Katrina Barron

TWISTED SECTORS FOR TENSOR PRODUCT VERTEX OPERATOR ALGEBRAS ASSOCIATED TO PERMUTATION GROUPS

Abstract: Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V⊗k. We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V⊗k are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V⊗k-modules from weak V-modules. For an arbitrary permutation automorphism g of V⊗k the category of weak admissible g-twisted modules for V⊗k is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V⊗k-modules for γ a general automorphism of V acting diagonally on V⊗k and g a permutation automorphism of V⊗k.

A supergeometric interpretation of vertex operator superalgebras

Conformal field theory (or more specifically, string theory) and related theories (cf. [BPZ], [FS], [V], and [S]) are the most promising attempts at developing a physical theory that combines all fundamental interactions of particles, including gravity. The geometry of this theory extends the use of Feynman diagrams, describing the interactions of point particles whose propagation in time sweeps out a line in space-time, to one-dimensional “particles” (strings) whose propagation in time sweeps out a two-dimensional surface. For genus zero holomorphic conformal field theory, algebraically, these interactions can be described by products of vertex operators or more precisely, by means of vertex operator algebras (cf. [Bo] and [FLM]). However, until 1990 a rigorous mathematical interpretation of the geometry and algebra involved in the “sewing” together of different particle interactions, incorporating the analysis of general analytic coordinates, had not been realized. In [H1] and [H2], motivated by the geometric notions arising in conformal field theory, Huang gives a precise geometric interpretation of the notion of vertex operator algebra by considering the geometric structure consisting of the moduli space of genus zero Riemann surfaces with punctures and local coordinates vanishing at the punctures, modulo conformal equivalence, together with the operation of sewing two such surfaces, defined by cutting discs around one puncture from each sphere and appropriately identifying the boundaries. Important aspects of this geometric structure are the concrete realization of the moduli space in terms of exponentials of a representation of the Virasoro algebra and a precise analysis of sewing using these resulting exponentials. Using this geometric structure, Huang then introduces the notion of geometric vertex operator algebra with central charge c ∈ C, and proves that the category of geometric vertex operator algebras is isomorphic to the category of vertex operator algebras. In [F], Friedan describes the extension of the physical model of conformal field theory to that of superconformal field theory and the notion of a superstring whose propagation in time sweeps out a supersurface. Whereas conformal field theory attempts to describe the interactions of bosons, superconformal field theory attempts to describe the interactions of boson-fermion pairs. This, in particular, requires an operator D such that D 2 = @ @z . Such an operator arises naturally in supergeometry. In [BMS], Beilinson, Manin and Schechtman study some aspects of superconformal symmetry, i.e., the Neveu-Schwarz algebra, from the viewpoint of algebraic geometry. In this work, we will take a differential geometric approach, extending Huang’s geometric interpretation of vertex operator algebras to a supergeometric interpretation of vertex operator superalgebras. Within the framework of supergeometry (cf. [D], [R] and [CR]) and motivated by superconformal field theory, we define the moduli space of super-Riemann surfaces with genus zero “body”, punctures, and local superconformal coordinates vanishing at the punctures, modulo superconformal equivalence. We announce the result that any local superconformal coordinates can be expressed in terms of exponentials of certain superderivations, and that these superderivations give a representation of the Neveu-Schwarz algebra with zero central charge. We define a

The Notion of N=1 Supergeometric Vertex Operator Superalgebra and the Isomorphism Theorem

We introduce the notion of N=1supergeometric vertex operator superalgebra motivated by the geometry underlying genus-zero, two-dimensional, holomorphic N=1 superconformal field theory. We then show, assuming the convergence of certain projective factors, that the category of such objects is isomorphic to the category of N=1 Neveu–Schwarz vertex operator superalgebras.

