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.     

An operator formulation of orbifold conformal field theory

In many applications of conformal field theory one encounters twisted conformal fields, i.e. fields which have branch cut singularities on the relevant Riemann surfaces. We present a geometrical framework describing twisted conformal fields on Riemann surfaces of arbitrary genus which is alternative to the standard method of coverings. We further illustrate the theory of twisted Grassmannians and its relation with the representation theory of the twisted oscillator algebras. As an application of the above, we expound an operator formalism for orbifold strings.     

A KN Operator Formalism for a Bosonic System

We present a KN operator formalism for a non-chiral bosonic system on higher genus Riemann surfaces using the KN bases and construct a Fock space in terms of infinitely many conserved charges.     

Deformations of complex structures and representations of Krichever-Novikov algebras

By applying the conformal Ward identities we study the representations of the Krichever-Novikov algebras associated to conformal field theories on compact Riemann surfaces. We compute the matrix elements between primary states of the KN generators corresponding to deformations of the complex structure. We show that these matrix elements depend on the derivatives of the partition function with respect to the moduli. The effects of this dependence on the highest weight representations is discussed.     

On the structure of open–closed topological field theory in two dimensions

I discuss the general formalism of two-dimensional topological field theories defined on open–closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which are responsible for the relation between the closed and open sectors. The fact that these correspondences need not be injective nor surjective has interesting implications for the problem of classifying ‘boundary conditions’. In particular, I give a clear geometric derivation of the (topological) boundary state formalism and point out some of its limitations. Finally, I formulate the problem of classifying (on-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss reducibility of boundary extensions.     

Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and n-point correlation functions for a vertex operator superalgebra on a genus two Riemann surface formed by sewing two tori together. For the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus Szegő kernels. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties. Using the bosonized formalism, a new genus two Jacobi product identity is described for the Riemann theta series. We compute and discuss the modular properties of the generating function for all n-point functions in terms of a genus two Szegő kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.     

Coulomb-gas representation on higher-genus surfaces

In this paper we use the Coulomb-gas approach to construct the minimal-model conformal blocks of higher-genus Riemann surfaces. We define the higher-genus blocks by sewing, and write them in terms of the rational blocks of a compactified scalar field. We show that spurious states decouple, which implies that the blocks degenerate correctly. As an example, we compute the genus-two partition function, and verify modular invariance for the subset of minimal models which only require one type of screening charge.     

Matrix models as CFT: Genus expansion

We show how the formulation of the matrix models as conformal field theories on a Riemann surfaces can be used to compute the genus expansion of the observables. Here we consider the simplest example of the Hermitian matrix model, where the classical solution is described by a hyperelliptic Riemann surface. To each branch point of the Riemann surface we associate an operator which represents a twist field dressed by the modes of the twisted boson. The partition function of the matrix model is computed as a correlation function of such dressed twist fields. The perturbative construction of the dressing operators yields a set of Feynman rules for the genus expansion, which involve vertices, propagators and tadpoles. The vertices are universal, the propagators and the tadpoles depend on the Riemann surface. As a demonstration we evaluate the genus-two free energy using the Feynman rules.     

Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.     

Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I

We define the partition and n-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories.     

Operator formalism on general algebraic curves

The usual Laurent expansion of the analytic tensors on the complex plane is generalized to any closed and orientable Riemann surface represented as an affine algebraic curve. As an application, the operator formalism for the b − c systems is developed. The physical states are expressed by means of creation and annihilation operators as in the complex plane and the correlation functions are evaluated starting from simple normal ordering rules. The Hilbert space of the theory exhibits an interesting internal structure, being splitted into n (n is the number of branches of the curve) independent Hilbert spaces. In this way we are able to realize new kinds of conformal field theories at genus zero with symmetry group Virⁿ ⊗ G, Vir being the Virasoro group and G denoting a discrete and nonabelian crystallographic group. Exploiting the operator formalism a large collection of explicit formulas of string theory is derived. Finally, we develop as an important byproduct new methods in order to handle differential equations related to monodromy, like the Riemann monodromy problem.     

Homotopy algebra morphism and geometry of classical string field theories

We discuss general properties of classical string field theories with symmetric vertices in the context of deformation theory. For a given conformal background, there are many string field theories corresponding to different decomposition of moduli space of Riemann surfaces. It is shown that any classical open string field theories on a fixed conformal background are A_∞-quasi-isomorphic to each other. This indicates that they have isomorphic moduli space of classical solutions. The minimal model theorem in A_∞-algebras plays a key role in these results. Its natural and geometric realization on formal supermanifolds is also given. The same results hold for classical closed string field theories, whose algebraic structures are governed by L_∞-algebras.     

The group of local biholomorphisms of ℂ1 and conformal field theory in the operator formalism

Motivated by the operator formulation of conformal field theory on Riemann surfaces, we study the properties of the infinite dimensional group of local biholomorphic transformations (conformal reparametrizations) of ¹ and develop elements of its representation theory.