Riemann-Hilbert treatment of Liouville theory on the torus

Pietro Menotti

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Riemann-Hilbert treatment of Liouville theory on the torus

Pietro Menotti

High Energy Physics - Theory · Submitted 24 Oct 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.51 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.11 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We apply a perturbative technique to study classical Liouville theory on the torus.	contribution 0.87	0.95	0.09	asserted 0.99
s1	After mapping the problem on the cut-plane we give the perturbative treatment for a weak source.	method 0.97	0.93	0.06	asserted 0.99
s2	When the torus reduces to the square the problem is exactly soluble by means of a quadratic transformation in terms of hypergeometric functions.	result 0.72	0.51	0.13	asserted 1.00
s3	We give general formulas for the deformation of a torus and apply them to the case of the deformation of the square.	contribution 0.93	0.94	0.11	asserted 1.00
s4	One can compute the Heun parameter to first order and express the solution in terms of quadratures.	result 0.59	0.75	0.13	hedged possibility 0.80
s5	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.	result 0.63	0.94	0.27	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

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.

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How the literature uses it

14 arXiv papers cite this in 48 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 34 sentences without a confident label

Builds on it 1 paper

In this paper we have extended the treatment of a previous paper [ref] to the general case in which the distribution of charges is not symmetric under reflections with respect to the origin.

Riemann-Hilbert treatment of Liouville theory on the torus: The general case · Conclusions · 1.00

Uses its method or tool 3 papers

The first [ref] is to exploit the symmetry for $z_t=0$ of the equation ([ref]) and of its solution under the inversion $z→-z$.

Accessory parameters for Liouville theory on the torus · The differential equation and the monodromy conditions · 0.85
The deformation technique developed in [ref] can be applied to both of these cases.

Riemann-Hilbert treatment of Liouville theory on the torus: The general case · Introduction · 0.92
For achieving rigorous lower bounds on such a radius of convergence, methods similar to those developed in [ref] for the convergence in the coupling strength should be applied.

On the monodromy problem for the four-punctured sphere · Introduction · 0.88

Cites it as background 9 papers

In the case of the $SU(2)$ theory with $N_f = 4$ the generalized prepotential is related to the logarithm of the conformal blocks of four Liouville operators on a sphere with the bare gauge coupling constant being associated with the cross-ratio of the punctures where the four operators are inserted [ref] .

Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_{f}=4$ flavours · Introduction and results · 0.84
Several conjectures and proposals [ref] , [ref] , [ref] , [ref] , [ref] for the computation of such accessory parameters have been put forward e.g. taking the semiclassical limit of the quantum correlation functions in conformal theories, the 4-point function on the sphere and the 1-point function on the torus .

Hyperbolic deformation of the strip-equation and the accessory parameters for the torus · Introduction · 0.96
For the previous studies of the toroidal conformal blocks in the framework of CFT see [ref] .

Holographic interpretation of 1-point toroidal block in the semiclassical limit · Introduction · 1.00

What it cites · 1 sentence without a confident label

Cites as background 7 papers

Recently some notable progress have been performed for the torus topology [ref] .

Recursion relations in CFT and N=2 SYM theory · Introduction · 0.86
Recently some notable progress have been performed for the torus topology [ref] .

Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks · Introduction · 0.86
Recently some notable progress have been performed for the torus topology [ref] .

Multipoint correlation functions in Liouville field theory and minimal Liouville gravity · Introduction · 0.86

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