Fermion Vertex Operator Research Articles

The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,Z)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.

Categorical Bernstein operators and the Boson-Fermion correspondence

We prove a conjecture of Cautis and Sussan providing a categorification of the Boson-Fermion correspondence as formulated by Frenkel and Kac. We lift the Bernstein operators to infinite chain complexes in Khovanov’s Heisenberg category $${\mathcal {H}}$$ and from them construct categorical analogues of the Kac-Frenkel fermionic vertex operators. These fermionic functors are then shown to satisfy categorical Clifford algebra relations, solving a conjecture of Cautis and Sussan. We also prove another conjecture of Cautis and Sussan demonstrating that the categorical Fock space representation of $${\mathcal {H}}$$ is a direct summand of the regular representation by showing that certain infinite chain complexes are categorical Fock space idempotents. In the process, we enhance the graphical calculus of $${\mathcal {H}}$$ by lifting various Littlewood-Richardson branching isomorphisms to the Karoubian envelope of $${\mathcal {H}}$$ .

Twisted Frobenius Identities from Vertex Operator Superalgebras

In consideration of the continuous orbifold partition function and a generating function for all n-point correlation functions for the rank two free fermion vertex operator superalgebra on the self-sewing torus, we introduce the twisted version of Frobenius identity.

Ward identities from recursion formulas for correlation functions in conformal field theory

A conformal block formulation for the Zhu recursion procedure in conformal field theory which allows to find $n$-point functions in terms of the lower correlations functions is introduced. Then the Zhu reduction operators acting on a tensor product of VOA modules are defined. By means of these operators we show that the Zhu reduction procedure generates explicit forms of Ward identities for conformal blocks of vertex operator algebras. Explicit examples of Ward identities for the Heisenberg and free fermionic vertex operator algebras are supplied.

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szego kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

Intertwining operators and fusion rules for vertex operator algebras arising from symplectic fermions

We determine fusion rules (dimensions of the spaces of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.

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 Szeg\"o 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 Szeg\"o kernel determinant. We also show that the Virasoro vector one point function satisfies a genus two Ward identity.

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.

Torus n-Point Functions for $${\mathbb{R}}$$ -graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds

We consider genus one n-point functions for a vertex operator superalgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orbifold n-point functions are given and we describe a generalization of Fay's trisecant identity for elliptic functions.

Composite-fermion wave functions as correlators in conformal field theory

It is known that a subset of fractional quantum Hall wave functions has been expressed as conformal field theory (CFT) correlators, notably the Laughlin wave function at filling factor $\nu=1/m$ ($m$ odd) and its quasiholes, and the Pfaffian wave function at $\nu=1/2$ and its quasiholes. We develop a general scheme for constructing composite-fermion (CF) wave functions from conformal field theory. Quasiparticles at $\nu=1/m$ are created by inserting anyonic vertex operators, $P_{\frac{1}{m}}(z)$, that replace a subset of the electron operators in the correlator. The one-quasiparticle wave function is identical to the corresponding CF wave function, and the two-quasiparticle wave function has correct fractional charge and statistics and is numerically almost identical to the corresponding CF wave function. We further show how to exactly represent the CF wavefunctions in the Jain series $\nu = s/(2sp+1)$ as the CFT correlators of a new type of fermionic vertex operators, $V_{p,n}(z)$, constructed from $n$ free compactified bosons; these operators provide the CFT representation of composite fermions carrying $2p$ flux quanta in the $n^{\rm th}$ CF Landau level. We also construct the corresponding quasiparticle- and quasihole operators and argue that they have the expected fractional charge and statistics. For filling fractions 2/5 and 3/7 we show that the chiral CFTs that describe the bulk wave functions are identical to those given by Wen's general classification of quantum Hall states in terms of $K$-matrices and $l$- and $t$-vectors, and we propose that to be generally true. Our results suggest a general procedure for constructing quasiparticle wave functions for other fractional Hall states, as well as for constructing ground states at filling fractions not contained in the principal Jain series.

Picture-changing operators and space-time supersymmetry

We explore the geometrical properties of fermionic vertex operators for a NSR superstring in order to establish the connection between world-sheet and target space supersymmetries. The mechanism of picture changing is obtained as a result of imposing certain constraints on a world-sheet gauge group of the NSR superstring theory. We find that picture-changing operators of different integer ghost numbers form a polynomial ring. By using properties of the picture-changing formalism, we establish a relation between the NSR and GS string theories. We show that, up to picture-changing transformations, the stress-energy tensor of the N = 1 NSR superstring theory can be obtained from the stress-energy tensor of the N = 1 GS superstring theory in a flat background by a simple field redefinition. The equations of motion of a GS superstring are shown to be fulfilled in the NSR operator formalism; they are also shown to be invariant under κ-symmetry, in terms of operator products in the NSR theory. This allows us to derive the space-time supersymmetry transformation laws for the NSR string theory. Then, we explore the properties of the κ-symmetry in the NSR formalism and find that it leads to some new relations between bosonic and fermionic correlation functions.

