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.     

Regular connections among generalized connections

The properties of the space of regular connections as a subset of the space of generalized connections in the Ashtekar framework are studied. For every choice of compact structure group and smoothness category for the paths, it is determined whether is dense in or not. Moreover, it is proven that has Ashtekar–Lewandowski measure zero for every non-trivial structure group and every smoothness category. The analogous results hold for gauge orbits instead of connections.     

Projective and affine connections on S and integrable systems

It is known that the Korteweg–de Vries (KdV) equation is a geodesic flow of an L² metric on the Bott–Virasoro group. This can also be interpreted as a flow on the space of projective connections on S¹. The space of differential operators Δ⁽ⁿ⁾=∂ⁿ+u₂∂ⁿ⁻²++u_n form the space of extended or generalized projective connections. If a projective connection is factorizable Δ⁽ⁿ⁾=(∂−((n+1)/2−1)p₁)(∂+(n−1)/2p_n) with respect to quasi primary fields p_i’s, then these fields satisfy ∑_i=1ⁿ((n+1)/2−i)p_i=0. In this paper we discuss the factorization of projective connection in terms of affine connections. It is shown that the Burgers equation and derivative non-linear Schrödinger (DNLS) equation or the Kaup–Newell equation is the Euler–Arnold flow on the space of affine connections.     

An identification of the connections of Quillen and Beilinson-Schechtman

Given a family of Riemann surfaces and a holomorphic vector bundle Beilinson and Schechtman construct a canonical connection on the associated determinant bundle. We prove the conjecture which states that their connection coincides with the Quillen connection. This is done by reducing to the case where along fibers are invertible. Both connection forms become more accessible in this case.Supported in part by N.S.F. Grant No. DMS-9201022Supported in part by National Science Council of Republic of China Grant No. NSC 82-0208-M-002-125-T, and NSERC of Canada Grant No. OGP 0121883     

The orbit space of the action of gauge transformation group on connections

The behaviour of orbits of the action of the group of smooth gauge transformations on connections for a principal bundle P(M, G) is discussed with and without compactness assumption on M and G. In the case of compact M and with suitable conditions on G a stratification structure for the space of orbits is established. A natural tame weak Riemannian metric is given on each stratum.     

Flat connections and Wigner-Yanase-Dyson metrics

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the α-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that in both cases, the existence of such a pair of connections is possible if and only if the metric is given by the Wigner-Yanase-Dyson skew information.     

Lax connections and Solutions of the Hybrid Superstring on AdS2×S2

In this paper, we show that the Lax connections can yield new classical solutions with a spectral parameter of the hybrid formulism for the Type IIB superstring in an AdS ₂ × S ² background with Ramond-Ramond flux. This series of classical solutions have the same infinite set of classically conserved charges.      

Structure of the space of reducible connections for Yang-Mills theories

The geometrical structure of the gauge equivalence classes of reducible connections are investigated. The general procedure to determine the set of orbit types (strata) generated by the action of the gauge group on the space of gauge potentials is given. In the so obtained classification, a stratum, containing generically certain reducible connections, corresponds to a class of isomorphic subbundles given by an orbit of the structure and gauge group. The structure of every stratum is completely clarified. A nonmain stratum can be understood in terms of the main stratum corresponding to a stratification at the level of a subbundle.     

Abelian Chern-Simons theory, Stokes’ theorem, and generalized connections

Generalized connections and their calculus have been developed in the context of quantum gravity. Here we apply them to abelian Chern-Simons theory. We derive the expectation values of holonomies in U(1) Chern-Simons theory using Stokes’ theorem, flux operators and generalized connections. A framing of the holonomy loops arises in our construction, and we show how, by choosing natural framings, the resulting expectation values nevertheless define a functional over gauge invariant cylindrical functions.The abelian theory considered in the present article is the test case for our method. It can also be applied to the non-abelian theory. Results will be reported in a companion article.     

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.     

Geodesics on a supermanifold and projective equivalence of superconnections

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 associated with the connection. Our (super) geodesics possess the same properties as in the classical case: there exists a unique (super) geodesic satisfying a given initial condition and when the connection is metric, our supergeodesics coincide with the trajectories of a free particle with unit mass. Moreover, using our definition, we are able to establish Weyl’s characterization of projective equivalence in the super context: two torsion-free (super) connections define the same geodesics (up to reparametrizations) if and only if their difference tensor can be expressed by means of a (smooth, even, super) 1-form.     

Quantum gauge fields and flat connections in 2-dimensional BF theory

The 2-dimensional BF theory is both a gauge theory and a topological Poisson σ-model corresponding to a linear Poisson bracket. In [3], Torossian discovered a connection which governs correlation functions of the BF theory with sources for the B-field. This connection is flat, and it is a close relative of the KZ connection in the WZW model. In this Letter, we show that flatness of the Torossian connection follows from (properly regularized) quantum equations of motion of the BF theory.     

Existence and uniqueness of solutions to superdifferential equations

We state and prove the theorem of existence and uniqueness of solutions to ordinary superdifferential equations on supermanifolds. It is shown that any supervector field, X = X₀ + X₁, has a unique integral flow, Г: ^1¦1 x (M, A_M) → (M, A_M), satisfying a given initial condition. A necessary and sufficient condition for this integral flow to yield an ^1¦1-action is obtained: the homogeneous components, X₀, and, X₁, of the given field must define a Lie superalgebra of dimension (1, 1). The supergroup structure on ^1¦1, however, has to be specified: there are three non-isomorphic Lie supergroup structures on ^1¦1, all of which have addition as the group operation in the underlying Lie group . On the other extreme, even if X₀, and X₁ do not close to form a Lie superalgebra, the integral flow of X is uniquely determined and is independent of the Lie supergroup structure imposed on ^1¦1. This fact makes it possible to establish an unambiguous relationship between the algebraic Lie derivative of supergeometric objects (e.g., superforms), and its geometrical definition in terms of integral flows. It is shown by means of examples that if a supergroup structure in ^1¦1 is fixed, some flows obtained from left-invariant supervector fields on Lie supergroups may fail to define an ^1¦1-action of the chosen structure. Finally, necessary and sufficient conditions for the integral flows of two supervector fields to commute are given.     

Physics beyond the standard model and cosmological connections: A summary from LCWS06

The international linear collider (ILC) is likely to provide us important insights into physics sector that may supersede our current paradigm, viz., the standard model. In anticipation of the possibility that the ILC may come up in the middle of the next decade, several groups are vigourously investigating its potential to explore this new sector of physics. The Linear Collider Workshop in Bangalore (LCWS06) had several presentations of such studies which looked at supersymmetry, extra dimensions and other exotic possibilities which the ILC may help us discover or understand. Some papers also looked at the understanding of cosmology that may emerge from studies at the ILC. This paper summarises these presentations.      

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.     

The moduli space of flat SU (2) and SO (3) connections over surfaces

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.     

Integration on supermanifolds and a generalized Cartan calculus

A suggestion by Berezin for a method of integration on supermanifolds is given a precise differential geometric meaning by assuming that a supermanifold is the total space of a fibre bundle with connection. The relevant objects for integration are identified as suitable horizontal/vertical projections of hyperforms. The latter are generalizations of differential forms having both covariant and contravariant indices. The exterior calculus of these projected hyperforms is developed, analogously to the Cartan calculus, by introducing appropriate derivations and determining their commutators, respectively anticommutators.