Vertex Operators Research Articles

A Kazhdan–Lusztig Correspondence for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ , the unrolled restricted quantum group of $$\mathfrak {sl}_3$$ at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of $$W^0_{\!A_2}(2)$$ -modules. Here, $$W^0_{\!A_2}(2)$$ is an orbifold of the octuplet vertex algebra $$W_{\!A_2}(2)$$ of Semikhatov, the latter being the natural $$\mathfrak {sl}_3$$ -analogue of the well known triplet algebra. Moreover, $$W^0_{\!A_2}(2)$$ is the parafermionic coset of the affine vertex algebra $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ . We formulate an explicit conjecture relating the representation theory of $$W^0_{\!A_2}(2)$$ and $$\overline{\mathcal {U}}_{\textsf{i}}^H\!(\mathfrak {sl}_3)$$ and work out the resulting structures of the corresponding $$L_{-\frac{3}{2}}(\mathfrak {sl}_3)$$ -modules. In particular, we obtain conjectural Loewy diagrams for the latter’s projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde’s formula. Finally, we give analogous results for $$W_{\!A_2}(2)$$ .

Mini-Workshop: Recent Developments in Representation Theory and Mathematical Physics

This mini-workshop was devoted to foster the interactions between mathematicians and mathematical physicists who are working on questions related to representation theory. This includes for example the representation theory of supergroups, vertex operator algebras and quantum groups. Another focus was on link and manifold invariants and TQFTs.

Worldsheet computation of heavy-light correlators

We compute a large collection of string worldsheet correlators describing light probes interacting with heavy black hole microstates. The heavy states consist of NS5 branes carrying momentum and/or fundamental string charge. In the fivebrane decoupling limit, worldsheet string theory on a family of such backgrounds is given by exactly solvable null-gauged WZW models. We construct physical vertex operators in these cosets, including all massless fluctuations. We compute a large class of novel heavy-light-light-heavy correlators in the AdS3 limit, where the light operators include those dual to chiral primaries of the holographically dual CFT. We compare a subset of these correlators to the holographic CFT at the symmetric product orbifold point, and find precise agreement in all cases, including for light operators in twisted sectors of the orbifold CFT. The agreement is highly non-trivial, and includes amplitudes that describe the analogue of Hawking radiation for these microstates. We further derive a formula for worldsheet correlators consisting of n light insertions on these backgrounds, and discuss which subset of these correlators are likely to be protected. As a test, we compute a heavy-light five-point function, obtaining precisely the same result both from the worldsheet and the symmetric orbifold CFT. This paper is a companion to and extension of [1].

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.

(0,2) Mirror Symmetry on Homogeneous Hopf Surfaces

Abstract In this work, we find the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. For this, we follow Borisov’s approach to mirror symmetry using vertex algebras and the chiral de Rham complex. Our examples of (0,2) mirrors are given by pairs of Hopf surfaces endowed with a Bismut-flat pluriclosed metric. Requiring that the geometry is homogeneous, we reduce the problem to the study of Killing spinors on a quadratic Lie algebra and the construction of embeddings of the $N=2$ superconformal vertex algebra in the superaffine vertex algebra, combined with topological T-duality.

More on Seiberg-Witten theory and Monstrous moonshine: A new simple method of calculating the prepotential

We continue the study of a relationship between the instanton expansion of the Seiberg-Witten (SW) prepotential of D=4, N=2SU(2) SUSY gauge theory and the Monstrous moonshine. As was done in [10], we expand the inverse of the modular j-function in u−1 around u=∞, where u is the familiar u parameter for the respective SW curves, and compute the complex gauge coupling τ as a function of the scalar vev a by using the Fourier expansion of j(τ) and the relation between u and a obtained by the Picard-Fuchs equation. In this way, the instanton expansion of the prepotential is related to the dimensions of representations of the Monster group. We show that, for the cases of Nf=2 and 3, q again has an expansion whose coefficients are all integer-coefficient polynomials of the moonshine coefficients if the expansion variables are appropriately chosen. This hints some unknown relation between the Liouville CFT and the vertex operator algebra CFT with different central charges. We also demonstrate that this new method of calculating the SW prepotential developed here is useful by performing some explicit computations.

Meromorphic cosets and the classification of three-character CFT

We investigate the admissible vector-valued modular forms having three independent characters and vanishing Wronskian index and determine which ones correspond to genuine 2d conformal field theories. This is done by finding bilinear coset-type relations that pair them into meromorphic characters with central charges 8, 16, 24, 32 and 40. Such pairings allow us to identify some characters with definite CFTs and rule out others. As a key result we classify all unitary three-character CFT with vanishing Wronskian index, excluding c = 8, 16. The complete list has two infinite affine series Br,1, Dr,1 and 45 additional theories. As a by-product, at higher values of the total central charge we also find constraints on the existence or otherwise of meromorphic theories. We separately list several cases that potentially correspond to Intermediate Vertex Operator Algebras.

Unitary forms for holomorphic vertex operator algebras of central charge 24

We prove that all holomorphic vertex operator algebras of central charge $24$ with non-trivial weight one subspaces are unitary. The main method is to use the orbifold construction of a holomorphic VOA $V$ of central charge $24$ directly from a Niemeier lattice VOA $V_N$. We show that it is possible to extend the unitary form for the lattice VOA $V_N$ to the holomorphic VOA $V$ by using the orbifold construction and some information of the automorphism group $\mathrm{Aut}(V)$.

Primitive axial algebras are of Jordan type

The notion of axial algebra is closely related to 3-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.

