Vertex algebras related to regular representations of SL2

Abstract We investigate a one-point restriction of conformal blocks on $$(\mathbb {P}^1,\infty ,1,0)$$ ( P 1 , ∞ , 1 , 0 ) associated with modules over a vertex operator algebra V . By restricting the module attached to the point $$\infty $$ ∞ to its bottom degree, we obtain a new formula for computing fusion rules in terms of a new left A ( V )-module $$M^1 \odot M^2$$ M 1 ⊙ M 2 over the Zhu algebra A ( V ), constructed from two V -modules $$M^1$$ M 1 and $$M^2$$ M 2 . As a consequence, for strongly rational vertex operator algebras, the construction of $$M^1 \odot M^2$$ M 1 ⊙ M 2 induces the fusion tensor product on the module category $$\textsf{Mod}(A(V))$$ Mod ( A ( V ) ) .

Let [Formula: see text] be a moonshine-type vertex operator algebra generated by Virasoro vectors [Formula: see text] of central charge [Formula: see text], where [Formula: see text], such that the Virasoro vertex operator algebra generated by [Formula: see text] is simple, [Formula: see text], and [Formula: see text] for distinct [Formula: see text]. We prove that if [Formula: see text] is simple, then [Formula: see text] is uniquely determined by its Griess algebra [Formula: see text]. If [Formula: see text], we further prove that [Formula: see text] is simple. As a result, we deduce that the subalgebra [Formula: see text] of the parafermion vertex algebra [Formula: see text] generated by [Formula: see text], for [Formula: see text], is simple and isomorphic to [Formula: see text], where [Formula: see text] is the root system of [Formula: see text].

A bstract We study the bosonic VOA associated with the 3D $$ \mathcal{N}=4 $$ N = 4 abelian linear quiver gauge theories arising from compactifying 4D $$ \mathcal{N}=2 $$ N = 2 Argyres-Douglas theories of ( A 1 , A 2 n −1 ) and ( A 1 , D 2 n ) types. These VOAs are obtained by canceling the gauge anomaly of the H-twisted 3D theory on the half-space by Heisenberg algebras on the boundary. We particularly conjecture a complete set of strong generators of these bosonic VOAs, which contains more than the Virasoro stress tensor and those arising from Higgs branch operators. We also find that these bosonic VOAs contain copies of the W 3 vertex algebra at c = −2 as sub vertex algebras.

Abstract Schur functions has been shown to satisfy certain plethysm stability properties and recurrence relations. In this paper, use vertex operator methods to study analogous stability properties of Schur’s Q -functions. Although the two functions have similar stability properties, we find a special case where the plethysm of Schur’s Q -functions exhibits linear increase.

The seminal discovery in the theory of q-series and combinatorics analysis has been theBailey lemma which has since made a revolutionary impact on the study of partition identities,modular forms and special functions since it was presented by W.N. Bailey in the middle ofthe 20th century. This review article has tried to give the historical details of the Bailey lemmaand recounts the origin, evolution and subsequent generalizations of the lemma in noting themassive contribution to a wide range of mathematics including number theory, representationtheory, and the physics of mathematics. This paper starts with an introduction to the classicalversion of the Bailey lemma, including how it was used in the first few years of its discovery toprovide systematic proofs of Rogers-Ramanujan-type identities and other forms of partitions. Itlater explores the methodological development of the lemma which was laid down by importantauthors like George Andrews and Basil Gordon, who introduced the notion of Bailey chainsas well as Bailey pairs as expansions of the work of Bailey. The developments have allowed usto discover new infinite families of identities and also achieve a better understanding of theunderlying combinatorial structures. Recent progress, including elliptic and multidimensionalextensions of the Bailey lemma and their application to vertex operator algebras and conformalfield theory are also examined in the paper. In a bid to rekindle the perpetual relevance andapplicability of the Bailey lemma as a unifying tool in mathematical studies, this paper aimsat uniting historical and modern perspectives of the same. The envisaged outcomes are theimproved insight into the flexibility of the lemma, the interconnections with other branchesof mathematics, and many more opportunities to be pursued in the further studies, especiallyfollowing the explosion of interdisciplinary applications of the lemma.

