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

Total Quantum Zeno Effect beyond Zeno Time

In this work we show that is possible to obtain Total Quantum Zeno Effect in an unstable systems for times larger than the correlation time of the bath. The effect is observed for some particular systems in which one can chose appropriate observables which frequent measurements freeze the system into the initial state. For a two level system in a squeezed bath one can show that there are two bath dependent observables displaying Total Zeno Effect when the system is initialized in some particular states. We show also that these states are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath.Comment: 6 pages, 3 figures, Contributed to Quantum Optics III, Pucon, Chile, November 200

Total Quantum Zeno effect and Intelligent States for a two level system in a squeezed bath

In this work we show that by frequent measurements of adequately chosen observables, a complete suppression of the decay in an exponentially decaying two level system interacting with a squeezed bath is obtained. The observables for which the effect is observed depend on the the squeezing parameters of the bath. The initial states which display Total Zeno Effect are intelligent states of two conjugate observables associated to the electromagnetic fluctuations of the bath.Comment: 5 pages, 3 figure