ALINEACIÓN DEL PMD 2019-2021 CON EL PED 2015-2021
REGLAMENTACIÓN MUNICIPAL
In the weakly coupled Type II string theories, a (static) D-brane can be viewed as a hyperplane in the spacetime M where the open strings end. If the open string with Neumann boundary conditions for coordinates (X0, X1, ..., Xp) in the 10D spacetime and Dirichlet boundary con- dition for (Xp+1, ..., X9), then D-branes are defined as a hyperplane in (X0, .., Xp) that hosts the (open) fundamental strings. Depending on their spatial dimension, we sometimes denote them as Dp-branes. Before exploring the D-brane physics, let us first set up some conventions. Similar to the description of the string using the world-sheet, one can use the world-volume to embed the Dp-branes into the 10D spacetime M. Denoting the p + 1 dimensional world-volume as W swept out by the Dp-brane propagating through the target space M, parametrizing by the coordinates ξa, a = 0, ..., p, one can define the embedding map as ı : W → M, ξa → Xµ(ξa).
Sector State SO(p − 1) (p + 1) fields NS ba
−1/2|pi Vector Aa(ξa)
NS bm−1/2|pi Scalar Xm(ξa)
R BC Spinor fermions λα
Table 1.1.: The spectra assoicated with the Dp-brane in type II strings.
Now let us start with the analysis of perturbative open string spectrum in flat spacetime. The massless bosonic spectra of open strings ending on a Dp-brane with world-volume coordinates contain a massless gauge field Aa, a = 0, 1, .., p and scalar fields Xm(ξa), m = p + 1, .., 9, which we listed in ??. The scalar fields Xm actually describe the transverse positions of the Dp-brane
embedded in the spacetime M. The gauge fields Aa(ξa), on the other hand, can be viewed as tangent space of the hyperplane of Dp-branes, which describe shapes of the D-branes as a "soliton" background, i.e. as a fixed topological defect in spacetime. As we said above, D-branes are viewed as dynamical extended objects by themselves, hence the hyperplanes are not rigid and instead, the shapes and positions can fluctuate. The low-energy theories of world-volume of a Dp-brane (or more general (p, q)-branes) should capture the dynamics of the above massless spectra generated by the open strings, and it turns out it reduce to super Yang Mills (SYM) theory7.
results, see, for example, Witten’s work on topological field theories.
7
The case with M5-branes in M-theory is subtle, as the fundamental objects would be M2-branes in M-theory. Hence the low-energy limit of world-volume of M5-brane should be describing the dynamics of strings, which
1.5. Branes and Gauge Theory
To be more precise, the kinetic term for the low energy effective theory of D-brane, describing the above open string massless spectrum, is given by the Dirac-Born-Infeld (DBI) action [32]
Sp= −µp
Z
dp+1ξe−φ(ξ)p−det(gab+ Bab+ 2πα0Fab), (1.83)
where gab:= ı∗gµν = gµν∂aXµ∂bXν, Bab:= ı∗Bµν = Bµν∂aXµ∂Xbν refer to the pullbacks of the
corresponding 10D bulk ones, F is the field strength of the world-volume U (1) gauge field Aa.
The prefactor of the dilaton e−φ(ξ) arises from the open string tree level, i.e. the disk. Turning down the field strength F and Kalb-Ramond 2-form field B, one can find that the DBI action is the p + 1 dimensional generalization of the 2D Polyakov action and can be justified by checking that the equation of motion of DBI satisfies conformal invariance for the open string in the D-brane background. Thereby the tension Tp of a Dp-brane is given by8
Tp= 2π gs`p+1s = |µp| gs , (1.84)
where the coefficient µp will be discussed later. The D-brane tension Tp scales with the string
coupling as Tp ' 1/gs, reflecting the non-perturbative nature of these states, and we will see it leads to a lot of peculiar properties of D-branes 9 . At the weak coupling limit, T
P tends to
TP → ∞ and it can be viewed as the rigid hyperplane in the weak coupling regime. In some
senses, one can view the tension Tp as the NS-NS charge.
