No-ghost Theorem Research Articles

Monstrous Moonshine for Integral Group Rings

Abstract We propose a conjecture that is a substantial generalization of the genus zero assertions in both Monstrous Moonshine and Modular Moonshine. Our conjecture essentially asserts that if we are given any homomorphism to the complex numbers from a representation ring of a group ring for a subgroup of the Monster, we obtain a Hauptmodul by applying this homomorphism to a self-dual integral form of the Moonshine module. We reduce this conjecture to the genus-zero problem for “quasi-replicable” functions, by applying Borcherds’ integral form of the Goddard–Thorn no-ghost theorem together with some analysis of the Laplacian on an integral form of the Monster Lie algebra. We prove our conjecture for cyclic subgroups of the Monster generated by elements in class 4A, and we explicitly determine the multiplicities for a decomposition of the integral Moonshine Module into indecomposable modules of the integral group rings for these groups.

On unitarity of tree-level string amplitudes

Four-particle tree-level scattering amplitudes in string theory are magically consistent with unitarity, reflected in the non-trivial fact that beneath the critical dimension, the residues of the amplitudes on massive poles can be expanded in partial waves with all positive coefficients. While this follows (rather indirectly) from the no-ghost theorem, the simplicity of the statement and its fundamental importance for the physical consistency of string theory begs for a more direct and elementary understanding. In this note we take a step in this direction by presenting a new expression for the partial wave coefficients of string amplitudes, given by surprisingly simple double/triple contour integrals for open/closed strings. This representation allows us to directly prove unitarity of all superstring theories in D ≤ 6 spacetime dimensions, and can also be used to determine various asymptotics of the partial waves at large mass levels.

BRST cohomology of timelike Liouville theory

We compute the Hermitian sector of the relative BRST cohomology of the spacelike and timelike Liouville theories with generic real central charge cL in each case, coupled to a spacelike Coulomb gas and a generic transverse CFT. This paper is a companion of [1], and its main goal is to completely characterize the cohomology of the timelike theory with cL≤ 1 which was defined there. We also apply our formulas to revisit the BRST cohomology of the spacelike Liouville theory with cL> 1, which includes generalized minimal gravity. We prove a no-ghost theorem for the Hermitian sector in the timelike theory and for some spacelike models.

Quantum gravity from timelike Liouville theory

A proper definition of the path integral of quantum gravity has been a long- standing puzzle because the Weyl factor of the Euclidean metric has a wrong-sign kinetic term. We propose a definition of two-dimensional Liouville quantum gravity with cos- mological constant using conformal bootstrap for the timelike Liouville theory coupled to supercritical matter. We prove a no-ghost theorem for the states in the BRST cohomology. We show that the four-point function constructed by gluing the timelike Liouville three- point functions is well defined and crossing symmetric (numerically) for external Liouville energies corresponding to all physical states in the BRST cohomology with the choice of the Ribault-Santachiara contour for the internal energy.

Light-cone reduction of Witten\u2019s open string field theory

We elucidate some exact relations between light-cone and covariant string field theories on the basis of the homological perturbation lemma for A∞. The covariant string field splits into the light-cone string field and trivial excitations of BRST quartets: the latter generates the gauge symmetry and covariance. We first show that the reduction of gauge degrees can be performed by applying the lemma, which gives a refined version of the no-ghost theorem of covariant strings. Then, we demonstrate that after the reduction, gauge-fixed theory can be regarded as a kind of effective field theory and it provides an exact gauge-fixing procedure taking into account interactions. As a result, a novel light-cone string field theory is obtained from Witten’s open string field theory.

Fricke Lie algebras and the genus zero property in Moonshine

We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in generalized Monstrous Moonshine, using a Borcherds–Kac–Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group.

String theory in target space

It is argued that the complete S-matrix of string theory at tree level in a flat background can be obtained from a small set of target space properties, without recourse to the worldsheet description. The main non-standard inputs are (generalised) Britto-Cachazo-Feng-Witten shifts, as well as the monodromy relations for open string theory and the Kawai-Lewellen-Tye relations for closed string theory. The roots of the scattering amplitudes and especially their appearance in the residues at the kinematic poles are central to the story. These residues determine the amplitudes through on-shell recursion relations. Several checks of the formalism are presented, including a computation of the Koba-Nielsen amplitude in the bosonic string. Furthermore the question of target space unitarity is (re-)investigated. For the Veneziano amplitude this question is reduced by Poincare invariance, unitarity and locality to that of positivity of a particular numerical sum. Interestingly, this analysis produces the main conditions of the no-ghost theorem on dimension and intercept from the first three poles of this amplitude.

Covariant approach to the no-ghost theorem in massive gravity

We discuss the no-ghost theorem in the massive gravity in a covariant manner. Using the BRST formalism and St\"{u}ckelberg fields, we first clarify how the Boulware-Deser ghost decouples in the massive gravity theory with Fierz-Pauli mass term. Here we find that the crucial point in the proof is that there is no higher (time) derivative for the St\"{u}ckelberg `scalar' field. We then analyze the nonlinear massive gravity proposed by de Rham, Gabadadze and Tolley, and show that there is no ghost for general admissible backgrounds. In this process, we find a very nontrivial decoupling limit for general backgrounds. We end the paper by demonstrating the general results explicitly in a nontrivial example where there apparently appear higher time derivatives for St\"{u}ckelberg scalar field, but show that this does not introduce the ghost into the theory.

