On density state of quantized Willmore surfaces-a way to quantized extrinsic string in R 3

Recently I quantized an elastica with Bernoulli-Euler functional in two-dimensional space using the modified KdV hierarchy. In this article, I will quantize a Willmore surface, or equivalently a surface with the Polyakov extrinsic curvature action, using the modified Novikov-Veselov (MNV) equation. In other words, I show that the density of state of the partition function for the quantized Willmore surface is expressed by volume of a subspace of the moduli of the MNV equation. §1. Introduction In series of works [1-7], I have considered the correspondence between an immersed object and the Dirac operator confined there. The Dirac operator confined in an immersed object is uniquely determined by the procedure which I proposed [1-4] and can be regarded as the representation matrix of the symmetry of the immersed object [1-7]. I had been studying it mainly on an elastica in a plane [1-6] and showed that the Dirac operator confined in an elastica is identified with the Lax operator of the modified KdV equation while the mathematical deformation of the elastica obeys the modified KdV hierarchy [6]. By investigating other quantum equations [8-9], I conjectured that such correspondence between the Dirac operator and geometry can be extended to higher dimensional immersed objects [2,3,4]. Couple years ago Konopelchenko [10,11] discovered that a conformal surface S immersed in three dimensional flat space R 3 obeys the Dirac equation, which I will call Konopelchenko-Kenmotsu-Weierstrass-Enneper (KKWE) [10-15] equation here, ∂f 1 = V f 2 , ¯ ∂f 2 = −V f 1 , (1-1) where V := 1 2 √ ρH, (1-2) H is the mean curvature of the surface S parameterized by complex z and ρ is the conformal metric induced from R 3. The KKWE equation completely exhibits the immersed geometry as the old Weierstrass-Enneper equation expresses the minimal surface [10-15]. In ref.[16] I showed that it is identified with the Dirac operator confined in the surface S and by quantizing the Dirac field I found that the quantized symmetry of the Dirac operator is also in agreement with the symmetry of the surface itself [17]. In other words, this KKWE equation is the equation which I conjectured before [2,3] and I had been searching for. Even though for more general surface, which is not conformal, the KKWE type equation was discovered by Burgress and Jensen [18] following my prescriptions [1], their equation is not easy to deal with and I …