Super line bundles

Super line bundles over supermanifolds are introduced as natural generalizations of line bundles over smooth manifolds. Their classification in terms of their obstruction class and the representation of their Chern class in terms of a connection on the super line bundle are discussed. The case where the base supermanifold is De Witt is analyzed in detail, both in the supersmooth and complex superanalytic case.     

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M ₀, respectively. The nonuniqueness of the local injections t: M ₀M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, C ^k fields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.Research work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR) and by the Italian Ministry of Public Education through the research project Geometria e Fisica.     

Gauge Invariance of Banks-Peskin differential forms (flat background)

We discuss some aspects of the gauge invariance of Banks-Peskin differential forms on a flat background.     

Quaternionic,quaternionic Kähler,and hyper-Kähler supermanifolds

Almost quaternionic, quaternionic, hyper-Kähler, and quaternionic Kähler supermanifolds are introduced and studied.     

The Egorov theorem for transverse Dirac-type operators on foliated manifolds

Egorov’s theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first-order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.     

Higher trace and Berezinian of matrices over a Clifford algebra

We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded commutative associative algebra A$A$. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, we recover the classical Dieudonné determinant of quaternionic matrices, but in general our quaternionic determinant is different. We show that the graded determinant of purely even (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded matrices of degree 0$0$ is polynomial in its entries. In the case of the algebra A=H$A=\mathbb{H}$ of quaternions, we calculate the formula for the Berezinian in terms of a product of quasiminors in the sense of Gelfand, Retakh, and Wilson. The graded trace is related to the graded Berezinian (and determinant) by a (_Z2)ⁿ${\left({\mathbb{Z}}_2\right)}^n$-graded version of Liouville’s formula.     

A generalization of Hamilton’s gradient estimate

In this short note, we obtain a “Φ$\Phi$-gradient estimate” for bounded solutions to the heat equation on compact Riemannian manifolds, which generalizes the well-known Hamilton gradient estimate [R.S. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993) 113–126]. We derive as applications some useful estimates for the Φ$\Phi$-entropy and the associated heat kernel.     

On m-accretive Schrödinger-type operators with singular potentials on Riemannian manifolds

We consider a Schrödinger-type differential expression _HV=∇^∗∇+V$H_V={\nabla }^{\ast }\nabla +V$, where ∇$\nabla$ is a Hermitian connection on a Hermitian vector bundle E$E$ over a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$ and positive smooth measure dμ$d\mu$, and V$V$ is a locally integrable section of the bundle of endomorphisms of E$E$. We give a sufficient condition for m$m$-accretivity of a realization of _HV$H_V$ in L²(E)$L^2\left(E\right)$.     

The analytic torsion of a cone over an odd dimensional manifold

We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W$W$. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Müller theorem  and  for a manifold with boundary, according to Brüning and Ma (2006) [5]. We also prove Poincaré duality for the analytic torsion of a cone.     

Unique continuation results for Ricci curvature and applications

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. Related to this issue, an isometry extension property is proved: continuous groups of isometries at conformal infinity extend into the bulk of any complete conformally compact Einstein metric. Relations of this property with the invariance of the Gauss–Codazzi constraint equations under deformations are also discussed.     

Schlesinger transforms for the discrete painlevé IV equation

We derive an auto-Bäcklund transformation for the discrete Painlevé IV equation and use it in order to derive Schlesinger transformations for the same equation as well as particular solutions in perfect analogy to the continuous case.     

Symmetries of sub-Riemannian surfaces

We obtain some results on symmetries of sub-Riemannian surfaces. In case of a contact sub-Riemannian surface we base on invariants found by Hughen [15]. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.     

A study of the discrete Pv equation: Miura transformations and particular solutions

We derive the form of the Miura transformation of the discrete P_v equation and show that it is indeed an auto-Bäcklund transformation, i.e. it relates the discrete P_v to itself. Using this auto-Bäcklund, we obtain the Schlesinger transformations of discrete P_v which relate the solution for one set of the parameters of the equation to that of another set of neighbouring parameters. Finally, we obtain particular solutions of the discrete P_v (i.e. solutions that exist only for some specific values of the parameters). These solutions are of two types: solutions involving the confluent hypergeometric function (on codimension-one submanifold of parameters) and rational solutions (on codimension-two submanifold of parameters).