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This work is to understand structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors. In this paper, we shall describe the varieties of semi-conformal vectors of a class of even lattice vertex operator algebras with rank two to characterize these vertex operator algebras. Moreover, we shall give the G-orbit decomposition of the fixed vertex operator subalgebra V L + of a rank-two lattice vertex operator algebra V L , where G is the automorphism group preserving the conformal vector of V L +. Based on this decomposition, we classify all maximal semi-conformal subalgebras of V L +. [ABSTRACT FROM AUTHOR]

We study a class of simple vertex operator algebras of OZ-type provided that each vertex operator algebra is generated by its Griess algebra V2, where V2 is linearly spanned by Virasoro elements ωij, 1 ≤ i < j ≤ n, which satisfy some conditions and that each vertex subalgebra generated by the Virasoro vector wij is isomorphic to the unitary simple vertex operator algebra L(cm, 0), m ∈ ℤ+. [ABSTRACT FROM AUTHOR]

We prove that if V is a unitary simple holomorphic vertex operator algebra of CFT-type, then V is rational, that is, all N -gradable V -modules are direct sums of copies of V. [ABSTRACT FROM AUTHOR]

In this paper, we study simple weak and ordinary twisted modules of the extended Heisenberg-Virasoro vertex operator algebra V L ~ F (ℓ 123 , 0) . We first determine the full automorphism groups of V L ~ F (ℓ 123 , 0) for all ℓ 1 , ℓ 2 , ℓ 3 , F ∈ C . They are isomorphic to certain subgroups of the general linear group GL 2 (C) . Then for a family of finite order automorphisms σ r 1 , r 2 of V L ~ F (ℓ 123 , 0) , we show that weak σ r 1 , r 2 -twisted V L ~ F (ℓ 123 , 0) -modules are in one-to-one correspondence with restricted modules of certain Lie algebras of level ℓ 123 , where r 1 , r 2 ∈ N . By this identification and vertex algebra theory, we give complete lists of simple ordinary σ r 1 , r 2 -twisted modules over V L ~ F (ℓ 123 , 0) . The results depend on whether F or ℓ 2 is zero or not. Furthermore, simple weak σ r 1 , r 2 -twisted V L ~ F (ℓ 123 , 0) -modules are also investigated. For this, we introduce and study restricted modules (including Whittaker modules) of a new Lie algebra L r 1 , r 2 which is related to the mirror Heisenberg-Virasoro algebra. [ABSTRACT FROM AUTHOR]

We study the simplest example of mirror symmetry for 3d = 4 SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello–Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the β γ VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig–Kanade–McRae and Creutzig–McRae–Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a nontrivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations. [ABSTRACT FROM AUTHOR]

E 7 + 1 / 2 is an intermediate Lie algebra filling a hole between E 7 and E 8 in the Deligne–Cvitanović exceptional series. It was found independently by Mathur, Muhki, Sen in the classification of 2d RCFTs via modular linear differential equations (MLDE) and by Deligne, Cohen, de Man in representation theory. In this paper we propose some new vertex operator algebras (VOA) associated with E 7 + 1 / 2 and give some useful information at small levels. We conjecture that the affine VOA (E 7 + 1 / 2) k is rational if and only if the level k is at most 5, and provide some evidence from the viewpoint of MLDE. We propose a conjectural Weyl dimension formula for infinitely many irreducible representations of E 7 + 1 / 2 , which generates almost all irreducible representations of E 7 + 1 / 2 with level k ≤ 4 . More concretely, we propose the affine VOA E 7 + 1 / 2 at level 2 and the rank-two instanton VOA associated with E 7 + 1 / 2 . We compute the VOA characters and provide some coset constructions. These generalize the previous works of Kawasetsu for affine VOA E 7 + 1 / 2 at level 1 and of Arakawa–Kawasetsu at level - 5 . We then predict the conformal weights of affine VOA E 7 + 1 / 2 at level 3, 4, 5. [ABSTRACT FROM AUTHOR]

