Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models

M. Matone

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Quantum Riemann surfaces, 2D gravity and the geometrical origin of minimal models

M. Matone

High Energy Physics - Theory · Submitted 17 Sep 1993

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

The authors' main claim Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.91 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.85 · compares with prior results 0.32 · mentions code or data availability 0.01.

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arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere $\Sigma_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $\Sigma_{0,m+n}$ implies a relation between conformal weights and ramification indices. This formulation works for arbitrary $d$ and admits a standard representation only for $d\le 1$. Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT.

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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	Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere $\Sigma_{0,m+n}$ with both parabolic and elliptic points.	contribution 1.00	0.85	0.13	asserted 0.85
s1	The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $\Sigma_{0,m+n}$ implies a relation between conformal weights and ramification indices.	result 0.50	0.61	0.14	asserted 0.99
s2	This formulation works for arbitrary $d$ and admits a standard representation only for $d\le 1$.	limitation 0.88	0.70	0.10	asserted 1.00
s3	Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT.	result 0.97	0.79	0.59	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.90 · evidence from proof or derivation 0.73.

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arXiv listing abstract v3

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S sections
T body sentences
C citations
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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

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere $\Sigma_{0,m+n}$ with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on $\Sigma_{0,m+n}$ implies a relation between conformal weights and ramification indices. This formulation works for arbitrary $d$ and admits a standard representation only for $d\le 1$. Furthermore, it turns out that the integerness of the ramification number constrains $d=1-24/(n^2-1)$ that for $n=2m+1$ coincides with the unitary minimal series of CFT.

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1 arXiv paper cite this in 2 sentences

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Cites it as background 1 paper

In the so-called geometric approach to quantum Liouville theory, developed by Takhtajan in [ref] based on the original proposal by Polyakov [ref] ,[See references [ref] for details regarding the relation between geometric and standard approaches to Liouville CFT.] the un-normalized correlation functions of Liouville vertex operators with “charges” $α_i$ on the Riemann surface $X$ are defined by (we set $ħ=1$) [equation] where $𝒞ℳ_α(X)$ is the space of all smooth conformal metrics $e^ψ(u,u̅) |du|^2$ on $X\{x_1,…,x_n}$ which have conical singularities of angles $2π(1- α_i)$ at the insertion points, and $_α[ψ(u,u̅)]$ is the Liouville action functional which we intend to study.

Classical Liouville Action and Uniformization of Orbifold Riemann Surfaces · Classical Liouville Action and Uniformization Theory · 0.84

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