scholarly journals LIOUVILLE THEORY: QUANTUM GEOMETRY OF RIEMANN SURFACES

1993 ◽  
Vol 08 (37) ◽  
pp. 3529-3535 ◽  
Author(s):  
LEON A. TAKHTAJAN
Keyword(s):  
Riemann Surfaces ◽  
Quantum Correction ◽  
Central Charge ◽  
Perturbation Expansion ◽  
Functional Integral ◽  
Liouville Theory ◽  
Conformal Anomaly ◽  
Quantum Geometry ◽  
Loop Contribution ◽  
Classical Expression

Inspired by Polyakov’s original formulation1,2 of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that the total quantum correction to the central charge of Liouville theory is given by one-loop contribution, which is equal to 1. Applied to the bosonic string, this result ensures the vanishing of total conformal anomaly along the lines different from those presented by KPZ3 and Distler-Kawai.4

scholarly journals LIOUVILLE THEORY: WARD IDENTITIES FOR GENERATING FUNCTIONAL AND MODULAR GEOMETRY

1994 ◽  
Vol 09 (25) ◽  
pp. 2293-2299 ◽  
Author(s):  
LEON A. TAKHTAJAN
Keyword(s):  
Moduli Spaces ◽  
Correlation Functions ◽  
Riemann Surfaces ◽  
Central Charge ◽  
Perturbation Expansion ◽  
Liouville Theory ◽  
Energy Tensor ◽  
Ward Identities ◽  
Generating Functional ◽  
Local Index

We continue the study of quantum Liouville theory through Polyakov’s functional integral,1,2 started in Ref. 3. We derive the perturbation expansion for Schwinger’s generating functional for connected multi-point correlation functions involving stress-energy tensor, give the “dynamical” proof of the Virasoro symmetry of the theory and compute the value of the central charge, confirming previous calculation in Ref. 3. We show that conformal Ward identities for these correlation functions contain such basic facts from Kähler geometry of moduli spaces of Riemann surfaces, as relation between accessory parameters for the Fuchsian uniformization, Liouville action and Eichler integrals, Kähler potential for the Weil-Petersson metric, and local index theorem. These results affirm the fundamental role that universal Ward identities for the generating functional play in Friedan-Shenker modular geometry.4

scholarly journals FRACTAL INDEX, CENTRAL CHARGE AND FRACTONS

2000 ◽  
Vol 15 (31) ◽  
pp. 1931-1939 ◽  
Author(s):  
WELLINGTON DA CRUZ ◽  
ROSEVALDO DE OLIVEIRA
Keyword(s):  
Hausdorff Dimension ◽  
Central Charge ◽  
Statistical Parameter ◽  
Conformal Anomaly ◽  
Fermi Velocity ◽  
Conformal Field ◽  
Fractal Index ◽  
Fractal Statistics ◽  
Universal Classes ◽  
Dilogarithm Function

We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν↔1/ν, for integer-value ν of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as [Formula: see text]. The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.

ON N = 4 INTEGRABLE MODELS

1994 ◽  
Vol 09 (06) ◽  
pp. 891-913 ◽  
Author(s):  
E. H. SAIDI ◽  
M. B. SEDRA
Keyword(s):  
Integrable Models ◽  
Conformal Algebra ◽  
Liouville Theory ◽  
Conformal Anomaly ◽  
Conserved Currents

Using the FS and HST versions of the free N = 4 matter multiplet (O4, (1/2)4), we construct two N = 4 SU(2) conformal superfield models. The corresponding N = 4 conserved currents are given. We find that no N = 4 SU(2) Liouville model exists as long as the SU(2) KM symmetry is manifestly preserved. However allowing an explicit breaking of the SU(2) KM subsymmetry of the N = 4 conformal algebra down to U(1) KM, we obtain a Feigin–Fuchs extension of the N = 4 supercurrent showing that N = 4 Liouville theory and its Toda generalizations could exist. Quantization and the N = 4 conformal anomaly are studied.