Let V be an ℕ-graded vertex algebra over a field 픽, and let g be an automorphism of V with a finite order T. Assume that the field 픽 contains a Tth primitive root of unity. We construct two associative algebras, Ag(V ) and Rg(V ), and investigate their properties and interconnections. As applications, we establish the following results: (1) If Rg(V ) is finite-dimensional, then Ag(V ) is also finite-dimensional; (2) If V is strongly finitely generated, then Ag(V ) is stably noetherian; (3) If V is strongly finitely generated and M is an irreducible admissible g-twisted V-module, then the endomorphism algebra EndV(M) is finite-dimensional. [ABSTRACT FROM AUTHOR]

In this paper, we study Z -graded vertex superalgebras over a general field of an odd prime characteristic. In particular, we study affine vertex superalgebras V g ˆ (ℓ , 0) and a family of quotient vertex superalgebras V g ˆ χ (ℓ , 0) with g assumed to be a restricted Lie superalgebra. Among the main results, we obtain a necessary and sufficient condition for a g ˆ -module of level ℓ to be a V g ˆ χ (ℓ , 0) -module. Furthermore, we study the special case with g = g 1 ¯ and introduce a quotient vertex superalgebra L g ˆ (ℓ , 0) of V g ˆ (ℓ , 0). As the main results, we prove that for ℓ ≠ 0 , vertex superalgebra L g ˆ (ℓ , 0) is simple, the adjoint module is the only irreducible N -graded module up to equivalence, and every L g ˆ (ℓ , 0) -module is isomorphic to the direct sum of some copies of L g ˆ (ℓ , 0). [ABSTRACT FROM AUTHOR]

This paper surveys the long-standing connections and impact between Vaughan Jones's theory of subfactors and various topics in mathematical physics, namely statistical mechanics, quantum field theory, quantum information, and two-dimensional conformal field theory. [ABSTRACT FROM AUTHOR]

Using the bases of principal subspaces for twisted affine Lie algebras except A 2 l (2) by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight kΛ0 and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in Hatayama et al. (2001). [ABSTRACT FROM AUTHOR]

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ 1 , σ 2 be commuting finite-order automorphisms of V and let σ = (σ 1 σ 2) − 1. Among the main results, for any σ -twisted V -module W and any nonzero complex number z , we construct a weak σ 1 ⊗ σ 2 -twisted V ⊗ V -module D σ 1 , σ 2 (z) (W) inside W ⁎. Let W 1 , W 2 be σ 1 -twisted, σ 2 -twisted V -modules, respectively. We show that P (z) -intertwining maps from W 1 ⊗ W 2 to W ⁎ are the same as homomorphisms of weak σ 1 ⊗ σ 2 -twisted V ⊗ V -modules from W 1 ⊗ W 2 into D σ 1 , σ 2 (z) (W). We also show that a P (z) -intertwining map from W 1 ⊗ W 2 to W ⁎ is equivalent to an intertwining operator of type ( W ′ W 1 W 2 ) , which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ -twisted V -module M with τ any finite-order automorphism of V , the coefficients of the q -graded trace function lie in D τ , τ − 1 (− 1) (V) and generate a τ ⊗ τ − 1 -twisted V ⊗ V -submodule isomorphic to M ⊗ M ′. [ABSTRACT FROM AUTHOR]

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 End V (M) is algebraic over F and that End V (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. [ABSTRACT FROM AUTHOR]

We classify rational simple self-dual vertex operator algebras V of CFT and of finite type with positive central charges, where d V = dim ⁡ V 1 is either 8 or 16, under the condition that the space spanned by characters of V -modules is equal to the solution space of some monic modular linear differential equation of the third order. It is shown that central charges are c = 8 for d V = 8 , and c = 4 , 16 for dim ⁡ d V = 16 , respectively. In fact, under the condition that V 1 equipped with the 0th product of V are abelian Lie algebras for d V = 8 , c = 8 and d V = 16 , c = 16 , the vertex operator algebras V are isomorphic to V 2 E 8 and V Λ 16 , respectively. For d V = 16 and c = 4 , V is isomorphic to the lattice vertex operator algebra V A 2 ⊕ A 2 . [ABSTRACT FROM AUTHOR]

