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 Sbrane.

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

S ≈ S + S . (195 ) SUGRA brane
In string theory, low energies means energies E satisfying √ -- E α′ ≪ 1. 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., gs ≪ 1, 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 (18View Equation)

|T background| ≫ |T brane|, (196 ) mn mn
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., E ℓ(10) ≪ 1 p, is naturally satisfied since ( ) E ℓ(10) ∼ E √ α′ g1∕4 ≪ 1 p s. The analogous condition for 11-dimensional supergravity is Eℓp ≪ 1.

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

  1. conditions on the derivatives of brane degrees of freedom, both geometrical Xm and world volume gauge fields, such as the value of the electric field;
  2. the reliability of the supergravity background;
  3. 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 𝒩 = 2 type IIA/IIB supergravity configuration satisfying the conditions described above, must also avoid

ϕ (10) 2 e ∼ 1, ℛ (ℓp ) ≃ 1. (197 )
Since the string coupling constant gs is defined as the expectation value of ϕ e, 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
ℛℓ2 ≃ 1, (198 ) p
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 ℓ p.

Since the strong coupling limit of type IIA string theory is M-theory, which at low energies is approximated by 𝒩 = 1 d = 11 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 R is comparable to ℓp. Using the relations (55View Equation), one learns

R ∼ ℓp ⇐ ⇒ gs ∼ 1. (199 )
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 (54View Equation).

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 (58View Equation) does, using Eq. (56View Equation).

Finally, the strong coupling limit of type IIB may also allow a geometrical description given the SL (2,ℝ) invariance of its supergravity effective action, which includes the S-duality transformation

eϕ → e−ϕ. (200 )
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 ℛ (ℓ(p10))2 ≪ 1 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 (149View Equation), the microscopic derivation of the DBI action assumed the world volume field strength Fμν was constant. Thus, kappa symmetric invariant D-brane effective actions ignore corrections in derivatives of this field strength, i.e., terms like ∂ρFμν or higher in number of derivatives. Interestingly, these corrections map to acceleration and higher-order derivative corrections in the scalar fields Xm under T-duality, see Eq. (80View Equation). 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

√ -- α′∂2X ≪ ∂X, (201 )
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 E can manifestly become ill defined for values above a certain critical electric field Ecrit 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 E > Ecrit as one where the string tension can no longer hold the string together.29


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