Branes as a generalization of charged particles

As a "theory of everything" (TOE), string/M-theory contains all good ideas in physics. During the 20th century, people realized that the framework of quantum field theory is a "theory of nearly everything" (TONE).

Not surprisingly, quantum field theories play an important role in string theory. They approximate its dynamics at long distances (effective field theories in the spacetime), exactly describe the ultimate long-distance limit of various theories on branes, and underlie the only "truly well-defined" systems of equations that define string/M-theory itself (two-dimensional world sheet CFTs, CFTs from AdS/CFT, gauge theories used as matrix models, string field theory as a QFT with infinitely many fields, and so on).

Whenever we try to deduce the stringy predictions for spacetime physics, we start by analyzing the spectrum – the fields (especially those that produce light or massless particles) whose presence in spacetime is implied by string/M-theory. These fields are conveniently classified by their spin, i.e. the maximum allowed value of e.g. \(J_{12}\) in a multiplet.




It turns out that fields with the spin \(J\gt 2\) – including \(J=2.5\) – have to be massive (as heavy as the string scale). The multiplets contain lots of components of a negative norm – the components of the field with an odd number of temporal indices such as those in \(T_{0\mu\nu}\) – but these would make many predicted probabilities negative and need to be erased by a gauge symmetry.

But the generators of a gauge symmetry that could erase the components \(T_{0\mu\nu}\) would need to carry the \(\mu\nu\) indices as well: they would have to have a nonzero spin. As the Coleman-Mandula theorem and its followups revealed, interesting quantum field theories (and therefore interesting vacua of string/M-theory) cannot have such symmetries. If there were so many conserved quantities that they could form such a large tensor, they would constrain the dynamics so severely that interactions would be pretty much banned.

So \(J=2\) is the maximum spin that light or massless fields may have. It turns out that all light or massless \(J=2\) fields must be "modes of the gravitational field" and the corresponding particles have to be "gravitons of a sort". To erase the negative-norm components such as \(h_{0\mu}\), one needs a conserved vector \(P_\mu\) but this has to be the energy-momentum vector associated with translations in spacetime.

Similarly, \(J=3/2\) fields produce a negative-norm component \(\chi_{0\alpha}\) with a time-like vector index and a spinor index. Those require a spinor of conserved quantities \(Q_\alpha\). The anticommutators of such spinors have to include a vector but we have said it has to be the stress-energy tensor producing diffeomorphisms. It follows that light or massless \(J=3/2\) fields have to be accompanied by the local supersymmetry – they have to be gravitino fields.

Now, \(J=1\) fields also need a gauge symmetry. The \(U(1)\) gauge symmetry of electromagnetism may be extended to a non-Abelian, Yang-Mills group. String theory predicts their appearance in many vacua, using many cute mechanisms.

Fields with \(J=1/2\) and \(J=0\) don't require any gauge symmetry to get rid of negative-norm components because the corresponding fields don't produce any negative-norm components. That's why we use the word "matter" for these \(J=1/2\) and \(J=0\) fields. But let's return to the \(J=1\) fields.

Electromagnetism: returning to the 19th century

Electromagnetism is the simplest model of \(J=1\) fields. These fields essentially transform as spacetime vectors. That's true for \(\vec E\) and \(\vec B\), the electric and magnetic vectors. Relativity merged these fields to an antisymmetric tensor\[

F_{\mu\nu} = -F_{\nu\mu},\quad F_{0i}=E_i,\quad F_{ij}=\epsilon_{ijk}B_k.

\] Because 1/2 of Maxwell's equations say \(dF=0\), using the language of \(p\)-forms, we may write \(F\), the 2-form of the field strength, locally as \(F=dA\) in terms of a 1-form electromagnetic potential whose components are \(A_\mu\). It still has a vector index, confirming that the corresponding particles have \(J=1\).

