22,723 research outputs found
Central extensions of current groups in two dimensions
In this paper we generalize some of these results for loop algebras and
groups as well as for the Virasoro algebra to the two-dimensional case. We
define and study a class of infinite dimensional complex Lie groups which are
central extensions of the group of smooth maps from a two dimensional
orientable surface without boundary to a simple complex Lie group G. These
extensions naturally correspond to complex curves. The kernel of such an
extension is the Jacobian of the curve. The study of the coadjoint action shows
that its orbits are labelled by moduli of holomorphic principal G-bundles over
the curve and can be described in the language of partial differential
equations. In genus one it is also possible to describe the orbits as conjugacy
classes of the twisted loop group, which leads to consideration of difference
equations for holomorphic functions. This gives rise to a hope that the
described groups should possess a counterpart of the rich representation theory
that has been developed for loop groups. We also define a two-dimensional
analogue of the Virasoro algebra associated with a complex curve. In genus one,
a study of a complex analogue of Hill's operator yields a description of
invariants of the coadjoint action of this Lie algebra. The answer turns out to
be the same as in dimension one: the invariants coincide with those for the
extended algebra of currents in sl(2).Comment: 17 page
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
So
The Landau electron problem on a cylinder
We consider the quantum mechanics of an electron confined to move on an
infinite cylinder in the presence of a uniform radial magnetic field. This
problem is in certain ways very similar to the corresponding problem on the
infinite plane. Unlike the plane however, the group of symmetries of the
magnetic field, namely, rotations about the axis and the axial translations, is
{\em not} realized by the quantum electron but only a subgroup comprising
rotations and discrete translations along the axial direction, is. The basic
step size of discrete translations is such that the flux through the `unit
cylinder cell' is quantized in units of the flux quantum. The result is derived
in two different ways: using the condition of projective realization of
symmetry groups and using the more familiar approach of determining the
symmetries of a given Hamiltonian.Comment: 26 pages, revtex file, no figures. In version 2, introduction is
expanded to explain our approach and references are updated. Results and
conclusions are unchange
Algebraic conformal quantum field theory in perspective
Conformal quantum field theory is reviewed in the perspective of Axiomatic,
notably Algebraic QFT. This theory is particularly developped in two spacetime
dimensions, where many rigorous constructions are possible, as well as some
complete classifications. The structural insights, analytical methods and
constructive tools are expected to be useful also for four-dimensional QFT.Comment: Review paper, 40 pages. v2: minor changes and references added, so as
to match published versio
From Operator Algebras to Superconformal Field Theory
We make a review on the recent progress in the operator algebraic approach to
(super)conformal field theory. We discuss representation theory, classification
results, full and boundary conformal field theories, relations to supervertex
operator algebras and Moonshine, connections to subfactor theory and
noncommutative geometry
The extension problem for partial Boolean structures in Quantum Mechanics
Alternative partial Boolean structures, implicit in the discussion of
classical representability of sets of quantum mechanical predictions, are
characterized, with definite general conclusions on the equivalence of the
approaches going back to Bell and Kochen-Specker. An algebraic approach is
presented, allowing for a discussion of partial classical extension, amounting
to reduction of the number of contexts, classical representability arising as a
special case. As a result, known techniques are generalized and some of the
associated computational difficulties overcome. The implications on the
discussion of Boole-Bell inequalities are indicated.Comment: A number of misprints have been corrected and some terminology
changed in order to avoid possible ambiguitie
Higher Genus Affine Lie Algebras of Krichever -- Novikov Type
Classical affine Lie algebras appear e.g. as symmetries of infinite
dimensional integrable systems and are related to certain differential
equations. They are central extensions of current algebras associated to
finite-dimensional Lie algebras g. In geometric terms these current algebras
might be described as Lie algebra valued meromorphic functions on the Riemann
sphere with two possible poles. They carry a natural grading. In this talk the
generalization to higher genus compact Riemann surfaces and more poles is
reviewed. In case that the Lie algebra g is reductive (e.g. g is simple,
semi-simple, abelian, ...) a complete classification of (almost-) graded
central extensions is given. In particular, for g simple there exists a unique
non-trivial (almost-)graded extension class. The considered algebras are
related to difference equations, special functions and play a role in Conformal
Field Theory.Comment: 9 pages, Talk presented at the International Conference on Difference
Equations, Special Functions, and Applications, Munich, July 200
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