Immersion anomaly of Dirac operator on surface in R-3

In previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dira c operator confined in a surface immersed in R-3 by means of a mass type po tential completely exhibits the surface itself and is identified with that of the generalized Weierstrass equation. In this article, I quantized the D irac field and calculated the gauge transformation which exhibits the gauge freedom of the parameterization of the surface. Then using the Ward-Takaha shi identity, I showed that the expectation value of the action of the Dira c field is expressed by the Willmore functional and area of the surface, or the action of Polyakov's extrinsic string.