I need to emphasize that even though \(F_{\mu\nu}\) has two indices, it doesn't mean that we should assign it with \(J=2\), something we correctly do with the metric tensor \(g_{\mu\nu}\). Why? I have mentioned that the right definition of "spin" is the maximum value of the component \(J_z=J_{12}\) of the angular momentum i.e. the \(z\)-component. I only picked the \(z\)-component to be specific enough. Of course that the \(x\), \(y\), or a tilted component would be equally good. But I talk about specific components so that you know that they matter and you don't make the mistake of counting the indices too naively.

Now, how can a component of a tensor have a nonzero value of \(J_{12}\), the generator rotating the 1st and 2nd axes into one another? Well, only the components of the tensor that contain some indices equal to either \(1\) or \(2\) have a nonzero \(J_{12}\) and the maximum \(J_{12}\) is pretty much the number of indices that are equal to either \(1\) or \(2\). The reason would be particularly clear if we were using the basis \(1+i2\) and \(1-i2\) instead of the real axes \(1\) and \(2\).

But the component \(F_{12}\) actually has \(J=0\). Why? Because it's \(B_z\), after all, and the value of \(B_z\) doesn't change if you rotate your objects around the \(z\)-axis, i.e. by the generator of rotations \(J_{12}\). In other words, \(F_{ij}\) for \(i,j=1,2\) has to be proportional to \(\epsilon_{ij}\) which is invariant under the \(SO(2)\) group of the rotations rotating \(1\) and \(2\) into each other. It means that only the components \(F_{1\mu}\) or \(F_{2\mu}\) for \(\mu\neq\{1,2\}\) carry a nonzero spin according to our definition and the spin is (up to the sign we remove) equal to one because the values of indices \(1,2\) may only appear once in the whole tensor, otherwise the component of the tensor either vanishes or is invariant under the \(SO(2)\) rotations. (Of course, the components \(F_{11}\) or \(F_{22}\) are also bad because they vanish due to the antisymmetry.)

This lesson holds much more generally: the totally antisymmetric tensors, whether we talk about \(F_{\kappa\lambda\dots \mu\nu}\) or \(A_{\kappa\lambda\dots\mu\nu}\), only carry \(J=1\), regardless of their number of indices. Such a general insight is relevant for our later discussion because we are going to talk about antisymmetric tensors with many indices.

If \(A_\mu\), the electromagnetic potential, is considered the fundamental field and \(F_{\mu\nu}\) is defined as\[

F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu,

\] then those Maxwell's equations that have no sources on the right hand side hold tautologically. \[

\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0

\] The other Maxwell's equations, those with sources\[

\partial_\mu F^{\mu\nu} = j^\nu

\] (this is their relativistic form) may be derived from the action\[

S = S_{\rm Maxwell}+S_{\rm sources},\\
S_{\rm Maxwell}=\int \dd^4 x \,{\mathcal L}_{\rm Maxwell},\quad L_{\rm Maxwell} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu},\\
S_{\rm sources} = \sum_{i} Q_i \int \dd x^\mu A_\mu (x^\nu).

\] This is a way to write the "sources" part of the action if we assume that the charged sources are given by discrete world lines in the spacetime. We just integrate over the lines – the world lines are divided to the \(dx^\mu\) elements which produces a natural integration measure – and the index \(\mu\) from the measure is contracted with the \(\mu\) index in the electromagnetic potential.

If we imagine that the charged sources are continuously diluted rather than "concentrated to the points and their world lines", we must replace the "charges" part of the action by the hypervolume integral of \(j^\mu A_\mu\). That form may be obtained from the previous one by dividing a continuous charge cloud to "lots of infinitesimal charges", summing over them, and rewriting the sum in the limit as an integral.

Also, we may generalize this \(j^\mu A_\mu\) form of the action to the case where the charged fields are quanta of charged fields, e.g. the Dirac field. If it's so, \(j^\mu\) may be simply replaced by the appropriate formula for the current as a function of the charged field, e.g. \[

j^\mu = e\cdot \bar\Psi \gamma^\mu \Psi.

