On the significance of polarization charge and isomagnetic surface in the interaction between conducting fluid and magnetic field

From the frozen-in field lines concept, a highly conducting fluid can move freely along, but not traverse to, magnetic field lines. We discuss this topic and find that in the study of the frozen-in field lines concept, the effects of inductive and capacitive reactance have been omitted. When admitted, the relationships among the motional electromotive field, the induced electric field, the eddy electric current, and the magnetic field becomes clearer and the frozen-in field line concept can be reconsidered. We emphasize the importance of isomagnetic surfaces and polarization charges, and show analytically that whether a conducting fluid can freely traverse magnetic field lines or not depends solely on the magnetic gradient in the direction of fluid motion. If a fluid does not change its density distribution and shape (can be regarded as a quasi-rigid body), and as long as it is moving along an isomagnetic surface, it can freely traverse magnetic field lines without any magnetic resistance no matter how strong the magnetic field is. When our analysis is applied, the origin of the magnetic field of sunspots can be interpreted easily. In addition, we also present experimental results to support our analysis.Comment: 12 pages, 12 figures, 4 table

Modular invariance for conformal full field algebras

Let V^L and V^R be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let F be a conformal full field algebra over the tensor product of V^L and V^R. We prove that the q_\tau-\bar{q_\tau}-traces (natural traces involving q_\tau=e^{2\pi i\tau} and \bar{q_\tau}=\bar{e^{2\pi i\tau}}) of geometrically modified genus-zero correlation functions for F are convergent in suitable regions and can be extended to doubly periodic functions with periods 1 and \tau. We obtain necessary and sufficient conditions for these functions to be modular invariant. In the case that V^L=V^R and F is one of those constructed by the authors in \cite{HK}, we prove that all these functions are modular invariant.Comment: 54 page