Factorization of Formal Exponentials and Uniformization

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g = g − ⊕ g + , and let s and t be commuting formal variables commuting with g. We prove that the map C: sg − [[s, t]] × tg + [[s, t]] → sg − [[s, t]] ⊕ tg + [[s, t]] defined by the Campbell–Baker–Hausdorff formula and given by esg − etg + = eC(sg − , tg + ) for g ± ∈ g ± [[s, t]] is a bijection, as is well known when g is finite-dimensional over R or C, by geometry. It follows that there exist unique Ψ ± ∈ g ± [[s, t]] such that etg + esg − = esΨ − etΨ + (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra g consisting of certain formal infinite series with coefficients in a Z-graded Lie algebra p, for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N = 1 Neveu–Schwarz superalgebra. For p the Virasoro algebra, the result was first proved by Huang as a step in the construction of a geometric formulation of the notion of vertex operator algebra, and for p a Grassmann envelope of the Neveu–Schwarz superalgebra, it was first proved by Barron as a corresponding step in the construction of a supergeometric formulation of the notion of vertex operator superalgebra. In the special case of the Virasoro (resp., N = 1 Neveu–Schwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in two-dimensional genus-zero holomorphic conformal (resp., N = 1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context.

THE MODULI SPACE OF N = 2 SUPER-RIEMANN SPHERES WITH TUBES

Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N = 2 superconformal field theory, we define the moduli space of N = 2 super-Riemann spheres with oriented and ordered half-infinite tubes (or equivalently, oriented and ordered punctures, and local superconformal coordinates vanishing at the punctures), modulo N = 2 superconformal equivalence. We develop a formal theory of infinitesimal N = 2 superconformal transformations based on a representation of the N = 2 Neveu–Schwarz algebra in terms of superderivations. In particular, via these infinitesimals we present the Lie supergroup of N = 2 superprojective transformations of the N = 2 super-Riemann sphere. We give a reformulation of the moduli space in terms of these infinitesimals. We introduce generalized N = 2 super-Riemann spheres with tubes and discuss some group structures associated to certain moduli spaces of both generalized and non-generalized N = 2 super-Riemann spheres. We define an action of the symmetric groups on the moduli space. Lastly we discuss the nonhomogeneous (versus homogeneous) coordinate system associated to N = 2 superconformal structures and the corresponding results in this coordinate system.

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.

On Twisted Modules for N=2 Supersymmetric Vertex Operator Superalgebras

The classification of twisted modules for N = 2 supersymmetric vertex operator superalgebras with twisting given by vertex operator superalgebra automorphisms which are lifts of a finite automorphism of the N = 2 Neveu–Schwarz Lie superalgebra representation is presented. These twisted modules include the Ramond-twisted sectors and mirror-twisted sectors for N = 2 vertex operator super algebras, as well as twisted modules related to more general “spectral flow” representations of the N = 2 Neveu–Schwarz algebra.

On permutation-twisted free fermions and two conjectures

We conjecture that the category of permutation-twisted modules for a multi-fold tensor product vertex operator superalgebra and a cyclic permutation of even order is isomorphic to the category of parity-twisted modules for the underlying vertex operator superalgebra. This conjecture is based on our observations of the cyclic permutation-twisted modules for free fermions as we discuss in this work, as well as previous work of the first author constructing and classifying permutation-twisted modules for tensor product vertex operator superalgebras and a permutation of odd order. In addition, we observe that the transposition isomorphism for two free fermions corresponds to a lift of the −1 isometry of the integral lattice vertex operator superalgebra corresponding to two free fermions under boson-fermion correspondence. We conjecture that all even order cyclic permutation automorphisms of free fermions can be realized as lifts of lattice isometries under boson-fermion correspondence. We discuss the role of parity stability in the construction of these twisted modules and prove that in general, parity-unstable weak twisted modules for a vertex operator superalgebras come in pairs that form orthogonal invariant subspaces of parity-stable weak twisted modules, clarifying their role in many other settings.

On the Correspondence Between Mirror-Twisted Sectors for N = 2 Supersymmetric Vertex Operator Superalgebras of the Form V ⊗ V and N = 1 Ramond Sectors of V

Using recent results of the author along with Vander Werf, we present the classification and construction of mirror-twisted modules for N = 2 supersymmetric vertex operator superalgebras of the form \(V \otimes V\) for the signed transposition mirror map automorphism. In particular, we show that the category of such mirror-twisted sectors for \(V \otimes V\) is isomorphic to the category of N = 1 Ramond sectors for V.

PERMUTATION-TWISTED MODULES FOR EVEN ORDER CYCLES ACTING ON TENSOR PRODUCT VERTEX OPERATOR SUPERALGEBRAS

We construct and classify