173 research outputs found
Integrable mappings and polynomial growth
We describe birational representations of discrete groups generated by
involutions, having their origin in the theory of exactly solvable
vertex-models in lattice statistical mechanics. These involutions correspond
respectively to two kinds of transformations on matrices: the
inversion of the matrix and an (involutive) permutation of the
entries of the matrix. We concentrate on the case where these permutations are
elementary transpositions of two entries. In this case the birational
transformations fall into six different classes. For each class we analyze the
factorization properties of the iteration of these transformations. These
factorization properties enable to define some canonical homogeneous
polynomials associated with these factorization properties. Some mappings yield
a polynomial growth of the complexity of the iterations. For three classes the
successive iterates, for , actually lie on elliptic curves. This analysis
also provides examples of integrable mappings in arbitrary dimension, even
infinite. Moreover, for two classes, the homogeneous polynomials are shown to
satisfy non trivial non-linear recurrences. The relations between
factorizations of the iterations, the existence of recurrences on one or
several variables, as well as the integrability of the mappings are analyzed.Comment: 45 page
Symmetry Decomposition of Chaotic Dynamics
Discrete symmetries of dynamical flows give rise to relations between
periodic orbits, reduce the dynamics to a fundamental domain, and lead to
factorizations of zeta functions. These factorizations in turn reduce the labor
and improve the convergence of cycle expansions for classical and quantum
spectra associated with the flow. In this paper the general formalism is
developed, with the -disk pinball model used as a concrete example and a
series of physically interesting cases worked out in detail.Comment: CYCLER Paper 93mar01
Holography, Matrix Factorizations and K-stability
Placing D3-branes at conical Calabi-Yau threefold singularities produces many
AdS/CFT duals. Recent progress in differential geometry has produced a
technique (called K-stability) to recognize which singularities admit conical
Calabi-Yau metrics. On the other hand, the algebraic technique of
non-commutative crepant resolutions, involving matrix factorizations, has been
developed to associate a quiver to a singularity. In this paper, we put
together these ideas to produce new AdS/CFT duals, with special
emphasis on non-toric singularities.Comment: 59 pages, 11 figures, 2 appendices; v2: typos fixed, expanded
discussion in section 2.5 and 4.
Strictly transversal slices to conjugacy classes in algebraic groups
We show that for every conjugacy class O in a connected semisimple algebraic
group G over a field of characteristic good for G one can find a special
transversal slice S to the set of conjugacy classes in G such that O intersects
S and dim O = codim S.Comment: 38 pages; minor modification
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