We now describe an important application of Dp-branes: Intersecting Brane Worlds.2
The general idea can already be understood even without compactifying the extra six dimensions. Various Dp-branes can extend along different dimensions and intersect along some subspace that contains R1,3. This way, interesting gauge theories and matter content arise along the dimensions common to all branes. In fact, the structure we find is naturally that of the Standard Model of Particle Physics! Thus, Intersecting Brane Worlds are important ingredients of string phenomenology, the subfield of string theory that tries to make contact between string theory in 10 dimensions and our 4-dimensional world.
We will first be working in R1,9 and consider configurations of branes intersecting along R1,3,
temporarily ignoring complications due to compactification. Among the various possibilities we choose as the probably simplest example a configuration of intersecting D6-branes in Type IIA theory. Let DAand DB be two such D6-branes which fill the following dimensions:
Dimension 0 1 2 3 4 5 6 7 8 9
DA X X X X X - X - X -
DB X X X X - X - X - X
Remarks:
• The two branes intersect along R1,3 × xi= 0, i = 4, . . . , 9. We will be interested in the
physical theory along these common dimensions.
• Even though the two branes intersect along R1,3, this setup is not yet a satisfactory ef-
fectively 4-dimensional theory. This is because all states propagating along DA and DB
propagate not only in 4 dimensions, but also in the remaining dimensions of the brane. The effect of this is negligible to first order only if the extra dimensions are small. Indeed, generalisations to models with 6 compact dimensions are simple.
• The above setup corresponds to an angle of π
2 in the three planes spanned by x 4− x5,
x6− x7, x8− x9. This can be generalised to angles ϕ
i, i = 1, 2, 3 in the three planes.
• In the presence of a Dp-brane, Poincar´e invariance of R1,9is obviously broken in the normal
directions. This corresponds to a spontaneous breakdown in the vacuum described by the branes. Since Poincar´e symmetry and supersymmetry are related via {Q, ¯Q} ' γ P , also
2For further reading we suggest e.g. [Z], Chapter 21, or the pedagogical review ”Toward Realistic Intersecting
D-Brane Models”, http://arXiv.org/abs/hep-th/0502005. A more advanced and very comprehensive text is also ”Four-dimensional String Compactifications with D-Branes, Orientifolds and Fluxes”, http://arXiv.org/abs/hep- th/0610327.
some amount of SUSY must be broken. It turns out that a single Dp-brane in Type II theories (of the appropriate type) preserves only 12 of the original amount of supercharges. • If 2 branes intersect as above, they will in general preserve different supercharges. The total amount of supersymmetry is then generated by the supercharges preserved by both of them. The amount of SUSY depends on the sum of angles ϕi between the branes.
You can find a detailed discussion e.g. in [P], Chapter 13.4. In the case considered here, supersymmetry is broken completely. This is just one example of how the theory in 4 dimensions - here arising as the common locus of the intersecting branes - can enjoy much less supersymmetry than the original theory in 10 dimensions!
• We have not taken into account any of the various subtle consistency conditions that arise at the quantum level and that are comparable to the tadpole cancellation conditions of Type I theory. For compact models, these conditions severely constrain the allowed brane setups. Unlike in pure field theory model building it is not possible to simply assemble all ingredients one would like for phenomenological reasons into a model. Rather one must show that the string equations of motion are satisfied. In other words, each model corresponds to a new effectively 4-dimensional vacuum of the unique 10-dimensional theory. Consider now stacks of NAand NB coincident branes of type DA and DB respectively. We have
the following 2 different sectors in the open string spectrum:
1) Strings starting and ending on the same brane (A − A sector and B − B sector) These contain the massless gauge bosons of gauge group U (NA) and U (NB), respectively (plus
their superpartners, depending on the amount of SUSY). The important feature is that along the dimensions common to DA and DB, both types of gauge bosons propagate! Therefore along
the common R1,3× xi= 0, i = 4, . . . , 9 the gauge group is U (N
A) × U (NB).
2) Strings stretched between different brane stacks
This sector is due to strings starting on A and ending on B (i.e. in the A → B sector) as well as strings starting on B and ending on A (i.e. in the B → A sector).
• Due to their tension, these are localised at the intersection of the branes, i.e. they propagate only along R1,3.
• As can be seen from their Chan-Paton factors, they transforms as bi-fundamentals of U (NA) × U (NB). The convention is that strings in A → B sector transform as ( ¯NA, NB).
The difference between the fundamental and and the anti-fundamental is that they are complex conjugates. We take ¯NAto have charge −1Aunder the diagonal U (1)Ain U (NA) =
SU (NA) × U (1)A(and NAto have charge +1A.) Then, the states in the sector B → A are
in representation (NA, ¯NB).