An irreducible module for the parafermion vertex operator algebra K (s l 2 , k) is said to be of σ-type if an automorphism of the fusion algebra of K (s l 2 , k) of order k is trivial on it. For any integer k ≥ 3, we show that there exists an automorphism of order 2 of the subalgebra of the fusion algebra of K (s l 2 , k) 〈 θ 〉 spanned by the irreducible direct summands of σ-type irreducible K (s l 2 , k) -modules, where θ is an involution of K (s l 2 , k). We discuss some examples of such an automorphism as well. [ABSTRACT FROM AUTHOR]

This text is a short and elementary introduction to the 'monstrous moonshine' aiming to be as accessible as possible. We first review the classification of finite simple groups out of which the monster naturally arises and the latter's features that are needed to state the moonshine conjecture of Conway and Norton. Then, we motivate modular functions and forms from the classification of complex tori, with the definitions of the J-invariant and its q-expansion as a goal. We eventually provide evidence for the monstrous moonshine correspondence, state the conjecture, and then introduce the ideas that led to its proof. Lastly, we give a brief account of some recent developments and current research directions in the field. [ABSTRACT FROM AUTHOR]

Motivated by a work of Li, we study nonlocal vertex algebras and their smash products over fields of positive characteristic. Through smash products, modular vertex algebras associated with positive definite even lattices are reconstructed. This gives a different construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess in lattice vertex operator algebras over a field of characteristic zero.

In this paper we mainly study the vertex operator algebra C VA (e , f) generated by two Ising vectors e and f with 〈 e , f 〉 = 5 2 10 . We prove C VA (e , f) is isomorphic to the 6 A -algebra U 6 A constructed in [20]. We also discuss the cases 〈 e , f 〉 = 1 4 , 1 2 5 , 0 and the case 〈 e , f 〉 = 1 2 8 with (τ e τ f) 4 = 1. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the diagonal coset vertex operator algebra C (L g (k + 2 , 0) , L g (k , 0) ⊗ L g (2 , 0)) is rational and C 2 -cofinite in case g = s o (2 n) , n ≥ 3 and k is an admissible number for g ˆ. It is also shown that the diagonal coset vertex operator algebra C (L s l 2 (k + 4 , 0) , L s l 2 (k , 0) ⊗ L s l 2 (4 , 0)) is rational and C 2 -cofinite in case k is an admissible number for s l 2 ˆ. Furthermore, irreducible modules of C (L s l 2 (k + 4 , 0) , L s l 2 (k , 0) ⊗ L s l 2 (4 , 0)) are classified in case k is a positive odd integer. [ABSTRACT FROM AUTHOR]

For a fixed positive integer k , any element g of the permutation group S k 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). [ABSTRACT FROM AUTHOR]

This paper is to study vertex superalgebras over a field of prime characteristic. For a general 1 2 Z -graded vertex superalgebra V , Zhu's A (V) theory is examined with some suitable modifications. Among the main results, Zhu's algebras of certain affine vertex superalgebras and Neveu-Schwarz vertex superalgebras, and irreducible modules for certain quotient vertex superalgebras associated to p -center, are determined. Irreducible modules for Clifford vertex superalgebras are also determined and a complete reducibility theorem is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated with a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; and (iii) the lattice VOA associated with a positive definite even lattice. We prove that in all these cases, the first cohomology H 1 (V , W) consists of zero-mode derivations for every N -graded V-module W (where the grading is not necessarily given by the L(0) operator). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. The relationship between the first cohomology of the VOA and that of the associated Zhu’s algebra is also discussed. [ABSTRACT FROM AUTHOR]

This paper aims to understand Heisenberg vertex algebras in terms of their semi-conformal structures. First, we study the moduli space of conformal and semi-conformal structures on Heisenberg vertex algebras that have the standard fixed conformal gradation by describing their automorphism groups. We describe its semi-conformal vectors as pairs consisting of regular subspaces and the projections of a fixed vector space in these regular subspaces. Then by automorphism groups G of Heisenberg vertex operator algebras, we get all G -orbits of varieties of their semi-conformal vectors and give some characterizations of Heisenberg vertex operator algebras. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra and g = (1 2 ⋯ k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V⊗k. It is proved that Dong-Li-Mason's associated associative algebra Ag(V⊗k) is isomorphic to Zhu's algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V⊗k-modules and irreducible V-modules. [ABSTRACT FROM AUTHOR]

