arXiv subjects

Katrina Barron

Publications and source records attributed to Katrina Barron.

At least 19 recordsLinked to original sources

arXiv2026

On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts

We study the mode transition algebras and Zhu algebras in the setting of $\mathbb{Z}$-graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the $\beta\gamma$-system). We show that the mode transition algebras of the Weyl vertex algebra at central charge 2 admit unity elements that form a family of strong unities in the sense of Damiolini-Gibney-Krashen. The existence of unities for the mode transition algebra of the Weyl vertex algebra at central charge 2 allows us to explicitly construct all higher level Zhu algebras of the Weyl vertex algebra at central charge 2. We further analyze weak modules of the Weyl vertex algebra at central charge 2 induced from Zhu algebras, proving that every such module is already induced from the level-zero Zhu algebra. We then prove that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, and hence not strongly interlocked in the sense of Barron-Batistelli-Orosz Hunziker-Yamskulna. More generally, we show that the property of being weakly interlocked is preserved under the action of an invertible Li's $\mathbf{\Delta}$ operator. As an application, we prove that all indecomposable reducible weight modules of the Weyl vertex algebra at central charge 2 obtained via spectral flow of Zhu-induced modules are likewise not weakly interlocked. These results clarify the role of being weakly interlocked in the modularity properties of bosonic ghost modules previously studied by Ridout-Wood and Allen-Wood.

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arXiv2023

Graded pseudo-traces for strongly interlocked modules for a vertex operator algebra and applications

We define the notion of {\it strongly interlocked} for indecomposable generalized modules for a vertex operator algebra, and show that the notion of graded pseudo-trace is well defined for modules which satisfy this property in certain settings. We prove that in these settings the graded pseudo-trace is a symmetric linear operator that satisfies the logarithmic derivative property. As an application, we prove that all the indecomposable reducible generalized modules for the rank one Heisenberg (one free boson) vertex operator algebras are strongly interlocked, independent of the choice of conformal vector and have well-defined graded pseudo-traces. We also completely characterize which indecomposable reducible generalized modules for the universal Virasoro vertex operator algebras induced from the level zero Zhu algebra are strongly interlocked. In particular, we prove that the universal Virasoro vertex operator algebra with central charge $c$ has modules induced from the level zero Zhu algebra with conformal weight $h$ that are strongly interlocked if and only if either $(c,h)$ is outside the extended Kac table, or the central charge is either $c = 1$ or $25$, the conformal weight satisfies a certain property, and the level zero Zhu algebra module being induced is determined by a Jordan block of size less than a certain specified parameter. We prove that all these modules for the universal Virasoro vertex operator algebra that are strongly interlocked have well-defined graded pseudo-traces. We give several examples of graded pseudo-traces for these Heisenberg and Virasoro strongly interlocked modules.

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arXiv2023

On generators and relations for higher level Zhu algebras and applications

We give some general results about the generators and relations for the higher level Zhu algebras for a vertex operator algebra. In particular, for any element $u$ in a vertex operator algebra $V$, such that $u$ has weight greater than or equal to $-n$ for $n \in \mathbb{N}$, we prove a recursion relation in the $n$th level Zhu algebra $A_n(V)$ and give a closed formula for this relation. We use this and other properties of $A_n(V)$ to reduce the modes of $u$ that appear in the generators for $A_n(V)$ as long as $u \in V$ has certain properties (properties that apply, for instance, to the conformal vector for any vertex operator algebra or if $u$ generates a Heisenberg vertex subalgebra), and we then prove further relations in $A_n(V)$ involving such an element $u$. We present general techniques that can be applied once a set of reasonable generators is determined for $A_n(V)$ to aid in determining the relations of those generators, such as using the relations of those generators in the lower level Zhu algebras and the zero mode actions on $V$-modules induced from those lower level Zhu algebras. We prove that the condition that $(L(-1) + L(0))v$ acts as zero in $A_n(V)$ for $n \in \mathbb{Z}_+$ and for all $v$ in $V$ is a necessary added condition in the definition of the Zhu algebra at level higher than zero. We discuss how these results on generators and relations apply to the level $n$ Zhu algebras for the Heisenberg vertex operator algebra and the Virasoro vertex operator algebras at any level $n \in \mathbb{N}$.

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arXiv2023

The level two Zhu algebra for the Heisenberg vertex operator algebra

We determine the level two Zhu algebra for the Heisenberg vertex operator algebra $V$ for any choice of conformal element. We do this using only the following information for $V$: the internal structure of $V$; the level one Zhu algebra of $V$ already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for $V$ with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in "On generators and relations for higher level Zhu algebras and applications", by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level $n$ Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level $n=2$. We also give a conjecture for the structure of the level $n$ Zhu algebra for the Heisenberg vertex operator algebra, for any $n >2$.