Vertex algebras and quantum master equation

We contribute to study the two dimensional chiral conformal field theories that arise from chiral deformations of free theories such as the $\beta \gamma$‑system, bc‑system and chiral bosons. Our method is based on the effective Batalin–Vilkovisky quantization theory. We establish an exact correspondence between renormalized quantum master equations for chiral deformations and Maurer–Cartan equations for the Lie algebra of modes of chiral vertex algebras. The generating functions on the torus are proven to have modular property with mild holomorphic anomaly. As an application, we construct an exact solution of quantum B‑model (BCOV theory) in complex one dimension that solves the full higher genus mirror symmetry conjecture on elliptic curves.

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∈N, we prove a recursion relation in the nth level Zhu algebra An(V) and give a closed formula for this relation. We use this and other properties of An(V) to reduce the modes of u that appear in the generators for An(V) as long as u∈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 An(V) involving such an element u. We present general techniques that can be applied once a set of reasonable generators is determined for An(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 An(V) for n∈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∈N.

A proof for string three-point functions in AdS3

Correlation functions of the SL(2,ℝ)-WZW model involving spectrally flowed vertex operators are notoriously difficult to compute. An explicit integral expression for the corresponding three-point functions was recently conjectured in [1]. In this paper, we provide a proof for this conjecture. For this, we extend the methods of [2] based on the so-called SL(2,ℝ) series identifications, which relate vertex operators belonging to different spectral flow sectors. We also highlight the role of holomorphic covering maps in this context. Our results constitute an important milestone for proving this instance of the AdS3/CFT2 holographic duality at finite ’t Hooft coupling.

Orbifolds of Gaiotto-Rapčák Y-algebras

The universal two-parameter W∞-algebra is a classifying object for vertex algebras of type W(2,3,…,N) for some N. Gaiotto and Rapčák recently introduced a large family of such vertex algebras called Y-algebras, which includes many known examples such as the principal W-algebras of type A. These algebras admit an action of Z2, and in this paper we study the structure of their orbifolds. Aside from the extremal cases of either the Virasoro algebra or the W3-algebra, we show that these orbifolds are generated by a single field in conformal weight 4, and we give strong finite generating sets.

Charge imbalance resolved Rényi negativity for free compact boson: Two disjoint interval case

In this paper, we study the symmetry decomposition of Rényi negativity into charge imbalance sectors for the 1+1 dimensional free compact boson field with a global U(1) symmetry in the ground state for the case of two disjoint intervals. We obtain multicharged and charged Rényi negativity moments by computing the four-point correlator of flux-generating vertex operators on the Riemann surface. We then obtain charge imbalance resolved Rényi negativity by taking the Fourier transform of the charged moments. Finally, we match our results against the tight-binding model as a numerical check.

A Proof of the Fusion Rules Theorem

We prove that the space of intertwining operators associated with certain admissible modules over vertex operator algebras is isomorphic to a quotient of the vector space of conformal blocks on a three-pointed rational curve defined by the same data. This provides a new proof and alternative version of Frenkel and Zhu’s fusion rules theorem, in terms of the dimension of certain bimodules over Zhu’s algebra, without the assumption of rationality.

Endomorphism property of vertex operator algebras over arbitrary fields

In this paper, we study the endomorphism properties of vertex operator algebras over an arbitrary field F, with Char(F)≠2. Let V be a strongly finitely generated vertex operator algebra over F, and M be an irreducible admissible V-module. We prove that every element in EndV(M) is algebraic over F and that EndV(M) is also finite-dimensional. As an application, we prove Schur's lemma for strongly finitely generated vertex operator algebras over arbitrary algebraically closed fields, and we give a test for absolute irreducibility of V-modules.

A worldsheet description of instant folded strings

Time-like linear dilaton backgrounds admit a classical solution that describes a closed folded string that is created at an instant. We refer to such strings as Instant Folded Strings (IFS). We study an exact worldsheet CFT description of an IFS that involves two vertex operators which describe two open string modes that propagate on a time-like FZZT-brane, which plays the role of a regulator to the IFS. We take advantage of this description to calculate the most basic quantity associated with IFSs — their production rate. Some implications of this calculation to stringy cosmology and black hole interior are briefly discussed.

Linear Transformations of Vertex Operators of Hall–Littlewood Polynomials

We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall–Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas, and generating functions. We prove that specializations of linearly transformed Hall–Littlewood polynomials describe all polynomial tau-functions of the KP and BKP hierarchies. Examples of linear transformations are related to multiparameter symmetric functions, Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature.

Extended plethystic vertex operators and plethystic universal characters

Extended plethystic vertex operators and plethystic universal characters

Cosets of Free Field Algebras via Arc Spaces

AbstractUsing the invariant theory of arc spaces, we find minimal strong generating sets for certain cosets of affine vertex algebras inside free field algebras that are related to classical Howe duality. These results have several applications. First, for any vertex algebra ${{\mathcal {V}}}$, we have a surjective homomorphism of differential algebras $\mathbb {C}[J_{\infty }(X_{{{\mathcal {V}}}})] \rightarrow \text {gr}^{F}({{\mathcal {V}}})$; equivalently, the singular support of ${{\mathcal {V}}}$ is a closed subscheme of the arc space of the associated scheme $X_{{{\mathcal {V}}}}$. We give many new examples of classically free vertex algebras (i.e., this map is an isomorphism), including $L_{k}({{\mathfrak {s}}}{{\mathfrak {p}}}_{2n})$ for all positive integers $n$ and $k$. We also give new examples where the kernel of this map is nontrivial but is finitely generated as a differential ideal. Next, we prove a coset realization of the subregular ${{\mathcal {W}}}$-algebra of ${{\mathfrak {s}}}{{\mathfrak {l}}}_{n}$ at a critical level that was previously conjectured by Creutzig, Gao, and the 1st author. Finally, we give some new level-rank dualities involving affine vertex superalgebras.