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.

q-Equivalent Hamiltonian Operators

1972 ◽  
Vol 50 (17) ◽  
pp. 2037-2047 ◽  
Author(s):  
M. Razavy
Keyword(s):  
Quantum Mechanics ◽  
Classical Limit ◽  
First Integral ◽  
Canonical Commutation Relation ◽  
Equation Of Motion ◽  
First Integrals ◽  
Position Operator ◽  
Integrals Of Motion ◽  
Adjoint Operators ◽  
Hamiltonian Operators

From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.

scholarly journals The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
V. N. Grebenev ◽  
A. N. Grishkov ◽  
M. Oberlack
Keyword(s):  
Correlation Function ◽  
Asymptotic Expansion ◽  
Lie Algebra ◽  
Degrees Of Freedom ◽  
Isotropic Turbulence ◽  
Central Charge ◽  
Riemannian Metrics ◽  
Equivalence Transformation ◽  
Velocity Correlation ◽  
Correlation Tensor

The extended symmetry of the functional of length determined in an affine spaceK3of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variablet) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metricsdl2(t)(Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation spaceK3and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic formdlD22(t)generated bydl2(t)which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric formcas for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame ofdlD22(t).

scholarly journals Correlation functions and vertex operators of Liouville theory

2004 ◽  
Vol 581 (1-2) ◽  
pp. 133-140 ◽  
Author(s):  
George Jorjadze ◽  
Gerhard Weigt
Keyword(s):  
Correlation Functions ◽  
Liouville Theory ◽  
Vertex Operators

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 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 A LECTURE ON THE LIOUVILLE VERTEX OPERATORS (REVIEW)

2004 ◽  
Vol 19 (supp02) ◽  
pp. 436-458 ◽  
Author(s):  
J. TESCHNER
Keyword(s):  
Free Field ◽  
Liouville Theory ◽  
Vertex Operators ◽  
Continuous Family ◽  
Matrix Elements ◽  
Crossing Symmetry ◽  
The Matrix

We reconsider the construction of exponential fields in the quantized Liouville theory. It is based on a free-field construction of a continuous family or chiral vertex operators. We derive the fusion and braid relations of the chiral vertex operators. This allows us to simplify the verification of locality and crossing symmetry of the exponential fields considerably. The calculation of the matrix elements of the exponential fields leads to a constructive derivation of the formula proposed by Dorn/Otto and the brothers Zamolodchikov.

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 BLACK HOLES, ENTROPY BOUND AND CAUSALITY VIOLATION

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3584-3591 ◽  
Author(s):  
I. P. NEUPANE
Keyword(s):  
Black Hole ◽  
Classical Limit ◽  
Gauge Theories ◽  
Black Hole Entropy ◽  
Einstein Gravity ◽  
Entropy Density ◽  
Causality Violation ◽  
High Energies ◽  
Gravity Duality ◽  
Black Hole Solutions

The gauge theory - gravity duality has provided us a way of studying QCD at high energies or short distances from straightforward calculations in classical general relativity. Among numerous results obtained so far, one of the most striking is the universality of the ratio of the shear viscosity to the entropy density. For all gauge theories with Einstein gravity dual this ratio has been found to be η/s = 1/4π. In this note, we consider higher curvature-corrected black hole solutions for which η/s can be smaller than 1/4π, thus violating the conjecture bound. Here we shall argue that the Gauss-Bonnet gravity and (Riemann)2 gravity theories, in particular, provide concrete examples in which inconsistency of a theory, such as a violation of microcausality at short distances, and a classical limit on black hole entropy are correlated.

SOME INTEGRAL FORMULAS FOR THE SOLUTIONS OF THE sl2 dKZ EQUATION WITH LEVEL −4

1994 ◽  
Vol 09 (32) ◽  
pp. 5673-5687 ◽  
Author(s):  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Correlation Function ◽  
Integral Formula ◽  
Time Integration ◽  
Direct Proof ◽  
Vertex Operators ◽  
Type I ◽  
Xxz Model ◽  
Integral Formulas ◽  
Xxx Model

A direct proof is given for the fact that the integral formula for the XXX limit of the trace of the type I q-vertex operators satisfies the deformed Knizhnik-Zamolodchikov (dKZ) equation with level −4. We have also carried out one-time integration by taking the residue at infinity. As a corollary of these we can construct a family of the integral formulas for solutions to the dKZ equation. Another corollary is the integral formula for the correlation function of the inhomogeneous XXX model, whose number of integrals is less than that of the previously obtained correlator for the XXZ model.

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