BOSONIZATION AND FERMION VERTICES ON AN ARBITRARY GENUS RIEMANN SURFACE BY USING A GLOBAL OPERATOR FORMALISM

1989 ◽  
Vol 04 (24) ◽  
pp. 2349-2362 ◽  
Author(s):  
JORGE RUSSO
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Explicit Computation ◽  
Operator Formalism ◽  
Central Extensions ◽  
Arbitrary Topology ◽  
Global Operator ◽  
Arbitrary Genus ◽  
Higher Genus ◽  
Spin Fields

Fermi-Bose equivalence is studied with the use of a global operator formalism on Riemann surfaces of arbitrary topology. The quantization of a scalar field on a circle is performed in detail, globally, at arbitrary genus. A new algebra of the Krichever-Novikov type naturally emerges. This admits three central extensions and generalizes standard algebras of the sphere to higher genus. It is shown by explicit computation that the central terms, as well as correlation functions, corresponding to the Bose and Fermi models agree. Spin fields and fermion vertices are defined within this framework and their conformal properties are investigated.

A NOTE ON KRICHEVER-NOVIKOV-KAC-MOODY ALGEBRA ON RIEMANN SURFACES

1990 ◽  
Vol 05 (27) ◽  
pp. 2215-2221 ◽  
Author(s):  
SHUICHI OJIMA
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Riemann Surfaces ◽  
Primary Field ◽  
Affine Algebra ◽  
Moody Algebra ◽  
Operator Formalism ◽  
Partial Differential ◽  
Global Operator ◽  
Higher Genus

On higher genus Riemann surfaces we define a primary field for affine algebra and vacuum with the use of the prescription by global operator formalism proposed by Krichever and Novikov. From the analytic viewpoint, the Ward-Takahashi identity for current insertion and a partial differential equation for correlators of primary fields are derived.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals SU(2)k×SU(2)l/SU(2)k+l COSET CONFORMAL FIELD THEORY AND TOPOLOGICAL MINIMAL MODEL ON HIGHER GENUS RIEMANN SURFACE

1993 ◽  
Vol 08 (31) ◽  
pp. 5537-5561 ◽  
Author(s):  
HITOSHI KONNO
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Riemann Surface ◽  
Vertex Operator ◽  
Correlation Functions ◽  
Minimal Model ◽  
The Other ◽  
Operator Formalism ◽  
Conformal Field ◽  
Higher Genus

We consider the Feigin-Fuchs-Felder formalism of the SU (2)k× SU (2)l/ SU (2)k+l coset minimal conformal field theory and extend it to higher genus. We investigate a double BRST complex with respect to two compatible BRST charges, one associated with the parafermion sector and the other associated with the minimal sector in the theory. The usual screened vertex operator is extended to the BRST-invariant screened three-string vertex. We carry out a sewing operation of these vertices and derive the BRST-invariant screened g-loop operator. The latter operator characterizes the higher genus structure of the theory. An analogous operator formalism for the topological minimal model is obtained as the limit l=0 of the coset theory. We give some calculations of correlation functions on higher genus.

scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

OPERATOR FORMALISM FOR SELF-DUAL LATTICE COMPACTIFICATION ON RIEMANN SURFACES

1988 ◽  
Vol 03 (14) ◽  
pp. 1401-1409
Author(s):  
I.G. KOH ◽  
H.J. SHIN
Keyword(s):  
String Theory ◽  
Generating Function ◽  
Generating Functions ◽  
Riemann Surfaces ◽  
Transformation Property ◽  
Dual Lattice ◽  
Operator Formalism ◽  
Heterotic String Theory ◽  
Grassmannian Manifold ◽  
Higher Genus

We apply the universal Grassmannian manifold approach to the heterotic string theory compactified on self-dual lattice. The generating functions on higher genus Riemann surfaces are explicitly constructed starting from that of non-chiral bosonic theory and adding the contributions of instanton sectors. The modular transformation property and its equivalent fermionic formulation of the generating function are also discussed.

