Generalized Twisted Modules Associated to General Automorphisms of a Vertex Operator Algebra

We introduce a notion of a strongly ${\mathbb{C}^{\times}}We introduce a notion of a strongly \mathbbC^×{\mathbb{C}^{\times}}-graded, or equivalently, \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{Z}}-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of a strongly \mathbbC{\mathbb{C}}-graded generalized g-twisted V-module if V admits an additional \mathbbC{\mathbb{C}}-grading compatible with g. Let V=\coprod_{n ? \mathbbZ}V_(n){V=\coprod_{n\in \mathbb{Z}}V_{(n)}} be a vertex operator algebra such that V₍₀₎=\mathbbC1{V_{(0)}=\mathbb{C}\mathbf{1}} and V _(n) = 0 for n < 0 and let u be an element of V of weight 1 such that L(1)u = 0. Then the exponential of 2p?{-1}  Res_x Y(u, x){2\pi \sqrt{-1}\; {\rm Res}_{x} Y(u, x)} is an automorphism g _u of V. In this case, a strongly \mathbbC{\mathbb{C}}-graded generalized g _u -twisted V-module is constructed from a strongly \mathbbC{\mathbb{C}}-graded generalized V-module with a compatible action of g _u by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{Z}} or \mathbbC{\mathbb{C}} and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.     

Modular-Invariance of Trace Functions¶in Orbifold Theory and Generalized Moonshine

The goal of the present paper is to provide a mathematically rigorous foundation to certain aspects of the theory of rational orbifold models in conformal field theory, in other words the theory of rational vertex operator algebras and their automorphisms. Under a certain finiteness condition on a rational vertex operator algebra V which holds in all known examples, we determine the precise number of g-twisted sectors for any automorphism g of V of finite order. We prove that the trace functions and correlation functions associated with such twisted sectors are holomorphic functions in the upper half-plane and, under suitable conditions, afford a representation of the modular group of the type prescribed in string theory. We establish the rationality of conformal weights and central charge. In addition to conformal field theory itself, where our conclusions are required on physical grounds, there are applications to the generalized Moonshine conjectures of Conway–Norton–Queen and to equivariant elliptic cohomology. Received: 7 January 1999 / Accepted: 14 March 2000     

Vertex Operator Algebras Associated to Admissible Representations of

The Kac-Wakimoto admissible modules for are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l,0) associated to irreducible highest weight modules at admissible level is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l,0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions. Received: 4 April 1996/Accepted: 1 August 1996     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Representations of Vertex Operator Algebra V L + for Rank One Lattice L

We classify the irreducible modules for the fixed point vertex operator subalgebra V_L⁺ of the vertex operator algebra V_L associated to a positive definite even lattice of rank 1 under the automorphism lifted from the у isometry of L.     

$${\phi}$$ -Coordinated Quasi-Modules for Quantum Vertex Algebras

We develop a theory of f{\phi} -coordinated (quasi-) modules for a general nonlocal vertex algebra where f{\phi} is what we call an associate of the one-dimensional additive formal group. By specializing f{\phi} to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of Li (Selecta Mathematica (New Series) 11:349–397, 2005). As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and f{\phi} -coordinated modules to a certain quantum βγ-system explicitly.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Modularity in Orbifold Theory for Vertex Operator Superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C₂-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C₂-cofinite and g-rational for any g ∈ G.Supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz     

A Functional-Analytic Theory¶of Vertex (Operator) Algebras, I

This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary ℤ-graded finitely-generated vertex algebra (V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from $V⊗V to (the algebraic completion of V) realizing linearly the conformal equivalence class of a genus-zero Riemann surface with analytically parametrized boundary obtained by deleting two ordered disjoint disks from the unit disk and by giving the obvious parametrizations to the boundary components. We extend such a linear map to a linear map from $H\tilde{\otimes} H$ ( being the completed tensor product) to H, and prove the continuity of the extension. For any finitely-generated ℂ-graded V-module (W, Y _W ) satisfying the standard grading-restriction axioms, the same method also gives a topological completion H _W of W and gives the continuous extensions from to H _W of the linear maps from to realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries. Received: 15 August 1998 / Accepted: 13 January 1999     

On the classification of simple vertex operator algebras

Inspired by a recent work of Frenkel-Zhu, we study a class of (pre-)vertex operator algebras (voa) associated to the self-dual Lie algebras. Based on a few elementary structural results we propose thatV, the category of Z₊-graded prevoasV in whichV[0] is one-dimensional, is a proper setting in which to study and classify simple objects. The categoryV is organized into what we call the minimalk ^th types. We introduce a functor —which we call the Frenkel-Lepowsky-Meurman functor—that attaches to each object inV a Lie algebra. This is a key idea which leads us to a (relative) classification of thesimple minimal first type. We then study the set of all Virasoro structures on a fixed minimal first typeV, and show that they are in turn classified by the orbits of the automorphism group Aut((V)) in cent((V)). Many new examples of voas are given. Finally, we introduce a generalized Kac-Casimir operator and give a simple proof of the irreducibility of the prolongation modules over the affine Lie algebras.     

