S. Matsutani, Dirac operator on a conformal surface immersed in R-4: A way to further generalized Weierstrass equation, REV MATH PH, 12(3), 2000, pp. 431-444
In the previous report (J. Ph3/a. A30 (1997) 4019-4029), I showed that the
Dirac operator defined over a conformal surface immersed in R-3 by means of
confinement procedure is identified with the differential operator of the
generalized Weierstrass equation and the Lax operator of the modified Novik
ov-Veselov (MNV) equation. In this article, using the same procedure, I det
ermine the Dirac operator defined over a conformal surface immersed in R-4,
which is for a Dirac field confined in the surface. Then it is reduced to
the Lax operators of the nonlinear Schrodinger and the MNV equations by tak
ing appropriate limits. It means that the Dirac operator is related to the
further generalized Weierstrass equation for a surface in R-4.