Vertex Operator Algebras in Mathematics and Physics

Description of two three-dimensional topological quantum field theories of\nWitten type as twisted supersymmetric theories is presented. Low-energy\neffective action and a corresponding topological invariant of three-dimensional\nmanifolds are considered.\n

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $\sigma$ of order $T$ with $M$ as a $\sigma$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a $\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.

Binary Codes and Vertex Operator (Super)Algebras

Vertex Operator Algebras and the Monster

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.

Elements of the annihilating ideal of a standard module

We initiate the study of boundary Vertex Operator Algebras (VOAs) of topologically twisted 3d \mathcal{N}=4 𝒩 = 4 rank-0 SCFTs. This is a recently introduced class of \mathcal{N}=4 𝒩 = 4 SCFTs that by definition have zero-dimensional Higgs and Coulomb branches. We briefly explain why it is reasonable to obtain rational VOAs at the boundary of their topological twists. When a rank-0 SCFT is realized as the IR fixed point of a \mathcal{N}=2 𝒩 = 2 Lagrangian theory, we propose a technique for the explicit construction of its topological twists and boundary VOAs based on deformations of the holomorphic-topological twist of the \mathcal{N}=2 𝒩 = 2 microscopic description. We apply this technique to the B B twist of a newly discovered family of 3d \mathcal{N}=4 𝒩 = 4 rank-0 SCFTs {\mathcal T}_r 𝒯 r and argue that they admit the simple affine VOAs L_r(\mathfrak{osp}(1|2)) L r ( 𝔬 𝔰 𝔭 ( 1 | 2 ) ) at their boundary. In the simplest case, this leads to a novel level-rank duality between L_1(\mathfrak{osp}(1|2)) L 1 ( 𝔬 𝔰 𝔭 ( 1 | 2 ) ) and the minimal model M(2,5) M ( 2 , 5 ) . As an aside, we present a TQFT obtained by twisting a 3d \mathcal{N}=2 𝒩 = 2 QFT that admits the M(3,4) M ( 3 , 4 ) minimal model as a boundary VOA and briefly comment on the classical freeness of VOAs at the boundary of 3d TQFTs.

We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras A k (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type.

Decomposition of the lattice vertex operator algebra [formula omitted

The varieties of semi-conformal vectors of affine vertex operator algebras

In this paper, we study McKay’s E7 observation on the Baby Monster. By investigating the so-called derived c=7/10 Virasoro vectors, we show that there is a natural correspondence between dihedral subgroups of the Baby Monster and certain subalgebras of the Baby Monster vertex operator algebra which are constructed by the nodes of the affine E7 diagram. This allows us to reinterpret McKay’s E7 observation via the theory of vertex operator algebras. For a class of vertex operator algebras including the Moonshine module, we will show that the product of two Miyamoto involutions associated to derived c=7/10 Virasoro vectors in certain commutant vertex operator algebras is an element of order at most 4. For the case of the Moonshine module, we obtain the Baby monster vertex operator algebra as the commutant and we can identify the group generated by these Miyamoto involutions with the Baby Monster and recover the {3,4}-transposition property of the Baby Monster in terms of vertex operator algebras.

The structure of spacetime duality and discrete worldsheet symmetries of compactified string theory is examined within the framework of noncommutative geometry. The full noncommutative string spacetime is constructed using the Frohlich-Gawedzki spectral triple which incorporates the vertex operator algebra of the string theory. The duality group appears naturally as a subgroup of the automorphism group of the vertex operator algebra and spacetime duality is shown to arise as the possibility of associating two independent Dirac operators, arising from the chiral structure of the worldsheet theory, to the noncommutative geometry.

On some generalizations of Batalin-Vilkovisky algebras

In recent work, Wang and Tener defined a class of “extremal” vertex operator algebras (VOAs), consisting of those with at least two simple modules and conformal dimensions as large as possible for the central charge. In this article, we show that there are exactly 15 character vectors of extremal VOAs with two simple modules. All but one of the 15 character vectors are realized by a previously known VOA. The last character vector is realized by a new VOA with central charge 33.

The two major approaches to chiral conformal field theory - one based on operator algebras and one based on vertex algebras - both lead to representation categories which are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor cat- egories. In this Arbeitsgemeinschaft, we have studied algebraic structures related to tensor categories arising in conformal field theory. The notion of a module category over this tensor category is central in the construction of a full local conformal field theory in various frameworks.