\] More generally, this current and its coupling to \(A_\mu\) may be derived by defining a phase-changing gauge transformation for the charged fields and replacing the partial derivatives by the covariant ones.

But let's return to the form of the "sources" action that is integrated over the world lines. We may see that in this form, the action is invariant under the gauge transformation\[

A_\mu\to A_\mu + \partial_\mu \lambda.

\] The Maxwell term in the action only depends on \(F_{\mu\nu}\) which is invariant under the gauge transformation by itself, so any function of it is also invariant. (In the non-Abelian gauge theories, \(F_{\mu\nu}\) isn't quite invariant but it is "covariant" and one may obtain invariant scalars by summing all of its squared components, for example.)

The "sources" term in the action is also invariant because the integral of \(\partial_\mu \lambda\) over the world line reduces to the difference of \(\lambda\) at the beginning and the end of the world line. And because we assume the end points to be at infinity where \(\lambda\) may be assumed to vanish (trivial transformations at infinity), the gauge invariance is guaranteed. We may also include "merging and splitting" world lines. As long as the total charge of the particles propagating along the world lines that meet in each vertex vanishes (the charge is conserved), the surface terms from these vertices will cancel, too.

Higher-dimensional generalization

Now, we should notice that the integral over the world lines – one-dimensional curves in the spacetime – may easily be generalized to an integral over \((p+1)\)-dimensional surfaces where \(p\) is anything between \(-1\) and the spacetime dimension minus one, \(d-1\). The original case of the charged point-like particles is the \(p=0\) case. There exists a case with fewer dimensions, \(p=-1\), which leads us to add terms "integrated over a point" (a 0-dimensional integration doesn't require any true integration, of course) and the point in spacetime is some kind of an "instanton", something localized not only in space but also in an instant of time (that's where the name comes from).

And there are strings \(p=1\) where the integral goes over 2-dimensional world sheets as well as higher-dimensional branes with \(p\geq 2\) that have their world volumes. In all these cases, the "sources" term has the form\[

S_{\rm sources} = \int \dd^{p+1}\Sigma^{\mu_1\mu_2\cdots \mu_{p+1}} A_{\mu_1\mu_2\cdots \mu_{p+1}}

\] where the \(A\) field is a \((p+1)\)-form. Note that the integral beautifully contracts the indices once again. The \((p+1)\)-dimensional submanifolds in the spacetime may be divided to infinitesimal units whose "area" may be expressed by an antisymmetric tensor – which generalizes \(\vec{dS}\) in the area integrals from the proofs of Gauss' law – and its indices may be naturally contracted with the indices of a generalized "electromagnetic" potential.

The action produces equations that are analogous to Maxwell's equations, too. Again, the identity \(dF=0\) holds automatically because \(F\) is defined by \(F=dA\). The other equations are \[

*d*F = J

\] which is an equation for a \((p+1)\)-form. You should always realize that this is just a multi-dimensional, braney generalization of those Maxwell's equations that are the special case for \(p=0\). For example, if you consider a flat, infinitely stretched string or another \(p\)-brane, the component \(J^{\mu 12\dots p}\) (where \(1,2,\dots,p\) are the extra spatial directions in which the brane's world volume is extended) behaves exactly like the component of \(J^\mu\) of the current in ordinary electromagnetism with a point-like electric source in which you simply erase all the dimensions \(1,2,\dots,p\).

People would probably say to be "familiar" with this generalization of electromagnetism 20 years ago but only in the mid 1990s, during the Second Superstring Revolution, they started to systematically think about all the possible "charged objects" and the corresponding "generalized electromagnetic fields" that appear in physics, especially in string/M-theory. In fact, some of these realizations were the breakthroughs that sparked – or co-sparked – the revolution.

