932 research outputs found
Einstein metrics on tangent bundles of spheres
We give an elementary treatment of the existence of complete Kahler-Einstein
metrics with nonpositive Einstein constant and underlying manifold
diffeomorphic to the tangent bundle of the (n+1)-sphere.Comment: 9 page
Representations of the quantum doubles of finite group algebras and solutions of the Yang--Baxter equation
Quantum doubles of finite group algebras form a class of quasi-triangular
Hopf algebras which algebraically solve the Yang--Baxter equation. Each
representation of the quantum double then gives a matrix solution of the
Yang--Baxter equation. Such solutions do not depend on a spectral parameter,
and to date there has been little investigation into extending these solutions
such that they do depend on a spectral parameter. Here we first explicitly
construct the matrix elements of the generators for all irreducible
representations of quantum doubles of the dihedral groups . These results
may be used to determine constant solutions of the Yang--Baxter equation. We
then discuss Baxterisation ans\"atze to obtain solutions of the Yang--Baxter
equation with spectral parameter and give several examples, including a new
21-vertex model. We also describe this approach in terms of minimal-dimensional
representations of the quantum doubles of the alternating group and the
symmetric group .Comment: 19 pages, no figures, changed introduction, added reference
Singular Monopoles and Supersymmetric Gauge Theories in Three Dimensions
According to the proposal of Hanany and Witten, Coulomb branches of N=4 SU(n)
gauge theories in three dimensions are isometric to moduli spaces of BPS
monopoles. We generalize this proposal to gauge theories with matter, which
allows us to describe the metrics on their spaces of vacua by means of the
hyperk\"ahler quotient construction. To check the identification of moduli
spaces a comparison is made with field theory predictions. For SU(2) theory
with k fundamental hypermultiplets the Coulomb branch is expected to be the D_k
ALF gravitational instanton, so our results lead to a construction of such
spaces. In the special case of SU(2) theory with four or fewer fundamental
hypermultiplets we calculate the complex structures on the moduli spaces and
compare them with field-theoretical results. We also discuss some puzzles with
brane realizations of three-dimensional N=4 theories.Comment: 26 pages, LaTeX, 6 figures; exposition improve
SU(3) monopoles and their fields
Some aspects of the fields of charge two SU(3) monopoles with minimal
symmetry breaking are discussed. A certain class of solutions look like SU(2)
monopoles embedded in SU(3) with a transition region or ``cloud'' surrounding
the monopoles. For large cloud size the relative moduli space metric splits as
a direct product AH\times R^4 where AH is the Atiyah-Hitchin metric for SU(2)
monopoles and R^4 has the flat metric. Thus the cloud is parametrised by R^4
which corresponds to its radius and SO(3) orientation. We solve for the
long-range fields in this region, and examine the energy density and rotational
moments of inertia. The moduli space metric for these monopoles, given by
Dancer, is also expressed in a more explicit form.Comment: 17 pages, 3 figures, latex, version appearing in Phys. Rev.
Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras
The Perk--Schultz model may be expressed in terms of the solution of the
Yang--Baxter equation associated with the fundamental representation of the
untwisted affine extension of the general linear quantum superalgebra
, with a multiparametric co-product action as given by
Reshetikhin. Here we present analogous explicit expressions for solutions of
the Yang-Baxter equation associated with the fundamental representations of the
twisted and untwisted affine extensions of the orthosymplectic quantum
superalgebras . In this manner we obtain generalisations of the
Perk--Schultz model.Comment: 10 pages, 2 figure
Solutions to the Yang-Baxter Equation and Casimir Invariants for the Quantised Orthosymplectic Superalgebra
For the last fifteen years quantum superalgebras have been used to model
supersymmetric quantum systems. A class of quasi-triangular Hopf superalgebras,
they each contain a universal -matrix, which automatically satisfies the
Yang--Baxter equation. Applying the vector representation to the left-hand side
of a universal -matrix gives a Lax operator. These are of significant
interest in mathematical physics as they provide solutions to the Yang--Baxter
equation in an arbitrary representation, which give rise to integrable models.
In this thesis a Lax operator is constructed for the quantised
orthosymplectic superalgebra for all where
is even. This can then be used to find a solution to the Yang--Baxter
equation in an arbitrary representation of , with the example of
the vector representation given in detail.
In studying the integrable models arising from solutions to the Yang--Baxter
equation, it is desirable to understand the representation theory of the
superalgebra. Finding the Casimir invariants of the system and exploring their
behaviour helps in this understanding. In this thesis the Lax operator is used
to construct an infinite family of Casimir invariants of and to
calculate their eigenvalues in an arbitrary irreducible representation.Comment: Approx. 120 pages, no figures, PhD thesi
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