We study the structure and representation theory of the principal W-algebra Wkpr of Vk(psl2|2). The defining operator product expansions are computed, as is the Zhu algebra, and these results are used to classify irreducible highest-weight modules. In particular, for k = +/- 12 , Wkpr is not simple and the corresponding simple quotient is the symplectic fermion vertex algebra. We use this fact, along with inverse Hamiltonian reduction, to study relaxed highest-weight and logarithmic modules for the small N = 4 superconformal algebra at central charges-9 and-3.

Quantum hamiltonian reduction is a fundamental tool of conformal field theory and vertex algebra representation theory. It has traditionally been applied to study highest-weight modules. On the other hand, inverse quantum hamiltonian reduction lends itself to the study of fully relaxed highest-weight modules and their spectral flows, sometimes called the standard modules. This is the first of several papers that study the composition of reduction and inverse-reduction functors. A general formalism is presented and exemplified with the simplest example, thereby computing the action of reduction on the standard modules of the affine vertex-operator algebra associated with 𝔰𝔩_2. The appearence of unbounded spectral sequences in this formalism may be of independent interest.

Abstract There has been a lot of recent work addressing the representation theory that underlies logarithmic conformal field theories. A full understanding of these models will however also need analytic data, in particular the correlation functions. Here, we explore the correlators of one of the most fundamental of all logarithmic models: the bosonic ghost system. In this first part, we use differential equations to show that certain correlation functions may be expressed using hypergeometric functions. Our main result is the consequent verification that there are four-point functions with logarithmic singularities. In a sequel, we will employ Coulomb gas and bootstrap methods to further refine the results presented here.

We use the newly developed technique of inverse quantum hamiltonian reduction to investigate the representation theory of the simple affine vertex algebra _2(,2) associated to 𝔰𝔩_3 at level = -3+/2 , for ⩾ 3 odd. Starting from the irreducible modules of the corresponding simple Bershadsky-Polyakov vertex operator algebras, we show that inverse reduction constructs all irreducible lower-bounded weight _2(,2) -modules. This proceeds by first constructing a complete set of coherent families of fully relaxed highest-weight _2(,2) -modules and then noting that the reducible members of these families degenerate to give all remaining irreducibles. Using this fully relaxed construction and the degenerations, we deduce modular S-transforms for certain natural generalised characters of these irreducibles and their spectral flows. With this modular data in hand, we verify that the (conjectural) standard Verlinde formula predicts Grothendieck fusion rules with nonnegative-integer multiplicities.

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_-3/2(𝔰𝔩_3) via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of 𝒰_^H(𝔰𝔩_3) , the unrolled restricted quantum group of 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 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_-3/2(𝔰𝔩_3) . We formulate an explicit conjecture relating the representation theory of W^0_A_2(2) and 𝒰_^H(𝔰𝔩_3) and work out the resulting structures of the corresponding L_-3/2(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) .

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-(3)/(2 )correspondence. We first investigate the representation theory of U-i(H)(sl(3)), the unrolled restricted quantum group of sl3 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)A(2)(0)(2)-modules. Here, (W)A(2)(0)(2) is an orbifold of the octuplet vertex algebra WA2(2) of Semikhatov, the latter being the natural sl(3)-analogue of the well known triplet algebra. Moreover, (W)A(2)(0)(2) is the parafermionic coset of the affine vertex algebra L- (3)/(2) (sl(3)) via a logarithmic Kazhdan-Lusztig (sl3). We formulate an explicit conjecture relating the representation theory of (W)A(2)(0)(2) and U-i (H)(sl(3)) and work out the resulting structures of the corresponding L-(3)/(2) 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). (sl3)-modules.

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 $\overline{U}^H_i(\mathfrak{sl}_3)$, 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 $W_{A_2}^0(2)$-modules. Here, $W_{A_2}^0(2)$ 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, $W_{A_2}^0(2)$ 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 $W_{A_2}^0(2)$ and $\overline{U}^H_i(\mathfrak{sl}_3)$ 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 $W_{A_2}(2)$.

This is the second of a series of papers devoted to the study of relaxed highest-weight modules over affine vertex algebras and W-algebras. The first [K. Kawasetsu and D. Ridout, Relaxed highest-weight modules I: Rank [Formula: see text] cases, Commun. Math. Phys. 368 (2019) 627–663, arXiv:1803.01989 [math.RT]] studied the simple “rank-[Formula: see text]” affine vertex superalgebras [Formula: see text] and [Formula: see text], with the main results including the first complete proofs of certain conjectured character formulae (as well as some entirely new ones). Here, we turn to the question of classifying relaxed highest-weight modules for simple affine vertex algebras of arbitrary rank. The key point is that this can be reduced to the classification of highest-weight modules by generalizing Olivier Mathieu’s coherent families [O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier [Formula: see text]Grenoble[Formula: see text] 50 (2000) 537–592]. We formulate this algorithmically and illustrate its practical implementation with several detailed examples. We also show how to use coherent family technology to establish the non-semisimplicity of category [Formula: see text] in one of these examples.