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arXiv2022

On rationality of $\mathbb{C}$-graded vertex algebras and applications to Weyl vertex algebras under conformal flow

Using the Zhu algebra for a certain category of $\mathbb{C}$-graded vertex algebras $V$, we prove that if $V$ is finitely $Ω$-generated and satisfies suitable grading conditions, then $V$ is rational, i.e. has semi-simple representation theory, with one dimensional level zero Zhu algebra. Here $Ω$ denotes the vectors in $V$ that are annihilated by lowering the real part of the grading. We apply our result to the family of rank one Weyl vertex algebras with conformal element $ω_μ$ parameterized by $μ\in \mathbb{C}$, and prove that for certain non-integer values of $μ$, these vertex algebras, which are non-integer graded, are rational, with one dimensional level zero Zhu algebra. In addition, we generalize this result to appropriate $\mathbb{C}$-graded Weyl vertex algebras of arbitrary ranks.

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arXiv2019

Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra $V$, we study the relationship between various types of $V$-modules and modules for the higher level Zhu algebras for $V$, denoted $A_n(V)$, for $n \in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n-1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable $V$-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are $V$-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the $V$-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when $V$ is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable $V$-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.

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arXiv2019

Classification of screening systems for lattice vertex operator algebras

We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra with an associated vertex operator (super)algebra. Screening pairs arising from weight one primary vectors acting commutatively on a lattice vertex operator algebra (the vacuum module) are classified into four general types, one type of which has been shown to play an important role in the construction and study of certain important families of $\mathcal{W}$-vertex algebras. These types of screening pairs we go on to study in detail through the notion of a system of screeners, which are lattice elements or `screening momenta' which give rise to screening pairs. We classify screening systems for all positive definite integral lattices of rank two, and for all positive definite even lattices of arbitrary rank when these lattices are generated by a screening system.

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arXiv2014

Permutation-twisted modules for even order cycles acting on tensor product vertex operator superalgebras

We construct and classify $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ even and $V$ a vertex operator superalgebra. In particular, we show that the category of weak $(1 \; 2 \; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ even is isomorphic to the category of weak parity-twisted $V$-modules. This result shows that in the case of a cyclic permutation of even order, the construction and classification of permutation-twisted modules for tensor product vertex operator superalgebras is fundamentally different than in the case of a cyclic permutation of odd order, as previously constructed and classified by the first author. In particular, in the even order case it is the parity-twisted $V$-modules that play the significant role in place of the untwisted $V$-modules that play the significant role in the odd order case.

math.QA↗

arXiv2014

On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form $V \otimes V$ 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 $V$.

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arXiv2013

Twisted modules for tensor product vertex operator superalgebras and permutation automorphisms of odd order

We construct and classify $(1 \; 2\; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ odd and for $V$ a vertex operator superalgebra. This extends previous results of the author, along with Dong and Mason, classifying all permutation-twisted modules for tensor product vertex operator algebras, to the setting of vertex operator superalgebras for odd order permutations. We show why this construction does not extend to the case of permutations of even order in the superalgebra case and how the construction and classification in the even order case is fundamentally different than that for the odd order permutation case. We present a conjecture made by the author and Nathan Vander Werf concerning the classification of permutation twisted modules for permutations of even order.

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arXiv2013

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.

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arXiv2013

Twisted modules for N=2 supersymmetric vertex operator superalgebras arising from finite automorphisms of the N=2 Neveu-Schwarz algebra

Twisted modules for N=2 supersymmetric vertex operator superalgebras are classified for the vertex operator superalgebra automorphisms which are lifts of a finite automorphism of the N=2 Neveu-Schwarz Lie superalgebra representation. These include the Ramond-twisted sectors and mirror-twisted sectors for N=2 vertex operator superalgebras, as well as twisted modules related to more general "spectral flow" representations of the N=2 Neveu-Schwarz algebra. We present the construct of twisted modules for free N=2 vertex operator superalgebras for all of the N=2 Neveu-Schwarz Lie superalgebra automorphisms of finite order. We show how to extend these to lattice N=2 vertex operator superalgebras. As a consequence, we also construct the Ramond-twisted sectors for free and lattice N=1 supersymmetric vertex operator superalgebras. We show that the lifting of the mirror automorphism for the N=2 Neveu-Schwarz algebra to an N=2 vertex operator superalgebra is not unique and that different mirror map vertex operator superalgebra automorphisms of an N=2 vertex operator superalgebra can lead to non-isomorphic categories of mirror-twisted modules, as in the case of free and lattice N=2 vertex operator superalgebras.