• To determine the details of the string spectrum we need to quantised an open string with mixed boundary conditions. For example, for the setup at hand, these are in the A → B sector: σ = 0 ∂σXn(τ, σ = 0) = 0, n = 0, . . . 3, 4, 6, 8, ∂τXm(τ, σ = 0) = 0, m = 5, 7, 9, σ = ` ∂σXn(τ, σ = `) = 0, n = 0, . . . 3, 5, 7, 9, ∂τXm(τ, σ = `) = 0, m = 4, 6, 8. (7.53)
This corresponds to DN boundary conditions in dimensions 4, . . . , 9 and be generalised to arbitrary angles ϕi between the branes.
• The mixed boundary conditions modify the oscillator modings. For DN strings this has been discussed. For general angles one arrives at fractional modings by shifting the NN moding to
n → n +ϕ
π. (7.54)
Consider now the A → B sector:
• The massless Ramond sector contains one fermion corresponding to a Dirac spinor, i.e. one chiral and one anti-chiral Weyl spinor. This is just the fermionic ground state along the extended 4 dimensions. The GSO projection will keep only one of the two, say the chiral one (R, +). Thus we have one ψαAB, where α = 1, 2 denotes a 4-dimensional Weyl spinor index.
• In the NS sector the sign of M2 of the lowest state is determined by the normal ordering
constant. This in turn depends on the angle between the branes as these shift the modings of the fields. For the above DN boundary conditions we find one boson of positive M2.
This reflects the fact the brane intersection breaks supersymmetry completely so that the massless fermion has no massless superparter.
The B → A sector follows by letting σ → ` − σ. This is just worldsheet parity. As discussed in the context of T-duality this flips chirality.
We therefore obtain the following massless spectrum (after GSO projection)
ψABα : ( ¯NA, NB), ψBAα˙ : (NA, ¯NB) (7.55)
The two fermions correspond to particle and anti-particle and thus describe the same degrees of freedom.
Let us summarise our findings:
A stack of two branes DA and DB intersecting along R1,3× pt. gives rise to a U (NA) × U (NB)
Yang-Mills theory plus one chiral fermion transforming in the bi-fundamental ( ¯NA, NB).
But wait a minute - this is just the structure of the Standard Model of Particle Physics (SM)! Namely, the SM gauge group is the product SU (3) × SU (2) × U (1)Y and the particle content is
given by 3 generations of chiral fermions in various bifundamentals: Particle SU (3) SU (2) U (1)Y QL 3 2 16 uc R ¯3 1 − 2 3 dc R ¯3 1 1 3 L 1 2 −1 2 ec R 1 1 +1 νc R 1 1 0
Our notation is that uc
R etc. are the charge conjugate of the right-handed fields and thus left-
handed.
Very crudely, this can be realised by in terms of 3 intersecting brane stacks with NA= 3, NB = 2,
Remarks:
• One notices that in the SM only SU (N ) groups appear, not U (N ) (apart from U (1)Y). In
Intersecting Brane Worlds, the diagonal U (1) ⊂ U (N ) turns out to decouple - its gauge boson is massive. In suitable configurations precisely one linear combination of U (1)s is massless. This must be identified with U (1)Y.
• The remaining U(1)s remain as perturbative global symmetries and account for the presence of accidental symmetries such as baryon and lepton number in the SM. This is particularly attractive because in the SM no explanation for the existence of these symmetries can be given.
• To account for the correct charges of all particles, more complicated configurations than just the above 3-stack model are required. Indeed suitable brane setups can be classified. Toroidal Intersecting Brane Worlds
So far we have been working in R1,9. To obtain a truly 4 dimensional effective field theory, the
extra six dimensions must be compact. The logic behind this compactification will be discussed in a more general context in the next chapter. Here let us focus on the simplest possibility and make a toroidal compactification ansatz
M1,9= R1,3× T6 (7.56)
with T6a six-torus.
• It is convenient to represent T6 as a factorisable 6-torus of the form T6= T2× T2× T2.
If we embed the brane configuration presented at the beginning of this section into such a compact model, the two D6-branes fill R1,3 and wrap a 1-cycle in each of the 2-tori given
by one of the axes.
• The 2 branes DA and DB now intersect in 3 points on T6. At each intersection point one
chiral bifundamental fermion ”lives”. Thus, in the effective theory in R1,3 we now find 3
chiral bifundamental fermions. This gives a beautiful way to think about family replication - the fact that we have 3 chiral generations of bifundamental matter in the SM.
As alluded to above, not every brane configuration leads to a fully consistent CFT. As in Type I theory one must check that the tadpoles of all 1-loop amplitudes cancel. This implies that we must actually consider unoriented open strings (such that the M¨obvious amplitudes can cancel the tadpoles of the annulus amplitude). Such models are called orientifolds. Finding consistent solutions which are compatible with the physics of the SM is an active field of modern day string theory.