Geometrically Induced Gauge Structure on Manifolds Embedded in a Higher Dimensional Space

Abstract

We explain in a context different from that of Maraner the formalism for describing motion of a particle, under the influence of a confining potential, in a neighbourhood of an n-dimensional curved manifold M^n embedded in a p-dimensional Euclidean space R^p with p >= n+2. The effective Hamiltonian on M^n has a (generally non-Abelian) gauge structure determined by geometry of M^n. Such a gauge term is defined in terms of the vectors normal to M^n, and its connection is called the N-connection. In order to see the global effect of this type of connections, the case of M^1 embedded in R^3 is examined, where the relation of an integral of the gauge potential of the N-connection (i.e., the torsion) along a path in M^1 to the Berry's phase is given through Gauss mapping of the vector tangent to M^1. Through the same mapping in the case of M^1 embedded in R^p, where the normal and the tangent quantities are exchanged, the relation of the N-connection to the induced gauge potential on the (p-1)-dimensional sphere S^{p-1} (p >= 3) found by Ohnuki and Kitakado is concretely established. Further, this latter which has the monopole-like structure is also proved to be gauge-equivalent to the spin-connection of S^{p-1}. Finally, by extending formally the fundamental equations for M^n to infinite dimensional case, the present formalism is applied to the field theory that admits a soliton solution. The resultant expression is in some respects different from that of Gervais and Jevicki.

52 pages, PHYZZX. To be published in Int. J. Mod. Phys. A

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