Global properties of supermanifolds

We construct new examples of supermanifolds, and determine the vector bundle structure of the supermanifolds commonly used in physics. We show that any supermanifold admits a foliation whose leaves are locally tangent to the soul directions in the coordinate charts, and which is one of a nested sequence of foliations. We point out that the existence of these foliations implies restrictions on the possible topologies of supermanifolds. For example, a compact supermanifold with a single even dimension must have vanishing Euler characteristic. We also show that a globally defined superfield on a nice compact supermanifold must be constant along the leaves of the foliations. By this mechanism, the global topology of a supermanifold can be used to impose physically interesting constraints on superfields. As an example, we exhibit a supermanifold which has the local geometry of flat superspace but is such that all globally defined superfields are chiral.Enrico Fermi Fellow. Research supported by the NSF: PHY 83-01221, and the Department of Energy: DE AC 02-82-ER-40073     

Berezin integration on non-compact supermanifolds

We investigate the Berezin integral of non-compactly supported quantities. In the framework of supermanifolds with corners, we give a general, explicit and coordinate-free representation of the boundary terms introduced by an arbitrary change of variables. As a corollary, a general Stokes’s theorem is derived—here, the boundary integral contains transversal derivatives of arbitrarily high order.     

Structure of supermanifolds and supersymmetry transformations

After giving a global, constraint-free Lagrangian formulation of theN=1 superspace supergravity in terms of super fibre bundles and differential forms over a supermanifold, we show that the concept of body manifold of a supermanifold provides a natural manner to reduce the theory to spacetime. This reduction, however, is not canonical, and the various ways in which it can be done give rise to transformations of the field variables which generalise the known invariances of theN=1 spacetime supergravity under supersymmetry transformations and spacetime diffeomorphisms.Research partly supported by the Gruppo Nazionale per la Fisica Matematica of the Italian Research Council and by the Italian Ministry of Public Education through the research project Geometria e Fisica     

On the structure of DeWitt supermanifolds

We show that for a wide and most natural class of (possibly infinite-dimensional) Grassmannian algebras of coefficients, the structure sheaf of every smooth DeWitt supermanifold is acyclic (i.e. its cohomology vanishes in positive degree). This result was previously known for finite-dimensional ground algebras and is new even for the original DeWitt algebra of supernumbers /GL∞. From here we deduce that (equivalence classes of) smooth DeWitt supermanifolds over a fixed ground algebra and of graded smooth manifolds are in a natural bijection with each other. However, contrary to what was stated previously by some authors, this correspondence fails to be functorial; so it happens, for instance, for Rogers' ground algebra B∞. Finally, we observe that every DeWitt super Lie group is a deformation of a graded Lie group over the spectrum Spec /GL of the ground algebra.     

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model (A-model) in the case when the target space is an (m ₀|m ₁)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m ₀–m ₁)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Research supported in part by NSF grant No. DMS-9201366.     

Projective synchronization of spatiotemporal chaos in a weighted complex network

<正>Projective synchronization of a weighted complex network is studied in which nodes are spatiotemporal chaos systems and all nodes are coupled not with the nonlinear terms of the system but through a weighted connection.The range of the linear coefficient matrix of separated configuration,when the synchronization is implemented,is determined according to Lyapunov stability theory.It is found that projective synchronization can be realized for unidirectional star-connection even if the coupling strength between the nodes is a given arbitrary weight value.The Gray-Scott models having spatiotemporal chaos behaviours are taken as nodes in the weighted complex network,and simulation results of spatiotemporal synchronization show the effectiveness of the method.     

Berezin integration on general fermionic supermanifolds

Recent results on the global structure of supermanifolds are used to define a notion of Berezin integration on any purely fermionic Rogers supermanifold. This leads to an integration theory on a large class of supermanifolds having both bosonic and fermionic coordinates. The existence of global functions and forms on such supermanifolds is discussed, as is some elementary cohomology of supermanifolds.Enrico Fermi Fellow. Research supported by the NSF (PHY 83-01221) and DOE (DE-AC02-82-ER-40073)     

Quillen superconnections and connections on supermanifolds

Given a supervector bundle $E=E_0\oplus E_1\to M$, we exhibit a parametrization of Quillen superconnections on $E$ by graded connections on the Cartan–Koszul supermanifold $(M,\Omega \left(M\right))$. The relation between the curvatures of both kind of connections, and their associated Chern classes, is discussed in detail. In particular, we find that Chern classes for graded vector bundles on split supermanifolds can be computed through the associated Quillen superconnections.     

