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Radical relations in orthogonal groups
AbstractAny relation between simple isometries is a consequence of relations of lengths ⩽4. This extends earlier results which deal with relations between reflections
Topological Defects on the Lattice I: The Ising model
In this paper and its sequel, we construct topologically invariant defects in
two-dimensional classical lattice models and quantum spin chains. We show how
defect lines commute with the transfer matrix/Hamiltonian when they obey the
defect commutation relations, cousins of the Yang-Baxter equation. These
relations and their solutions can be extended to allow defect lines to branch
and fuse, again with properties depending only on topology. In this part I, we
focus on the simplest example, the Ising model. We define lattice spin-flip and
duality defects and their branching, and prove they are topological. One useful
consequence is a simple implementation of Kramers-Wannier duality on the torus
and higher genus surfaces by using the fusion of duality defects. We use these
topological defects to do simple calculations that yield exact properties of
the conformal field theory describing the continuum limit. For example, the
shift in momentum quantization with duality-twisted boundary conditions yields
the conformal spin 1/16 of the chiral spin field. Even more strikingly, we
derive the modular transformation matrices explicitly and exactly.Comment: 45 pages, 9 figure
Partner symmetries and non-invariant solutions of four-dimensional heavenly equations
We extend our method of partner symmetries to the hyperbolic complex
Monge-Amp\`ere equation and the second heavenly equation of Pleba\~nski. We
show the existence of partner symmetries and derive the relations between them
for both equations. For certain simple choices of partner symmetries the
resulting differential constraints together with the original heavenly
equations are transformed to systems of linear equations by an appropriate
Legendre transformation. The solutions of these linear equations are
generically non-invariant. As a consequence we obtain explicitly new classes of
heavenly metrics without Killing vectors.Comment: 20 pages, 1 table, corrected typo
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