28 research outputs found
Graded Differential Geometry of Graded Matrix Algebras
We study the graded derivation-based noncommutative differential geometry of
the -graded algebra of complex -matrices
with the ``usual block matrix grading'' (for ). Beside the
(infinite-dimensional) algebra of graded forms the graded Cartan calculus,
graded symplectic structure, graded vector bundles, graded connections and
curvature are introduced and investigated. In particular we prove the
universality of the graded derivation-based first-order differential calculus
and show, that is a ``noncommutative graded manifold'' in a
stricter sense: There is a natural body map and the cohomologies of and its body coincide (as in the case of ordinary graded manifolds).Comment: 21 pages, LATE
The antifield Koszul-Tate complex of reducible Noether identities
A generic degenerate Lagrangian system of even and odd fields is examined in
algebraic terms of the Grassmann-graded variational bicomplex. Its
Euler-Lagrange operator obeys Noether identities which need not be independent,
but satisfy first-stage Noether identities, and so on. We show that, if a
certain necessary and sufficient condition holds, one can associate to a
degenerate Lagrangian system the exact Koszul-Tate complex with the boundary
operator whose nilpotency condition restarts all its Noether and higher-stage
Noether identities. This complex provides a sufficient analysis of the
degeneracy of a Lagrangian system for the purpose of its BV quantization.Comment: 23 page
Semistability vs. nefness for (Higgs) vector bundles
According to Miyaoka, a vector bundle E on a smooth projective curve is
semistable if and only if a certain numerical class in the projectivized bundle
PE is nef. We establish a similar criterion for the semistability of Higgs
bundles: namely, such a bundle is semistable if and only if for every integer s
between 0 and the rank of E, a suitable numerical class in the scheme
parametrizing the rank s locally-free Higgs quotients of E is nef. We also
extend this result to higher-dimensional complex projective varieties by
showing that the nefness of the above mentioned classes is equivalent to the
semistability of the Higgs bundle E together with the vanishing of the
discriminant of E.Comment: Comments: 20 pages, Latex2e, no figures. v2 includes a generalization
to complex projective manifolds of any dimension. To appear in Diff. Geom.
App
Noether's second theorem for BRST symmetries
We present Noether's second theorem for graded Lagrangian systems of even and
odd variables on an arbitrary body manifold X in a general case of BRST
symmetries depending on derivatives of dynamic variables and ghosts of any
finite order. As a preliminary step, Noether's second theorem for Lagrangian
systems on fiber bundles over X possessing gauge symmetries depending on
derivatives of dynamic variables and parameters of arbitrary order is proved.Comment: 31 pages, to be published in J. Math. Phy
Graded infinite order jet manifolds
The relevant material on differential calculus on graded infinite order jet
manifolds and its cohomology is summarized. This mathematics provides the
adequate formulation of Lagrangian theories of even and odd variables on smooth
manifolds in terms of the Grassmann-graded variational bicomplex.Comment: 30 page
Modular classes of skew algebroid relations
Skew algebroid is a natural generalization of the concept of Lie algebroid.
In this paper, for a skew algebroid E, its modular class mod(E) is defined in
the classical as well as in the supergeometric formulation. It is proved that
there is a homogeneous nowhere-vanishing 1-density on E* which is invariant
with respect to all Hamiltonian vector fields if and only if E is modular, i.e.
mod(E)=0. Further, relative modular class of a subalgebroid is introduced and
studied together with its application to holonomy, as well as modular class of
a skew algebroid relation. These notions provide, in particular, a unified
approach to the concepts of a modular class of a Lie algebroid morphism and
that of a Poisson map.Comment: 20 page
An example of the Langlands correspondence for irregular rank two connections on P^1
Special kinds of rank 2 vector bundles with (possibly irregular) connections
on P^1 are considered. We construct an equivalence between the derived category
of quasi-coherent sheaves on the moduli stack of such bundles and the derived
category of modules over a TDO ring on certain non-separated curve. We identify
this curve with the coarse moduli space of some parabolic bundles on P^1. Then
our equivalence becomes an example of the categorical Langlands correspondence.Comment: Section 5 was shortened by referring to results of Hernandez Ruiperez
et al. The reader might want to look at the previous (2nd) version for a more
self-contained exposition. Other minor change