Abstract Suppose a Lie group G acts on a vertex algebra $$\mathcal {V}$$ V . In this article we construct a vertex algebra $${\tilde{V}}$$ V ~ , which is an extension of $$\mathcal {V}$$ V by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $$\mathfrak {g}$$ g -connections $$(\textrm{d}+A)$$ ( d + A ) . The category of representations of $${\tilde{\mathcal {V}}}$$ V ~ fibres over the set of connections, and the fibres should be viewed as $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V , generalizing the familiar notion of g -twisted modules. In fact, another application of our result is that it proposes an explicit definition of $$(\textrm{d}+A)$$ ( d + A ) -twisted modules of $$\mathcal {V}$$ V in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. In particular, we have in mind limits of the generalized quantum Langlands kernel, in which case G is the Langland dual and $$\mathcal {V}$$ V is conjecturally the Feigin-Tipunin vertex algebra and the extension $${\tilde{\mathcal {V}}}$$ V ~ is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.

Vertex $F$-algebras are a deformation of the concept of an ordinary vertex algebra in which the additive formal group law is replaced by an arbitrary formal group law $F$. The main theorem of this paper constructs a Lie algebra from a vertex $F$-algebra - for the additive formal group law, this extends Borcherds' well-known construction for ordinary vertex algebras. Our construction involves the new concept of an $F$-residue and some other new algebraic concepts, which are deformations of familiar concepts for the special case of an additive formal group law.

A bstract Supergravity solutions describing supersymmetric rotating bound states of NS fivebranes, fundamental strings and momentum can sometimes have an ergoregion but no horizon. In such supersymmetric ergoregions, there is an unusual feature that there exist BPS “excitations” that cost negative (or zero) energy as measured from asymptotic infinity. We study the spectrum of supergravity and string excitations of these backgrounds using their worldsheet description as gauged Wess-Zumino-Witten (WZW) models, and construct their holographic map to transitions in the dual spacetime CFT. The backgrounds generically contain orbifold singularities, and we exhibit vertex operators corresponding to twisted sector ground states localized at the orbifold fixed points. The gauged WZW model thus provides a valuable tool to explore the stringy structure of such heavy BPS states in AdS 3 / CFT 2 .

A bstract Sen’s action for chiral bosons in 2 dimensions describes two chiral scalars, one of which couples to the physical metric and one of which couples to a flat metric. It has a generalisation in which the flat metric is replaced by an arbitrary second metric and so can be formulated on any curved world-sheet. When the two metrics are equal, the theory reduces to a βγ system, giving a non-unitary c = 2 conformal field theory. We argue that the relation between this and the theory of two chiral bosonic scalars of the same chirality can be viewed as a ‘bosonisation’. We show that the standard vertex operators for the chiral scalars are vertex operators and line operators in the Sen formulation and derive the formulation in the Sen theory of correlation functions in the chiral scalar theory. The flat space Sen theory can be coupled to two different world-sheet metrics in such a way that one scalar couples to one metric and the other to the other metric, so obtaining the general formulation with two metrics. In d = 4 k + 2 dimensions, the bi-metric action for a 2 k -form gauge field with self-dual field strength reduces, when the two metrics are equal, to a conformal field theory with a BF -type action, except that B is a self-dual d /2-form and F is a d /2-form field strength, F = dP . The self-duality of B means that this is not a topological theory but instead represents two self-dual gauge fields. This has a generalisation to a democratic action for p -form gauge fields in any dimension.

Abstract Let $\mathbb {k}$ be a field, and let $\mathcal {C}$ be a Cauchy complete $\mathbb {k}$ -linear braided category with finite-dimensional morphism spaces and . We call an indecomposable object X of $\mathcal C$ non-negligible if there exists $Y\in \mathcal {C}$ such that is a direct summand of $Y\otimes X$ . We prove that every non-negligible object $X\in \mathcal {C}$ such that $\dim \operatorname {End}(X^{\otimes n})<n!$ for some n is automatically rigid. In particular, if $\mathcal {C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal {C}$ is rigid. As applications, we simplify Huang’s proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal {C}$ , the data of a $\mathcal {C}$ -modular functor is equivalent to a modular fusion category structure on $\mathcal {C}$ , answering a question of Bakalov and Kirillov. Furthermore, we show that if $\mathcal {C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal {C}$ is zero. Hence $\mathcal {C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.

