Modular Properties of 3D Higher Spin Theory

In the three-dimensional sl(N) Chern-Simons higher-spin theory, we prove that the conical surplus and the black hole solution are related by the S-transformation of the modulus of the boundary torus. Then applying the modular group on a given conical surplus solution, we generate a 'SL(2,Z)' family of smooth constant solutions. We then show how these solutions are mapped into one another by coordinate transformations that act non-trivially on the homology of the boundary torus. After deriving a thermodynamics that applies to all the solutions in the 'SL(2,Z)' family, we compute their entropies and free energies, and determine how the latter transform under the modular transformations. Summing over all the modular images of the conical surplus, we write down a (tree-level) modular invariant partition function.Comment: 51 pages; v2: minor corrections and additions; v3: final version, to appear in JHE

The neutral top-pion and the lepton flavor violating ZZ decays Z→liljZ\to l_{i}l_{j}

Taking into account the constraints of the present experimental limit of the process $\mu\to e\gamma$ on the free parameters of topcolor-assisted technicolor(TC2) models, we calculate the contributions of the neutral top-pion to the lepton flavor violating(LFV) $Z$ decays $Z\to l_{i}l_{j}$. Our results show that the value of the branching ratio $Br(Z\to \tau l)$ is larger than that of $Br(Z\to \mu e)$ in all of the parameter space and there is $Br(Z\to \tau\mu)\approx Br(Z\to \tau e)\simeq 2\times 10^{-14}$, which is far from the reach of present or future experiments.Comment: 12 pages and 3 figures. To be published in J. Phys.