High-spin states of<Superscript>125</Superscript>Sb: Particle-core excitation coupling

Excited states of¹²⁵Sb have been studied using in-beam γ spectroscopy techniques via the¹²⁴Sn(⁷Li, α2n) reaction at a beam energy of 32 MeV. A high-spin level scheme including 21 new γ-transitions and 14 new excited states have been established. Three isomers have been identified at 1970, 2110 and 2471 keV and the ranges of their half-lives have been estimated from the delayed coincidence data. The level structure of¹²⁵Sb is discussed in terms of particle-core excitation coupling. With the help of empirical shell model calculations the three isomers are proposed to have three-quasiparticle πg_7/2⊗v(h _11/2 s _1/2)₅₋, πg_7/2⊗v(h _11/2 d _3/2)₇₋ and πg_7/2⊗v(h ₁₁^2/2)₁₀ ⁺ configurations, respectively.     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

High-spin states of rotational bands built upon νi13/2 in 166Lu

High-Spin states of odd-odd ¹⁶⁶Lu were populated using the ¹³⁹La(³⁰Si,3nγ)¹⁶⁶Lu at a beam energy of 120 MeV. Twelve new γ-rays were placed on top of the previously known two rotational bands built upon πg _7/2⊗νi _13/2 and πh _11/2⊗νi _13/2. Extending high-spin states up to 21⁺ and 25⁻ for each band, we have observed the onset of band crossing near ħω _c ≈ 0.35 MeV. The band crossing frequency of the yrast πh _11/2⊗νi _13/2 band is consistent with the neutron BC band crossing observed in lighter odd-odd Lu isotopes.     

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Rotational structures in 123Cs

High-spin states in ¹²³Cs, populated via the ¹⁰⁰Mo ( ²⁸Si, p4n) fusion-evaporation reaction at E _lab = 130 MeV, have been investigated employing in-beam γ-ray spectroscopic techniques. Rotational bands, built on πg _7/2, πg _9/2 and the unique-parity πh _11/2 orbitals, have been extended and evolve into bands involving rotationally aligned ν(h _11/2)² and π(h _11/2)² quasiparticles. A three-quasiparticle band based on the high-K πh _11/2 ⊗ νg _7/2 ⊗ νh _11/2 configuration has also been observed. Total Routhian Surface (TRS) calculations have been used to predict the nuclear shape parameters ( β₂, β₄, γ) for the various assigned configurations. The assigned configurations have been discussed in the framework of a microscopic theory based on the deformed Hartree-Fock (HF) and angular-momentum projection techniques.     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

Rotational structures in the 125Cs nucleus

The collective band structures of the ¹²⁵Cs nucleus have been investigated by in-beam γ-ray spectroscopic techniques following the ¹¹⁰Pd ( ¹⁹F, 4n) reaction at 75MeV. The previously known level scheme, with rotational bands built on πg_7/2, πg_9/2 and πh_11/2 orbitals, has been extended and evolves into bands involving rotationally aligned ν(h_11/2)² and π(h_11/2)² quasiparticles. A strongly coupled band has been reassigned a high-K πh_11/2 ⊗ νg_7/2 ⊗ νh_11/2 three-quasiparticle configuration and a new side band likely to be its chiral partner has been identified. Configurations assigned to various bands are discussed in the framework of Principal/Tilted Axis Cranking (PAC/TAC) model calculations.     

A Functional-Analytic Theory of Vertex (Operator) Algebras,II

For a finitely-generated vertex operator algebra V of central charge c, a locally convex topological completion H ^V is constructed. We construct on H ^V a structure of an algebra over the operad of the power Det ^c/2 of the determinant line bundle Det over the moduli space of genus-zero Riemann surfaces with ordered analytically parametrized boundary components. In particular, H ^V is a representation of the semi-group of the power Det ^c/2 (1) of the determinant line bundle over the moduli space of conformal equivalence classes of annuli with analytically parametrized boundary components. The results in Part I for -graded vertex algebras are also reformulated in terms of the framed little disk operad. Using Mays recognition principle for double loop spaces, one immediate consequence of such operadic formulations is that the compactly generated spaces corresponding to (or the k-ifications of) the locally convex completions constructed in Part I and in the present paper have the weak homotopy types of double loop spaces. We also generalize the results above to locally-grading-restricted conformal vertex algebras and to modules.     

Decays of the mesons and the singlet photon

In an attempt to explain the recent measurements on the radiative decays of the vector-mesons (V→Pγ), we study the consequences of introducing a small admixture of SU (3) singlet piece in the electromagnetic current. We find that this leads to an excellent fit of the theory with the new measurements on theV→Pγ decays. However, this addition adversely affects the fit of the leptonic decays of the vector mesons (V→e ⁺ e ⁻) and of the radiative decay of the pion (π→2γ). We conclude that the overall fit to the available data does not favour a large (>10%) admixture of the SU(3) singlet. The decay rates have been calculated in the vectormeson dominance model. At the hadronic vertex (VVP), we assume asymptotic nonet symmetry. The electromagnetic couplings (V−γ) are the ones appropriate to vector-mixing.