Topological restrictions for circle actions and harmonic morphisms

Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is nonnegative and even. In particular, $M^m$ has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example $CP^{n}$, $K3$ surfaces, $S^{2n}\times P_g$ ($n\geq2$) where $P_g$ is the closed surface of genus $g\geq2$ can never be the domain of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and $\phi:(M^4,g)\to(N^3,h)$ a non-constant harmonic morphism. Suppose that one of the following assertions holds: (1) $(M^4,g)$ is half-conformally flat and its scalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat, (3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemannian coverings, $\phi$ is the canonical projection $T^4\to T^3$ between flat tori.Comment: 18 pages; Minor corrections to Proposition 3.1 and small changes in Theorem 2.8, proof of Theorem 3.3 and Remark 3.

Gauge-Fermion Unification and Flavour Symmetry

After we study the 6-dimensional ${\cal N} = (1, 1)$ supersymmetry breaking and $R$ symmetry breaking on $M^4\times T^2/Z_n$, we construct two ${\cal N} = (1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_3$ where $E_6$ is broken down to $SO(10)\times U(1)_X$ by orbifold projection. In Model I, three families of the Standard Model fermions arise from the zero modes of bulk vector multiplet, and the $R$ symmetry $U(1)_F^{I} \times SU(2)_{{\bf 4}_-}$ can be considered as flavour symmetry. This may explain why there are three families of fermions in the nature. In Model II, the first two families come from the zero modes of bulk vector multiplet, and the flavour symmetry is similar. In these models, the anomalies can be cancelled, and we have very good fits to the SM fermion masses and mixings. We also comment on the ${\cal N}=(1, 1)$ supersymmetric $E_6$ models on $M^4\times T^2/Z_4$ and $M^4\times T^2/Z_6$, SU(9) models on $M^4\times T^2/Z_3$, and SU(8) models on $T^2$ orbifolds.Comment: Latex, 33 pages, minor change