315 research outputs found
Exploring the SO(32) Heterotic String
We give a complete classification of Z_N orbifold compactification of the
heterotic SO(32) string theory and show its potential for realistic model
building. The appearance of spinor representations of SO(2n) groups is analyzed
in detail. We conclude that the heterotic SO(32) string constitutes an
interesting part of the string landscape both in view of model constructions
and the question of heterotic-type I duality.Comment: 21 pages, 5 figure
Iwahori-Hecke algebras for Kac-Moody groups over local fields
We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over
a local non-archimedean field. We use the hovel associated to this situation,
which is the analogue of the Bruhat-Tits building for a reductive group. The
fixer K of some chamber in the standard apartment plays the role of the Iwahori
subgroup. We can define the Iwahori-Hecke algebra as the algebra of some
K-bi-invariant functions on the group with support consisting of a finite union
of double classes. As two chambers in the hovel are not always in a same
apartment, this support has to be in some large subsemigroup of the Kac-Moody
group. In the split case, we prove that the structure constants of the
multiplication in this algebra are polynomials in the cardinality of the
residue field, with integer coefficients depending on the geometry of the
standard apartment. We give a presentation of this algebra, similar to the
Bernstein-Lusztig presentation in the reductive case, and embed it in a greater
algebra, algebraically defined by the Bernstein-Lusztig presentation. In the
affine case, this algebra contains the Cherednik's double affine Hecke algebra.
Actually, our results apply to abstract "locally finite" hovels, so that we can
define the Iwahori-Hecke algebra with unequal parameters.Comment: Version 2: Section on the extended affine case added, containing the
relationship with the DAHAs, to appear in Pacific Journal of Mathematic
The Role of Spin(9) in Octonionic Geometry
Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some
interactions between Spin(9) and geometries related to octonions. Several
topics are discussed in this respect: explicit descriptions of the Spin(9)
canonical 8-form and its analogies with quaternionic geometry as well as the
role of Spin(9) both in the classical problems of vector fields on spheres and
in the geometry of the octonionic Hopf fibration. Next, we deal with locally
conformally parallel Spin(9) manifolds in the framework of intrinsic torsion.
Finally, we discuss applications of Clifford systems and Clifford structures to
Cayley-Rosenfeld planes and to three series of Grassmannians.Comment: 25 page
A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions
State sum constructions, such as Kuperberg's algorithm, give partition
functions of physical systems, like lattice gauge theories, in various
dimensions by associating local tensors or weights, to different parts of a
closed triangulated manifold. Here we extend this construction by including
matter fields to build partition functions in both two and three space-time
dimensions. The matter fields introduces new weights to the vertices and they
correspond to Potts spin configurations described by an -module
with an inner product. Performing this construction on a triangulated manifold
with a boundary we obtain the transfer matrices which are decomposed into a
product of local operators acting on vertices, links and plaquettes. The vertex
and plaquette operators are similar to the ones appearing in the quantum double
models (QDM) of Kitaev. The link operator couples the gauge and the matter
fields, and it reduces to the usual interaction terms in known models such as
gauge theory with matter fields. The transfer matrices lead to
Hamiltonians that are frustration-free and are exactly solvable. According to
the choice of the initial input, that of the gauge group and a matter module,
we obtain interesting models which have a new kind of ground state degeneracy
that depends on the number of equivalence classes in the matter module under
gauge action. Some of the models have confined flux excitations in the bulk
which become deconfined at the surface. These edge modes are protected by an
energy gap provided by the link operator. These properties also appear in
"confined Walker-Wang" models which are 3D models having interesting surface
states. Apart from the gauge excitations there are also excitations in the
matter sector which are immobile and can be thought of as defects like in the
Ising model. We only consider bosonic matter fields in this paper.Comment: 52 pages, 58 figures. This paper is an extension of arXiv:1310.8483
[cond-mat.str-el] with the inclusion of matter fields. This version includes
substantial changes with a connection made to confined Walker-Wang models
along the lines of arXiv:1208.5128 and subsequent works. Accepted for
publication in JPhys
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