IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ³

Abstract

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 KonopelchenkoKenmotsu-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.

Electron wave functions on in a static magnetic field of arbitrary direction

“…There is a wealth of literature concerning curvature effects when a particle is constrained to a two-dimensional surface in three-space [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38], including some dealing with the torus specifically [39], but the scope of this work will remain restricted to study of the Hamiltonian given by Eq. (9).…”

Quantum effects for extrinsic geometry of strings via the generalized Weierstrass representation

“…The generalized Weierstrass representation (1)-(2) has allowed already to obtain several interesting results in differential geometry of surfaces where the extrinsic Polyakov action is known for a long time as the Willmore functional [21] (see [22]- [24]), in the theory of liquid membranes [25] and in the string theory [26], [16]- [17]. The representation (1)- (2) gives also the possibility to define an infinite class of integrable deformations of surfaces generated by the modified Veselov-Novikov hierarchy [15].…”

“…A characteristic feature of these deformations is that they preserve the extrinsic Polyakov action [20], [22]. This circumstance has been used in [16]- [17] to quantize the Willmore surface (surfaces which provide extremum to the Willmore functional (Polyakov action)).…”

“…Note that the generalized Weierstrass representation has been used recently within a different technique [16]- [17] to evaluate quantum effects in the string theory.…”

Induced Surfaces and Their Integrable Dynamics Ii. Generalized Weierstrass Representations in 4‐d Spaces and Deformations via Ds Hierarchy

“…In differential geometry, its use has allowed us to obtain several interesting results both of local and global character, in particular, for the Willmore functional W = H 2 dS , where H is the mean curvature of the surface (see e.g., [14][15][16][17][18][19][20][21][22]). In physics, it has been applied to study of various problems in the theory of liquid membranes, 2-D gravity and string theory [17], [23][24][25][26][27]. In the string theory, the functional W = H 2 dS is known as the Polyakov extrinsic action, and in membrane theory, it is the Helfrich free energy [7][8][9].…”