The Dirac operator on surfaces immersed into 4d manifolds
by
V.V. Varlamov
Siberia State University of Industry, Novokuznetsk

As known, there exists a close relationship between a theory of the Dirac operator on surfaces immersed into 3d manifolds and so-called spinor representations of surfaces (SRS) conformally immersed into 3d Euclidean space [1]. The present paper is an extension of the Friedrich's work onto a 4-dimensional case. Previously, the Dirac operator and SRS for surfaces conformally immersed into 4d complex space are considered in the framework of a generalized Weierstrass representation [2, 3]. Let \Psi be a spinor field on the 4d pseudo-riemannian manifold (M^{p, q}, g), p+q=4. At this point the spinor field \Psi is understood as an element of a minimal left ideal of the Clifford algebra \cl_{p, q}. For example, in the case of the Lorentzian manifold M^{1, 3} we have \Psi in I_{1, 3}=\cl_{1, 3}e₁₃=\cl⁺_{1, 3}\frac12(1+\gamma₀) =~ \C₂\frac12(1+i\sigma₁₂), and also \Psi in I_{2, 2}=\cl_{2, 2}e₂₂=\cl⁺_{2, 2}\frac12(1+\cE₁₃)\frac12(1+\cE₂₄) =~ \Om_{2, 0}\frac12(1-i\Upsilon₁₂) for the Kleinian manifold M^{2, 2}. Further, for the every Clifford algebra \cl_{p, q} of a tangent space of M^{p, q} there exists a decomposition \cl_{p, q} =~ \cl_{r, s}\otimes\cl_{k, t}, where r+s=k+t=2. This decomposition allows to represent a general element of \cl_{p, q} in the form \sum³_i=0\clⁱ_{r, s}\zeta_i. The restricted spinor field on the surface S^{r, s} takes a form \Psi_{|S^{r, s}}=(\epsilon⁺_rs\Psi, \epsilon^-_rs\Psi), where \epsilon ^+/- _rs=\frac12(1 +/- \epsilon\zeta₁\zeta₂) are mutually orthogonal idempotents.
The main result is
Theorem Let \Psi in I_{p, q} be a real Killing spinor field on the 4d pseudo-riemannian manifold M^{p, q} and let \Psi|S^{r, s}=\psi = \psi⁺\oplus\psi^- be a restricted spinor field on the surface immersed into the manifold M^{p, q}. Then a Dirac operator on the surface S^{r, s}\hookrightarrow M^{p, q} defined as follows

D(\psi+) = (\alpha-\frac12\epsilon\betaH)\psi-, D(\psi-) = (\alpha+\frac12\epsilon\betaH)\psi+,

where \varepsion = 1 for the immersions S^{0, 2}\hookrightarrow M^{1, 3}, S^{2, 0}\hookrightarrow M^{3, 1}, S^{1, 1}\hookrightarrow M^{2, 2}, and \epsilon = i for the immersions S^{2, 0}\hoorightarrow M^{4, 0}, S^{2, 0}\hookrightarrow M^{0, 4}, S^{1, 1}\hookrightarrow M^{1, 3}, S^{1, 1}\hookrightarrow M^{3, 1}, S^{2, 0}\hookrightarrow M^{2, 2}, s^{0, 2}\hookrightarrow M^{2, 2}. H is a mean curvature of the surface, \alpha = \lambda₁g₁₁+\lambda₂g₂₂, \beta = g₁₁+g₂₂.

Date received: February 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadw-48.