Odd symplectic geometry types on supermanifolds

Generalization of symplectic geometry on manifolds in a supersymmetric case is examined in the present work. In the even case, this leads either to even symplectic geometry, that is, the geometry on supermanifolds with the nondegenerate Poisson bracket, or to the geometry on the Fedosov even supermanifolds. In the odd case, two different scalar symplectic structures exist (namely, the odd closed differential 2-form and antibracket), which can be used to construct various symplectic geometry types on supermanifolds. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 52–57, February, 2008.     

A remark on a new category of supermanifolds

In a previous paper a new category of supermanifolds, called -supermanifolds, was introduced. The objects of that category are pairs (M, ), with M a topological space and a suitably defined sheaf of ₂ -graded commutative B_L - algebras, B_L being a Grassmann algebra with L generators. In this note we complete the analysis of that category by showing that is isomorphic with the sheaf of - maps M → B_L.     

Projective synchronization of a complex network with different fractional order chaos nodes

Based on the stability theory of the linear fractional order system, projective synchronization of a complex network is studied in the paper, and the coupling functions of the connected nodes are identified. With this method, the projective synchronization of the network with different fractional order chaos nodes can be achieved, besides, the number of the nodes does not affect the stability of the whole network. In the numerical simulations, the chaotic fractional order Lü system, Liu system and Coullet system are chosen as examples to show the effectiveness of the scheme.     

Quasi-riemannian structures on supermanifolds and characteristic classes

The notion of a quasi-Riemannian metric being an alternative to generalization of the Riemann metrics to supermanifolds is introduced. Unlike standard supermetrics, the quasi-Riemannian metrics exist on arbitrary supermanifolds, though they are not supersymmetric under the permutation of indices. The application of the quasi-Riemannian structures to the theory of characteristic classes of supermanifolds is considered.     

How different are the supermanifolds of Rogers and DeWitt?

A DeWitt supermanifold always has the structure of a vector bundle over an ordinary spacetime manifold, whereas a Rogers supermanifold is not so restricted. Corresponding to the vector space fibers of the DeWitt supermanifold, a Rogers supermanifold has a foliation by submanifolds, or leaves, parametrized by soul coordinates only. We show that the universal covering space of any leaf always admits a flat metric. If the covering space is complete in this metric, it must in fact be a vector space. We combine this result with known theorems about foliations to give conditions under which a compact Rogers supermanifold with a single even dimension is necessarily a quotient space of flat superspace. We also show that a supermanifold defined by a polynomial equation in flat superspace is always of the DeWitt type. Finally, we exhibit new supermanifold structures forR ² and the 2-torus which show that the foliation of a Rogers supermanifold can be quite exotic.Enrico Fermi Fellow. Research supported by the NSF: PHY 83-01221, and the Department of Energy: DE AC02-82-ER-40073     

Graded manifolds,supermanifolds and infinite-dimensional Grassmann algebras

A new approach to superdifferentiable functions of Grassmann variables is developed, which avoids ambiguities in odd derivatives. This is used to give an improved definition of supermanifold over a finite-dimensional Grassmann algebra. A natural embedding of super-manifolds over Grassmann algebras with increasing number (L) of generators is developed, and thus a limit asL tends to infinity is possible. A correspondence between graded manifolds and supermanifolds is constructed, extending results of [5] and [8].Research supported by the SERC under advanced research fellowship number B/AF/687     

Super Toeplitz operators and non-perturbative deformation quantization of supermanifolds

The purpose of this paper is to construct non-perturbative deformation quantizations of the algebras of smooth functions on Poisson supermanifolds. For the examplesU ^1¦1 andC ^m¦n , algebras of super Toeplitz operators are defined with respect to certain Hilbert spaces of superholomorphic functions. Generators and relations for these algebras are given. The algebras can be thought of as algebras of quantized functions, and deformation conditions are proven which demonstrate the recovery of the super Poisson structures in a semi-classical limit.Supported in part by the Department of Energy under grant DE-FG02-88ER25065Supported in part by the Italian National Institute for Nuclear Physics (INFN)