Meromorphic Open-String Vertex Algebras and Riemannian Manifolds

A bstract We present a prescription for computing tree-level scattering amplitudes in 10D super-Yang-Mills (SYM) theory using the pure spinor worldline formalism. The pure spinor formalism has proven to be a powerful framework for studying supersymmetric field theories, providing manifestly covariant and BRST-invariant formulations of amplitudes. By incorporating the worldline approach, we construct a first-quantized representation of SYM amplitudes in 10D, where interactions are encoded through the insertion of vertex operators along the particle’s trajectory. We explicitly compute the N-point function, demonstrating an agreement with the α ′ → 0 limit of open superstring amplitudes and confirming that the kinematic numerators satisfy the expected BRST relations. Our results establish the pure spinor worldline formalism as a tool for studying scattering amplitudes and suggest further applications to 11D supergravity.

We find that multiple vertex algebras can arise from a single 4d \mathcal{N}=2 𝒩 = 2 superconformal field theory (SCFT). The connection is given by the BPS monodromy operator M M , which is a wall-crossing invariant quantity that captures the BPS spectrum on the Coulomb branch. For a class of low-rank Argyres-Douglas theories, we find that the trace of the multiple powers of the monodromy operator Tr M^N M N yield modular functions that can be identified with the vacuum characters of certain vertex algebra for each N N . In particular, we realize unitary VOAs of the Deligne-Cvitanović exceptional series type (A_2)_1 ( A 2 ) 1 , (G_2)_1 ( G 2 ) 1 , (D_4)_1 ( D 4 ) 1 , (F_4)_1 ( F 4 ) 1 , (E_6)_1 ( E 6 ) 1 from Argyres-Douglas theories. We also find the modular invariant characters of the ‘intermediate vertex algebras’ (E_{7\frac{1}{2}})_1 ( E 7 1 2 ) 1 and (X_1)_1 ( X 1 ) 1 . Our analysis allows us to construct 3d \mathcal{N}=2 𝒩 = 2 gauge theories that flow to \mathcal{N}=4 𝒩 = 4 SCFTs in the IR, whose specialized half-index can be identified with these modular invariant characters.

Abstract A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\textrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disk. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions through its associated Clifford algebra and study its twisted module with respect to irregular $\mathfrak{s}\mathfrak{l}_{2}$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara-type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, in particular work by Witten [51], and more generally in higher-dimensional context. Our original motivation in [24] comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections (similar to the affine Lie algebra at critical level) and as such gives a family of deformations of the Feigin–Tipunin algebra. Our present article now describes, in the smallest case, the fibres and their representation categories over irregular connections.

Abstract Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan–Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of $$s_{\nu }$$ s ν in the plethysm product $$(p_{n}\circ h_{k})s_{\mu }$$ ( p n ∘ h k ) s μ . Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.

Abstract In this paper we review recent applications of cohomology methods associated with vertex operator algebras to problems in conformal field theory and differential geometry.

Orbifolds of pointed vertex operator algebras I

We compute tree-level gluon amplitudes as worldline correlators of vertex operators in a bosonic spinning particle model. In this framework, the particle’s position degrees of freedom are extended by complex bosonic variables that encode its spin. In the free theory, the model exhibits a first-class constraint algebra whose gauging ensures the unitarity of the quantum theory. This algebra is a contraction of the s l ( 2 , R ) algebra, which is by itself a subalgebra of the Virasoro algebra. In string theory, gauging the Virasoro algebra plays a similar role. Our model admits a consistent truncation to describe a pure spin-1 particle. Non-Abelian interactions are introduced by using Becchi-Rouet-Stora-Tyutin (BRST) techniques, which allow us to extract the vertex operators of the theory as suitable deformations of the BRST charge. BRST invariance is central to ensuring the consistency of the tree-level amplitudes analyzed in this work. We discuss connections with similar worldline constructions and comment on the potential relevance of this framework for uncovering the structures underlying the double-copy program in gauge and gravitational theories.

A bstract The chiral algebra of 4D $$\mathcal{N}=4$$ SU( N ) super-Yang-Mills theory is an $$\mathcal{N}=4$$ superconformal vertex operator algebra. We analyse the structure of this algebra by studying recursively the constraints that are required by the associativity of the operator product expansion. We find that the algebra is uniquely characterized by the central charge (which can take an arbitrary value), without any additional free parameter. Furthermore, the truncation pattern of the OPE coefficients suggests that the algebra cannot arise from the symmetric orbifold.