Twisted Quantum Affine Algebras and Equivariant $$\phi $$-Coordinated Modules for Quantum Vertex Algebras

In this paper we propose two sets of nonlinear integral equations (NLIE) for describing the thermodynamics in the sine-Gordon model, when higher Lorentz spin conserved charges are also coupled to the Gibbs ensemble. We call them NLIE I and II. The derivation of the equations, is based on T−Q relations given by the equivalent thermodynamic Bethe ansatz (TBA) formulation of the problem in the repulsive regime. Though the equations are derived in the repulsive regime at discrete values of the coupling constant, a straightforward analytical continuation ensures their validity within the whole repulsive regime of the theory. For the NLIE I formulation, appropriate analytical continuation makes the penetration into the attractive regime also possible. However, the magnitude of this penetration is restricted by the spin of the largest spin conserved charge contained in the Gibbs ensemble. Within their range of validity, these NLIE formulations provide efficient theoretical frameworks for computing expectation values of conserved charge densities, their associated currents, and vertex operators and their descendants, with respect to the generalized Gibbs ensemble.

Modularity of vertex operator algebra correlators with zero modes

Modular toroidal vertex algebras and their modules

We study sheaves on holomorphic spaces of loops (also referred to as arcs ) and apply this to the study of the complex, defined in [2], governing deformations of the Poisson vertex algebra structure on the space of holomorphic loops into a Poisson variety. We describe this complex in terms of the (continuous) de Rham–Lie cohomology of an associated Lie algebroid object in locally linearly compact topological (alias Tate ) sheaves of modules on ℒ + M . In particular this allows us to easily compute the cohomology of the above in the case where π is symplectic, we obtain de Rham cohomology of M .

In this paper, we study the effects of vertex operators on soliton solutions of the modified Kadomtsev–Petviashvili (mKP) equation based on the Grassmannian $$Gr(N, M)$$ . We present a classification theorem for the solutions of the mKP equation and demonstrate the connection among vertex operators, theta functions, and the tau-function of the mKP equation. We also express the soliton solution on $$Gr(N,M)$$ of the mKP equation in terms of the $$M$$ -theta function and free fermions. We show that the new tau-function obtained by the action of the vertex operator on the tau-function of the mKP equation can still form a solution of the mKP equation. We study all kinds of solutions of the mKP equation obtained through the action of vertex operators. We find that when the mKP equation solutions obtained via the action of the vertex operator satisfy the regularity condition, the chord diagram exhibits non-crossing between the graph related to the parameters of the vertex operators and the graph related to the parameters of the original solution.

On Rota-Baxter vertex operator algebras

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 vertex operator algebra, [Formula: see text] be in [Formula: see text] and [Formula: see text] for [Formula: see text] be a [Formula: see text]-twisted module, where [Formula: see text] are commuting automorphisms of [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text]. Suppose [Formula: see text] is an intertwining operator of type [Formula: see text]. We construct an [Formula: see text]-[Formula: see text]-bimodule [Formula: see text] which determines the action of [Formula: see text] from the bottom level of [Formula: see text] to the bottom level of [Formula: see text], and we explore its connections with fusion rules.

Abstract We construct a new family of affine $W$-algebras $W^{k}(\lambda ,\mu )$ parameterized by partitions $\lambda $ and $\mu $ associated with the centralizers of nilpotent elements in $\mathfrak{g}\mathfrak{l}_{N}$. The new family unifies a few known classes of $W$-algebras. In particular, for the column-partition $\lambda $ we recover the affine $W$-algebras $W^{k}(\mathfrak{g}\mathfrak{l}_{N},f)$ of Kac, Roan, and Wakimoto, associated with nilpotent elements $f\in \mathfrak{g}\mathfrak{l}_{N}$ of type $\mu $. Our construction is based on a version of the BRST complex of the quantum Drinfeld–Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras $W^{k}(\lambda ,\mu )$ yields a family of generalized finite $W$-algebras $U(\lambda ,\mu )$, which we also describe independently as associative algebras.

A bstract The scattering equations relate massless scattering kinematics to marked points on a Riemann sphere, and underpin remarkable formulae for the full tree-level S-matrices of many interesting QFTs, including cubic biadjoint scalars, Yang-Mills theory and general relativity. The scattering equations arise from worldsheet correlators of ambitwistor string theories, which has enabled their generalisation to anti-de Sitter (AdS) space in certain cases. In this paper, we use the scattering equations and ambitwistor strings to prove the correspondence between an appropriate flat limit of boundary correlators in AdS and Carrollian scattering amplitudes — massless amplitudes written in position space on the null conformal boundary — for any number of external states and spacetime dimensions in tree-level, cubic scalar theories. We first derive the Carrollian version of the scattering equations in Minkowski space and their associated Carrollian amplitude formulae, by direct Fourier transform from momentum space and from ambitwistor strings with a Carrollian basis of vertex operators. We then take the flat limit of known formulae for all tree-level boundary correlators of cubic scalar theories in AdS, recovering the Carrollian amplitudes in flat space. In the special case of AdS 3 , we also make some comments on the flat space limit of spinning boundary correlators.

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].

We suggest that the chiral (e_{8})_{1} theory-in many senses the simplest vertex operator algebra-may have Haagerup symmetry H_{i} for i=1, 2, 3. Likewise, we suggest that the nonchiral (E_{8})_{1} Wess-Zumino-Witten model may have H_{i}×H_{i}^{op} symmetry, and that gauging the diagonal symmetry gives a c=8 theory with Z(H_{3}) symmetry, which is the theory predicted in Evans and Gannon [Commun. Math. Phys. 307, 463 (2011)CMPHAY0010-361610.1007/s00220-011-1329-3]. Along the way, we show that (E_{8})_{1} also has a Fib×Fib^{op} symmetry, and that gauging the diagonal symmetry gives the (G_{2})_{1}×(F_{4})_{1} Wess-Zumino-Witten model, explaining the well-known conformal embedding (G_{2})_{1}×(F_{4})_{1}⊂(E_{8})_{1}. Finally, we suggest a relation to theories with H_{3} symmetry at c=2, 6, complimenting the discussion with new modular bootstrap results.

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.

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 Let be a ‐cofinite vertex operator algebra, not necessarily rational or self‐dual. In this paper, we establish various versions of the sewing‐factorization (SF) theorems for conformal blocks associated to grading‐restricted generalized modules of (where ). In addition to the versions announced in the Introduction of the first part of this series of papers by Gui and Zhang, we prove the following coend version of the SF theorem. Let be a compact Riemann surface with incoming and outgoing marked points, and let be another compact Riemann surface with incoming and outgoing marked points. Assign and to the incoming marked points of and , respectively. For each , assign and its contragredient to the outgoing marked points of and , respectively. Denote the corresponding spaces of conformal blocks by and . Let be the ‐pointed surface obtained by sewing along their outgoing marked points. Then, the sewing of conformal blocks — proved to be convergent in the second part of this series of papers by Gui and Zhang — yields an isomorphism of vector spaces We also discuss the relationship between conformal blocks and the modular functors defined using Lyubashenko's coend/construction.

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.

Full Universal Enveloping Vertex Algebras from Factorisation

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.