Embedding Branes in Minkowski Space-Time Dimensions To Decipher Them As Particles Or Otherwise


The physics treatment of Dirichlet branes in terms of boundary conditions is very analogous to that of the “bulk” quantum field theory, and the next step is again to study the renormalization group. This leads to equations of motion for the fields which arise from the open string, namely the data (M, E, ∇). In the supergravity limit, these equations are solved by taking the submanifold M to be volume minimizing in the metric on X, and the connection ∇ to satisfy the Yang-Mills equations.

Like the Einstein equations, the equations governing a submanifold of minimal volume are highly nonlinear, and their general theory is difficult. This is one motivation to look for special classes of solutions; the physical arguments favoring supersymmetry are another. Just as supersymmetric compactification manifolds correspond to a special class of Ricci-flat manifolds, those admitting a covariantly constant spinor, supersymmetry for a Dirichlet brane will correspond to embedding it into a special class of minimal volume submanifolds. Since the physical analysis is based on a covariantly constant spinor, this special class should be defined using the spinor, or else the covariantly constant forms which are bilinear in the spinor.

The standard physical arguments leading to this class are based on the kappa symmetry of the Green-Schwarz world-volume action, in which one finds that the subset of supersymmetry parameters ε which preserve supersymmetry, both of the metric and of the brane, must satisfy

φ ≡ Re εt Γε|M = Vol|M —– (1)

In words, the real part of one of the covariantly constant forms on M must equal the volume form when restricted to the brane.

Clearly dφ = 0, since it is covariantly constant. Thus,

Z(M) ≡ ∫φ —– (2)

depends only on the homology class of M. Thus, it is what physicists would call a “topological charge”, or a “central charge”.

If in addition the p-form φ is dominated by the volume form Vol upon restriction to any p-dimensional subspace V ⊂ Tx X, i.e.,

φ|V ≤ Vol|V —– (3)

then φ will be a calibration in the sense of implying the global statement

φ ≤ ∫Vol —– (4)

for any submanifold M . Thus, the central charge |Z (M)| is an absolute lower bound for Vol(M).

A calibrated submanifold M is now one satisfying (1), thereby attaining the lower bound and thus of minimal volume. Physically these are usually called “BPS branes,” after a prototypical argument of this type due, for magnetic monopole solutions in nonabelian gauge theory.

For a Calabi-Yau X, all of the forms ωp can be calibrations, and the corresponding calibrated submanifolds are p-dimensional holomorphic submanifolds. Furthermore, the n-form Re eΩ for any choice of real parameter θ is a calibration, and the corresponding calibrated submanifolds are called special Lagrangian.

This generalizes to the presence of a general connection on M, and leads to the following two types of BPS branes for a Calabi-Yau X. Let n = dimR M, and let F be the (End(E)-valued) curvature two-form of ∇.

The first kind of BPS D-brane, based on the ωp calibrations, is (for historical reasons) called a “B-type brane”. Here the BPS constraint is equivalent to the following three requirements:

  1. M is a p-dimensional complex submanifold of X.
  2. The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
  3. In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
  4. F satisfies Im e(ω|M + ils2F)p = 0 for some real constant φ, where ls is the correction.

The second kind of BPS D-brane, based on the Re eΩ calibration, is called an “A-type” brane. The simplest examples of A-branes are the so-called special Lagrangian submanifolds (SLAGs), satisfying

(1) M is a Lagrangian submanifold of X with respect to ω.

(2) F = 0, i.e., the vector bundle E is flat.

(3) Im e Ω|M = 0 for some real constant α.

More generally, one also has the “coisotropic branes”. In the case when E is a line bundle, such A-branes satisfy the following four requirements:

(1)  M is a coisotropic submanifold of X with respect to ω, i.e., for any x ∈ M the skew-orthogonal complement of TxM ⊂ TxX is contained in TxM. Equivalently, one requires ker ωM to be an integrable distribution on M.

(2)  The 2-form F annihilates ker ωM.

(3)  Let F M be the vector bundle T M/ ker ωM. It follows from the first two conditions that ωM and F descend to a pair of skew-symmetric forms on FM, denoted by σ and f. Clearly, σ is nondegenerate. One requires the endomorphism σ−1f : FM → FM to be a complex structure on FM.

(4)  Let r be the complex dimension of FM. r is even and that r + n = dimR M. Let Ω be the holomorphic trivialization of KX. One requires that Im eΩ|M ∧ Fr/2 = 0 for some real constant α.

Coisotropic A-branes carrying vector bundles of higher rank are still not fully understood. Physically, one must also specify the embedding of the Dirichlet brane in the remaining (Minkowski) dimensions of space-time. The simplest possibility is to take this to be a time-like geodesic, so that the brane appears as a particle in the visible four dimensions. This is possible only for a subset of the branes, which depends on which string theory one is considering. Somewhat confusingly, in the type IIA theory, the B-branes are BPS particles, while in IIB theory, the A-branes are BPS particles.


Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = vawa = 0. By definition, their null tangent vectors have the two-component spinor form λAπA, where λA is varying and πA is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λAπA μAπA = 0 = η(v, v) = λAπA λAπA . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field πA is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP3. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.