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.


Superstrings as Grand Unifier. Thought of the Day 86.0


The first step of deriving General Relativity and particle physics from a common fundamental source may lie within the quantization of the classical string action. At a given momentum, quantized strings exist only at discrete energy levels, each level containing a finite number of string states, or particle types. There are huge energy gaps between each level, which means that the directly observable particles belong to a small subset of string vibrations. In principle, a string has harmonic frequency modes ad infinitum. However, the masses of the corresponding particles get larger, and decay to lighter particles all the quicker.

Most importantly, the ground energy state of the string contains a massless, spin-two particle. There are no higher spin particles, which is fortunate since their presence would ruin the consistency of the theory. The presence of a massless spin-two particle is undesirable if string theory has the limited goal of explaining hadronic interactions. This had been the initial intention. However, attempts at a quantum field theoretic description of gravity had shown that the force-carrier of gravity, known as the graviton, had to be a massless spin-two particle. Thus, in string theory’s comeback as a potential “theory of everything,” a curse turns into a blessing.

Once again, as with the case of supersymmetry and supergravity, we have the astonishing result that quantum considerations require the existence of gravity! From this vantage point, right from the start the quantum divergences of gravity are swept away by the extended string. Rather than being mutually exclusive, as it seems at first sight, quantum physics and gravitation have a symbiotic relationship. This reinforces the idea that quantum gravity may be a mandatory step towards the unification of all forces.

Unfortunately, the ground state energy level also includes negative-mass particles, known as tachyons. Such particles have light speed as their limiting minimum speed, thus violating causality. Tachyonic particles generally suggest an instability, or possibly even an inconsistency, in a theory. Since tachyons have negative mass, an interaction involving finite input energy could result in particles of arbitrarily high energies together with arbitrarily many tachyons. There is no limit to the number of such processes, thus preventing a perturbative understanding of the theory.

An additional problem is that the string states only include bosonic particles. However, it is known that nature certainly contains fermions, such as electrons and quarks. Since supersymmetry is the invariance of a theory under the interchange of bosons and fermions, it may come as no surprise, post priori, that this is the key to resolving the second issue. As it turns out, the bosonic sector of the theory corresponds to the spacetime coordinates of a string, from the point of view of the conformal field theory living on the string worldvolume. This means that the additional fields are fermionic, so that the particle spectrum can potentially include all observable particles. In addition, the lowest energy level of a supersymmetric string is naturally massless, which eliminates the unwanted tachyons from the theory.

The inclusion of supersymmetry has some additional bonuses. Firstly, supersymmetry enforces the cancellation of zero-point energies between the bosonic and fermionic sectors. Since gravity couples to all energy, if these zero-point energies were not canceled, as in the case of non-supersymmetric particle physics, then they would have an enormous contribution to the cosmological constant. This would disagree with the observed cosmological constant being very close to zero, on the positive side, relative to the energy scales of particle physics.

Also, the weak, strong and electromagnetic couplings of the Standard Model differ by several orders of magnitude at low energies. However, at high energies, the couplings take on almost the same value, almost but not quite. It turns out that a supersymmetric extension of the Standard Model appears to render the values of the couplings identical at approximately 1016 GeV. This may be the manifestation of the fundamental unity of forces. It would appear that the “bottom-up” approach to unification is winning. That is, gravitation arises from the quantization of strings. To put it another way, supergravity is the low-energy limit of string theory, and has General Relativity as its own low-energy limit.

Philosophy of Dimensions: M-Theory. Thought of the Day 85.0


Superstrings provided a perturbatively finite theory of gravity which, after compactification down to 3+1 dimensions, seemed potentially capable of explaining the strong, weak and electromagnetic forces of the Standard Model, including the required chiral representations of quarks and leptons. However, there appeared to be not one but five seemingly different but mathematically consistent superstring theories: the E8 × E8 heterotic string, the SO(32) heterotic string, the SO(32) Type I string, and Types IIA and IIB strings. Each of these theories corresponded to a different way in which fermionic degrees of freedom could be added to the string worldsheet.

Supersymmetry constrains the upper limit on the number of spacetime dimensions to be eleven. Why, then, do superstring theories stop at ten? In fact, before the “first string revolution” of the mid-1980’s, many physicists sought superunification in eleven-dimensional supergravity. Solutions to this most primitive supergravity theory include the elementary supermembrane and its dual partner, the solitonic superfivebrane. These are supersymmetric objects extended over two and five spatial dimensions, respectively. This brings to mind another question: why do superstring theories generalize zero-dimensional point particles only to one-dimensional strings, rather than p-dimensional objects?