As we said, the low-energy limit of D-brane should be captured by a super Yang Mills (SYM) theory. How can DBI action be linked to the SYM theory? Well, based on two derivative form of SYM, one should extract the two derivative leading order of DBI action. To do that, one should invoke the formula
det(1 + M ) = 1 + tr(M ) −1 2tr(M
2) + ..., (1.85)
and expand the DBI action (1.83) at the almost flat space gµν = ηµν with slowly-varying fields
to order F4, (∂X)4 and then we have
gab∼ ηab+ ∂aXµ∂bXµ+ O((∂X)4), Bµν = 0. (1.86)
If we further assume that 2πα0Fαβ and ∂X are small and of the same order, then we can easily
see that the DBI action reduces to Sp = TpVp− 1 4g2 Y M Z dp+1x(FabFab+ ( 2 2πα0) 2∂ aXm∂aXm) + O(F4), (1.87)
arise from the boundary of M2-branes on M5-branes, this necessarily involves higher structures of gauge theories, the proper mathematics would be gerbes rather than the vector bundles/sheaves. For details on this aspect, we refer to [31] and references therein.
8
Here we already evaluated the e−φ(ξ) at the asymptotic value φ0 which gives rise to string coupling gs−1. 9 One may wonder what would be the back-reaction on the geometry since D-brane carries tension. To see
this, one can easily check that N coincident branes in a D dimensional spacetime give rise to a gravitational potential (classically) V ∼ GNN Tp/rD−p−3with Newton’s constant GN given by GN∼ `8sgs2, hence for N
coincident Dp-branes the strength of the gravitational potential is gsN . This implies that the back-reaction of
the D-branes on the spacetime geometry can be neglected at distances larger than the string scale if gsN 1,
which corresponds to the limit of the weakly coupled string theory. Hence this suggests that it is reasonable to view the D-branes at the weakly coupled string theory as a hyperplane in the flat Minkowski space where open fundamental string ends.
where the Vp is the Dp-brane volume and the coupling gY M2 is given by
g2Y M = gs`p−3s . (1.88)
Note that for D3-branes, the string coupling gscoincidents with the square of the SYM coupling.
The first term contributes to the vacuum energy and can be omitted, thereby the DBI action reduces to the SYM action! Note that the above limit we’ve taken turns out to be equivalent to sending the string length `s→ 0, namely we have decoupled the gravity and also massive string
modes as their mass scales as m ∼ `1
s. In such a limit, in order to keep gY M fixed, one needs to
impose gs→ 0 for p < 3 and gs→ ∞ for p > 3.
Furthermore, a Dp-brane (with suitable dimension, which will be explained shortly) can be described as a Bogomol’ny-Prasad-Sommerfeld (BPS) saturated state preserving half of the 32 supercharges of type II string theory, as the open string boundary conditions are invariant under only half of the transformations. In other words, the Type II vacua without D-branes preserve all 32 supersymmetries, but the states containing a D-brane are invariant under only half the supersymmetries. Being BPS states implies that parallel Dp-branes do not exert forces on each other, since roughly speaking, the repulsion forces from the charge exactly cancel their gravitational(and dilaton) attractive force. As a result of this, a stack of Dp-branes can be placed on top of each other without any repulsive or attractive forces and the individual Dp-branes are indistinguishable. This essentially promotes the Chan-Paton labels to be a non-abelian U (Nc)
matrix. Correspondingly, the low-energy world-volume of a stack of Nc Dp-branes turns out
to be a SYM with the non-abelian gauge group U (NC) 10 and 16 supercharges. The massless spectra Xm(ξa), Aa then transform under the adjoint representation of U (Nc) gauge group,
where the diagonal components of Xm and Cartan generators of Aa arise from the states generated by open strings whose endpoints are on the same Dp-brane. On the other hand, the off-diagonal components of Xm and charged gauge bosons come from open strings ending on different Dp-branes. The Lagrangian for the U (Nc) SYM reads
L = − 1 gSY M2 Tr( 1 4FabF ab+ 1 `4 s DaXmDaX + V ), (1.89)
where the covariant derivative acts as DaXm = ∂
aXm − i[Aa, Xm]. Flat directions of the
potential V, defined as
V ∝X 1
g2Y M`8 s
Tr[Xm, Xn]2, (1.90)
are parametrized by the diagonal Xm, which describes the Coulomb branch of the U (Nc) theory. The vev of Xm then represents the transverse position of these NcDp-branes. As an aside, it
turns out that one can also describe the low energy dynamics of a stack NcDp-branes in flat
space in static gauge by the dimensional reduction to p + 1 dimensions of 10D N = 1 SYM. The above analysis can also carried over to the cases that D-branes have a curved world-volume. In such cases, the gauge groups would not necessarily be U (N ). It turns out, in perturbative Type II string theories, the gauge groups can be realized by D-branes are only three options: SU(N), SO(N) and Sp(N), which we will mention in 1.9. For other options, especially the exceptional ones such as En gauge groups, they necessarily invoke non-perturbative degrees of
freedom, while certain (p, q) strings become light and build up the necessary components for exceptional groups.
