Analytic regularity of CR maps into spheres

Let $M$ be a connected real-analytic hypersurface in \C^N and $\S$ the unit real sphere in \C^{N'}, $N'> N\geq 2$. Assume that $M$ does not contain any complex-analytic hypersurface of \C^N and that there exists at least one strongly pseudoconvex point on $M$. We show that any CR map $f\colon M\to \S$ of class $C^{N'-N+1}$ extends holomorphically to a neighborhood of $M$ in \C^N.Comment: 11 page

Superloop Space

In this paper will construct and analyse the superloop space formulation of a $\mathcal{N} =1$ supergauge theory in three dimensions. We will obtain expressions for the connection and the curvature in this superloop space in terms of ordinary supergauge fields. This curvature will vanish, unless there is a monopole in the spacetime. We will also construct a quantity which will give the monopole charge in this formalism. Finally, we will show how these results even hold for a deformed superspace.Comment: 10 pages, 0 figures, accepted for publication in EP

Clustering of Galaxies with Dynamical Dark Energy

In this paper, we study thermodynamics of the cluster of galaxies under the effect of dynamical dark energy. We evaluate the configurational integral for interacting system of galaxies in an expanding universe by including the effects produced by the varying $\Lambda$. The gravitational partition function is obtained using this configuration integral. We obtain thermodynamics quantities in canonical ensemble which depend on time and investigate the second law of thermodynamics. We also calculate the distribution function in grand canonical ensemble. The time evolution of the clustering parameter of galaxies is investigated for the time dependent (dynamical) dark energy. We conclude that the second law of thermodynamics is valid for the total system of cluster of galaxies and dynamical dark energy. We calculate correlation function and show that our model is very close to Peebles's power law, in agreement with the N-body simulation. It is observed that thermodynamics quantities depend on the modified clustering parameter for this system of galaxies.Comment: 8 pages, 15 figure