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 GENUS-ZERO MODULAR FUNCTORS AND INTERTWINING OPERATOR ALGEBRAS

1998 ◽  
Vol 09 (07) ◽  
pp. 845-863 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Central Charge ◽  
Vertex Operator Algebras ◽  
Main Step ◽  
Group Representations ◽  
Genus Zero ◽  
Intertwining Operator

In [7] and [9], the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras was solved implicitly earlier in [5]. In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [3, 4, 11, 12, 8] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge [Formula: see text] is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying a certain generalized meromorphicity property. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [7]. One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and, in particular, from representations of suitable vertex operator algebras.

Complex powers of eta-function

2018 ◽  
Vol 14 (06) ◽  
pp. 1619-1625
Author(s):  
Alexander Zuevsky
Keyword(s):  
Riemann Surface ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
The Self ◽  
Genus Two ◽  
Complex Powers

We derive a formula for complex powers of the [Formula: see text]-function using the identities for a vertex operator algebra correlation functions in terms of [Formula: see text]-functions obtained in the self-sewing procedure of the torus to form a genus two Riemann surface.

scholarly journals Foliation of a space associated to vertex operator algebra

2021 ◽  
Author(s):  
A. Zuevsky
Keyword(s):  
Riemann Surface ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Riemann Surfaces ◽  
Vertex Operator Algebra ◽  
Operator Algebras ◽  
Vertex Operator Algebras

In this paper, we construct the foliation of a space associated to correlation functions of vertex operator algebras, considered on Riemann surfaces. We prove that the computation of general genus g correlation functions determines a foliation on the space associated to these correlation functions a sewn Riemann surface. Certain further applications of the definition are proposed.

scholarly journals THE MOONSHINE MODULE FOR CONWAY’S GROUP

2015 ◽  
Vol 3 ◽  
Author(s):  
JOHN F. R. DUNCAN ◽  
SANDER MACK-CRANE
Keyword(s):  
Automorphism Group ◽  
Fixed Points ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Fourier Expansion ◽  
Vertex Operator Algebra ◽  
Natural Analogue ◽  
Genus Zero ◽  
Leech Lattice ◽  
Special Case

We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

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 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.

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