10
Note there are other choices such as SO(N ) or Sp(N ), but that depends on further configurations, especially the O-planes, which will be introduced momentarily.
1.5. Branes and Gauge Theory
Having said the Dp-branes are BPS states of the Type II strings, one would wonder which conserved charges, determined entirely by their mass/tension in the corresponding supersym- metric algebra, are carried by the Dp-branes. Well, it turns out there is only one set of charges for Dp-branes with the correct Lorentz transformation properties- Ramond-Ramond charges. Indeed, the Dp-branes carry charges with respect to the RR gauge symmetries by coupling to the R-R p + 1 potential fields Cp+1 through the integral µp
R
Wı ∗C
p+111. This fact is consistent
with the statement we listed in Insert 2.1 that the perturbative string states cannot see the charges of the RR fields, as the Dp-brane, whose tension is proportional to 1/gs, is genuinely
non-perturbative. Since they carry the conserved charge, they are hence stable.
As we said earlier, Cp+1, as a p + 1-form potential field, is hodge dual to C7−p by dC7−p= ∗dCp+1. Based on the discussion in Insert 2.1, we then say that p + 1 form Cp+1 is electrically
coupled to a Dp-brane and magnetically coupled to a D(6-p)-brane in Type II strings. We see that in type IIA theory with p in RR potentials Cp+1 being even, we thus have stable Dp-branes with p = 0, 2, 4, 6, 8. In type IIB string theory, we can have stable Dp-branes with p = −1, 1, 3, 5, 7, (9) 12. Dp-branes with wrong dimension in type II strings, for example a Dp-brane with even value of p in Type IIB, can also host the open strings. However, they do not carry the conserved charge as there are no suitable RR fields. As a consequence, they are not stable and break all spacetime supersymmetry. They are essentially the same objects as the ones in the bosonic strings, and generate tachyons in their spectrum.
In terms of the charges, as a generalization of 6D version eg = 2πn, they similarly are subject to the Dirac quantization
µpµ6−p∈ 2πZ, (1.91)
where the charge µp is measured by the generalized Gauss law Z
S8−p
∗Fp+2= µp. (1.92)
However, it turns out that Dp-branes not only carry the charge of Cp+1, but also of lower form Cn fields with n < p + 1. There are various ways to see that. For example one can use T-duality to see that Dp-branes also couple to Cp−1 RR fields. The full action of RR fields
coupling had been derived in [34] by invoking the anomaly inflow mechanism13, which is given by the Chern-Simons action, sometime also known as Wess-Zumino action.
The Chern-Simons action for the Dp-branes takes the form
SDp= −µp 2 Z W X 2p ı∗C2p∧ Tr(eiF) ∧ v u u t b A(TDp) b A(NDp) , (1.93) 11
This point can be also reflected by the fact that the Dp-branes can be viewed as the soliton solutions of classical equation of motion in the supergravity [33], as we will discuss shortly
12
As we have stressed, there are only closed string propagating on the bulk of type II strings. Including D9-branes in the Type IIB then implies that open strings can also propagate on the bulk, which would leads to a contradiction. Indeed, this is only consistent in Type I theory, which can be viewed as type IIB with 32 D9-branes and an orientifold (to be discussed later). D8-branes is also subtle, It is actually a domain-wall coupling to a non-dynamical 9-form field C9.
13
The anomaly inflow mechanism, firstly introduced in the context of gauge theory [35], could usually be applied to the anomalous theory such that it can be coupled/embedded in a higher dimensional theory whose anomalous variation of the classical action localizes at ("flow" to) the world-volume of the original anomalous theory and cancels its anomaly.
where the charge is given by µp = 2π
`p+1s
αp 14. TDp and NDp denote the tangent and normal
space to the Dp-brane along W and the conventions of Chern Character and the A-roof genus can be found in the appendix B.1.1. Note furthermore that we are writing the brane action in terms of Tr = λ1trfund, where the Dynkin index λ is given in Table 4.1.
P
2pı∗C2p denotes
the formal sum of RR fields ı∗C2p, which are pull-backs of the bulk RR fields15. Since we are
working in the democratic formulation, where each RR gauge potential Cp+1 is accompanied
by its magnetic dual, the Chern-Simons action has to include a factor of 12 [40], which we are making manifest in (A.3). Note that the Chern-Simons action does not involve the metric and is thus of topological nature.
The gauge invariant field strength F above is defined as
F = i(`2sF + 2πı∗B2I) . (1.94)
Compared to expressions oftentimes used in the literature we have absorbed a factor of −12π in the definition of F . The minus "-" here is in order to be consistent with the other conventions for anomaly in A.2.