RATIONAL HOPF ALGEBRAS: POLYNOMIAL EQUATIONS, GAUGE FIXING, AND LOW-DIMENSIONAL EXAMPLES

1995 ◽  
Vol 10 (24) ◽  
pp. 3431-3476 ◽  
Author(s):  
JÜRGEN FUCHS ◽  
ALEXANDER GANCHEV ◽  
PETER VECSERNYÉS
Keyword(s):  
Hopf Algebras ◽  
Field Theories ◽  
Polynomial Equations ◽  
Fusion Rules ◽  
Gauge Freedom ◽  
Rational Field ◽  
All Solutions ◽  
Gauge Fixing ◽  
Level 3 ◽  
Low Dimensional

Rational Hopf algebras, i.e. certain quasitriangular weak quasi-Hopf algebras whose representations form a tortile modular C* category, are expected to describe the quantum symmetry of rational field theories. In this paper the essential structure (hidden by a large gauge freedom) of rational Hopf algebras is revealed. This allows one to construct examples of rational Hopf algebras starting only from the corresponding fusion ring. In particular we classify all solutions for fusion rules with not more than three sectors, as well as for the level 3 affine [Formula: see text] fusion rules.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

scholarly journals ON TWO-DIMENSIONAL QUASITOPOLOGICAL FIELD THEORIES

2009 ◽  
Vol 24 (32) ◽  
pp. 6105-6121 ◽  
Author(s):  
P. TEOTONIO-SOBRINHO ◽  
C. MOLINA ◽  
N. YOKOMIZO
Keyword(s):  
Hopf Algebras ◽  
Gauge Theories ◽  
Local Symmetry ◽  
Two Dimensions ◽  
Field Theories ◽  
Partition Functions ◽  
Wilson Loops ◽  
Two Dimensional ◽  
Lattice Field ◽  
Expected Values

We study a class of lattice field theories in two dimensions that includes gauge theories. We show that in these theories it is possible to implement a broader notion of local symmetry, based on semisimple Hopf algebras. A character expansion is developed for the quasitopological field theories, and partition functions are calculated with this tool. Expected values of generalized Wilson loops are defined and studied with the character expansion.

scholarly journals TOWERS OF ALGEBRAS IN RATIONAL CONFORMAL FIELD THEORIES

1991 ◽  
Vol 06 (12) ◽  
pp. 2045-2074 ◽  
Author(s):  
CÉSAR GOMEZ ◽  
GERMAN SIERRA
Keyword(s):  
Sufficient Conditions ◽  
Field Theories ◽  
Partition Functions ◽  
Polynomial Equations ◽  
Conformal Field Theories ◽  
Modular Invariant ◽  
Conformal Field ◽  
Ade Classification

Jones fundamental construction is applied to rational conformal field theories. The Jones algebra which emerges in this application is realized in terms of duality operations. The generators of the algebra are an open version of Verlinde’s operators. The polynomial equations appear in this context as sufficient conditions for the existence of Jones algebra. The ADE classification of modular invariant partition functions is put in correspondence with Jones classification of subfactors.

scholarly journals Quantum groups and conformal field theories

1989 ◽  
Vol 329 (1605) ◽  
pp. 343-347
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Conformal Theory ◽  
Conformal Field

Rational conformal field theories can be interpreted as defining quasi-triangular Hopf algebras. The Hopf algebra is determined by the duality properties of the conformal theory.

scholarly journals On Fusion Rules in Logarithmic Conformal Field Theories

1997 ◽  
Vol 12 (10) ◽  
pp. 1943-1958 ◽  
Author(s):  
Michael A. I. Flohr
Keyword(s):  
Quantum Group ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Minimal Models ◽  
Group Representations ◽  
Fusion Rules ◽  
Conformal Field ◽  
Almost All

We find the fusion rules for the cp,1 series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c3p,3. This leads to the conjecture that (almost) all minimal models with c = cp,q, gcd (p,q) > 1, belong to the class of rational logarithmic conformal field theories.

scholarly journals GEOMETRICALLY RELATING MOMENTUM CUT-OFF AND DIMENSIONAL REGULARIZATION

2012 ◽  
Vol 10 (02) ◽  
pp. 1250081 ◽  
Author(s):  
SUSAMA AGARWALA
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Hopf Algebras ◽  
Dimensional Regularization ◽  
Mass Scale ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Coupling Constant ◽  
Regularization Scheme ◽  
Β Function

The β function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the β functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.

scholarly journals The study on general cubic equations over p-adic fields

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1115-1131
Author(s):  
Mansoor Saburov ◽  
Mohd Ahmad ◽  
Murat Alp
Keyword(s):  
Polynomial Equation ◽  
Rational Numbers ◽  
Lower Degree ◽  
System Of Equations ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Diophantine Problem ◽  
Cubic Equation ◽  
All Solutions ◽  
General Number

A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field Qp belongs to domains Z*p, Zp \ Z*p, Qp \ Zp, Qp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.

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