scholarly journals Logarithmic intertwining operators and genus-one correlation functions

2016 ◽  
Vol 18 (06) ◽  
pp. 1650026 ◽  
Author(s):  
Francesco Fiordalisi
Keyword(s):  
Differential Equations ◽  
Correlation Functions ◽  
Modular Group ◽  
Vertex Operator Algebra ◽  
Formal Series ◽  
Positive Energy ◽  
Intertwining Operators ◽  
System Of Differential Equations ◽  
Logarithmic Intertwining Operators ◽  
Genus One

This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and [Formula: see text]-cofinite vertex operator algebra [Formula: see text]. We consider grading-restricted generalized [Formula: see text]-modules which admit a right action of some associative algebra [Formula: see text], and intertwining operators among such modules which commute with the action of [Formula: see text] ([Formula: see text]-intertwining operators). We obtain duality properties, i.e. suitable associativity and commutativity properties, for [Formula: see text]-intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal [Formula: see text]-traces of products of [Formula: see text]-intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal [Formula: see text]-traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group.

scholarly journals DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

2005 ◽  
Vol 07 (05) ◽  
pp. 649-706 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Differential Equations ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Modular Invariance ◽  
Genus Zero ◽  
Modular Invariant ◽  
The One ◽  
Genus One

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

scholarly journals VIRASORO CORRELATION FUNCTIONS FOR VERTEX OPERATOR ALGEBRAS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250106 ◽  
Author(s):  
DONNY HURLEY ◽  
MICHAEL P. TUITE
Keyword(s):  
Graph Theory ◽  
Vertex Operator ◽  
Generating Functions ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Genus Zero ◽  
Genus One

We consider all genus zero and genus one correlation functions for the Virasoro vacuum descendants of a vertex operator algebra. These are described in terms of explicit generating functions that can be combinatorially expressed in terms of graph theory related to derangements in the genus zero case and to partial permutations in the genus one case.

scholarly journals DIFFERENTIAL EQUATIONS AND INTERTWINING OPERATORS

2005 ◽  
Vol 07 (03) ◽  
pp. 375-400 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Differential Equations ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Tensor Category ◽  
Extension Property ◽  
Singular Points ◽  
Intertwining Operators ◽  
Natural Structure ◽  
Systems Of Differential Equations

We show that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W)<∞, where C1(W) is the subspace of W spanned by elements of the form u-1w for u ∈ V+ = ∐n>0 V(n) and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.

scholarly journals KNIZHNIK–ZAMOLODCHIKOV-TYPE EQUATIONS FOR GAUGED WZNW MODELS

1998 ◽  
Vol 13 (08) ◽  
pp. 1345-1367 ◽  
Author(s):  
IAN I. KOGAN ◽  
ALEX LEWIS ◽  
OLEG A. SOLOVIEV
Keyword(s):  
Black Hole ◽  
Differential Equations ◽  
Correlation Functions ◽  
Striking Similarity ◽  
Minimal Models ◽  
System Of Differential Equations ◽  
The Given ◽  
Zamolodchikov Equation

We study correlation functions of coset constructions by utilizing the method of gauge dressing. The given method results in a system of differential equations which are generalizations of the Knizhnik–Zamolodchikov equation. As an example, we apply this method to the minimal models and to the Witten 2D black hole. We exhibit a striking similarity between the latter and the gravitational dressing. In particular, we look for logarithmic operators in the 2D black hole.

scholarly journals Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

2017 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Category Theory ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebra ◽  
Intertwining Operators ◽  
Matrix Elements ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

scholarly journals Outflowing Envelopes From Stars at Arbitrary Optical Depths a New Approach to the Problem

1998 ◽  
Vol 11 (1) ◽  
pp. 381-381
Author(s):  
A.V. Dorodnitsyn
Keyword(s):  
Differential Equations ◽  
Massive Star ◽  
System Of Differential Equations ◽  
Continuous Transition ◽  
Satisfactory Description ◽  
Sonic Point ◽  
New Approach ◽  
Star Evolution ◽  
Matter Flux ◽  
Massive Star Evolution

We have considered a stationary outflowing envelope accelerated by the radiative force in arbitrary optical depth case. Introduced approximations provide satisfactory description of the behavior of the matter flux with partially separated radiation at arbitrary optical depths. The obtained systemof differential equations provides a continuous transition of the solution between optically thin and optically thick regions. We analytically derivedapproximate representation of the solution at the vicinity of the sonic point. Using this representation we numerically integrate the system of equations from the critical point to the infinity. Matching the boundary conditions we obtain solutions describing the problem system of differential equations. The theoretical approach advanced in this work could be useful for self-consistent simulations of massive star evolution with mass loss.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

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