3.7 Regime of validity
After thoroughly discussing the kinematic structure of the effective action describing the propagation of
single branes in arbitrary on-shell backgrounds, I would like to reexamine the regime of validity under
which the dynamics of the full string (M-) theory reduces to
.
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.
Validity of the background description:
Whenever the supergravity approximation is not reliable, the brane
description will also break down. Assuming no extra massless degrees of freedom arise, any on-shell
type IIA/IIB supergravity configuration satisfying the conditions described above, must also avoid
Since the string coupling constant
is defined as the expectation value of
, the first condition
determines the regions of spacetime where string interactions become strongly coupled. The second
condition, or any dimensionless scalar quantity constructed out of the Riemann tensor, determines the
regions of spacetime where curvature effects cannot be neglected. Whenever there are points in our classical
geometry where any of the two conditions are satisfied, the assumptions leading to the classical equations of
motion being solved by the background under consideration are violated. Thus, our approximation is not
self-consistent in these regions.
Similar considerations apply to 11-dimensional supergravity. In this case, the first natural
condition comes from the absence of strong curvature effects, which would typically occur whenever
where once more the scalar curvature can be replaced by other curvature invariants constructed out of
the 11-dimensional Riemann tensor in appropriate units of the 11-dimensional Planck scale
.
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.
Validity of the brane description:
Besides the generic low energy and weak coupling requirements applying
to D-brane effective actions (149), the microscopic derivation of the DBI action assumed the world volume
field strength
was constant. Thus, kappa symmetric invariant D-brane effective actions ignore
corrections in derivatives of this field strength, i.e., terms like
or higher in number of derivatives.
Interestingly, these corrections map to acceleration and higher-order derivative corrections in the scalar
fields
under T-duality, see Eq. (80). Thus, there exists the further requirement that all dynamical
fields in brane effective actions are slowly varying. In Minkowski, this would correspond to conditions like
or similar tensor objects constructed with the derivative operator in appropriate string units. In a general
curved background, these conditions must be properly covariantised, although locally, the above always
applies.
Notice these conditions are analogous to the ones we would encounter in the propagation of a point
particle in a fixed background. Any corrections to geodesic motion would be parameterised by an expansion
in derivatives of the scalar fields parameterising the particle position, this time in units of the mass
particle.
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.
Heuristically, one interprets the regime with
as one where the string tension can no longer hold the string
together.