Immersion anomaly of Dirac operator on surface in R 3

In previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac field confined in a surface immersed in R 3 by means of a mass type potential is governed by the Konopelchenko-Kenmotsu-Weierstrass-Enneper 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. §1. Introduction In the previous report [1], I showed that the Dirac field confined in a thin curved surface S immersed in three dimensional flat space R 3 obeys the Dirac equation which is discovered by Konopelchenko [2-4] ∂f 1 = pf 2 , ¯ ∂f 2 = −pf 1 , (1-1) where p := 1 2 √ ρH, (1-2) H is the mean curvature of the surface S parameterized by complex z and ρ is the factor of the conformal metric induced from R 3. This equation completely represents the immersed geometry as the old Weierstrass-Enneper equation expresses the minimal surface [2]. Even though the relation had been essentially found by Kenmotsu [4-7], the formulation as the Dirac type was performed by Konopelchenko and recently it is revealed that the Dirac operator has more physical and mathematical meanings; the Dirac operator is a translator between the geometrical and analytical objects [8] even in the arithmetic geometry of the number theory [9]. Thus although the Dirac type equation (1-1) has been called as the generalized Weierstrass equation, in this article I will call it Konopelchenko-Kenmotsu-Weierstrass-Enneper (KKWE) equation. The immersion geometry is currently studied in the various fields, e.g., soliton theory, the differential geometry, the harmonic map theory, string theory and so on. In the soliton theory, the question what is the integrable is the most important theme and one of its answers might be found in the immersed geometry. In fact, the static sine-Gordon equation was discovered by Euler, from the energy functional of the elastica given by Daniel Bernoulli in eighteenth century, as an elastica immersed in R 2 [10] and the net sine-Gordon equation was found in the last century as a surface immersed in R 3 [11]. Recently Goldstein and Petrich discovered the modified KdV (MKdV) hierarchy by considering one parameter deformation of a space curve immersed in …