Vertex Operators Research Articles

On the nilpotent orbits arising from admissible affine vertex algebras

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

Rationality and C2-cofiniteness of certain diagonal coset vertex operator algebras

In this paper, it is shown that the diagonal coset vertex operator algebra C(Lg(k+2,0),Lg(k,0)⊗Lg(2,0)) is rational and C2-cofinite in case g=so(2n), n≥3 and k is an admissible number for gˆ. It is also shown that the diagonal coset vertex operator algebra C(Lsl2(k+4,0),Lsl2(k,0)⊗Lsl2(4,0)) is rational and C2-cofinite in case k is an admissible number for sl2ˆ. Furthermore, irreducible modules of C(Lsl2(k+4,0),Lsl2(k,0)⊗Lsl2(4,0)) are classified in case k is a positive odd integer.

Formula omitted] at level [formula omitted

Let Ll=L(sl2l+1,−l−12) be the simple vertex operator algebra based on the affine Lie algebra slˆ2l+1 at boundary admissible level −l−12.We consider a lift ν of the Dynkin diagram involution of A2l=sl2l+1 to an involution of Ll. The ν-twisted Ll-modules are A2l(2)-modules of level −l−12 with an anti-homogeneous realization. We classify simple ν-twisted highest-weight (weak) Ll-modules using twisted Zhu algebras and singular vectors for slˆ2l+1 at level −l−12 obtained by Perše.We find that there are finitely many such modules up to isomorphism, and the ν-twisted (weak) Ll-modules that are in category O for A2l(2) are semi-simple.

Odd dimensional nonlocal Liouville conformal field theories

We construct Euclidean Liouville conformal field theories in odd number of dimensions. The theories are nonlocal and non-unitary with a log-correlated Liouville field, a mathcal{Q} -curvature background, and an exponential Liouville-type potential. We study the classical and quantum properties of these theories including the finite entanglement entropy part of the sphere partition function F, the boundary conformal anomaly and vertex operators’ correlation functions. We derive the analogue of the even-dimensional DOZZ formula and its semi-classical approximation.

Formal vertex laws associated to Lie conformal algebras

We introduce several definitions within the framework of vertex and conformal algebras that are analogous to some important concepts of the classical Lie theory. Most importantly, we define formal vertex laws, which correspond to the notion of formal group laws. We prove suitable vertex/conformal versions of a number of classical results, such as the Milnor–Moore theorem, Cartier duality, and the equivalence between formal group laws and Lie algebras.

Witten’s cubic open string field theory on multiple Dp-branes

We study Witten's cubic open string field theory on multiple $Dp$-branes. On multiple $Dp$-branes the string fields carry $U(N)$ group indices. Mapping the string world-sheet onto the upper half complex plane and evaluating the Polyakov string path integral of Witten's cubic open string, we obtain a Fock space representation of one, two and three-string vertex operators. In the low energy region, the system is described by massless gauge fields and massless scalar fields, carrying $U(N)$ group indices. The two-string vertex induces mass terms for the gauge fields and the scalar fields. The cubic interaction reduces correctly to the cubic term of the non-Abelian gauge fields, cubic interaction terms of the gauge fields, and the scalar fields on $(p+1)$ dimensional space. Thus, Witten's string field theory may be a useful tool to study the quantum dynamics of open strings on multiple $Dp$-branes.

Rationality and fusion rules of exceptional $\mathcal {W}$-algebras

First, we prove the Kac–Wakimoto conjecture on modular invariance of characters of exceptional affine \mathcal {W} -algebras. In fact more generally we prove modular invariance of characters of all lisse \mathcal {W} -algebras obtained through Hamiltonian reduction of admissible affine vertex algebras. Second, we prove the rationality of a large subclass of these \mathcal {W} -algebras, which includes all exceptional \mathcal {W} -algebras of type A and lisse subregular \mathcal {W} -algebras in simply laced types. Third, for the latter cases we compute S -matrices and fusion rules. Our results provide the first examples of rational \mathcal {W} -algebras associated with nonprincipal distinguished nilpotent elements, and the corresponding fusion rules are rather mysterious.

Schottky vertex operator cluster algebras

Using recursion formulas for vertex operator algebra higher genus characters with formal parameters identified with local coordinates around marked points on a Riemann surface of arbitrary genus, we introduce the notion of a vertex operator cluster algebra structure. Cluster elements and mutation rules are explicitly defined, and the simplest example of a vertex operator cluster algebra is presented.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

Super orbifold theory

Let V be a simple vertex operator superalgebra and G a finite automorphism group of V containing the canonical automorphism σ such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined, a global dimension formula for V in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition the S-matrix of VG is computed.

First cohomologies of affine, Virasoro, and lattice vertex operator algebras

In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For negative energy representations of Virasoro VOA, the same conclusion holds when $W$ is $L(0)$-graded with lowest weight greater or equal to $-3$. Relationship between the first cohomology of the VOA and that of the associated Zhu's algebra is also discussed.

