2,057 research outputs found
Supersymmetric quantum theory and non-commutative geometry
Classical differential geometry can be encoded in spectral data, such as
Connes' spectral triples, involving supersymmetry algebras. In this paper, we
formulate non-commutative geometry in terms of supersymmetric spectral data.
This leads to generalizations of Connes' non-commutative spin geometry
encompassing non-commutative Riemannian, symplectic, complex-Hermitian and
(Hyper-)Kaehler geometry. A general framework for non-commutative geometry is
developed from the point of view of supersymmetry and illustrated in terms of
examples. In particular, the non-commutative torus and the non-commutative
3-sphere are studied in some detail.Comment: 77 pages, PlainTeX, no figures; present paper is a significantly
extended version of the second half of hep-th/9612205. Assumptions in Sect.
2.2.5 clarified; final version to appear in Commun.Math.Phy
Conformal Field Theory and Geometry of Strings
What is quantum geometry? This question is becoming a popular leitmotiv in
theoretical physics and in mathematics. Conformal field theory may catch a
glimpse of the right answer. We review global aspects of the geometry of
conformal fields, such as duality and mirror symmetry, and interpret them
within Connes' non-commutative geometry. Extended version of lectures given by
the 2nd author at the Mathematical Quantum Theory Conference, Vancouver,
Canada, August 4 to 8, 1993Comment: 44 pages, latex file, 5 references adde
Tensor calculus on noncommutative spaces
It is well known that for a given Poisson structure one has infinitely many
star products related through the Kontsevich gauge transformations. These gauge
transformations have an infinite functional dimension (i.e., correspond to an
infinite number of degrees of freedom per point of the base manifold). We show
that on a symplectic manifold this freedom may be almost completely eliminated
if one extends the star product to all tensor fields in a covariant way and
impose some natural conditions on the tensor algebra. The remaining ambiguity
either correspond to constant renormalizations to the symplectic structure, or
to maps between classically equivalent field theory actions. We also discuss
how one can introduce the Riemannian metric in this approach and the
consequences of our results for noncommutative gravity theories.Comment: 17p; v2: extended version, to appear in CQ
Poisson Algebra of Wilson Loops and Derivations of Free Algebras
We describe a finite analogue of the Poisson algebra of Wilson loops in
Yang-Mills theory. It is shown that this algebra arises in an apparently
completely different context; as a Lie algebra of vector fields on a
non-commutative space. This suggests that non-commutative geometry plays a
fundamental role in the manifestly gauge invariant formulation of Yang-Mills
theory. We also construct the deformation of the loop algebra induced by
quantization, in the large N_c limit.Comment: 20 pages, no special macros necessar
Quantum Bi-Hamiltonian Systems
We define quantum bi-Hamiltonian systems, by analogy with the classical case,
as derivations in operator algebras which are inner derivations with respect to
two compatible associative structures. We find such structures by means of the
associative version of Nijenhuis tensors. Explicit examples, e.g. for the
harmonic oscillator, are given.Comment: 14 pages; the paper is posted for archival purpose
- …