During the “second superstring revolution” of the mid-nineties it was found that, in addition to the 1+1-dimensional string solutions, string theory contains soliton-like Dirichlet branes. These Dp-branes have p + 1-dimensional worldvolumes, which are hyperplanes in 9 + 1-dimensional spacetime on which strings are allowed to end. If a closed string collides with a D-brane, it can turn into an open string whose ends move along the D-brane. The end points of such an open string satisfy conventional free boundary conditions along the worldvolume of the D-brane, and fixed (Dirichlet) boundary conditions are obeyed in the 9 − p dimensions transverse to the D-brane.

D-branes make it possible to probe string theories non-perturbatively, i.e., when the interactions are no longer assumed to be weak. This more complete picture makes it evident that the different string theories are actually related via a network of “dualities.” T-dualities relate two different string theories by interchanging winding modes and Kaluza-Klein states, via R → α′/R. For example, Type IIA string theory compactified on a circle of radius R is equivalent to Type IIB string theory compactified on a circle of radius 1/R. We have a similar relation between E8 × E8 and SO(32) heterotic string theories. While T-dualities remain manifest at weak-coupling, S-dualities are less well-established strong/weak-coupling relationships. For example, the SO(32) heterotic string is believed to be S-dual to the SO(32) Type I string, while the Type IIB string is self-S-dual. There is a duality of dualities, in which the T-dual of one theory is the S-dual of another. Compactification on various manifolds often leads to dualities. The heterotic string compactified on a six-dimensional torus T6 is believed to be self-S-dual. Also, the heterotic string on T4 is dual to the type II string on four-dimensional K3. The heterotic string on T6 is dual to the Type II string on a Calabi-Yau manifold. The Type IIA string on a Calabi-Yau manifold is dual to the Type IIB string on the mirror Calabi-Yau manifold.

This led to the discovery that all five string theories are actually different sectors of an eleven-dimensional non-perturbative theory, known as M-theory. When M-theory is compactified on a circle S1 of radius R11, it leads to the Type IIA string, with string coupling constant gs = R3/211. Thus, the illusion that this string theory is ten-dimensional is a remnant of weak-coupling perturbative methods. Similarly, if M-theory is compactified on a line segment S1/Z2, then the E8 × E8 heterotic string is recovered.

Just as a given string theory has a corresponding supergravity in its low-energy limit, eleven-dimensional supergravity is the low-energy limit of M-theory. Since we do not yet know what the full M-theory actually is, many different names have been attributed to the “M,” including Magical, Mystery, Matrix, and Membrane! Whenever we refer to “M-theory,” we mean the theory which subsumes all five string theories and whose low-energy limit is eleven-dimensional supergravity. We now have an adequate framework with which to understand a wealth of non-perturbative phenomena. For example, electric-magnetic duality in D = 4 is a consequence of string-string duality in D = 6, which in turn is the result of membrane-fivebrane duality in D = 11. Furthermore, the exact electric-magnetic duality has been extended to an effective duality of non-conformal N = 2 Seiberg-Witten theory, which can be derived from M-theory. In fact, it seems that all supersymmetric quantum field theories with any gauge group could have a geometrical interpretation through M-theory, as worldvolume fields propagating on a common intersection of stacks of p-branes wrapped around various cycles of compactified manifolds.

In addition, while perturbative string theory has vacuum degeneracy problems due to the billions of Calabi-Yau vacua, the non-perturbative effects of M-theory lead to smooth transitions from one Calabi-Yau manifold to another. Now the question to ask is not why do we live in one topology but rather why do we live in a particular corner of the unique topology. M-theory might offer a dynamical explanation of this. While supersymmetry ensures that the high-energy values of the Standard Model coupling constants meet at a common value, which is consistent with the idea of grand unification, the gravitational coupling constant just misses this meeting point. In fact, M-theory may resolve long-standing cosmological and quantum gravitational problems. For example, M-theory accounts for a microscopic description of black holes by supplying the necessary non-perturbative components, namely p-branes. This solves the problem of counting black hole entropy by internal degrees of freedom.