Searcharxiv⌕ Search

arXiv subjects

Pietro Menotti

Publications and source records attributed to Pietro Menotti.

At least 19 recordsLinked to original sources
arXiv2025

Convergence of classical conformal blocks

We give a recursive method to compute the classical conformal blocks in Liouville field theory. The values of the expansion coefficients are given by an algebraic scheme which works to all orders. The algebraic expression of the intervening matrices are explicitly given. With regard to the problem of the convergence of the series we rigorously prove that it has a finite (non zero) convergence radius. We then comment on the relation of the conformal block problem with the Riemann-Hilbert problem.

hep-th↗
arXiv2023

Liouville field theory on genus 2 surfaces

Exploiting the generalization of the Weierstrass $\wp$ function to genus $2$ given by Komori, we give the exact connection of the related monodromy problem for genus $2$ and the classical weak $n$-point correlation functions. We also provide the Green function of an Helmholtz operator on genus $2$ surfaces and prove the real analyticity of the higher genus Green function. This gives at the non perturbative level, through the continuation method, the real analyticity of the conformal factor and of the accessory parameters in presence of sources.

hep-th↗
arXiv2022

The continuation method and the real analyticity of the accessory parameters: the parabolic case

We give the proof of the real analyticity of the accessory parameters in Liouville field theory as a function of the position of the sources in the case in which in addition to elliptic sources, parabolic sources are present. The method is a non trivial extension of the elliptic case as it requires in an intermediate step the introduction of a regulator. The treatment holds also in the case of the torus. A discussion is given of the extension to higher genus surfaces.

hep-th↗
arXiv2021

The continuation method and the real analyticity of the accessory parameters: the general elliptic case

We apply the Le Roy-Poincaré continuation method to prove the real analytic dependence of the accessory parameters on the position of the sources in Liouville theory in presence of any number of elliptic sources. The treatment is easily extended to the case of the torus with any number of elliptic singularities. A discussion is given of the extension of the method to parabolic singularities and higher genus surfaces.

hep-th↗
arXiv2020

Real analyticity of accessory parameters

We consider the problem of the real analytic dependence of the accessory parameters of Liouville theory on the moduli of the problem, for general elliptic singularities. We give a simplified proof of the almost everywhere real analyticity in the case of a single accessory parameter as it occurs e.g. in the sphere topology with four sources or for the torus topology with a single source by using only the general analyticity properties of the solution of the auxiliary equation. We deal then the case of two accessory parameters. We use the obtained result for a single accessory parameter to derive rigorous properties of the projection of the problem on lower dimensional planes. We derive the real analyticity result for two accessory parameters under an assumption of irreducibility.

hep-th↗
arXiv2018

Torus classical conformal blocks

After deriving the classical Ward identity for the variation of the action under a change of the modulus of the torus we map the problem of the sphere with four sources to the torus. We extend the method previously developed for computing the classical conformal blocks for the sphere topology to the tours topology. We give the explicit results for the classical blocks up to the third order in the nome included and compare them with the classical limit of the quantum conformal blocks. The extension to higher orders is straightforward.

hep-th↗
arXiv2016

On the solution of Liouville equation

We give a short and rigorous proof of the existence and uniqueness of the solution of Liouville equation with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.

math-ph↗
arXiv2016

Polyakov relation for the sphere and higher genus surfaces

The Polyakov relation, which in the sphere topology gives the changes of the Liouville action under the variation of the position of the sources, in the case of higher genus is related also to the dependence of the action on the moduli of the surface. We write and prove such a relation for genus 1 and for all hyperelliptic surfaces.

hep-th↗
arXiv2016

Classical conformal blocks

We give a simple iterative procedure to compute the classical conformal blocks on the sphere to all order in the modulus.

hep-th↗
arXiv2014

On the monodromy problem for the four-punctured sphere

We consider the monodromy problem for the four-punctured sphere in which the character of one composite monodromy is fixed, by looking at the expansion of the accessory parameter in the modulus $x$ directly, without taking the limit of the quantum conformal blocks for infinite central charge. The integrals which appear in the expansion of the Volterra equation, involve products of two hypergeometric functions to first order and up to four hypergeometric functions to second order. It is shown that all such integrals can be computed analytically. We give the complete analytical evaluation of the accessory parameter to first and second order in the modulus. The results agree with the evaluation obtained by assuming the exponentiation hypothesis of the quantum conformal blocks in the limit of infinite central charge. Extension to higher orders is discussed.