Motivated by a work of Li, we study nonlocal vertex algebras and their smash products over fields of positive characteristic. Through smash products, modular vertex algebras associated with positive definite even lattices are reconstructed. This gives a different construction of the modular vertex algebras obtained from integral forms introduced by Dong and Griess in lattice vertex operator algebras over a field of characteristic zero. [ABSTRACT FROM AUTHOR]

We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).

We prove the Feigin–Tipunin conjecture (Feigin and Tipunin in Logarithmic CFTs connected with simple Lie algebras. arXiv:1002.5047) on the geometric construction of the logarithmic W-algebras W (p) Q associated with a simply-laced simple Lie algebra g and p ∈ Z ≥ 2 . [ABSTRACT FROM AUTHOR]

We study a simple, self-dual, rational, and C 2 -cofinite vertex operator algebra of CFT-type whose simple current modules are graded by Z 2 k. Based on those simple current modules, a vertex operator algebra associated with a Z 2 k -code is constructed. The classification of irreducible modules for such a vertex operator algebra is established. Furthermore, all the irreducible modules are realized in a module for a certain lattice vertex operator algebra. [ABSTRACT FROM AUTHOR]

The structure of 3 C -algebra and 5 A -algebra constructed by Lam-Yamada-Yamauchi is studied and the uniqueness of the vertex operator algebra structure of these two algebras is established. We also give the fusion rules for these two algebras. [ABSTRACT FROM AUTHOR]

The restricted S -matrix of V G is determined for any regular vertex operator algebra V and finite automorphism group G of V. As an application, the S -matrices for cyclic permutation orbifolds of prime orders are computed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that (V # H) # H ⁎ ≅ M (n , V) as S -local vertex operator algebras, where V is a simple vertex operator algebra, and H is the group algebra of a finite subgroup of Aut (V) , and n = dim ⁡ H. [ABSTRACT FROM AUTHOR]

We prove that for any type F 4 unitary affine VOA V 𝔣 4 l , sufficiently many intertwining operators satisfy polynomial energy bounds. This finishes the Wassermann type analysis of intertwining operators for all WZW-models. [ABSTRACT FROM AUTHOR]

In this paper, under the assumption that the diagonal coset vertex operator algebra C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is rational and C 2 -cofinite, the global dimension of C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is obtained and the quantum dimensions of multiplicity spaces viewed as C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) -modules are also obtained. As an application, a method to classify irreducible modules of C (L g (k + l , 0) , L g (k , 0) ⊗ L g (l , 0)) is provided. As an example, we prove that the diagonal coset vertex operator algebra C (L E 8 (k + 2 , 0) , L E 8 (k , 0) ⊗ L E 8 (2 , 0)) is rational, C 2 -cofinite, and classify irreducible modules of C (L E 8 (k + 2 , 0) , L E 8 (k , 0) ⊗ L E 8 (2 , 0)) . [ABSTRACT FROM AUTHOR]

We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated with the integral over the quiver variety. [ABSTRACT FROM AUTHOR]

Let L be an even (positive definite) lattice and g ∈ O (L) . In this article, we prove that the orbifold vertex operator algebra V L g ^ has group-like fusion if and only if g acts trivially on the discriminant group D (L) = L ∗ / L (or equivalently (1 - g) L ∗ < L ). We also determine their fusion rings and the corresponding quadratic space structures when g is fixed point free on L. By applying our method to some coinvariant sublattices of the Leech lattice Λ , we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using the orbifold vertex operator algebra V Λ g g ^ . [ABSTRACT FROM AUTHOR]

In this paper, extensions of nonunitary rational Virasoro vertex operator algebras corresponding to some exceptional modular invariants are constructed. The uniqueness of these extensions is also established. [ABSTRACT FROM AUTHOR]

