scholarly journals QUANTUM RIEMANN SURFACES, 2-D GRAVITY AND THE GEOMETRICAL ORIGIN OF MINIMAL MODELS

1994 ◽  
Vol 09 (31) ◽  
pp. 2871-2878 ◽  
Author(s):  
MARCO MATONE
Keyword(s):  
Quantum Gravity ◽  
Classical Limit ◽  
Riemann Surfaces ◽  
Ward Identity ◽  
Riemann Sphere ◽  
Projective Connection ◽  
Minimal Models ◽  
Standard Representation ◽  
Basic Object ◽  
Liouville Action

Based on a recent paper by Takhtajan, we propose a formulation of 2-D quantum gravity whose basic object is the Liouville action on the Riemann sphere Σ0, 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 Σ0, 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 ≤ 1. Furthermore, it turns out that the integerness of the ramification number constrains d = 1 − 24/(n2 − 1) that for n = 2m + 1 coincides with the unitary minimal series of CFT.

scholarly journals Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

2021 ◽  
Vol 24 (3) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Ulrike Bücking
Keyword(s):  
Error Estimate ◽  
Branch Point ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Convergence Result ◽  
Edge Length ◽  
Ramified Covering ◽  
Compact Riemann Surfaces ◽  
Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

Tadpole operator for minimal models on arbitrary genus Riemann surfaces

1990 ◽  
Vol 237 (3-4) ◽  
pp. 379-385 ◽  
Author(s):  
G. Cristofano ◽  
G. Maiella ◽  
R. Musto ◽  
F. Nicodemi
Keyword(s):  
Riemann Surfaces ◽  
Minimal Models ◽  
Arbitrary Genus

A COMPACT BOSON COUPLED TO TWO-DIMENSIONAL GRAVITY

1991 ◽  
Vol 06 (15) ◽  
pp. 2743-2754 ◽  
Author(s):  
NORISUKE SAKAI ◽  
YOSHIAKI TANII
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Riemann Surfaces ◽  
Partition Functions ◽  
Two Dimensional ◽  
Dimensional Gravity ◽  
The Continuum ◽  
Continuum Field

The radius dependence of partition functions is explicitly evaluated in the continuum field theory of a compactified boson, interacting with two-dimensional quantum gravity (noncritical string) on Riemann surfaces for the first few genera. The partition function for the torus is found to be a sum of terms proportional to R and 1/R. This is in agreement with the result of a discretized version (matrix models), but is quite different from the critical string. The supersymmetric case is also explicitly evaluated.

scholarly journals Cosmological constraints on a classical limit of quantum gravity

2005 ◽  
Vol 72 (4) ◽  
Author(s):  
Damien A. Easson ◽  
Frederic P. Schuller ◽  
Mark Trodden ◽  
Mattias N. R. Wohlfarth
Keyword(s):  
Quantum Gravity ◽  
Classical Limit ◽  
Cosmological Constraints

scholarly journals COSET REALIZATION OF UNIFYING ${\mathcal W}$ ALGEBRAS

1995 ◽  
Vol 10 (16) ◽  
pp. 2367-2430 ◽  
Author(s):  
R. BLUMENHAGEN ◽  
W. EHOLZER ◽  
A. HONECKER ◽  
R. HÜBEL ◽  
K. HORNFECK
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Classical Case ◽  
Quantum Level ◽  
Minimal Models ◽  
Classical Formula

We construct several quantum coset [Formula: see text] algebras, e.g. [Formula: see text] and [Formula: see text] and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying [Formula: see text] algebras of Casimir [Formula: see text] algebras. We show that it is possible to give coset realizations of various types of unifying [Formula: see text] algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp (2n) realize the unifying [Formula: see text] algebras which have previously been introduced as [Formula: see text]. In addition, minimal models of [Formula: see text] are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum [Formula: see text] algebras, we discuss orbifolding of [Formula: see text] algebras which on the quantum level has different properties than in the classical case. We demonstrate through some examples that the classical limit — according to Bowcock and Watts — of these finitely nonfreely generated quantum [Formula: see text] algebras probably yields infinitely nonfreely generated classical [Formula: see text] algebras.

scholarly journals Some results on Teichmüller spaces of Klein surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 65-76
Author(s):  
Pablo Arés Gastesi
Keyword(s):  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Topological Type ◽  
Uniformization Theorem ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces ◽  
Simply Connected Domain ◽  
Nonorientable Surfaces ◽  
Simply Connected ◽  
Orientable Surfaces

The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

SPURIONS IN FUNCTIONAL AND OPERATOR QUANTIZATION OF STRING THEORY AND HARMONIC GAUGE

1991 ◽  
Vol 06 (12) ◽  
pp. 1061-1068
Author(s):  
A.P. DEMICHEV ◽  
M.Z. IOFA
Keyword(s):  
String Theory ◽  
Path Integral ◽  
Riemann Surfaces ◽  
Ward Identity ◽  
Brst Quantization ◽  
Higher Genus ◽  
The Difference ◽  
Lagrange Approach

We discuss the difference between the Lagrange and the operator BRST quantization in string theory on Riemann surfaces of higher genus. An example of the harmonic gauge yielding the non-anomalous BRST Ward identity in the path integral Lagrange approach is studied in detail.

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