125 research outputs found
Linear independence of Gamma values in positive characteristic
We investigate the arithmetic nature of special values of Thakur's function
field Gamma function at rational points. Our main result is that all linear
independence relations over the field of algebraic functions are consequences
of the known relations of Anderson and Thakur arising from the functional
equations of the Gamma function.Comment: 51 page
On moment maps associated to a twisted Heisenberg double
We review the concept of the (anomalous) Poisson-Lie symmetry in a way that
emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop
turns out to be very useful for several applications: we prove that the left
and the right actions of a group on its twisted Heisenberg double
realize the (anomalous) Poisson-Lie symmetries and we explain in a
very transparent way the concept of the Poisson-Lie subsymmetry and that of
Poisson-Lie symplectic reduction. Under some additional conditions, we
construct also a non-anomalous moment map corresponding to a sort of
quasi-adjoint action of on . The absence of the anomaly of this
"quasi-adjoint" moment map permits to perform the gauging of deformed WZW
models.Comment: 52 pages, LaTeX, introduction substantially enlarged, several
explanatory remarks added, final published versio
Finite multiplicity theorems for induction and restriction
We find upper and lower bounds of the multiplicities of irreducible
admissible representations of a semisimple Lie group occurring in the
induced representations from irreducible representations
of a closed subgroup .
As corollaries, we establish geometric criteria for finiteness of the
dimension of (induction) and of
(restriction) by means of the real flag variety , and discover that
uniform boundedness property of these multiplicities is independent of real
forms and characterized by means of the complex flag variety.Comment: to appear in Advances in Mathematic
Superposition Formulas for Darboux Integrable Exterior Differential Systems
In this paper we present a far-reaching generalization of E. Vessiot's
analysis of the Darboux integrable partial differential equations in one
dependent and two independent variables. Our approach provides new insights
into this classical method, uncovers the fundamental geometric invariants of
Darboux integrable systems, and provides for systematic, algorithmic
integration of such systems. This work is formulated within the general
framework of Pfaffian exterior differential systems and, as such, has
applications well beyond those currently found in the literature. In
particular, our integration method is applicable to systems of hyperbolic PDE
such as the Toda lattice equations, 2 dimensional wave maps and systems of
overdetermined PDE.Comment: 80 page report. Updated version with some new sections, and major
improvements to other
Quantum Mechanics On Spaces With Finite Fundamental Group
We consider in general terms dynamical systems with finite-dimensional,
non-simply connected configuration-spaces. The fundamental group is assumed to
be finite. We analyze in full detail those ambiguities in the quantization
procedure that arise from the non-simply connectedness of the classical
configuration space. We define the quantum theory on the universal cover but
restrict the algebra of observables \O to the commutant of the algebra
generated by deck-transformations. We apply standard superselection principles
and construct the corresponding sectors. We emphasize the relevance of all
sectors and not just the abelian ones.Comment: 40 Pages, Plain-TeX, no figure
Weaving Worldsheet Supermultiplets from the Worldlines Within
Using the fact that every worldsheet is ruled by two (light-cone) copies of
worldlines, the recent classification of off-shell supermultiplets of
N-extended worldline supersymmetry is extended to construct standard off-shell
and also unidextrous (on the half-shell) supermultiplets of worldsheet
(p,q)-supersymmetry with no central extension. In the process, a new class of
error-correcting (even-split doubly-even linear block) codes is introduced and
classified for , providing a graphical method for classification of
such codes and supermultiplets. This also classifies quotients by such codes,
of which many are not tensor products of worldline factors. Also,
supermultiplets that admit a complex structure are found to be depictable by
graphs that have a hallmark twisted reflection symmetry.Comment: Extended version, with added discussion of complex and quaternionic
tensor products demonstrating that certain quotient supermultiplets do not
factorize over any ground fiel
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
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