Nonzero Ghost Number Research Articles

On ghost fermions

The path integral for ghost fermions, which is heuristically made use of in the Batalin-Fradkin-Vilkovisky approach to quantization of constrained systems, is derived from first principles. The derivation turns out to be rather different from that of physical fermions since the definition of Dirac states for ghost fermions is subtle. With these results at hand, it is then shown that the nonminimal extension of the Becchi-Rouet-Stora-Tyutin operator must be chosen differently from the notorious choice made in the literature in order to avoid the boundary terms that have always plagued earlier treatments. Furthermore it is pointed out that the elimination of states with nonzero ghost number requires the introduction of a thermodynamic potential for ghosts; the reason is that Schwarz's Lefschetz formula for the partition function of the time-evolution operator is not capable, despite claims to the contrary, to get rid of nonzero ghost number states on its own. Finally, we comment on the problems of global topological nature that one faces in the attempt to obtain the solutions of the Dirac condition for physical states in a configuration space of nontrivial geometry; such complications give rise to anomalies that do not obey the Wess-Zumino consistency conditions.

The Semi-Infinite Cohomology of Affine Lie Algebras

We study in detail the semi-infinite or BRST cohomology of general affine Lie algebras. This cohomology is relevant in the BRST approach to gauged WZNW models. We prove the existence of an infinite sequence of elements in the cohomology for non-zero ghost numbers. This will imply that the BRST approach to topological WZNW model admits many more states than a conventional coset construction. This conclusion also applies to some non-topological models. Our work will also contain results on the structure of Verma modules over affine Lie algebras. In particular, we generalize the results of Verma and Bernstein-Gel'fand-Gel'fand,for finite dimensional Lie algebras, on the structure and multiplicities of Verma modules.

Spacetime diffeomorphisms and topological w ∞ symmetry in two-dimensional topological string theory

This paper analyzes spacetime symmetries of topological string theory on a two-dimensional torus, and points out that the spacetime geometry of the model is that of the Batalin-Vilkovisky formalism. Previously I found an infinite symmetry algebra in the absolute BRST cohomology of the model. Here I find an analog of the BV Δ operator, and show that if defines a natural semirelative BRST cohomology. In the absolute cohomology, the ghost-number-zero symmetries form the algebra of all infinitesimal spacetime diffeomorphisms, extended at non-zero ghost numbers to the algebra of all odd-symplectic diffeomorphisms on a spacetime supermanifold. In the semirelative cohomology, the symmetries are reduced to w ∞ at ghost number zero, and to a topologically twisted N = 2 w ∞ superalgebra when all ghost numbers are included. I discuss deformations of the model that break parts of the spacetime symmetries while preserving the topological BRST symmetry on the worldsheet. In the absolute cohomology of the deformed model, another topological w ∞ superalgebra may emerge, while the semirelative cohomology leads to a bosonic w ∞ symmetry.

Two-dimensional string theory and the topological torus

We analyze topological string theory on a two-dimensional torus, focusing on symmetries in the matter sector. Even before coupling to gravity, the topological torus has an infinite number of point-like physical observables, which give rise via the BRST descent equations to an infinite symmetry algebra of the model. The point-like observables of ghost-number zero form a topological ground ring, whose generators span a spacetime manifold; the symmetry algebra represents all (ground ring valued) diffeomorphisms of the spacetime. At nonzero ghost numbers, the topological ground ring is extended to a surprising, the spacetime manifold becomes a supermanifold, and the symmetry algebra preserves a symplectic form on it. In a decompactified limit of cylindrical target topology, we find a nilpotent charge which behaves like a spacetime topological BRST operator. After coupling to topological gravity, this model might represent a topological phase of c = 1 string theory. We also point out some analogies to two-dimensional superstrings with the chiral GSO projection, and to string theory with c = −2.

INTERACTIONS OF DISCRETE STATES WITH NONZERO GHOST NUMBER IN c = 1 2D GRAVITY

We study the interactions of the discrete states with nonzero ghost number in c = 1 two-dimensional (2D) quantum gravity. By using the vertex operator representations, it is shown that their interactions are given by the structure constants of the group of the area preserving diffeomorphism similar to those of vanishing ghost number. The effective action for these states is also worked out. The result suggests that the whole system has a BRST-like symmetry.

STATES OF NONZERO GHOST NUMBER IN c<1 MATTER COUPLED TO 2D GRAVITY

We study c<1 matter coupled to gravity in the Coulomb gas formalism using the double cohomology of the string BRST and Felder BRST charges. We find that states outside the primary conformal grid are related to the states of nonzero ghost number by means of descent equations given by the double cohomology. Some aspects of the Virasoro structure of the Liouville-Fock space are studied. As a consequence, states of nonzero ghost number are easily constructed by “solving” these descent equations. This enables us to map ghost number conserving correlation functions involving nonzero ghost number states into those involving states outside the primary conformal grid.

Spectrum of Two-Dimensional (Super) Gravity

We review the BRST analysis of the system of a (super)conformal matter coupled to 2D (super)gravity. The spectrum and its operator realization are reported. In particular, the operators associated with the states of nonzero ghost number are given. We also discuss the ground ring structure of the super-Liouville coupled to c =1 matter. In appendices, hermiticities, states for c < 1 conformal matter coupled to gravity and the proof for the spectrum are discussed

Improved covariant quantization of the heterotic superstring

A generalized lagrangian BRST formalism is developed and used to study the covariant quantization of the heterotic superstring. The quantum action is by construction invariant under the off-shell nilpotent BRST transformations, which is achieved by introducing a minimal set of auxiliary variables with nonzero ghost number. The method relies on the classical gauge structure, and represents a natural lagrangian extension of the ideas existing in the hamiltonian BRST approach.