Principal subspaces for the quantum affine vertex algebra in type A1(1)

By using the ideas of Feigin and Stoyanovsky and Calinescu, Lepowsky and Milas we introduce and study the principal subspaces associated with the Etingof-Kazhdan quantum affine vertex algebra of integer level $k\geqslant 1$ and type $A_1^{(1)}$. We show that the principal subspaces possess the quantum vertex algebra structure, which turns to the usual vertex algebra structure of the principal subspaces of generalized Verma and standard modules at the classical limit. Moreover, we find their topological quasi-particle bases which correspond to the sum sides of certain Rogers-Ramanujan-type identities.

Celestial operator products from the worldsheet

We compute the operator product expansions of gluons and gravitons in celestial CFT from the worldsheet OPE of vertex operators of four-dimensional ambitwistor string theories. Remarkably, the worldsheet OPE localizes on the short-distance singularity between vertex operator insertions which in turn coincides with the OPE limit of operator insertions on the celestial sphere. The worldsheet CFT dynamically produces known celestial OPE coefficients — as well as infinite towers of SL(2, ℝ) descendant contributions to the celestial OPE — without any truncations or approximations. We obtain these results for all helicities and incoming/outgoing configurations. Furthermore, the worldsheet OPE encodes the infinite-dimensional symmetry algebras associated with the conformally soft sectors of gauge theory and gravity. We provide explicit operator realizations of the currents generating these symmetries on ambitwistor space in terms of vertex operators for soft gluons and gravitons, also computing their actions on hard particles of all helicities. Lastly, we show that the worldsheet OPE for momentum eigenstates produces the collinear splitting functions of gluons and gravitons.

A theory of tensor products of modules for 𝒮-local vertex operator algebra M(n,V )

In this paper, we introduce the notions of intertwining operator and tensor product for [Formula: see text]-modules, where [Formula: see text] is a vertex operator algebra, and prove that the tensor product is commutative and associative.

S-matrix in permutation orbifolds

For a fixed positive integer k, any element g of the permutation group Sk acts on the tensor product vertex operator algebra V⊗k in the obvious way. In this paper, we determine the S-matrix of (V⊗k)G if G=〈g〉 is the cyclic group generated by g=(1,2,⋯,k).

G-twisted associative algebras of vertex operator superalgebras

Let V be a vertex operator superalgebra and G a finite automorphism group of V. Let σ be the order 2 automorphism of V associated with the superstructure of V. A sequence of associative algebras AG,n(V) are constructed to study the twisted representations of V for nonnegative n∈1T′Z where T′ is the order of group generated by G and σ. This result which generalizes many previous results on Zhu's algebras is then used to investigate the super orbifold theory. If V is a simple vertex operator superalgebra and S is a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G, a duality theorem of Schur-Weyl type is obtained for the actions of certain finite dimensional semisimple associated algebra Aα(G,S) and VG on the direct sum of twisted V-modules in S. In particular, for any g∈G every irreducible g-twisted V-module is completely reducible VG-module.

BPS Algebras in 2D String Theory

We discuss a set of heterotic and type II string theory compactifications to 1+1 dimensions that are characterized by factorized internal worldsheet CFTs of the form V_1otimes bar{V}_2, where V_1, V_2 are self-dual (super) vertex operator algebras. In the cases with spacetime supersymmetry, we show that the BPS states form a module for a Borcherds–Kac–Moody (BKM) (super)algebra, and we prove that for each model the BKM (super)algebra is a symmetry of genus zero BPS string amplitudes. We compute the supersymmetric indices of these models using both Hamiltonian and path integral formalisms. The path integrals are manifestly automorphic forms closely related to the Borcherds–Weyl–Kac denominator. Along the way, we comment on various subtleties inherent to these low-dimensional string compactifications.

Modularity of Bershadsky–Polyakov minimal models

The Bershadsky–Polyakov algebras are the original examples of nonregular W-algebras, obtained from the affine vertex operator algebras associated with mathfrak {sl}_3 by quantum Hamiltonian reduction. In Fehily et al. (Comm Math Phys 385:859–904, 2021), we explored the representation theories of the simple quotients of these algebras when the level mathsf {k} is nondegenerate-admissible. Here, we combine these explorations with Adamović’s inverse quantum Hamiltonian reduction functors to study the modular properties of Bershadsky–Polyakov characters and deduce the associated Grothendieck fusion rules. The results are not dissimilar to those already known for the affine vertex operator algebras associated with mathfrak {sl}_2, except that the role of the Virasoro minimal models in the latter is here played by the minimal models of Zamolodchikov’s mathsf {W}_3 algebras.

The Singular Support of the Ising Model

Abstract We prove a new quasiparticle sum expression for the character of the Ising model vertex algebra, related to the Jackson–Slater $q$-series identity of Rogers–Ramanujan type and to Nahm sums for the matrix $\left (\begin {smallmatrix}8&3\\3&2 \end {smallmatrix}\right ) $. We find, as consequences, an explicit monomial basis for the Ising model and a description of its singular support. We find that the ideal sheaf of the latter, defining it as a subscheme of the arc space of its associated scheme, is finitely generated as a differential ideal. We prove three new $q$-series identities of the Rogers–Ramanujan–Slater type associated with the three irreducible modules of the Virasoro Lie algebra of central charge $1/2$. We give a combinatorial interpretation to the identity associated with the vacuum module.