They fully appreciated that M-theory in 11 dimensions had a 3-form potential \(C_{\lambda\mu\nu}\) that naturally had electrically charged sources, M2-branes (membranes) with \(p=2\). One may also construct the 4-form field strength \(F=dC\) with components \(F_{\kappa\lambda\mu\nu}\) and, much like any antisymmetric tensor, it may be Hodge-dualized to a dual 7-form field strength (\(11-4=7\)) to see that this may be locally (and away from the M2-branes) written as the exterior derivative of a dual 6-form electromagnetic potential that couples to M5-branes' world volumes.

A much newer insight, due to Joe Polchinski, revolutionized people's understanding of perturbative string theory, especially things like type I, IIA, and IIB string theory. It had been known that these theories had multiple massless antisymmetric tensor fields in the Ramond-Ramond sector. (Bosonic string theory and heterotic string theory don't have the Ramond-Ramond fields so they don't have the corresponding charges even though uncharged D-branes exist in bosonic string theory, too.) These Ramond-Ramond \(p\)-form fields may be obtained by tensor multiplying the spinors from the left-movers and the spinors from the right-movers on the type I/II superstring.

It turns out that type IIA string theory contains \((p+1)\)-forms \(A\) with all possible odd values of \((p+1)\) i.e. all allowed even values of \(p\). Similarly, type IIB – and its projection of a sort, type I – contains all (or, in the type I case, one-half of) allowed Ramond-Ramond potentials with all the even values of \((p+1)\) i.e. odd values of \(p\). The odd/even parity restriction results from the opposite/same chirality of the two spinors (left-moving, right-moving) that we are tensor-multiplying.

It's natural to write down the "sources" terms that are integrals over the world volumes of \(p\)-branes – with all allowed even values in type IIA and all allowed odd values in type IIB. If we phrase the situation in this way, you could always find it natural to assume that type I or type II string theory contains new charged objects - branes with an arbitrary even/odd number of spatial dimensions, respectively.

However, prior to the mid 1990s, it was believed that strings were the whole story. Literally. And one may prove that an arbitrary excited string state carries vanishing (no) Ramond-Ramond charges. So the world volumes that are being integrated over are not world volumes of any stringy excitations. If such "sources" terms where the Ramond-Ramond potentials \(A_{(p+1)}\) are being integrated over a world volume exist, they must be world volumes of completely new objects. And people didn't want to introduce new objects because they believed that this would ruin the consistency of string theory where everything has to be made of strings.

Nevertheless, Joe Polchinski realized that these objects do exist in string theory, anyway. They're the Dirichlet branes or D-branes for short. Are they new objects, intruders that spoil the consistency of string theory? Are they aliens who disrespect the rule that everything must be made of strings? Not at all! Instead, Joe Polchinski showed that all the D-branes have always been there and they are effectively made out of strings, too.

They may be imagined as "black \(p\)-branes", i.e. generalizations of black holes extended over additional dimensions, which are solutions to low-energy (generalized Einstein's) equations of string theory (the Ramond-Ramond fields are nonzero). But, as Polchinski realized, they may also be viewed as composites made out of open strings. In fact, every perturbation of the shape of a D-brane is equivalent to some (coherent) state of open strings that are attached to these D-branes with both of its end points! The same is true for any field – new world volume fields of the D-branes (different from the Ramond-Ramond fields that are always in the whole spacetime!) – which are identified with open strings, too.

Just like the spacetime geometry (a field-like property of the spacetime) is fully made out of closed strings in string theory, all the internal (field-like) properties of the D-branes are made out of open strings. In this way, Polchinski showed that these "chimeras", branes charged under the Ramond-Ramond charges (whose gauge potentials appear as fields in the Ramond-Ramond sector of the closed strings) exist, after all. When he did so, he also gave us a much more sensible and deeper interpretation of the open strings and their boundary conditions.

