superspaces and superstrings as almost real-complex supermanifolds

In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that mathematicians study are real or (almost) complex ones, while Minkowski superspaces are completely different objects. They are what we call almost real-complex supermanifolds, i.e., real supermanifolds with a non-integrable distribution, the collection of subspaces of the tangent space, and in every subspace a complex structure is given. An almost complex structure on a real supermanifold can be given by an even or odd operator; it is complex (without "always") if the suitable superization of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we define the circumcised analog of the Nijenhuis tensor. We compute it for the Minkowski superspaces and superstrings. The space of values of the circumcised Nijenhuis tensor splits into (indecomposable, generally) components whose irreducible constituents are similar to those of Riemann or Penrose tensors. The Nijenhuis tensor vanishes identically only on superstrings of superdimension 1|1 and, besides, the superstring is endowed with a contact structure. We also prove that all real forms of complex Grassmann algebras are isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to related recent work by Witten is adde

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them

Among the simple finite dimensional Lie algebras, only sl(n) possesses two automorphisms of finite order which have no common nonzero eigenvector with eigenvalue one. It turns out that these automorphisms are inner and form a pair of generators that allow one to generate all of sl(n) under bracketing. It seems that Sylvester was the first to mention these generators, but he used them as generators of the associative algebra of all n times n matrices Mat(n). These generators appear in the description of elliptic solutions of the classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't Hooft's work on confinement operators in QCD, and various other instances. Here I give an algorithm which both generates sl(n) and explicitly describes a set of defining relations. For simple (up to center) Lie superalgebras, analogs of Sylvester generators exist only for sl(n|n). The relations for this case are also computed.Comment: 14 pages, 6 figure

Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?

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 examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin-Kostant-Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the functionals on classical configurations considered in a previous paper are nothing but superfunctions on M. We present a programme for future mathematical work, which applies to any classical field model with fermion fields. This programme is (partially) implemented in successor papers.Comment: 46 pages, LateX2E+AMSLaTe

Minimal deformations of the commutative algebra and the linear group GL(n)

We consider the relations of generalized commutativity in the algebra of formal series $M_q (x^i )$, which conserve a tensor $I_q$-grading and depend on parameters $q(i,k)$ . We choose the $I_q$-preserving version of differential calculus on $M_q$ . A new construction of the symmetrized tensor product for $M_q$-type algebras and the corresponding definition of minimally deformed linear group $QGL(n)$ and Lie algebra $qgl(n)$ are proposed. We study the connection of $QGL(n)$ and $qgl(n)$ with the special matrix algebra \mbox{Mat} (n,Q) containing matrices with noncommutative elements. A definition of the deformed determinant in the algebra \mbox{Mat} (n,Q) is given. The exponential parametrization in the algebra \mbox{Mat} (n,Q) is considered on the basis of Campbell-Hausdorf formula.Comment: 14 page

Fedosov and Riemannian supermanifolds

Generalizations of symplectic and metric structures for supermanifolds are analyzed. Two types of structures are possible according to the even/odd character of the corresponding quadratic tensors. In the even case one has a very rich set of geometric structures: even symplectic supermanifolds (or, equivalently, supermanifolds with non-degenerate Poisson structures), even Fedosov supermanifolds and even Riemannian supermanifolds. The existence of relations among those structures is analyzed in some details. In the odd case, we show that odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor. However, odd Riemannian supermanifolds can only have constant curvature.Comment: 20 page

Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.Comment: v3: added a footnote on p.10 about a wrong derivation of the correct statemen

Degenerate Odd Poisson Bracket on Grassmann Variables

A linear degenerate odd Poisson bracket (antibracket) realized solely on Grassmann variables is presented. It is revealed that this bracket has at once three nilpotent $\Delta$-like differential operators of the first, the second and the third orders with respect to the Grassmann derivatives. It is shown that these $\Delta$-like operators together with the Grassmann-odd nilpotent Casimir function of this bracket form a finite-dimensional Lie superalgebra.Comment: 5 pages, LATEX. Corrections of misprints. The relation (23) is adde

On contractions of classical basic superalgebras

We define a class of orthosymplectic $osp(m;j|2n;\omega)$ and unitary $sl(m;j|n;\epsilon)$ superalgebras which may be obtained from $osp(m|2n)$ and $sl(m|n)$ by contractions and analytic continuations in a similar way as the special linear, orthogonal and the symplectic Cayley-Klein algebras are obtained from the corresponding classical ones. Casimir operators of Cayley-Klein superalgebras are obtained from the corresponding operators of the basic superalgebras. Contractions of $sl(2|1)$ and $osp(3|2)$ are regarded as an examples.Comment: 15 pages, Late

An improved design of an inductive fault current limiter based on a superconducting cylinder

The paper deals with basic designs of a fault current limiter of the transformer type which differ each other by the mutual location of a primary winding and a superconducting short-circuited cylinder. Theoretical study of the main parameters of the different designs is performed in the framework of the critical state model and shows that the most effective is a design in which the primary winding is divided to two sections with equal turn numbers. The sections are placed inside and outside of the cylinder and connected in series. Such arrangement of the windings leads to a substantial reduction of AC losses in the superconducting cylinder, an increase of the activation current and a decrease of the inductive reactance in the normal regime of a protected circuit. The experimental results obtained on the laboratory model with a BSSCO cylinder confirm the theoretical predictions.Comment: 17 pages, 9 figure