AN EXPLICIT COMPUTATION FOR THE BOSE-FERMI EQUIVALENCE ON RIEMANN SURFACES OF GENUS g

1989 ◽  
Vol 04 (17) ◽  
pp. 4437-4447
Author(s):  
NOUREDDINE CHAIR
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Riemann Surface ◽  
Partition Function ◽  
Quadratic Form ◽  
Riemann Surfaces ◽  
Partition Functions ◽  
Explicit Computation ◽  
Dimensional Torus ◽  
Conformal Field

The instanton sum in the partition function for D bosons on a Riemann surface of genus g, with values in a general D-dimensional torus, TD = RD/ΛD is given explicitly. When the rational metric Q of the lattice, ΛD, is the identity we get the bosonization formula of Alvarez-Gaumé et al. for SO( 2D ). If Q is orthogonal, in the bosonization formula, we get the theta function associated with the quadratic form Q, if Q is generic we get rational Conformal Field Theory. Also we look for conditions on a twisted spin bundle LE, which may ensure that our partition functions arise from some generalized bosonization formulas.

HIGHER GENUS WAVEFUNCTION OF ANYON SYSTEM

1991 ◽  
Vol 06 (02) ◽  
pp. 163-170
Author(s):  
JNANADEVA MAHARANA ◽  
SUDHAKAR PANDA
Keyword(s):  
Correlation Function ◽  
Riemann Surfaces ◽  
Scalar Fields ◽  
Vertex Operators ◽  
Arbitrary Genus ◽  
Higher Genus ◽  
Free Scalar

A method is proposed for constructing the wavefunction of anyons on Riemann surfaces of arbitrary genus. This has been carried out in the framework of computing the correlation function of the chiral vertex operators, involving free scalar fields, with appropriate external momenta. Our technique, as a check, reproduces the already known anyonic wavefunction on the sphere as well as that on the torus.

KRICHEVER — NOVIKOV BASES FOR THE BOSONIC STRING

1989 ◽  
Vol 04 (17) ◽  
pp. 4469-4474
Author(s):  
TAEJIN LEE ◽  
K. S. VISWANATHAN
Keyword(s):  
Riemann Surfaces ◽  
Spectral Representation ◽  
Bosonic String ◽  
Scalar Boson ◽  
Tensor Fields ◽  
Operator Formalism ◽  
Simple Manner ◽  
Arbitrary Genus ◽  
Boson Fields ◽  
Wick’S Theorem

The global operator formalism proposed recently by Krichever and Novikov using meromorphic tensor fields on general Riemann surfaces of arbitrary genus is applied to the bosonic string. The well-known Green's function of the scalar boson fields is obtained in a simple manner from the spectral representation. Proof of Wick's theorem is also given.

NONSTANDARD CONFORMAL FIELD THEORIES FROM THE b-c SYSTEMS ON GENERAL ALGEBRAIC CURVES

1996 ◽  
Vol 11 (12) ◽  
pp. 2213-2229 ◽  
Author(s):  
F. FERRARI ◽  
J. SOBCZYK
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Operator Formalism ◽  
Normal Ordering ◽  
Conformal Field ◽  
Complex Sphere ◽  
Free Scalar

In this paper we develop an operator formalism for the b–c systems with conformalweight λ=1 defined on a general closed and orientable Riemann surface. The advantageof our approach is that the Riemann surface is represented as an affine algebraic curve.In this way it is possible to show that the b–c systems at higher genera are equivalentto nonstandard conformal field theories on the complex sphere. The amplitudes of theseconformal field theories, rigorously computed using simple normal ordering prescriptions,are single-valued on the algebraic curve and coincide with the correlation functions of theoriginal b–c systems. Besides the obvious applications in string theories and conformalfield theories, (the b–c systems at λ=1 are intimately related to the free scalar fieldtheory), the operator formalism presented here also sheds some light on the quantizationof field theories on Riemann surfaces.

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