Let ℳ (p) (p = 2,3,...) be the singlet vertex operator algebra and ω its conformal vector. We classify the simple weak ℳ (p) -modules with a non-zero element u such that for some integer s ≥ 2, ω i u ∈ ℂu (i = ⌊s/2⌋ + 1,⌊s/2⌋ + 2,...,s − 1), ω s u ∈ ℂ × u , and ωiu = 0 for all i > s. [ABSTRACT FROM AUTHOR]

In this paper, twisted modules for modular affine vertex algebras V g ˆ (ℓ , 0) and for their quotient vertex algebras V g ˆ χ (ℓ , 0) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σ T = 1. It is proved that 1 T N -graded irreducible σ -twisted V g ˆ 0 (ℓ , 0) -modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u (g 0) , where g 0 is the subalgebra of σ -fixed points in g. It is also proved that when g = h is abelian, the twisted Heisenberg Lie algebra h ˆ [ σ ] is actually isomorphic to the untwisted Heisenberg Lie algebra h ˆ , unlike in the case of characteristic zero. Furthermore, for any nonzero level ℓ , irreducible σ -twisted L h ˆ (ℓ , 0) -modules are explicitly classified and the complete reducibility of every σ -twisted L h ˆ (ℓ , 0) -module is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure and representation of a 6 A -algebra which is a vertex operator algebra generated by two Ising vectors e , f with inner product 〈 e , f 〉 = 5 2 10 . In particular, we prove the uniqueness of the vertex operator algebra structure of this 6A-algebra, classify the irreducible modules, and determine the fusion rules. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra, M be an admissible V-module. An - -bimodule for any nonnegative integers m and n is constructed and studied. The fusion rules for admissible V-modules are determined in terms of bimodule. for irreducible V-module M is given explicitly if V is rational. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic p > 2. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the p -centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible N -graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible N -graded modules and they are C 2 -cofinite. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic p). We first introduce a bialgebra H and we then introduce a notion of H -module vertex algebra and a notion of (V , H) -module for an H -module vertex algebra V. Then we give a modular version of Frenkel–Huang–Lepowsky's theory and study invariant bilinear forms on an H -module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general H -module vertex algebra, and we apply our results to affine vertex algebras and Virasoro vertex algebras. [ABSTRACT FROM AUTHOR]

By using quantum vertex operators we study the invariance of the rank n free-fermion vertex algebra under the action of the group ℤ∕2ℤ and obtain its minimal generating set. When n = 1, it is well known that this subalgebra is isomorphic to the Virasoro vertex algebra with central charge 1∕2. In the n = 2 case we show that invariant subalgebra is isomorphic to a simple quotient of a certain W-algebra, which we explicitly construct. For n≥3, our approach leads to a rediscovery of the spinor representation of the affine vertex algebra associated to the Lie algebra 픰픬(n) of I. Frenkel. [ABSTRACT FROM AUTHOR]

We complete the classification of conformal embeddings of a maximally reductive subalgebra k into a simple Lie algebra g at non-integrable non-critical levels k by dealing with the case when k has rank less than that of g. We describe some remarkable instances of decomposition of the vertex algebra Vk(g) as a module for the vertex subalgebra generated by k. We discuss decompositions of conformal embeddings and constructions of new affine Howe dual pairs at negative levels. In particular, we study an example of conformal embeddings A1×A1↪C3 at level k=-1/2, and obtain explicit branching rules by applying certain q-series identity. In the analysis of conformal embedding A1×D4↪C8 at level k=-1/2 we detect subsingular vectors which do not appear in the branching rules of the classical Howe dual pairs. [ABSTRACT FROM AUTHOR]

The vertex operator algebra structure of a strongly regular holomorphic vertex operator algebra V of central charge 24 is proved to be uniquely determined by the Lie algebra structure of its weight one space V1 if V1 is a Lie algebra of the type A12 1,4, B6 2,2, B4 3,2, B³ 4,2, B² 6,2, B12,2, D² 4,2B4 2,1, D8,2B² 4,1, A43,2A41,1, D² 5,2A²3 ,1, D9,2A7,1, C4 4,1, or D6,2B² 3,1C4,1. [ABSTRACT FROM AUTHOR]