Prior to the Second Superstring Revolution, people would only consider open strings with Neumann boundary conditions because the Dirichlet conditions were "ugly" – they broke the translational and Lorentz symmetries in the spacetime. However, he realized – building on his insights and Petr Hořava's and others' insights from the late 1980s – that Dirichlet boundary conditions are equally natural and, in fact, related to the tolerated Neumann boundary conditions by T-dualities (so they must be equally legitimate). Indeed, the Dirichlet boundary conditions pick a "preferred" submanifold in the spacetime but it's a good thing because this "preferred" submanifold is nothing else than the world volume of the D-branes, the multi-dimensional objects charged under the Ramond-Ramond fields.

You may see that if you tried to push some of these ideas before the mid 1990s, you would encounter at least the following two problems: the objects charged under the Ramond-Ramond fields didn't seem to exist; and the addition of Dirichlet boundary conditions would produce some new unwanted submanifolds in the spacetime. You could decide that these are the two reasons to assume that nothing is charged under the Ramond-Ramond charges; and Dirichlet boundary conditions should be banned.

Instead, Polchinski showed that these two problems actually exactly cancel against one another! If you're an optimist, you may generalize this insight by believing that an even number of problems is like no problems at all. ;-) Indeed, the Dirichlet boundary conditions produce new "preferred" submanifolds in the spacetime – but these submanifolds are nothing else than the D-branes, the mysterious objects that carry the Ramond-Ramond charges! And open strings attached to these D-branes – by choosing the corresponding Dirichlet-and-Neumann mixed boundary conditions for individual dimensions – actually come in states that are able to deform the shape of the D-branes and modify the values of all the fields that are confined to the D-branes' world volume, too. The D-branes are fully dynamical and they're made out of strings, too, although one needs an "infinite number" of the strings to create a brane out of nothing, in the same sense in which solitons (e.g. magnetic monopoles) are composed of a large number of light excitations.

Polchinski has cancelled two problems against each other and allowed one to generalize the electromagnetic fields in many ways. People were forced to admit that these objects have always been part of string theory and they're actually needed for consistency (and T-duality, a symmetry) although they used to be overlooked. But that was very far from the last consequence of his breakthrough.

D-branes have become important players in string theory. In some respects, they are equally fundamental as the fundamental strings; in other respects, they remain less fundamental; but there are also respects in which D-branes are even more fundamental than strings. Although, in the weakly coupled regimes, they're "solitons made of strings", they may also be viewed as new independent degrees of freedom. That's particularly important in situations in which the D-branes become light. That occurs, for example, at the strong coupling: D-branes become light and "more fundamental" if you go to the opposite, infinitely strongly coupled limit of string theory.

D-branes may also be wrapped on "cycles" in a curved geometry. These wrapped D-branes have proven that topology of the spacetime manifold may smoothly change; that certain ALE-like manifolds automatically produce non-Abelian gauge symmetries, and many other things. D-branes became existentially important for string theory's ability to "microscopically" (using methods of statistical mechanics) prove that black holes indeed carry the entropy whose value was derived "macroscopically" or "thermodynamically" by Hawking and Bekenstein in the 1970s (the first successful proof that string theory passes these tests was completed by Strominger and Vafa). Finally, D-branes were used to define all of string theory – in Matrix theory as well as the AdS/CFT correspondence.

With the hindsight, everything makes sense. It is beautiful. There are some wonderful and apparently consistent possibilities that you would be surprised by if Nature wasted the opportunities they offer. And indeed, Nature – string theory – didn't waste the opportunities.

In the late 1990s, in a period that may already be classified as "years after the Second Superstring Revolution" (much like the current years), D-branes were used to extract many other new wonderful insights. Ashoke Sen studied unstable D-branes (typically D-branes that aren't charged under any generalized electromagnetic fields and, in this sense, they don't belong to this blog entry) and gave the world a brand new understanding of tachyons and instabilities. Tachyons on D-branes aren't just lethal signs of inconsistency that force you to abandon the theory as a taboo immediately. Instead, they're just signs of an instability that you may calmly study and if you do so, you will find out that the instability leads to a well-defined conclusion which involves the complete decay of the unstable D-brane.