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 first part of this work uses the algorithm recently detailed in Kawasetsu and Ridout (Commun Contemp Math 24:2150037, 2022. arXiv:1906.02935 [math.RT]) to classify the irreducible weight modules of the minimal model vertex operator algebra $${\textsf {L} }_{{\textsf {k} }}(\mathfrak {sl}_{3})$$ , when the level $${\textsf {k} }$$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $$\widehat{\mathfrak {sl}}_{3}$$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family’s parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in Kawasetsu (Adv Math 393:108079, 2021. arXiv:2003.10148 [math.RT]), these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $$\mathfrak {sl}_{3}$$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $${\textsf {k} }=-\frac{3}{2}$$ . This gives the first nontrivial test of this formalism for a nonrational vertex operator algebra of rank greater than 1 and confirms the expectation that the methodology developed here will apply in much greater generality.

The first part of this work uses the algorithm recently detailed in Kawasetsu and Ridout (Commun Contemp Math 24:2150037, 2022. arXiv:1906.02935 [math.RT]) to classify the irreducible weight modules of the minimal model vertex operator algebra Lk(sl3), when the level k is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms ofsl3 on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family’s parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in Kawasetsu (Adv Math 393:108079, 2021. arXiv:2003.10148 [math.RT]), these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level sl3 minimal models. The second part of this work applies the standard module formalism to compute these explicitly when k = − 3 2 . This gives the first nontrivial test of this formalism for a nonrational vertex operator algebra of rank greater than 1 and confirms the expectation that the methodology developed here will apply in much greater generality.

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\hat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.

The Nappi–Witten model is a Wess–Zumino–Witten model in which the target space is the nonreductive Heisenberg group $$H_4$$ . We consider the representation theory underlying this conformal field theory. Specifically, we study the category of weight modules, with finite-dimensional weight spaces, over the associated affine vertex operator algebra $$\mathsf {H}_4$$ . In particular, we classify the irreducible $$\mathsf {H}_4$$ -modules in this category and compute their characters. We moreover observe that this category is nonsemisimple, suggesting that the Nappi–Witten model is a logarithmic conformal field theory.

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 − 3 2 (sl3) via a logarithmic Kazhdan–Lusztig correspondence. We first investigate the representation theory of Ui (sl3), the unrolled restricted quantum group of sl3 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 A2(2)modules. Here, W 0 A2(2) is an orbifold of the octuplet vertex algebra WA2(2) of Semikhatov, the latter being the natural sl3-analogue of the well known triplet algebra. Moreover, W 0 A2 (2) is the parafermionic coset of the affine vertex algebra L − 3 2 (sl3). We formulate an explicit conjecture relating the representation theory of W 0 A2(2) and U H i (sl3) and work out the resulting structures of the corresponding L − 3 2 (sl3)-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 WA2(2).

The Bershadsky–Polyakov algebras are the minimal quantum hamiltonian reductions of the affine vertex algebras associated to $$\mathfrak {sl}_{3}$$ and their simple quotients have a long history of applications in conformal field theory and string theory. Their representation theories are therefore quite interesting. Here, we classify the simple relaxed highest-weight modules, with finite-dimensional weight spaces, for all admissible but nonintegral levels, significantly generalising the known highest-weight classifications (Arakawa in Commun Math Phys 323:627–633, 2013, Adamović and Kontrec in Classification of irreducible modules for Bershadsky–Polyakov algebra at certain levels). In particular, we prove that the simple Bershadsky–Polyakov algebras with admissible nonintegral $$\mathsf {k}$$ are always rational in category $$\mathscr {O}$$ , whilst they always admit nonsemisimple relaxed highest-weight modules unless $$\mathsf {k}+\frac{3}{2} \in \mathbb {Z}_{\geqslant 0}$$ .

We show that there is a braided tensor category structure on the category of C1-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge c=13−6(t+t−1), with t∉Q, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category.

The first part of this work uses the algorithm recently detailed in arXiv:1906.02935 to classify the irreducible weight modules of the minimal model vertex operator algebra $L_k(\mathfrak{sl}_3)$, when the level $k$ is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of $\hat{\mathfrak{sl}}_3$ on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family's parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in arXiv:2003.10148, these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level $\mathfrak{sl}_3$ minimal models. The second part of this work applies the standard module formalism to compute these explicitly when $k=-\frac32$. We expect that the methodology developed here will apply in much greater generality.

We investigate a class of reducible yet indecomposable modules over the N=2 superconformal algebras. These so-called staggered modules exhibit a non-diagonalisable action of the Virasoro mode L0. Using recent results on the coset construction of N=2 minimal models, we explicitly construct such modules for central charges c=−1 and c=−6. We also describe spectral-flow orbits and symmetries of the families of staggered modules which arise via the coset.