The low energy effective theories of Dp-branes are called Dirac-Born-Infeld (DBI) action.
People also study the modification of DBI action in NS-NS and R-R field backgrounds.
For the theory to be gauge invariant and anomaly free, we need to replace U(1) field strength F by B + F and add Wess-Zumino terms into the original DBI action. In fact, the NS-NS and R-R fields are the massless mode of the close string spectrum. They are the background fields of open string scattering amplitudes just like the gravitational background. When we use the open string scattering amplitudes to study the effective theories of D-brane in NS-NS and R-R field backgrounds, we may have different inter-pretations for these background fields. For example, the NS-NS background fields can be absorbed into the field strength of open-string oscillation mode or the open-string metric, then we get different effective theories of D-brane. For these effective theories, we call all of them to be D-brane in field backgrounds. However, this terminology is confusing in this thesis. The effective theory, which we want to talk in this thesis, is the effective theory without manifest background fields. In this case, the effects of background fields hide in the geometry and the symmetry algebra of effective theory. We try to distinguish them in next subsection.
1.1.1 Terminology Explanation
When we want to talk about a theory in some field backgrounds, we need to know what it really means.
Firstly, the meaning of background field is that we neglect the dynamic behavior of this background field. The simple case is that we study the matter field in electric-magnetic fields background, in this case, we neglect the dynamic contribution of EM gauge fields.
So the background fields what we means are constant fields. The terminology “theory in constant field background” is the same as the terminology “theory in field background”.
Secondly, when we talk about the effective field theory in field background, the ef-fective theory usually does not include the manifest background fields dependence. The
effect of background field can appear in effective mass, effective coupling or new geometry.
Hence we will have a new field theory, then we can study the equivalent phenomena in the two theories. We do not have a simple example in field theory. However, the phenomena appear frequently in string theory. For example, the effective description of Dp-brane is not unique, we have more than one effective description. The first example is the com-mutative and noncomcom-mutative gauge theory for Dp-brane in NS-NS B field background.
People [12] understand this phenomena as the result of different regularization method of open string scattering amplitude analysis. The effective field theory will be different in the different regularization method, they can be related by changing variables. In this case, this change of variables is called the Seiberg-Witten map [12]. However, the two different effective field theories are not really the same after Seiberg-Witten map, they are different by higher derivative terms and total derivative terms. Hence, they stand for the different parts of the full D-brane theory, while they can have overlap in the scal-ing limit (Appendix C). Hence, in order to distscal-inguish these two situations from other cases, we use the terminology of Dp-brane “with” NS-NS and R-R fields for originally well known DBI action. We use the terminology of Dp-brane “in” NS-NS and R-R fields
“background” for the case what we want to talk in this thesis. The effective field theory
“in” fields background does not have manifest background fields dependence.
Finally, we study the theory in large field background in the most part of this thesis.
In this limit, the effective field theory becomes simpler and easier to analyze.
1.1.2 Dirac-Born-Infeld Action and Yang-Mills Gauge Theory
In this subsection, we want to write down the explicit action form of effective field theory of D-brane. It is called the Dirac-Born-Infeld(DBI) action [4]. Roughly speaking, the DBI action comes from the calculation of open string scattering amplitude. When we calculate the β-function of open string scattering amplitude, because the theory has conformal invariance, the β-function must vanish. From these constraints, we can find the constraints of fields. These fields are the oscillation mode of open string. These constraints of fields can be understood as the equations of motion which are derived from corresponding effective field theory action. The effective action (DBI action) is described by p+1 coordinates ξa, a = 0, 1, . . . , p. The DBI action is written as1 [4]:
SDBI = Tp
Z
dp+1ξpdet(Gab+ 2πα′Fab), (1.1)
1In this chapter, we use the review paper of Dp-brane [6]
here Tp is defined by 1
(2π)pgsℓp+1s , which is the tension of Dp-brane. It is the generalization of the string tension TF 1 = 2πα1 ′. The p labels the number of spatial dimensions for Dp-brane. The gs is string coupling and ℓs =√
α′ is identified as string length. The Gab is the induced metric in Dp-brane, it is usually complex in the fermionic part. Here, we give the bosonic part of the induce metric:
Gab= ηM N∂aXM∂bXN, (1.2) where M is from 0 to p. We can choose gauge to let Xa = ξa. So, the remaining scalars in DBI action are the transverse coordinates in target spacetime, and we label them with 2πα′XI I = p + 1, . . . , 9. Here, we use the factor 2πα′ to make the mass dimension of XI equal to one. Hence, we can rewrite action as:
SDBI = Tp
Z
dp+1ξpdet(ηab+ 2πα′∂aXI∂aXI + 2πα′Fab). (1.3) The F is the field strength of one form gauge potential A, that is F = dA in Maxwell theory. We can regard the DBI action as the high energy version of Maxwell action. To take the low energy limit α′ → 0 and omit the scalar terms, we can get:
SDBI = Tp Z
dp+1ξpdet ηab(1−1
4FabFab+ O(α′)). (1.4) The low energy limit makes the D-brane theory to become simpler.
1.1.3 Dp-Branes with NS-NS and R-R Fields
The dynamics of Dp-Brane will be affected by background fields, which come from the closed string NS-NS and R-R sector. In NS-NS sector, we have graviton gM N which is symmetry rank-2 field, and NS-NS B-field 2πα′BM N which is antisymmetry two-form field. We also have dilaton field Φ, which is a scalar. All of them will modify the form of DBI action. For simplicity, here we only consider the effect of NS-NS B-field. The action of Dp-brane in NS-NS B field background can be written as:
SDBI = Tp
Z
dp+1ξpdet(ηab+ 2πα′∂aXI∂aXI + 2πα′(Fab+ Bab)), (1.5) which can be realized by modification of Gab, the induce metric, in following way:
Gab = (ηM N + 2πα′BM N)∂aXM∂bXN, (1.6)
the mixed terms of B and X will vanish for the antisymmetry of B field. The action form can have the gauge symmetry of two form field B with additional shift of one form field A:
B → B + dΛ, A→ A − Λ, (1.7)
such that B + F term do not transform.
The R-R sectors of close string are some higher ranks form. For example, the Dp-brane can have R-R (p+1)-form,(p-1)-form,. . .,1-form (or 0-form for odd p), we label them by Cp+1, Cp−1, . . . , C1 (or C0 for odd p).
The action of Dp-brane in R-R field background can be written as [13, 14]:
SDBI = Tp
Z
dp+1ξpdet(Gab+ 2πα′Fab) + SW Z, (1.8) here the new term is written by:
SW Z = µp electric charge of Dp-brane. In fact, the calculation of open-string scattering amplitude in R-R background is very difficult. People do not know how to quantize the nonlinear sigma model in curved spacetime. However, we can know the field contents, the gauge symmetry, and the supersymmetry from the flat space calculation. Hence, we can use these informations to analyze the effective worldvolume theory of Dp-brane with R-R fields. For example, the Wess-Zumino term (SW Z) is introduced to cancel the gauge anomaly in superstring theory.
While the DBI-like action of multiple Dp-branes is incomplete and unclear (the rel-evant papers [15, 16]), we can still use non-abelian Yang-Mills action to describe them.
Yang-Mills action is the leading term of multiple Dp-branes action after taking zero slope limit (α′ → 0).