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arXiv2011

On uniformization of N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2 superconformally (resp., N=1 superanalytically) equivalent to a manifold with transition functions containing no odd functions of the even variable if and only if a certain cohomology group is trivial, namely the first Cech cohomology group of the body Riemann surface with coefficients in the sheaf consisting of the reciprocal of a line bundle tensor the holomorphic vector fields over the body. In particular, this gives a general criteria for when a DeWitt N=1 superanalytic super-Riemann surface is N=1 superanalytically equivalent to a ringed-space (1,1)-supermanifold, as studied in the algebro-geometric setting. This general classification result implies there is a countably infinite family of N=2 superconformal equivalence classes of N=2 superconformal DeWitt super-Riemann surfaces with genus-zero compact body, and N=2 superconformal DeWitt super-Riemann surfaces with simply connected body are classified up to N=2 superconformal equivalence by conformal equivalence classes of holomorphic line bundles over the underlying body Riemann surface. In addition, N=2 superconformal DeWitt super-Riemann surfaces with compact genus-one body and transition functions which correspond to the trivial cocycle in the first Cech cohomology group of the body Riemann surface with coefficients in the reciprocal of a line bundle tensor the sheaf of holomorphic vector fields over the body are classified up to N=2 superconformal equivalence by holomorphic line bundles over the torus modulo conformal equivalence. The corresponding results for the uniformization of N=1 superanalytic DeWitt super-Riemann surfaces of genus zero or one are presented.

math.DG↗

arXiv2010

Automorphism groups of N=2 superconformal super-Riemann spheres

In previous work, the author proved that there is a countably infinite family of N=2 superconformal equivalence classes of DeWitt N=2 superconformal super-Riemann surfaces with closed, genus-zero body. In this paper, we determine the automorphism groups for these N=2 superconformal super-Riemann surfaces, and analyze the Lie structure of these groups. Under the correspondence between N=2 superconformal and N=1 superanalytic structures, the results extend to the determination of automorphism groups of N=1 superanalytic DeWitt super-Riemann surfaces with closed, genus-zero body.

math.QA↗

arXiv2007

On axiomatic aspects of N=2 vertex superalgebras with odd formal variables, and deformations of N=1 vertex superalgebras

The notion of "N = 2 vertex superalgebra with two odd formal variables" is presented, the main axiom being 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 algebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N=2 superconformal operators. The notion of N=2 Neveu-Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. 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. The notion of ``N=2 (Neveu-Schwarz) vertex (operator) superalgebra with one odd formal variable" is formulated. It is shown that this formulation naturally arises from alternate notions of N=1 superconformality and the continuous deformation of an N=1 (Neveu-Schwarz) vertex (operator) superalgebra with one odd formal variable. This notion is formulated to reflect the underlying N=1 superanalytic geometry, and it is shown that the equivalence of the notions of N=2 (Neveu-Schwarz) vertex (operator) superalgebra with one and with two odd formal variables reflects the equivalence of N=2 superconformal and N=1 superanalytic geometry. In particular we prove that the group of formal N=2 superconformal functions vanishing at zero and invertible in a neighborhood of zero is isomorphic to a certain subgroup of N=1 superanalytic functions vanishing at zero and invertible in a neighborhood of zero.

math.QA↗

arXiv2007

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.

math.QA↗

arXiv2006

An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice vertex operator algebra and an automorphism arising from an arbitrary lattice isometry. This construction, from a physical point of view, is related to the space-time geometry associated with the lattice in the sense of string theory. The other construction is due to the first author, jointly with C. Dong and G. Mason, in the case of a multi-fold tensor product of a given vertex operator algebra with itself and a permutation automorphism of the tensor factors. The latter construction is based on a certain change of variables in the worldsheet geometry in the sense of string theory. In the case of a lattice that is the orthogonal direct sum of copies of a given lattice, these two very different constructions can both be carried out, and must produce isomorphic twisted modules, by a theorem of the first author jointly with Dong and Mason. In this paper, we explicitly construct an isomorphism, thereby providing, from both mathematical and physical points of view, a direct link between space-time geometry and worldsheet geometry in this setting.

math.QA↗

arXiv2003

Superconformal change of variables for N=1 Neveu-Schwarz vertex operator superalgebras

Superconformal change of variables formulas for N=1 Neveu-Schwarz vertex operator superalgebras are presented for general invertible superconformal changes of variables. Using the underlying worldsheet supergeometry of propagating superstrings, geometric proofs of the change of variables formulas are given for the case of convergent superconformal changes of variables. More general formal algebraic proofs of the change of variables formulas in the case of formal superconformal changes of variables are then given. Finally, isomorphic families of N=1 NS-VOSAs are derived from the superconformal change of variables formulas.

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