Beltrami Parametrization Research Articles

Induced quantum gravity and quasiconformal mappings

Induced quantum gravity is known to break down in the strong-coupling regime 1<or=dmatter<or=25. A naive interpretation of the breakdown is given. The appearance of spiky structures is shown to be related to the anticonformal limit of quasiconformal mappings (the so called Beltrami parametrization). A generalization to the supersymmetric case (N=1, 2) is proposed. The local effective action is given in terms of the Beltrami parameterization for N=0, 1, 2. All results belong to spherical topology.

Stochastic quantization of 2D gravity and its link with 3D gravity and topological 4D gravity

We apply stochastic quantization to two-dimensional gravity. The Laplace operator acting on the space of all metrics takes a particularly simple form in terms of the Beltrami parametrization. We show the equivalence between the quantum theory defined by the standard Faddeev-Popov gauge fixing of the two-dimensional diffeomorphism invariance and the one defined by stochastic quantization. We do so by using the gauge freedom left in the Langevin equation of a diffeomorphism-invariant theory to adjust the drift force. Another choice of the drift force, comparable to that of Zwanziger for Yang-Mills theories, seems to avoid the analogue of the Gribov ambiguity, i.e. the necessity of the by-hand restriction to one fundamental domain. We relate the two-dimensional gravity to a three-dimensional theory, based on the three-dimensional gravitational Chern-Simons action for SL(2, C ), ISO(3) or SU(2) × SU(2) (depending on the genus of the two-dimensional Riemann surface), in which all fields of the stochastic quantization have been distributed as components of the gauge fields. To study the three-dimensional theory, stochastic quantization can be applied once more. This gives a theory with the action of topological gravity in four dimensions, namely the Pontrjagin invariant ∫ N 2× R× R tr R ∧ R , gauge fixed by self-duality conditions.

The gauged BRST symmetry

We develop a systematic method for constructing the gauged BRST symmetry for any theory. In this framework, the gauged BRST symmetry results as the combination of two basic symmetries of the gauge-fixed theory which one considers: the BRST symmetry and the ghost number symmetry, this latter being promoted to a local one. From this, we can derive a general relation between the BRST and the ghost number Noether currents. We then take advantage of the present method to elaborate on a geometrical algorithm leading to the obtainment of the gauged BRST symmetry for a large class of theories, in arbitrary dimensions of space. These involve systems of antisymmetric tensor gauge fields of arbitrary rank, eventually coupled to gravity. This algorithm allows us to derive algebraically the expressions for the possible consistent anomalies of the BRST Noether current algebras; various examples are explicitly discussed. The gauged BRST symmetry for the free bosonic string is also constructed and used to exhibit the link between the trace anomaly and the nilpotency anomaly of the BRST charge operator. In particular, when the Beltrami parametrization is introduced, we show that the corresponding BRST symmetry can be gauged in a way compatible with the holomorphic factorization. A further use of Ward-Slavnov identities constraining the BRST and ghost number current algebra allows us to recover the well-known local counterterm necessary, at the one-loop level, for rendering the BRST current a good conformal operator.

Beltrami differentials and 2d (super)gravity in the chiral gauge

We consider the Beltrami parametrization of a 2-dimensional metric and work out the basic transformation laws of the constituting fields. These results are then applied to investigate the geometric structure of 2-dimensional gravity in Polyakov's light-cone gauge. Subsequently, we elaborate on the supersymmetric generalization of this theory. The general relation to the superconformai gauge is presented in detail and the corresponding change of coordinates is shown to satisfy simple super Beltrami equations.

Beltrami Gauge Symmetry and 2D Induced Gravity11The project supported-in part by National Natural Science Foundation of China.

We introduce the Beltrami gauge symmetry related to the Beltrami parametrization of the metrics and the Beltrami equations. We explore its role in 2d induced gravity and show that Polyakov's-SL(2R) and KPZ's residual symmetries are subsymmetries of the Beltrami gauge symmetry in the light-cone gauge. We also find that the 2d induced gravity with or without matter can be reformulated as Beltrami-Liouville field theories on Riemann surfaces of higher genus.

Left-right decomposition of two-dimensional superspace geometry and its BRS structure

Two-dimensional (1, 1) superspace geometry is reviewed with special emphasis on its Beltrami parametrization and the factorization of the corresponding BRS transformations. Generic matter superfields of arbitrary chiral weights are discussed in this geometrical setting. Based upon a thorough discussion of the purely bosonic case these supersymmetric structures are presented in superfield formalism as well as in component field language.

Green-Schwarz superstring. Beltrami parametrization and world-sheet supersymmetric generalization

We study a generalization of the superstring theory in which both world-sheet and space-time supersymmetries are explicit. This hybrid superstring interpolates between the Green-Schwarz and the Ramond-Neveu-Schwarz formulations. We show that the Beltrami parametrization of the world-sheet provides a natural setting for the description of K-transformation already in the Green-Schwarz string. We construct the hybrid superstring action using the (1, 1) supersymmetric version of the Beltrami parametrization. We discuss the generalization of K-transformations.

TheB-C system in the Beltrami parametrization and the Slavnov-Taylor identity

TheB-C system with an arbitrary conformal weight by Friedan, Martinec and Shenker is formulated by using the Beltrami parametrization. The holomorphic factorization will be manifest. We quantize the system in the BRST formalism. It will be shown by an explicit calculation that the Slavnov-Taylor identity is afflicted by the holomorphic anomaly.

The relationship between the gauged BRST symmetry and the ghost number symmetry

A general method of gauging the BRST algebra by combining a (local) ghost number symmetry with the standard (global) BRST algebra, is displayed. This method enables us to construct an automatically nilpotent local BRST algebra and to obtain in a straightforward way the corresponding versions of the action and the eventual anomalies. To illustrate the procedure we study the Yang-Mills and the free bosonic string theories (including the “conformal” Beltrami parametrization) and show that it reproduces the results discussed in the literature. Two major outcomes of this scheme are briefly discussed: a possible connection between ghost number and BRST currents, arising from the Slavnov identities and the implications of the ghost number anomaly for the BRST localization program in theories such as string theory.

Beltrami parametrization and gauged BRST symmetry for the free bosonic string

The BRST symmetry for the free bosonic string in the Beltrami parametrization is gauged via a Noether procedure. The resulting local BRST symmetry is characterized by a (localized) differential algebra which is shown to be isomorphic to the (global) BRST algebra one starts from. The corresponding locally invariant gauge-fixed action which exhibits the factorization property is constructed, and a form for the (perturbative BRST current-algebra) anomaly, associated with the localized differential algebra is given. The equivalence between the Noether localization procedure presented here and a procedure which mimics the direct “geometrical” gauging of the BRST symmetry for the Yang-Mills theory is shown.