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On the Three-point Function in Minimal Liouville Gravity

Al. Zamolodchikov

TL;DR

The paper develops a complete framework to compute three-point functions in minimal Liouville gravity by combining Liouville theory with generalized minimal models. It derives the Liouville three-point function via bootstrap, formulates the GMM three-point function through continuous α-parameters and bootstrap equations, and then constructs minimal gravity correlators as a factorized product of Liouville and GMM pieces with explicit leg-factors and normalization against the Liouville partition function. It also analyzes analytic properties and the relation between Liouville and GMM under analytic continuation, and validates the results against matrix-model predictions for both the scaling Lee–Yang case and Φ_{1,3}-perturbed MG, including loop-gas interpretations. The outcome is explicit, normalized expressions for MG two- and three-point numbers and a clear link between continuum CFT methods and discrete matrix-model results in 2D quantum gravity.

Abstract

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity.

On the Three-point Function in Minimal Liouville Gravity

TL;DR

The paper develops a complete framework to compute three-point functions in minimal Liouville gravity by combining Liouville theory with generalized minimal models. It derives the Liouville three-point function via bootstrap, formulates the GMM three-point function through continuous α-parameters and bootstrap equations, and then constructs minimal gravity correlators as a factorized product of Liouville and GMM pieces with explicit leg-factors and normalization against the Liouville partition function. It also analyzes analytic properties and the relation between Liouville and GMM under analytic continuation, and validates the results against matrix-model predictions for both the scaling Lee–Yang case and Φ_{1,3}-perturbed MG, including loop-gas interpretations. The outcome is explicit, normalized expressions for MG two- and three-point numbers and a clear link between continuum CFT methods and discrete matrix-model results in 2D quantum gravity.

Abstract

The problem of the structure constants of the operator product expansions in the minimal models of conformal field theory is revisited. We rederive these previously known constants and present them in the form particularly useful in the Liouville gravity applications. Analytic relation between our expression and the structure constant in Liouville field theory is discussed. Finally we present in general form the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity.

Paper Structure

This paper contains 12 sections, 153 equations, 5 figures.

Table of Contents

  1. Liouville gravity
  2. Liouville three-point function
  3. Generalized minimal models
  4. Conformal bootstrap in GMM
  5. GMM three-point function
  6. Analytic continuation in $b^{2}$
  7. Minimal gravity
  8. Two-point function and normalization
  9. Matrix models
  10. Functional relations
  11. Double gamma and $\Upsilon$
  12. "Exponential" normalization in GMM

Figures (5)

  • Figure 1: Analytic properties of $\Upsilon_{b}(x)$ as a function of $b^{2}$
  • Figure 2: Analytic continuation of $\Upsilon_{b}$ (with $\operatorname*{Im}b^{2}<0$) to $\Upsilon_{\beta}$ with $\beta=ib$
  • Figure 3: Poles of the double gamma function $\Gamma_{2}(x|\omega_{1},\omega_{2})$.
  • Figure 4: Position of poles of the function $\Upsilon(x|\omega_{1},\omega_{2})$.
  • Figure 5: Zeros of the product $\Upsilon(x|\omega_{1},\omega_{2})\Upsilon (x-\omega_{1}|e^{i\pi}\omega_{1},\omega_{2})$ in the $x$-plane. Open circles are these of the first multiplier and filled ones are those of the second. Together they form the regular lattice of zeros of the theta function. Arrows show the "periods" $-\omega_{1}$ and $\omega_{2}$ of $\Upsilon(x-\omega _{1}|e^{i\pi}\omega_{1},\omega_{2})$.