• BMV mechanism
• anti de Sitter
• higher spin
• string theory
• string spectrum
• BRST
• free string field theory
• mixed symmetry fields

Why we need String Theory and we do not still understand it

The best theories that we have today to describe our Universe are the Standard Model of elementary particles and General Relativity. Both theories are today in striking agreement with experimental data, and yet they cannot be the end of the story since the Standard Model does not describe the Gravitational force, while General Relativity does not appear to combine nicely with Quantum Mechanics and the uncertainty principle.

The main framework currently available to overcome this problem is String Theory: if elementary particles are extended rather than pointlike objects, their collisions involve only small portions of them and scattering amplitudes are drastically softened at energies sized by the string length.

Particles emerge in String Theory as oscillation normal modes of a quantum relativistic string. Since a classical string can possess an arbitrarily high angular momentum in the center of mass frame if it rotates fast enough, these quantum relativistic strings can possess oscillations of arbitrary high spin. This is the way higher spin particles come into the game in String Theory.

However, our present understanding of String Theory is very far from complete. Any attempt to recover the Standard Model and General Relativity at low energies encounters serious conceptual problems: the string action is consistent (free of anomalies) only in ten spacetime dimensions (26 in the purely bosonic case), so that one has to get rid somehow of some extra dimensions. However, the theory does not suggest in any way how to perform this reduction, which is effected supposing that the extra dimensions be wrapped on some manifold so small that we have not been able to detect it up to now. Moreover, the theory poses rather loose constraints on the nature of this manifold: there is an enormous number of possible vacua, which give rise in principle to completely different low energy theories. At present, we still do not know whether this arbitrariness is an intrinsic feature of String Theory or is a reflection of our limited understanding of it.

String Theory is not unique even in ten dimensions, since it presents itself in five possible declinations (Type I, IIA, IIB, HE SO(32), HE E_8 x E_8). These superstring theories, however, emerge as particular limits of a unique theory defined in eleven, rather than ten, dimensions. We know very little about this so-called M-theory, but we do know that its low-energy limit is described by the eleven-dimensional supergravity and that it is not a theory of strings.

One cannot even claim that we have a good control of these five theories in ten dimensions. Indeed, we have at our disposal a neat action for a single string and a neat picture for the Hilbert space of single particle states, but we lack any geometrical characterization of the interactions. As a result, one is still bound to define the theory via the counterparts, in this context, of Feynman diagrams, representing the evolution of one or more strings in spacetime and ordered according to the number of handles, holes or crosscaps (the counterparts of loops in this more complicated setting). Although this loop expansion is neater than in Quantum Field Theory since it involves essentially one contribution at each order, in the bosonic string, where the structure of the series is relatively under control, it reveals an asymptotic nature, so that it does not converge and leaves out non-perturbative effects. Some of these have been associated, in recent years, to a rich zoo of extended objects (branes).




Strings and Higher Spins: the ideas behind this Thesis

Analyzing String Theory from the perspective of Quantum Field Theory could shed new light on both, since up to now we understand only a little corner of the possible field theories: at the quantum level we have some control only of scalars, Dirac fermions and vectors, to which one may add the gravitino and the graviton at the classical level. The study of string inspired higher-spin field theories goes in the direction of enlarging this little corner and of clarifying whether the nature of the mechanical string model deserves the central role that it has long had in the present formulation, despite the fact that the string tension lacks a dynamical origin.

Interactions of higher-spin gauge fields have always been problematic. For instance, in flat space they are constrained by a series of no-go theorems that forbid conventional interactions of massless particles of spin larger than two with anything that couples to gravity, and indeed the only theory of interacting higher-spin fields that is known at present, the Vasiliev theory, is naturally formulated around AdS space and has a singular flat limit. The particles that it describes possess the tensorial properties of the first Regge trajectory of the open bosonic string, and yet the Vasiliev theory yields an expansion around an AdS background that is not under control in String Theory.

It is therefore important to extend our understanding of detailed features of higher-spin theories to AdS space, since for one matter the remarkable Vasiliev construction is deeply rooted on algebraic properties and is largely implicit. Even with symmetric fields, the massless AdS case is more involved than the flat one, since the usual definition of masslessness related to the presence of on shell gauge symmetry allows also for intermediate (partially massless) cases, where only a subset of the flat-space gauge symmetry is preserved. However, while we have acquired by now a rather good control of free symmetric fields in AdS, one can hardly say the same for mixed-symmetry ones, which account for most massive string excitations. Here the massless limit is even more intricate due to the so-called BMV mechanism: unlike the symmetric case, it is impossible to maintain the full gauge symmetry present in flat space, and as a result what one could call an AdS massless particle of mixed symmetry has always more degrees of freedom than its flat-space counterpart. Hence, its flat limit gives rise to more than one type of particle in general.

