Perturbative Nambu-Goto strings exist in any dimension

It is part of the narrative of string theory that quantum bosonic strings only exist in D = 26 dimensions. As usual, it is not as simple as that. Even in the well-known covariant quantization scheme, the Nambu-Goto string can be consistently quantized in any dimension D \leq 26. In the critical case D=26 the quantization just becomes particularly nice.

Based on Pohlmeyer’s charges and using techniques from Loop Quantum Gravity, the closed string can even be quantized in any dimension. However, the non-separable Hilbert spaces used in the construction are difficult to handle, so one usually tries to avoid such spaces, if possible.

Another possible approach, the one I am primarily concerned in this post, is perturbative. A framework for such a perturbative treatment was developed in a joint paper, published in CMP, with Dorothea Bahns and Kasia Rejzner. It is a conservative approach, in that it uses well-established techniques of Quantum Field Theory (QFT). Also here it turns out that there are no anomalies, yielding a well-defined effective field theory in any dimension. It can be seen as a toy model for quantum gravity, sharing some its features (diffeomorphism invariance, perturbative non-renormalizability).

The fact that the Nambu-Goto string can be quantized perturbatively in any dimension seems to be largely unknown, even to experts. Let me thus describe this approach and in particular sketch the argument for the absence of anomalies.

So how does one quantize the Nambu-Goto string perturbatively? The point is to consider, well, perturbations; more precisely, perturbations around arbitrary non-degenerate classical solutions of the Nambu-Goto string (in the following also referred to as the background). So let \Sigma \subset M be a classical solution, i.e., a world-sheet (a submanifold) in the target space M. A perturbation can be conveniently described by a vector field \varphi^a on \Sigma, taking values in T M. For an illustration, consider this Figure:

Parametrization of the dynamical world-sheet.
How a vector field on the background \Sigma with values in TM describes the dynamical world-sheet \tilde \Sigma.

For a flat target space, the dynamical world-sheet is thus given by the embedding

\tilde X^a(x) = X^a(x) + \lambda \varphi^a(x),

where X^a is the embedding of \Sigma in M, and \lambda is a formal perturbation parameter. Expanding the Nambu-Goto action in \lambda yields an action for the perturbation \varphi.

Obviously, things are not as simple as that. Due to diffeomorphism or reparametrization invariance, the equations of motion for \varphi that one obtains in this way are not hyperbolic. A gauge fixing is necessary. This can be achieved as for ordinary gauge theories, in the BRST formalism.

An infinitesimal reparametrization is given by a vector field c^\mu on \Sigma, with values in T \Sigma, cf. the following Figure:

Action of a reparametrization
A diffeomorphism c of \Sigma maps the original parametrization \varphi of the dynamical world-sheet \tilde \Sigma into another one, \rho(c) \varphi. Infinitesimally, c becomes a vector field.

The infinitesimal reparametrization acts on the field \varphi as

\rho(c) \varphi^a = c^\mu \partial_\mu \tilde X^a = c^\mu \partial_\mu X^a + \lambda c^\mu \partial_\mu \varphi^a.

One promotes c to a ghost and introduces the BRST differential s defined by

s \varphi^a = c^\mu \partial_\mu \tilde X^a, \qquad s c^\mu = \lambda c^\nu \nabla_\nu c^\mu.

In the quantization of a gauge theory, anomalies may occur, i.e., the quantum theory may no longer be invariant under the symmetry. Such anomalies can be removed unless certain cohomology classes H^g(s) and H^g(s | d) at positive ghost number g are not trivial. Hence, if one can show that these cohomology classes are trivial, then the theory can be quantized consistently.

Now the free (\lambda independent) part s_0 of the BRST differential acts as

s_0 \varphi^a = c^\mu \partial_\mu X^a, \qquad s_0 c^\mu = 0.

By the assumption of non-degeneracy of the classical solution, \partial_\mu X^a has maximal rank. It follows that all the ghost are in the range of s_0, so H^g(s_0) and H^g(s_0 | d) are trivial at positive ghost number (in other gauge theories, derivatives of ghosts appear in s_0, so only these derivatives drop out of the cohomology). Cohomological perturbation theory implies that then also H^g(s_0) and H^g(s_0 | d) are trivial. Hence, the theory can be quantized without anomalies.

A gauge fixing still needs to be performed. Suitable choices would be harmonic gauge \nabla_\mu (\sqrt{\tilde g} \tilde g^{\mu \nu}) = 0, with \tilde g the metric induced by the embedding \tilde X, or transversal gauge \varphi^a \partial_\mu X_a = 0, i.e., setting all longitudinal modes to zero. The latter seems to have been first proposed by Kleinert, and it is particularly suitable for calculations, as it turns out that the longitudinal modes which are set to zero do not occur anyway in the free part of the action. Hence, on the level of the free theory, one can work entirely only with the physical (transversal) modes.

In the end, we thus arrive at a well-defined effective field theory living on the submanifold \Sigma. This is in general a curved space-time, and the field equations only depend on the geometric data of the background, i.e., its induced metric and the second fundamental form. It is thus natural to quantize using techniques from QFT on curved space-times. In particular, one performs renormalization locally, i.e., for the definition of a local quantity at x only the background geometric data at x can be used, c.f. also my previous post. Furthermore, Poincare covariance can be ensured.

The last thing to check is that the physical subspace of the representation space, as defined via the BRST charge, is positive definite, i.e., a Hilbert space. In the paper, we checked this for an open string with Dirichlet boundary conditions, but in more recent work, I showed that this is also the case for rotating open and closed strings. These results will be discussed in a future post.

The model has an interesting feature. We can easily think of observables that measure the position of the string in target space. Just pick a test function f on the target space, restrict it to the dynamical submanifold \tilde \Sigma, and integrate it. If this observable gives a non-zero value, then the string intersects the support of f. The construction is obviously reparametrization invariant, and also independent of the split into background X and perturbation \varphi. If we now pick a background \Sigma with a non-trivial compact intersection with the support of f and expand in \lambda, we obtain a localized observable on the background \Sigma, despite diffeomorphism invariance. Hence, it is not just diffeomorphism invariance which leads to the absence of local observables in gravity, it is the absence of external structures (the target space in the present case).

You may now legitimately ask: What can we do with the theory in the end? The theory is certainly not a fundamental theory. Apart from being perturbatively non-renormalizable, there is also no preferred classical solution around which to expand, i.e., no vacuum. The closest thing to a classical ground state of the Nambu-Goto string would be an event or a world-line, but these are degenerate (one may also debate on whether one should call these solutions, as these are configurations at which the action is not continuously differentiable).

However, if one fixes a non-zero angular momentum, then there are well-defined classical ground states, i.e., rotating string solutions. Doing perturbation theory around these solutions, one can compute semi-classical corrections to the Regge trajectory, i.e., the relation between energy and angular momentum. These are interesting from a phenomenological point of view, as the Nambu-Goto string can be seen as a model for the vortex line connecting the valence quarks in mesons. Furthermore, one can compare with the corresponding results of the non-perturbative quantization schemes, in order to check whether these have the correct semi-classical limit. This will be discussed in a future post.

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