As already stressed at the beginning of Section 3, working at low energies allows us to consider the action

In string theory, low energies means energies satisfying . This guarantees that no on-shell states will carry energies above that scale allowing one to write an effective action in terms of the fields describing massless excitations and their derivatives. The argument is valid for both the open and the closed string sectors. Furthermore, to ensure the validity of this perturbative description, one must ensure the weak coupling regime is satisfied, i.e., , to suppress higher loop world sheet contributions.Dynamically, all brane effective actions reviewed previously, describe the propagation of a brane in a fixed on-shell spacetime background solving the classical supergravity equations of motion. Thus, to justify neglecting the dynamics of the gravitational sector, focusing on the brane dynamics, one must guarantee condition (18)

but also to work in a regime where the effective Newton’s constant tends to zero. Given the low energy and weak coupling approximations, the standard lore condition for the absence of quantum gravity effects, i.e., , is naturally satisfied since . The analogous condition for 11-dimensional supergravity is .The purpose of this section is to spell out more precisely the conditions that make the above requirements not sufficient. As in any effective field theory action, one must check the validity of the assumptions made in their derivation. In our discussions, this includes

- conditions on the derivatives of brane degrees of freedom, both geometrical and world volume gauge fields, such as the value of the electric field;
- the reliability of the supergravity background;
- the absence of extra massless degrees of freedom emerging in string theory under certain circumstances.

I will break the discussion below into background and brane considerations.

Since the strong coupling limit of type IIA string theory is M-theory, which at low energies is approximated by supergravity, it is clear that there should exist further conditions. This connection involves a compactification on a circle, and it is natural to examine whether our approximations hold as soon as its size is comparable to . Using the relations (55), one learns

Thus, as soon as the M-theory circle explores subPlanckian eleven-dimensional scales, which would not allow a reliable eleven-dimensional classical description, the type IIA string coupling becomes weakly coupled, opening a possible window of reliable classical geometrical description in terms of the KK reduced configuration (54).The above discussion also applies to type IIA and IIB geometries. As soon as the scale of some compact submanifold, such as a circle, explores substringy scales, the original metric description stops being reliable. Instead, its T-dual description (58) does, using Eq. (56).

Finally, the strong coupling limit of type IIB may also allow a geometrical description given the invariance of its supergravity effective action, which includes the S-duality transformation

The latter maps a strongly coupled region to a weakly coupled one, but it also rescales the string metric. Thus, one must check whether the curvature requirements hold or not.It is important to close this discussion by reminding the reader that any classical supergravity description assumes the only relevant massless degrees of freedom are those included in the supergravity multiplet. The latter is not always true in string theory. For example, string winding modes become massless when the circle radius the string wraps goes to zero size. This is precisely the situation alluded to above, where the T-dual description, in which such modes become momentum modes, provides a T-dual reliable description in terms of supergravity multiplet fluctuations. The emergence of extra massless modes in certain classical singularities in string theory is far more general, and it can be responsible for the resolution of the singularity. The existence of extra massless modes is a quantum mechanical question that requires going beyond the supergravity approximation. What certainly remains universal is the geometrical breaking down associated with the divergence of scalar curvature invariants due to a singularity, independently of whether the latter is associated with extra massless modes or not.

Brane effective actions carrying electric fields can manifestly become ill defined for values
above a certain critical electric field for which the DBI determinant vanishes. It was
first noticed for the bosonic string in [120, 403] that such critical electric field is the value for
which the rate of Schwinger charged-string pair production [442] diverges. This divergence
captures a divergent density of string states in the presence of such critical electric field. These
calculations were extended to the superstring in [25]. The conclusion is the same, though in this
latter case the divergence applies to any pair of charge-conjugate states. Thus, there exists a
correlation between the pathological behaviour of the DBI action and the existence of string
instabilities.^{28}
Heuristically, one interprets the regime with as one where the string tension can no longer hold the string
together.^{29}

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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