The precise relation between String Theory and higher-spin field theory has long been a debated subject, since the first provides after all a key example of interacting (massive) particles of any spin, so that broken higher-spin symmetries seem to lie at the heart of its properties, although the link was never made precise. While the Vasiliev theory is naturally expanded around AdS, String Theory takes its simplest form around flat space, and can bypass the old no-go theorems simply because its higher-spin excitations are infinite in number and massive. A complete quantization of String Theory in AdS has not been attained yet, and the only results currently available provide indications on the purely bosonic case and in low-tension limit. Looking at string-inspired Lagrangians for higher spins in AdS can then provide a good vantage point to explore String Theory in a curved background, and at the same time it can shed some light on why the Vasiliev theory apparently describes massless excitations that only fill its first Regge trajectory.

The main purpose of this Thesis is to investigate Lagrangian formulations for higher-spin fields that are inspired by the BRST quantization of the bosonic string, in order to gain new insights on these points. Our main result is a conjecture on the spectrum of the tensile string in AdS, which rests on a motivated technical assumption of the BRST charge of the string, which provides a rationale as to why the first Regge trajectory can become naturally lighter than the rest of the spectrum, and altogether massless in a suitable limit that was anticipated long ago by Sezgin and Sundell.




Plan of the work and summary of the results

The main result of this Thesis is a conjecture on the asymptotic pattern of massive excitations that the bosonic string possesses in an AdS background, which was inspired by a detailed analysis of the tensile (and massive) string BRST charge.

The first part of the Thesis is devoted to a review of the BRST quantization of the bosonic open string, focusing on how one can extract from it higher-spin Lagrangians, with special emphasis on the emergence of "triplets", systems of three fields capable of accounting for reducible sets of excitations in the tensionless limit, in which the Weyl anomaly disappears together with the very notion of a critical dimension. After a brief review of higher-spin field theories, we come to the first part of our work, which concerns massive deformations of the BRST charge in flat space. Here we have obtained two main results. First, we have provided evidence that no obstruction is apparently encountered when trying to build a massive Q away from the critical dimension. The reason for its emergence in String Theory can be traced, in fact, to the requirement that the charge be cubic in the oscillators, while the BRST charges that we have constructed in some examples are of higher order. We have considered what happens to the BRST charge away from the critical dimension: part of the gauge symmetry is broken, and as a consequence the Lagrangian propagates extra degrees of freedom. Still, we have been able to recast this non-critical Lagrangian as a gauge fixing of a new Lagrangian enjoying an unbroken gauge symmetry, up to second mass level of the string, and we have gathered some evidence that this may be possible at any level. If one leaves aside the requirement of a cubic charge, the critical dimension D_c = 26 looses its special role, and indeed we were able to build Lagrangians with any value for D_c.

At any level starting from the third, more than one type of particle propagates. As can be seen already at level three, the fields describing them are linked together in the action. We have found a reason for this fact: cubic BRST charge forbids certain terms, and in order to avoid them the string is compelled to rotate into one another fields associated to different particles, making their decoupling highly nontrivial. In order to avoid non diagonal mass terms, the rotated fields are forced to have the same mass, and this fact provides somehow a rationale for why the string assigns to all particles at level N the mass
m^2 = (N-1) / alpha'.

The following part of the Thesis is devoted to AdS deformations of the BRST charge. In the flat case, our Lagrangians contain mass-shell terms of the form
phi (Box-m^2) phi,
and it is crucial for the nilpotency of the BRST charge that Box-m^2 commute with all possible operations that one can perform with the oscillators, namely symmetrized gradients, divergences, traces and operations on the indices. In AdS the D'Alembertian does not commute with covariant derivatives, and its most natural counterpart for totally symmetric fields is the Lichnerowicz operator:
Box_L phi = Box phi + (1 / L^2) (s(D+s-2)phi - 2g phi').
We have found a generalization of this operator for mixed-symmetry fields bearing M families of indices that has the form
Box_L = Box + (1 / L^2) ((D-M-1)S^i_i + S^i_j S^j_i) - (2 / L^2) g^{ij} T_{ij},
and we have verified at least in some examples that mass terms of the form
phi(Box_L-m_0^2)phi
are the only possible choice. In the Thesis we analyze in detail the two relatively simple and yet instructive cases of a spin 2 tensor phi_{mu nu} and of the hook, a field phi_{mu nu, rho} that is symmetric in the first two indices and whose total symmetrization vanishes identically, also taking a close look at their possible massless and partly massless limits and at the BMV mechanism.

The requirement of avoiding non-diagonal mass terms after the rotations now demands that the parameter m_0^2 be the same for all particles at the same level. But m_0^2 is not the actual mass of the excitations, which is defined through the unitarity bound, and the relation between them was actually shown to be
m^2 = m_0^2 - (1 / L^2) ((sum_i s_i (s_i+D-2i-1)) + (s_1-n-1) (s_1-n+D-2)).
As a consequence, the particles at a fixed level do not have the same mass, and the lightest are the ones belonging to the first Regge trajectory, with a mass gap at level N of at least
Delta m^2 >= (2 / L^2)(2N+D-5),
which grows linearly with N. This result completes and vindicates the original arguments of Sezgin and Sundell, since it provides a rationale for the decoupling of this trajectory from the rest of the string spectrum. Hence, it provides a justification for the very existence of the Vasiliev theory, the only known theory of interacting higher spin fields, as an effective field theory for this low-lying portion of the string spectrum.