99 research outputs found
Toric Genera
Our primary aim is to develop a theory of equivariant genera for stably
complex manifolds equipped with compatible actions of a torus T^k. In the case
of omnioriented quasitoric manifolds, we present computations that depend only
on their defining combinatorial data; these draw inspiration from analogous
calculations in toric geometry, which seek to express arithmetic, elliptic, and
associated genera of toric varieties in terms only of their fans. Our theory
focuses on the universal toric genus \Phi, which was introduced independently
by Krichever and Loeffler in 1974, albeit from radically different viewpoints.
In fact \Phi is a version of tom Dieck's bundling transformation of 1970,
defined on T^k-equivariant complex cobordism classes and taking values in the
complex cobordism algebra of the classifying space. We proceed by combining the
analytic, the formal group theoretic, and the homotopical approaches to genera,
and refer to the index theoretic approach as a recurring source of insight and
motivation. The resultant flexibility allows us to identify several distinct
genera within our framework, and to introduce parametrised versions that apply
to bundles equipped with a stably complex structure on the tangents along their
fibres. In the presence of isolated fixed points, we obtain universal
localisation formulae, whose applications include the identification of
Krichever's generalised elliptic genus as universal amongst genera that are
rigid on SU-manifolds. We follow the traditions of toric geometry by working
with a variety of illustrative examples wherever possible. For background and
prerequisites we attempt to reconcile the literature of east and west, which
developed independently for several decades after the 1960s.Comment: 35 pages, LaTeX. In v2 references made to the index theoretical
approach to genera; rigidity and multiplicativity results improved;
acknowledgements adde
Geometric aspects of gauge and spacetime symmetries
We investigate several problems in relativity and particle physics where symmetries play a central role; in all cases geometric properties of Lie groups and their quotients are related to physical effects. The first part is concerned with symmetries in gravity. We apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes. We then make group deformation local, generalising deformed special relativity (DSR) by describing gravity as a gauge theory of the de Sitter group. We find that in our construction Minkowski space has a connection with torsion; physical effects of torsion seem to rule out the proposed framework as a viable theory. A third chapter discusses a formulation of gravity as a topological BF theory with added linear constraints that reduce the symmetries of the topological theory to those of general relativity. We discretise our constructions and compare to a similar construction by Plebanski which uses quadratic constraints. In the second part we study CP violation in the electroweak sector of the standard model and certain extensions of it. We quantify fine-tuning in the observed magnitude of CP violation by determining a natural measure on the space of CKM matrices, a double quotient of SU(3), introducing different possible choices and comparing their predictions for CP violation. While one generically faces a fine-tuning problem, in the standard model the problem is removed by a measure that incorporates the observed quark masses, which suggests a close relation between a mass hierarchy and suppression of CP violation. Going beyond the standard model by adding a left-right symmetry spoils the result, leaving us to conclude that such additional symmetries appear less natural
Generic Linear Recurrent Sequences and Related Topics
The aim of the book is to introduce and develop the elementary theory of generic linear recurrent relations and to show how it provides a natural framework to put into a unified perspective many seemingly unrelated subjects. Among them: traces of an endomorphism and the Cayley-Hamilton theorem, Generic Linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (revisiting Putzer’s method), universal decomposition algebras of a polynomial into the product of two monic polynomials of fixed smaller degree, vertex operators obtained via Schubert calculus tools (Giambelli’s formula). Emphasis will be put on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank. The classical example of the Hirota bilinear form of the Kadomtsev-Petshiasvilii (KP) hierarchy, seen as equations of the Plu ̈cker embedding of an infinite-dimensional Grassmannian, will be included. The research that lead to the present paper was partially supported by a grant of the group GNSAGA of INdA
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