3,141 research outputs found
Symmetries of Spin Calogero Models
We investigate the symmetry algebras of integrable spin Calogero systems
constructed from Dunkl operators associated to finite Coxeter groups. Based on
two explicit examples, we show that the common view of associating one symmetry
algebra to a given Coxeter group is wrong. More precisely, the symmetry
algebra heavily depends on the representation of on the spins. We prove
this by identifying two different symmetry algebras for a spin Calogero
model and three for spin Calogero model. They are all related to the
half-loop algebra and its twisted versions. Some of the result are extended to
any finite Coxeter group.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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
On the support of the Ashtekar-Lewandowski measure
We show that the Ashtekar-Isham extension of the classical configuration
space of Yang-Mills theories (i.e. the moduli space of connections) is
(topologically and measure-theoretically) the projective limit of a family of
finite dimensional spaces associated with arbitrary finite lattices. These
results are then used to prove that the classical configuration space is
contained in a zero measure subset of this extension with respect to the
diffeomorphism invariant Ashtekar-Lewandowski measure.
Much as in scalar field theory, this implies that states in the quantum
theory associated with this measure can be realized as functions on the
``extended" configuration space.Comment: 22 pages, Tex, Preprint CGPG-94/3-
Enlarged symmetry algebras of spin chains, loop models, and S-matrices
The symmetry algebras of certain families of quantum spin chains are
considered in detail. The simplest examples possess m states per site (m\geq2),
with nearest-neighbor interactions with U(m) symmetry, under which the sites
transform alternately along the chain in the fundamental m and its conjugate
representation \bar{m}. We find that these spin chains, even with {\em
arbitrary} coefficients of these interactions, have a symmetry algebra A_m much
larger than U(m), which implies that the energy eigenstates fall into sectors
that for open chains (i.e., free boundary conditions) can be labeled by j=0, 1,
>..., L, for the 2L-site chain, such that the degeneracies of all eigenvalues
in the jth sector are generically the same and increase rapidly with j. For
large j, these degeneracies are much larger than those that would be expected
from the U(m) symmetry alone. The enlarged symmetry algebra A_m(2L) consists of
operators that commute in this space of states with the Temperley-Lieb algebra
that is generated by the set of nearest-neighbor interaction terms; A_m(2L) is
not a Yangian. There are similar results for supersymmetric chains with
gl(m+n|n) symmetry of nearest-neighbor interactions, and a richer
representation structure for closed chains (i.e., periodic boundary
conditions). The symmetries also apply to the loop models that can be obtained
from the spin chains in a spacetime or transfer matrix picture. In the loop
language, the symmetries arise because the loops cannot cross. We further
define tensor products of representations (for the open chains) by joining
chains end to end. The fusion rules for decomposing the tensor product of
representations labeled j_1 and j_2 take the same form as the Clebsch-Gordan
series for SU(2). This and other structures turn the symmetry algebra \cA_m
into a ribbon Hopf algebra, and we show that this is ``Morita equivalent'' to
the quantum group U_q(sl_2) for m=q+q^{-1}. The open-chain results are extended
to the cases |m|< 2 for which the algebras are no longer semisimple; these
possess continuum limits that are critical (conformal) field theories, or
massive perturbations thereof. Such models, for open and closed boundary
conditions, arise in connection with disordered fermions, percolation, and
polymers (self-avoiding walks), and certain non-linear sigma models, all in two
dimensions. A product operation is defined in a related way for the
Temperley-Lieb representations also, and the fusion rules for this are related
to those for A_m or U_q(sl_2) representations; this is useful for the continuum
limits also, as we discuss in a companion paper
Infinite index extensions of local nets and defects
Subfactor theory provides a tool to analyze and construct extensions of
Quantum Field Theories, once the latter are formulated as local nets of von
Neumann algebras. We generalize some of the results of [LR95] to the case of
extensions with infinite Jones index. This case naturally arises in physics,
the canonical examples are given by global gauge theories with respect to a
compact (non-finite) group of internal symmetries. Building on the works of
Izumi, Longo, Popa [ILP98] and Fidaleo, Isola [FI99], we consider generalized
Q-systems (of intertwiners) for a semidiscrete inclusion of properly infinite
von Neumann algebras, which generalize ordinary Q-systems introduced by Longo
[Lon94] to the infinite index case. We characterize inclusions which admit
generalized Q-systems of intertwiners and define a braided product among the
latter, hence we construct examples of QFTs with defects (phase boundaries) of
infinite index, extending the family of boundaries in the grasp of [BKLR16].Comment: 50 page
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