The q-Analogous Vertex Operator and Quantum Affine Algebras

The extended quantum affine algebras of q-analogous vertex operator have been constructed in this letter, and the similar algebra structure of q-analogous fermionic vertex operator is also analysed.

Level-one representations of the affine lie algebraB n (1)

We give an explicit description of the Kac-Peterson-Lepowsky construction of the basic representation for the affine Lie algebraB n (1). Using the conjugacy classes of the Weyl group ofB n, we describe all inequivalent maximal Heisenberg subalgebras of the corresponding affine Lie algebra. We associate to these Heisenberg subalgebras multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence for these fields provides us with fermionic vertex operators, whose ‘normal ordered products’ give the (twisted) vertex operators of the Kac-Peterson-Lepowsky construction.

Level one representations of the twisted affine algebrasa n (2) andd n (2)

We give an explicit description of the Kac-Peterson-Lepowsky construction of the level one representations for the twisted affine Lie algebrasan(2) anddn(2). We assign to any conjugacy class of the Weyl group ofan anddn 1) an equivalent class of maximal Heisenberg subalgebras of the corresponding twisted affine Lie algebras, and 2) multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence for these fields provides us with fermionic vertex operators, whose ‘normal ordered products’ give the (twisted) vertex operators of the Kac-Peterson-Lepowsky construction.

STRUCTURE OF IRREDUCIBLE SU(2) PARAFERMION MODULES DERIVED VIA THE FEIGIN-FUCHS CONSTRUCTION

We show how the Feigin-Fuchs Coulomb-gas construction, with two free Gaussian bosons, can be used to derive the representation theory of the SU(2) parafermion models. We identify the generators of the chiral algebra within the bosonic Fock space and derive the chiral algebra of the finitely reducible models, which correspond to the SU(2) and SU(1, 1) parafermion algebras. We then focus on the SU(2) highest-weight modules in the remainder of the paper. Unitarity of the modules requires that the states of the parafermion theory be independent of the zero modes of two fermionic vertex operators of the bosonic theory. The expressions for the Virasoro highest weights of the models are doubly degenerate in the bosonic Fock space. We formulate the correlation functions of these operators in the parafermion Hilbert space, and in particular, the fusion rules for the Virasoro highest weights are derived in an elegant way. Finally, the irreducible parafermion characters are derived. We discuss the connection between our analysis and previous work on representation theory based on BRST cohomology.

Auxiliary supergravity fields, Chern-Simons terms and the heterotic string

The generating functional of target space S-matrix amplitudes is generalized to include auxiliary field and fermionic vertex operators. Using this formalism, a new Green-Schwarz mechanism associated with the auxiliary vector field of the four-dimensional, N=1 supergravity multiplet of the heterotic string is derived.

Covariant fermionic and heterotic strings from the bosonic string

We discuss the embedding of the covariantly formulated superstring and heterotic string theories into the 26-dimensional bosonic string. The construction is based on equalities among bosonic and fermionic one-loop partition functions and, in particular, shows how the one-loop partition function of the superconformal ghost system can be obtained from the compactified bosonic string. It provides a natural expression for the covariant fermion vertex operator in the context of the bosonic string. We also discuss the necessary truncations which are needed to obtain the covariant heterotic string by compactification of the bosonic string on self-dual lattices.

(2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory

Criteria for unbroken N=1 space-time supersymmetry in the heterotic string theory in the presence of background fields are discussed. We make use of the construction of the fermion vertex operator in the Neveu-Schwarz-Ramond model. (2, 0) world-sheet supersymmetry is shown to be one of the necessary conditions for space-time supersymmetry in most cases. Constraints on the various background fields implied by (2, 0) world-sheet supersymmetry are derived, taking into account the effect of σ-model loop corrections. Special care is taken to study the effect of local Lorents and gauge anomaly on these constraints. Our analysis determines the constraints unambigously up to field redefinitions.

Covariant quantization of supersymmetric string theories: The spinor field of the Ramond-Neveu-Schwarz model

A brief review of superconformal field theory and superstrings is presented. The spacetime spinor contribution to the fermion vertex operator is constructed and the four-fermion amplitude is calculated using the differential equation method in the SO(1, 9) current algebra. The spinor field is also described as a vertex operator in the bosonization of the Ramond-Neveu-Schwarz fermions, and the two-cocycle for the fermion vertex is given.