q In this expression, ∇ is the covariant derivative and H = dB. Scale invariance is obtained through the conditions
βG =βB =βΦ = 0 . (4.4)
For the flat background spacetime theseβ-functions vanish, except for the first term inβΦ, which gives again the condition on the number of spacetime
dimensions D.
The most remarkable property of these equations is that they can be integrated. In fact it is possible to obtain them as variations of a spacetime action. This action is given by
S = 1 2κ2 0 ) dDx√−Ge−2Φ 2 −2(D−26) 3α( +R− 1 12H 2+ 4|∂µΦ|2+O(α() 3 . (4.5) We will encounter a technically similar situation, though for open strings, in the investigation of the topological theory. The string map is then re- stricted to constant maps, so that the integration of the β-functions yields an effective static spacetime potential.
4.3
Open string effective action
In the open string case similar methods can be applied to the σ-model action. The difference here is that now additional boundary fields are
4.3 Open string effective action 39
present. The boundary action which can be added has the form
)
dx0-T(X) +Aµ(X) ˙Xµ+· · ·. , (4.6)
whereT is the open string tachyon,Adenotes the photon field and the dots stand for massive fields. While this σ-model partition function approach was successful for the massless string modes leading to covariant expressions to all orders in powers of gravitons and dilatons in the closed string case and the vector field strength in the open string case, it produced unfamiliar expressions when applied to the tachyon field T. The expression for the partition functionZ[T] computed by expanding in derivatives ofT has the following structure in the critical bosonic string theory (both in the closed string case on 2-sphere and open string case on the disk):
Z =a0
)
dDXe−T *1 +a1α(∂2T +O(α(2)
+
. (4.7)
The constants a0 and a1 are renormalised constants, where a1 is scheme
dependent. Again, as in the closed string case, some extra input or guiding principle is necessary to fix an off-shell extension of scattering amplitudes. Reverting to the original boundaryσ-model with tachyonic and massless modes only, one notices that the model is renormalisable within the stan- dard derivative expansion, i.e. the space of boundary couplings is closed un- der renormalisation group operations. As has been argued in [132, 134, 130], the effective action for the massless fields should be given by the renor- malised partition function. This conjecture holds up to the first few orders [11]. However, when in addition to the photon field also a tachyon field is admitted, then one finds that the tachyon generates a potential. This requires but a modification of the conjecture S[T, A] =Z[T, A]ren, and we will see in chapter 8 how this can be implemented.
Part II
Closed string deformations in
open string field theory
Chapter 5
Boundary string field theory
5.1
Generalities
The formulation of string theory as it has been presented in the previous chapter is a perturbative formulation. Implicitly it is assumed that the theory can be formulated as expansion around a fixed configuration point, a consistent vacuum. This is most obvious when employing the language of conformal field theories as they, by definition, describe a renormalisation group fixed point. This immediately raises the question, whether it is possible to formulate, in the spirit of quantum field theory, a string field theory. Despite the success of the conformal field theory description of string theory it is clear, that in the end the theory can only be finalised by setting up a ‘second quantised’ version.
Such a string field theory would incorporate all possible vacua as classical configurations, would contain an understanding of non-perturbative phe- nomena like solitonic connections between distant points in moduli space and would possibly also contain M-theory as limit. In any case it is reason- able to assume that such a second quantised version should be evidently built upon fundamental principles of string theory – principles, which are certainly not (easily) accessible in perturbative formulations. Maybe the lacking of such basic insight is the greatest flaw in modern string research; or the greatest challenge. A big problem is, for example, to identify the overall dynamical degrees of freedom which are not only valid in a certain
44 5. Boundary string field theory
region of moduli space. In connection with this it seems that a subtle relation between open and closed strings play an important role.
Most excitingly a certain correspondence between them is obvious al- ready in the earliest calculations of scattering amplitudes. Although ini- tially open and closed strings are treated as different objects, it becomes clear that open string amplitudes contain closed strings as intermediate states as soon as one goes beyond tree-level. The heuristic picture of one- loop open strings which look like tree-level closed strings support this ob- servation. Over the years a number of evidences has been found from various areas in string theory pointing towards a certain duality of open and closed strings [67, 90, 127, 24, 79, 53, 118, 39, 123, 94, 78]. The reason for progress in this direction was a increasingly better understanding of non-perturbative open strings. At last the availability of open string field theories, which are in the centre of interest for such issues, has given var- ious new insights, among them the famous Sen conjecture (see [120] and references therein).
An open string field theory comprises the consistent truncation to the open string sector concerning the relevant degrees of freedom, at least non- perturbatively. In modern language it can be considered as the worldvolume theory that includes all open string modes living on a D-brane. There are basically two formulations available, cubic OSFT [145] and boundary string field theory (BSFT) [148, 150, 121, 122], the relation between both being not entirely clear. Both come with advantages and disadvantages, predestinating them for application in different realms.
In works of Sen it has been shown that BSFT provides an answer to the vacuum selection problem for open strings. Namely, in the limit of small string coupling constant the closed string background can be fixed and it then is possible to ask, what are the possible D-brane configurations for such a background. This question is answered byclassical open string field theory, whose equations of motion directly give the desired vacua. In this picture D-branes are viewed as solitons of the open string tachyon. Since it is possible to calculate the potential for the tachyonic degrees of freedom and to consistently truncate the theory to massless (and tachyonic) fields, the minima of the potential correspond to static vacua.