scholarly journals The non-rational limit of D-series minimal models

2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Sylvain Ribault
Keyword(s):  
Central Charge ◽  
Structure Constant ◽  
Liouville Theory ◽  
Two Dimensional ◽  
Minimal Models ◽  
Distribution Factor ◽  
Smooth Functions ◽  
Limit Theory ◽  
Conformal Field ◽  
Verlinde Formula

We study the limit of D-series minimal models when the central charge tends to a generic irrational value c\in (-\infty, 1)c∈(−∞,1). We find that the limit theory’s diagonal three-point structure constant differs from that of Liouville theory by a distribution factor, which is given by a divergent Verlinde formula. Nevertheless, correlation functions that involve both non-diagonal and diagonal fields are smooth functions of the diagonal fields’ conformal dimensions. The limit theory is a non-trivial example of a non-diagonal, non-rational, solved two-dimensional conformal field theory.

scholarly journals The large-c Virasoro identity block is a semi-classical Liouville correlator

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Gideon Vos
Keyword(s):  
Correlation Function ◽  
Liouville Equation ◽  
Classical Limit ◽  
Central Charge ◽  
Equation Of Motion ◽  
Einstein Gravity ◽  
Liouville Theory ◽  
Vertex Operators

Abstract It will be shown analytically that the light sector of the identity block of a mixed heavy-light correlator in the large central charge limit is given by a correlation function of light operators on an effective background geometry. This geometry is generated by the presence of the heavy operators. It is shown that this background geometry is a solution to the Liouville equation of motion sourced by corresponding heavy vertex operators and subsequently that the light sector of the identity block matches the Liouville correlation function in the semi-classical limit. This method effectively captures the spirit of Einstein gravity as a theory of dynamical geometry in AdS/CFT. The reason being that Liouville theory is closely related to semi-classical asymptotically AdS3 gravity.

VIRASORO ALGEBRA IN THE KN ALGEBRA: BOSONIC STRING WITH FERMIONIC GHOSTS ON RIEMANN SURFACES

1991 ◽  
Vol 06 (31) ◽  
pp. 2861-2870
Author(s):  
HIROSHI KOIBUCHI
Keyword(s):  
Clifford Algebra ◽  
Riemann Surfaces ◽  
Central Charge ◽  
Virasoro Algebra ◽  
String Model ◽  
Bosonic String ◽  
Combined System

The bosonic string model with fermionic ghosts is considered in the framework of the KN algebra. Our attentions are paid to representations of KN algebra and a Clifford algebra of the ghosts. We show that a Virasoro-like algebra is obtained from KN algebra when KN algebra has certain antilinear anti-involution, and that it is isomorphic to the usual Virasoro algebra. We show that there is an expected relation between a central charge of this Virasoro-like algebra and an anomaly of the combined system.

scholarly journals THERMODYNAMICS OF NONSPHERICAL BLACK HOLES FROM LIOUVILLE THEORY

2012 ◽  
Vol 27 (20) ◽  
pp. 1250111 ◽  
Author(s):  
FANG-FANG YUAN ◽  
YONG-CHANG HUANG
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Central Charge ◽  
Black Hole Entropy ◽  
Liouville Theory ◽  
Two Dimensional ◽  
Hamiltonian Approach ◽  
Black Rings ◽  
Dimensional Black Hole ◽  
Black Hole Temperature

A Liouville formalism was proposed many years ago to account for the black hole entropy. It was recently updated to connect thermodynamics of general black holes, in particular the Hawking temperature, to two-dimensional Liouville theory. This relies on the dimensional reduction to two-dimensional black hole metric. The relevant dilaton gravity model can be rewritten as a Liouville-like theory. We refine the method and give general formulas for the corresponding scalar and energy–momentum tensors in Liouville theory. This enables us to read off the black hole temperature using a relation which was found about three decades ago. Then the range of application is extended to include nonspherical black holes such as neutral and charged black rings, topological black hole and the case coupled to a scalar field. As for the entropy, following previous authors, we invoke the Lagrangian approach to central charge by Cadoni and then use the Cardy formula. The general relevant parameters are also given. This approach is more advantageous than the usual Hamiltonian approach which was used by the old Liouville formalism for black hole entropy.

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