0 Submanifold Differential Operators in D - Module Theory II

This article is one of a series of papers. For this decade, the Dirac operator on a submanifold has been studied as a restriction of the Dirac operator in n-dimensional euclidean space E n to a surface or a space curve as physical models. These Dirac operators are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case and completely represent the submanifolds. For example, the analytic index of Dirac operator of a space curve is identified with its writhing number. As another example, the operator determinants of the Dirac operators are closely related to invariances of the immersed objects, such as Euler-Bernoulli and Willmore functionals for a space curve and a conformal surface respectively. In this article, we will give mathematical construction of the Dirac operator by means of D-module and reformulate my recent results mathematically. In an earlier article in this series [I], we showed the construction of the submanifold Schrödinger operator in terms of D-module theory. In this article, we will apply the scheme to the spin bundle to construct the Dirac operator of submanifold. We will call the previous article [I] and its proposition or definition and reference like (I-2.1) and [I-Mat2], which means proposition or definition 2-1 and reference [Mat2] in [I] respectively. Applying the quantum mechanical scheme [I and its references] to Dirac operators for a restricted particle along a low-dimensional submanifold in n-dimensional euclidean space E n , we obtained natural Dirac operators on curves in E In this decade, I have been studying these Dirac operators and investigating their properties. From physical point of view, I showed that they exhibit the symmetry of corresponding submanifolds and found a non-trivial extension of Atiyah-Singer type index theorem to submanifold [Mat2,6]. The Dirac operators of curves in E n (n ≥ 2) are related to the Frenet-Serret relations and are identified with the Lax operators of (1 + 1)-dimensional soliton equations, e.g., modified Korteweg-de Vries equation [MT, Mat2,8], nonlinear Schrödinger equation [Mat1, 3, 6], complex modified Korteweg-de Vries equation [Mat15], and so on [Mat3]. The Dirac operators on conformal surfaces in E n (n = 3, 4) are concerned with the generalized Weierstrass equation representing a surface [Mat11, 14,16] and also identified with the Lax operators of modified Novikov-Veselov (MNV) equations [Ko1, 2, T1, 2]. The generalized Weierstrass equation is very interesting …