Batalin-Vilkovisky algebras and two-dimensional topological field theories

By a Batalin-Vilkovisky algebra, we mean a graded commutative algebraA, together with an operator :A A ₊₁ such that ² = 0, and [,a]–a is a graded derivation ofA for allaA. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF     

On the geometry of multi-Dirac structures and Gerstenhaber algebras

In a companion paper, we introduced a notion of multi-Dirac structures, a graded version of Dirac structures, and we discussed their relevance for classical field theories. In the current paper we focus on the geometry of multi-Dirac structures. After recalling the basic definitions, we introduce a graded multiplication and a multi-Courant bracket on the space of sections of a multi-Dirac structure, so that the space of sections has the structure of a Gerstenhaber algebra. We then show that the graph of a k$k$-form on a manifold gives rise to a multi-Dirac structure and also that this multi-Dirac structure is integrable if and only if the corresponding form is closed. Finally, we show that the multi-Courant bracket endows a subset of the ring of differential forms with a graded Poisson bracket, and we relate this bracket to some of the multisymplectic brackets found in the literature.     

Modularity of logarithmic parafermion vertex algebras

The parafermionic cosets \(\mathsf {C}_{k} = {\text {Com}} ( \mathsf {H} , \mathsf {L}_{k}(\mathfrak {sl}_{2}) )\) are studied for negative admissible levels k, as are certain infinite-order simple current extensions \(\mathsf {B}_{k}\) of \(\mathsf {C}_{k}\). Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to \(\mathsf {C}_{k}\), irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-modules are obtained from those of \(\mathsf {L}_{k}(\mathfrak {sl}_{2})\). Assuming the validity of a certain Verlinde-type formula likewise gives the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible \(\mathsf {B}_{k}\)-modules. The irreducible \(\mathsf {C}_{k}\)- and \(\mathsf {B}_{k}\)-characters are computed and the latter are shown, when supplemented by pseudotraces, to carry a finite-dimensional representation of the modular group. The natural conjecture then is that the \(\mathsf {B}_{k}\) are \(C_2\)-cofinite vertex operator algebras.     

The real topological vertex at work

We develop the real vertex formalism for the computation of the topological string partition function with D-branes and O-planes at the fixed point locus of an anti-holomorphic involution acting non-trivially on the toric diagram of any local toric Calabi–Yau manifold. Our results cover in particular the real vertex with non-trivial fixed leg. We give a careful derivation of the relevant ingredients using duality with Chern–Simons theory on orbifolds. We show that the real vertex can also be interpreted in terms of a statistical model of symmetric crystal melting. Using this latter connection, we also assess the constant map contribution in Calabi–Yau orientifold models. We find that there are no perturbative contributions beyond one-loop, but a non-trivial sum over non-perturbative sectors, which we compare with the non-perturbative contribution to the closed string expansion.     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

The vertex formulation of the Bazhanov-Baxter model

In this paper we formulate an integrable model on the simple cubic lattice. TheN-valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex-type tetrahedron equation. In the thermodynamic limit our model is equivalent to the Bazhanov-Baxter model. In the case whenN=2 we reproduce Korepanov's and Hietarinta's solutions of the tetrahedron equation as special cases.     

On the classification of simple vertex operator algebras

Inspired by a recent work of Frenkel-Zhu, we study a class of (pre-)vertex operator algebras (voa) associated to the self-dual Lie algebras. Based on a few elementary structural results we propose thatV, the category of Z₊-graded prevoasV in whichV[0] is one-dimensional, is a proper setting in which to study and classify simple objects. The categoryV is organized into what we call the minimalk ^th types. We introduce a functor —which we call the Frenkel-Lepowsky-Meurman functor—that attaches to each object inV a Lie algebra. This is a key idea which leads us to a (relative) classification of thesimple minimal first type. We then study the set of all Virasoro structures on a fixed minimal first typeV, and show that they are in turn classified by the orbits of the automorphism group Aut((V)) in cent((V)). Many new examples of voas are given. Finally, we introduce a generalized Kac-Casimir operator and give a simple proof of the irreducibility of the prolongation modules over the affine Lie algebras.     

Fermionic construction of vertex operators for twisted affine algebras

We construct vertex operator representations of the twisted affine algebras in terms of fermionic (or perafermionic in some cases) elementary fields. The folding method applied to the extended Dynkin diagrams of the affine algebras allows us to determine explicitly these fermionic fields as vertex operators.     

Classification of screening systems for lattice vertex operator algebras

Letters in Mathematical Physics - We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra...     

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello–Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta–gamma system using the method of effective BV quantization.     

Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

Letters in Mathematical Physics - Let L be an even (positive definite) lattice and $$gin O(L)$$. In this article, we prove that the orbifold vertex operator algebra $$V_{L}^{{hat{g}}}$$ has...     

Krichever-Novikov formulation of topological conformal field theory

The Krichever-Novikov (KN) global operator formalism is applied to construct a topological conformal field theory on a compact Riemann surface from an N=2 super-conformal field theory. The topological version of the KN algebra is derived and the BRST charge is shown to be genus-dependent in this formulation. This leads to an interesting cohomology structure for the physical subspace of the Hilbert space.     

Dynamical approximation entropies and topological entropy in operator algebras

Dynamical entropy invariants, based on a general approximation approach are introduced for C^*-and W^*-algebra automorphisms. This includes a noncommutative extension of topological entropy.Supported in part by Grant No. DMS 9204174 from the National Science Foundation.     

Quantum dimensions and irreducible modules of some diagonal coset vertex operator algebras

Letters in Mathematical Physics - In this paper, under the assumption that the diagonal coset vertex operator algebra $$C(L_{mathfrak {g}}(k+l,0),L_{mathfrak {g}}(k,0)otimes L_{mathfrak...     

New integrable hierarchies from vertex operator representations of polynomial Lie algebras

We give a representation–theoretic interpretation of recent discovered coupled soliton equations using vertex operators construction of affinization of not simple but quadratic Lie algebras. In this setup we are able to obtain new integrable hierarchies coupled to each Drinfeld–Sokolov of A, B, C, D hierarchies and to construct their soliton solutions.     

On the dilaton vertex in the covariant formulation of strings

We investigate the dilaton vertex operator in the covariant formulation of strings. We find that the Faddeev-Popov ghosts play an essential role to construct conformal fields.     

Gerstenhaber Algebras and BV-Algebras in Poisson Geometry

The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we find an explicit connection between the Koszul-Brylinski operator and the modular class of a Poisson manifold. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.