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) ≡ ∫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
∫M φ ≤ ∫M 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 eiθΩ 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:
- M is a p-dimensional complex submanifold of X.
- The 2-form F is of type (1, 1), i.e., (E, ∇) is a holomorphic vector bundle on M.
- In the supergravity limit, F satisfies the Hermitian Yang-Mills equation:ω|p−1M ∧ F = c · ω|pMfor some real constant c.
- F satisfies Im eiφ(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 eiθΩ 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 eiα 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 eiαΩ|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.