hep-th↗
arXiv2013

Hyperbolic deformation of the strip-equation and the accessory parameters for the torus

By applying an hyperbolic deformation to the uniformization problem for the infinite strip, we give a method for computing the accessory parameter for the torus with one source as an expansion in the modular parameter q. At O(q^0) we obtain the same equation for the accessory parameter and the same value of the semiclassical action as the one obtained from the b -> 0 limit of the quantum one point function. The procedure can be carried over to the full O(q^2) or even higher order corrections although the procedure becomes somewhat complicated. Here we compute to order q^2 the correction to the weight parameter intervening in the conformal factor and it is shown that the unwanted contribution O(q) to the accessory parameter equation cancel exactly.

hep-th↗
arXiv2013

Late-time expansion in the semiclassical theory of Hawking radiation

We give a detailed treatment of the back-reaction effects on the Hawking spectrum in the semiclassical approach to the Hawking radiation. We solve the exact system of non linear equations giving the action of the system, by a rigorously convergent iterative procedure. The first two terms of such an expansion give the O(omega/M) correction to the Hawking spectrum.

hep-th↗
arXiv2012

Accessory parameters for Liouville theory on the torus

We give an implicit equation for the accessory parameter on the torus which is the necessary and sufficient condition to obtain the monodromy of the conformal factor. It is shown that the perturbative series for the accessory parameter in the coupling constant converges in a finite disk and give a rigorous lower bound for the radius of convergence. We work out explicitly the perturbative result to second order in the coupling for the accessory parameter and to third order for the one-point function. Modular invariance is discussed and exploited. At the non perturbative level it is shown that the accessory parameter is a continuous function of the coupling in the whole physical region and that it is analytic except at most a finite number of points. We also prove that the accessory parameter as a function of the modulus of the torus is continuous and real-analytic except at most for a zero measure set. Three soluble cases in which the solution can be expressed in terms of hypergeometric functions are explicitly treated.

hep-th↗
arXiv2011

Late-time expansion in the semiclassical theory of the Hawking radiation

We give a detailed treatment of the back-reaction effects on the Hawking spectrum in the late-time expansion within the semiclassical approach to the Hawking radiation. We find that the boundary value problem defining the action of the modes which are regular at the horizon admits in general the presence of caustics. We show that for radii less that a certain critical value $r_c$ no caustic occurs for all values of the wave number and time and we give a rigorous lower bound on such a critical value. We solve the exact system of non linear equations defining the motion, by an iterative procedure rigorously convergent at late times. The first two terms of such an expansion give the $O(ω/M)$ correction to the Hawking spectrum.

hep-th↗
arXiv2011

Riemann-Hilbert treatment of Liouville theory on the torus: The general case

We extend the previous treatment of Liouville theory on the torus, to the general case in which the distribution of charges is not necessarily symmetric. This requires the concept of Fuchsian differential equation on Riemann surfaces. We show through a group theoretic argument that the Heun parameter and a weight constant are sufficient to satisfy all monodromy conditions. We then apply the technique of differential equation on a Riemann surface to the two point function on the torus in which one source is arbitrary and the other small. As a byproduct we give in terms of quadratures the exact Green function on the square and on the rhombus with opening angle 2 pi/6 in the background of the field generated by an arbitrary charge.

hep-th↗
arXiv2010

Riemann-Hilbert treatment of Liouville theory on the torus

We apply a perturbative technique to study classical Liouville theory on the torus. After mapping the problem on the cut-plane we give the perturbative treatment for a weak source. When the torus reduces to the square the problem is exactly soluble by means of a quadratic transformation in terms of hypergeometric functions. We give general formulas for the deformation of a torus and apply them to the case of the deformation of the square. One can compute the Heun parameter to first order and express the solution in terms of quadratures. In addition we give in terms of quadratures of hypergeometric functions the exact symmetric Green function on the square on the background generated by a one point source of arbitrary strength.

hep-th↗
arXiv2010

Consistent gravitational anomalies for chiral scalars

Starting from the Henneaux-Teitelboim action for a chiral scalar, which generalizes to curved space the Floreanini-Jackiw action, we give two simple derivations of the exact consistent gravitational anomaly. The first derivation is through the Schwinger-DeWitt regularization. The second exploits cohomological methods and uses the fact that in dimension two the diffeomorphism transformations are described by a single ghost which allows to climb the cohomological chain in a unique way.

hep-th↗