Null physical states in string models

This note is a brief addendum to my article Nucl. Phys. B 864 (2012) 285, [arXiv: 1110.5510], which discusses the noghost theorem in Ramond sectors of string models. In this addendum we derive additional information about the structure of null physical states in the Ramond-Neveu-Schwarz model.

Brane induced gravity: From a no-go to a no-ghost theorem

Numerous claims in the literature suggest that gravity induced on a higher codimensional surface violates unitarity in the weak coupling regime. However, it remained unclear, why a conserved source localized on this surface and giving rise to an induced gravity term at low energies would absorb and emit the associated ghost, given a consistent source-free theory. In this article it is shown that the appearance of the induced Einstein-Hilbert term does not threaten the unitarity of the theory. It is shown that the would-be ghost highlighted in previous works is nondynamical and therefore not associated with a state in the Hilbert space. The physics arguments behind this statement are presented in a semicovariant language, but the detailed proof is given using Dirac's constraint analysis. The Hamiltonian on the constraint surface of the linearized theory is derived and turns out to be manifestly positive definite. As a result of these investigations, brane-induced gravity goes without a ghost, opening an exciting window of opportunity for consistent deformations of gravity at the largest observable distances.

Improved proof of the no-ghost theorem for fermion states of the superstring

The purpose of this note is to extend the improved proof of the no-ghost theorem for the bosonic and Neveu–Schwarz dual resonance models, presented in my article [C.B. Thorn, Nucl. Phys. B 286 (1987) 61], to cover the Ramond fermion string. As in that paper, the improvement involves the identification of an efficient basis for string state space and a self-contained proof, based on the super-Virasoro algebra, of the linear independence of the basis elements. We use our results to calculate the BRST cohomology for this system.

UNITARITY FOR THE BOSONIC NAPPI–WITTEN STRING FROM BRST VIEW

In this paper, we present the proof of a no-ghost theorem for the bosonic string in Nappi–Witten spacetime by the BRST-technology. The key point in this proof is to compare the characters with the signature functions, where traces are taken over the whole state space. It includes the states of Nappi–Witten string, the states of some extra unitary CFT and the conformal ghost states. Only the signature function of the Nappi–Witten string part is not trivial. Here we obtain it in a new complete basis for the states space of Nappi–Witten string. For all real number b, which is a parameter of the Nappi–Witten metric, the characters and the signature functions are exactly equivalent. Then we can state that the physical string states in Nappi–Witten group are all of non-negative norm.

No-Ghost Theorem for Neveu-Schwarz String in 0-Picture via Similarity Transformation

A proof of the no-ghost theorem for Neveu-Schwarz string in 0-picture is outlined in a manner using a similarity transformation. It is shown that a nontrivial metric consistent with the BRST cohomology is needed to define a positive semidefinite norm in the physical Hilbert space. The one-to-one correspondence between physical states in 0-picture and those in the conventional (−1)-picture are confirmed. As a by-product, we find a new inverse picturechanging operator, which is noncovariant but has nonsingular operator product with itself. A possibility to construct a new gauge-invariant superstring field theory is discussed.

No-Ghost Theorem for Neveu-Schwarz String in 0-Picture

The no-ghost theorem for Neveu-Schwarz string is directly proved in 0-picture. The one-to-one correspondence between physical states in 0-picture and those in the conventional (-1)-picture are confirmed. It is shown that a non-trivial metric consistent with the BRST cohomology is needed to define a positive semi-definite norm in the physical Hilbert space. As a by-product, we find a new inverse picture changing operator, which is non-covariant but has non-singular operator product with itself. A possibility to construct a new gauge invariant superstring field theory is discussed.

No-ghost theorem for the bosonic Nappi-Witten string

We prove a no-ghost theorem for a bosonic string propagating in Nappi-Witten spacetime. This is achieved in two steps. We first demonstrate unitarity for a class of NW/U(1) modules: the norm of any ...

No-Ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model

A new realization of the fourth-order derivative Pais-Uhlenbeck oscillator is constructed. This realization possesses no states of negative norm and has a real energy spectrum that is bounded below. The key to this construction is the recognition that in this realization the Hamiltonian is not Dirac Hermitian. However, the Hamiltonian is symmetric under combined space reflection P and time reversal T. The Hilbert space that is appropriate for this PT-symmetric Hamiltonian is identified and it is found to have a positive-definite inner product. Furthermore, the time-evolution operator is unitary.

On the No-Ghost Theorem in String Theory

We give a simple proof of the no-ghost theorem in the critical bosonic string theory by using a similarity transformation.

The no-ghost theorem in curved backgrounds with a timelike u(1): NSR string

It is well known that the standard no-ghost theorem is valid as long as the background has the light-cone directions. We prove the no-ghost theorem for the NSR string when only the timelike direction is flat. This is done by the BRST quantization, using the technique of Frenkel, Garland and Zuckerman and our previous results for the bosonic string. The theorem actually applies as long as the timelike direction is written as a u(1) SCFT.

The BRST quantization and the no-ghost theorem forAdS3

In our previous papers, we prove the no-ghost theorem without light-cone directions (hep-th/0005002, hep-th/0303051). We point out that our results are valid for more general backgrounds. In particular, we prove the no-ghost theorem for AdS_3 in the context of the BRST quantization (with the standard restriction on the spin). We compare our BRST proof with the OCQ proof and establish the BRST-OCQ equivalence for AdS_3. The key in both approaches lies in the certain structure of the matter Hilbert space as a product of two Verma modules. We also present the no-ghost theorem in the most general form.

BRST quantization of string theory inAdS3

We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed.