Generalized Weierstrass-Enneper Inducing, Conformal Immersions, and Gravity

Abstract

Basic quantities related to 2D gravity. such as Polyakov extrinsic action, Nambu-Goto action, geometrical action and the Euler characteristic, are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces in R3. Connection of the GWE inducing with conformal immersion is made and various aspects of the theory are shown to be invariant under the modified Veselov-Novikov hierarchy of flows. The geometry of [Formula: see text] surfaces (h ~ mean curvature) is shown to be connected with the dynamics of infinite- and finite-dimensional integrable systems. Connections with Liouville-Beltrami gravity are indicated.

SusyCPN−1model and surfaces in

“…The subject of Weierstrass representations of surfaces immersed in multidimensional spaces was introduced few years ago by Konopelchenko et al 1,2 . This has generated interest 3,4 in looking at the properties of these surfaces and relating them to the solutions This approach involved writing the equation for the harmonic map as a conservation law and then observing that in this construction a special operator played a key role.…”

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

“…The generalized Weierstrass representation (1)- (2) is equivalent to another Weierstrass type representation which was proposed in [19] and has been used within the Gauss map approach in the papers [12]- [13]. The equivalence is established by simple formulae [20] …”

“…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)).…”

IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ3

“…His program has been studied in the framework of W-algebra [23] but recently was investigated by Carroll and Konopelchenko [24] and Viswanathan and Parthasarathy [25] using more direct method. Polyakov's extrinsic action in the classical level is the same as the Willmore functional [26,27],…”

“…This functional integral is known as the Willmore functional [26,27] and, recently, as the Polyakov's extrinsic action in the 2-dimensional gravity [22,24,25]. For later convenience, I will fix B 0 = 1 and introduce a quantity [1][2][3][4],…”