Immersion anomaly of Dirac operator on surface in R3

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