172 research outputs found
Integrable vortex-type equations on the two-sphere
We consider the Yang-Mills instanton equations on the four-dimensional
manifold S^2xSigma, where Sigma is a compact Riemann surface of genus g>1 or
its covering space H^2=SU(1,1)/U(1). Introducing a natural ansatz for the gauge
potential, we reduce the instanton equations on S^2xSigma to vortex-type
equations on the sphere S^2. It is shown that when the scalar curvature of the
manifold S^2xSigma vanishes, the vortex-type equations are integrable, i.e. can
be obtained as compatibility conditions of two linear equations (Lax pair)
which are written down explicitly. Thus, the standard methods of integrable
systems can be applied for constructing their solutions. However, even if the
scalar curvature of S^2xSigma does not vanish, the vortex equations are well
defined and have solutions for any values of the topological charge N. We show
that any solution to the vortex equations on S^2 with a fixed topological
charge N corresponds to a Yang-Mills instanton on S^2xSigma of charge (g-1)N.Comment: 14 pages; v2: clarifying comments added, published versio
Liouville bootstrap via harmonic analysis on a noncompact quantum group
The purpose of this short note is to announce results that amount to a
verification of the bootstrap for Liouville theory in the generic case under
certain assumptions concerning existence and properties of fusion
transformations. Under these assumptions one may characterize the fusion and
braiding coefficients as solutions of a system of functional equations that
follows from the combination of consistency requirements and known results.
This system of equations has a unique solution for irrational central charge
c>25. The solution is constructed by solving the Clebsch-Gordan problem for a
certain continuous series of quantum group representations and constructing the
associated Racah-coefficients. This gives an explicit expression for the fusion
coefficients. Moreover, the expressions can be continued into the strong
coupling region 1<c<25, providing a solution of the bootstrap also for this
region.Comment: 16 pages, typos removed incl. important one in (48
Causality and dispersion relations and the role of the S-matrix in the ongoing research
The adaptation of the Kramers-Kronig dispersion relations to the causal
localization structure of QFT led to an important project in particle physics,
the only one with a successful closure. The same cannot be said about the
subsequent attempts to formulate particle physics as a pure S-matrix project.
The feasibility of a pure S-matrix approach are critically analyzed and their
serious shortcomings are highlighted. Whereas the conceptual/mathematical
demands of renormalized perturbation theory are modest and misunderstandings
could easily be corrected, the correct understanding about the origin of the
crossing property requires the use of the mathematical theory of modular
localization and its relation to the thermal KMS condition. These new concepts,
which combine localization, vacuum polarization and thermal properties under
the roof of modular theory, will be explained and their potential use in a new
constructive (nonperturbative) approach to QFT will be indicated. The S-matrix
still plays a predominant role but, different from Heisenberg's and
Mandelstam's proposals, the new project is not a pure S-matrix approach. The
S-matrix plays a new role as a "relative modular invariant"..Comment: 47 pages expansion of arguments and addition of references,
corrections of misprints and bad formulation
Quantum Anti-de Sitter space and sphere at roots of unity
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined
which is covariant under U_q(so(2,D-1)), for q a root of unity. The
star-structure is studied in detail. The scalar fields have an intrinsic
high-energy cutoff, and arise most naturally as fields on orbifolds AdS_q^D
\times S^D/G if D is odd, and AdS_q^D \times S_{\chi}^{2D-1}/G if D is even.
Here G is a finite abelian group, and S_{\chi} is a certain ``chiral sector''
of the classical sphere. Hilbert spaces of square integrable functions are
discussed. Analogous results are found for the q-deformed sphere S_q^D.Comment: 45 pages, LaTeX, 2 figures using epsf. Slight change in notation
allows to obtain AdS^2, AdS^3 as special cases of the general schem
Magnetic monopoles and symmetries in noncommutative space
In this paper, we review the progress in the analysis of magnetic monopoles
as generalized states in quantum mechanics. We show that the considered model
contains rich algebraic structure that generates symmetries which have been
utilized in different physical contexts. Even though are we focused on quantum
mechanics in noncommutative space , the results can be
reconstructed in ordinary quantum mechanics in as well.Comment: 7 page
The beat of a fuzzy drum: fuzzy Bessel functions for the disc
The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure
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