There may also be "knots" on the tachyonic fields which prevent the D-brane from a complete annihilation. In that case, lower-dimensional branes may survive as a leftover of the decay. By realizing this fact around 2000, people were able to describe all lower-dimensional branes as some "knots" on higher-dimensional ones: they no longer classified D-branes by "homology of the spacetime" (cycles on which "infinitely thin" branes may be wrapped) but its "K-theory" (ways how the "thickened" shape of D-branes may arise from "knots" on a gauge field). Some heavily mathematically oriented string theorists pushed this line of reasoning further to argue that D-branes are really all about derived categories and similar fancy things.

There are lots of insights that combine into a beautiful whole. They fit together, much like a system of many keys and locks and combined key-locks and so on (I am thinking about a wild huge bloc of many building blocks that have keys and locks at various sides that are rotating in complex ways and always happen to hit the right counterpart of the lock/key). But there are also some overall lessons resulting from all these stories. People need lots of knowledge as well as imagination to think about many possibilities, many viable and important ways to generalize the concepts they have known, and so on. But it often happens – especially in string theory – that all these objects are "already there" in the theory; we just have to take notice. Careful analyses of string theory – which we had to treat exactly as if it were a new continent objectively discovered in the real, physical world – have often forced us to discover something that no one was able to "invent" before this "almost experimental, adventurous" excursion to the string-theoretical inland.

Ancient Greek and Roman philosophers would often believe that they may discover all the important facts about Nature by pure thought. In principle, they could have been right (well, a smart enough guy or babe could have written all conceivable systems of equations and pick at least QED by some comparison of the equations with known and rudimentary empirical facts we wouldn't even call experiments), but in practice, their attitude was far too arrogant and unrealistic. Instead, Galileo and Newton established the scientific method in which the pure thought was supplemented by something perhaps even more important and more characteristic for science, namely experimental and empirical tests of hypotheses.

It was still true that the "preparation of the hypotheses" and "guessing of the right mathematical structures and equations" was a purely theoretical enterprise, some kind of "pure arts" which should be done primarily by bold individuals with miraculous abilities who think that they can discover everything by a "divine intervention" combined with their ingenuity.

But what we learned in the mid 1990s – and maybe earlier than that – was something else. Even when we're preparing the hypotheses and looking for the right mathematical structures (the most viable "type of maths") and the right equations and the generalizations of all these concepts (we haven't found "all the maths" that may be relevant for a proper definition and understanding of string theory yet), we must sometimes be experimenters of a sort. I mean experimenters of a theoretical sort who make "experiments" with particular situations that may be described by a theory. These are theoretical experiments but they proceed much like real experiments. A theoretical consistency check works almost just like the classic experimental confirmation of a theory. And sometimes we're forced to discover an exotic behavior of the concepts in the theory and new phenomena, objects, and their relationships – and it's almost just like if we are discovering real new objects and phenomena by genuine old-fashioned experiments.

Physicists who understand this message know that they must be sufficiently humble – and appreciate their limits in comparison with Mother Nature and Her heart (or whatever organ it is) called mathematics. In practice, the right directions in which the maths of our theories should be generalized, the right new objects and terms and mathematical structures, and so on, are usually not discovered by self-described geniuses who claim to be seers and have the power to guess the right answers. Almost all the people in this category end up being hopeless cranks although many of them are surrounded by whole armies of journalists and other morons who don't want to see that they worship a crank. Instead, the right progress is being generated by the folks who lovably play with the theory and its various aspects, in mundane and especially extreme situations, and who are simply forced to discover the new concepts, objects, phenomena, and relationships because they have always been there! One still needs a lot of knowledge as well as creativity and a "big picture" perspective – but one also needs something else, something closer to the experimenters' job, some kind of hard work.

Even without direct experimentation, string theorists play the role of de facto experimenters throughout most of their working time and this is how most of the amazing progress in recent decades was generated.

And that's the memo.