Ashoke Sen and tachyon condensation

Some CERN and FNAL news: The latest Higgs paper by ATLAS has the significance level of 5.9 sigma. I find it silly to write articles about every new increased number of this sort. The significance level will clearly keep on increasing with the square root of the number of collisions. You could have learned about this simple fact as well as the proportionality factor on TRF since December 2011.

More interestingly, the Tevatron has published evidence for the Higgs boson in events involving bottom quark-antiquark pairs, a rare discipline in which the Tevatron could have competed with the LHC (or beat it for a while). This signal pretty much erases speculations that the July 4th Higgs-like particle refuses to interact with bottom quarks etc. It does interact and the available data make its properties more or less identical to the Standard Model Higgs predictions.

Ashoke Sen (yup, I took this Wikipedia picture of him in front of ex-Glashow, ex-my, and now-Randall office) is one of the nine winners of the inaugural Milner fundamental physics prize. And he deserves it a great deal.

You will find something like 250 papers he has written and 20,000 citations those papers have accumulated. The list includes 22 papers with at least 250 citations – and his most famous papers talk about the counting of black hole microstates in many contexts, D-brane actions, Matrix theory, S-duality, subtleties of heterotic string theory, and others.

Still, despite the fact that strong-weak duality was quoted as the main justification of his $3 million award (Sen figured out that the Montonen-Olive duality had to apply e.g. to heterotic compactifications to 4 dimensions as well), there's another theme in string theory he is the true father and main stockholder of: tachyon condensation. He started this minirevolution by his visionary 1998 paper Tachyon condensation on the brane-antibrane system.




Lots of technology and detailed insights were later added by others as well as Sen himself but the change of the paradigm he brought in the late 1990s was a testimony of his ingenuity. Edward Witten, who later refined some of those insights, once stated that he has no idea about the divine interventions that led Sen to his prophetic insights. (OK, I improved Witten's words a bit.)

Tachyons: the ugly guys

To appreciate how much he changed our perceptions of tachyons, we must recall some of the history of tachyons in physics. Einstein's special theory of relativity allows us to divide directions in spacetime (and energy-momentum vectors) to timelike ones, spacelike ones, and the intermediate null i.e. lightlike ones.

A basic consequence of special relativity is that no particle can move faster than light. Equivalently, \[

p^\mu p_\mu = E^2 - p^2 c^2 \gt 0

\] and all world lines of actual particles must be timelike (or at least null). This requirement arises from causality and the Lorentz symmetry: if the world lines were spacelike, these particles would be moving forward in time according to some reference frames and backward in time according to others. In the latter reference frames, the effect would precede the cause which shouldn't happen in a logically consistent world.

The word "tachyon" is derived from Greek τάχος ("tachos") for speed. That refers to the huge, superluminal speed of those particles. Some laymen like to say that special relativity predicts these particles. Well, it introduces the concept of tachyons but it also instantly says something about the new concept and it is not flattering: they aren't allowed to exist.



Tachos is not directly related to tacos.

If we switch from classical mechanics to quantum physics, we need to work with relativistic quantum physics to see what happens with the tachyons. Relativistic quantum physics requires quantum field theory or its refinements (such as string theory). What happens with tachyons in quantum field theory? Well, consider spinless tachyons (and for some technical reasons, both string theory and informal consistency arguments rooted in effective field theory actually allow spinless tachyons only). Their Lagrangian is a Klein Gordon Lagrangian,\[

\LL = \frac 12 \partial_\mu T \cdot \partial^\mu T + \frac 12 \mu^2 T^2.

\] The coefficient in front of the mass term is positive which actually means that the potential is \(-\mu^2 T^2/2\): it is negatively definite. This unusual sign of the mass term is exactly what is needed to switch from timelike vectors \(p^\mu\) to spacelike ones, assuming the picture with classical world lines. The squared mass of the tachyon is negative; you could say that the mass itself is imaginary.

The point \(T=0\) of the potential \[

V(T) = -\frac 12 T^2

\] is a stationary point but it is no longer a minimum: it is a maximum. So if Nature sits there for a while, it won't last forever. It will prefer to roll down and cause a catastrophic instability. Needless to say, the Higgs field with \(H\equiv T\) behaves as a tachyon near the point \(H=0\). That's the ultimate reason why the electroweak symmetry is broken: the Higgs field rolls down to another place with a lower value of \(V(H)\) than the value \(V(0)\).

This Higgs-like analogy sounds natural and it could lead us to look for a better point where the tachyons could live. But that wasn't how string theorists would be thinking about tachyons until the late 1990s.

Tachyons as an illness

Instead, string theorists would immediately declare every theory with tachyons to be sick. They would never ask further questions. The existence of a tachyon in a theory meant that a good physicist should never study such a theory in detail.

The old, bosonic string theory required \(D=26\) dimensions. It didn't contain any fermions – which was already bad enough – but it had one more annoying feature. The lowest-squared-mass particle actually had a negative squared mass; it was a tachyon. The squared mass was\[

m_\text{open tachyon}^2 =-\frac{1}{\alpha'},\quad m_\text{closed tachyon}^2 = -\frac{4}{\alpha'}.

\] Note that due to the factor of four, the (not squared) mass of the closed string tachyon is exactly equal to twice the (not squared) mass of the open string tachyon. You may literally think that the closed string tachyon is a package with one open-string tachyon for the left-moving excitations on the closed string; and one open-string tachyon for the right-movers. The constant \(\alpha'\), the Regge slope, is inversely proportional to the string tension \(T\) via \(2\pi T\alpha'=1\).

For decades, it would seem that this tachyon was a pathology of the bosonic string theory. It manifested itself by horrible infrared divergences in the loop diagrams. Consequently, bosonic string theory was just a toy model to learn some techniques and conformal field theory. But the truly physically consistent theory was superstring theory which contained fermions but no tachyon and which cancelled all those infrared divergences.

String theorists would instinctively dismiss any theory or any vacuum with any tachyons, too. But Ashoke Sen was able to think different. In some sense, it is not surprising that his first "modern understanding of a stringy tachyon" occurred in the brane-antibrane system. It's because it's the situation in which it seems most obvious that we can't deny that the building blocks may be arranged in this way. Type II string theory implies the existence of D-branes – and of course the existence of D-antibranes or anti-D-branes (which one do you prefer?), too. They're separately stable and consistent. And by the clustering property, it must be possible to place them near one another. So we must have an answer to what happens if those D-branes and anti-D-branes are close to each other.

Tachyons in brane-antibrane systems

However, the brane-antibrane system looked exactly as ill as other systems with tachyons. Why?

If you derive the usual superstring theory in the RNS (Ramond-Neveu-Schwarz) variables, you must impose something known as the GSO projections to get a spacetime-supersymmetric spectrum. All physical states in the single-string Hilbert space have to obey\[

(-1)^{GSO_{\rm left}} \ket\psi = +\ket\psi,\quad
(-1)^{GSO_{\rm right}} \ket\psi = +\ket\psi.

\] In a suitable basis of eigenstates of these two parity-like operators, the two projections above reduce the number of basis vectors to one-half per projection i.e. to one quarter in total. Only one quarter of the basis vectors survive this GSO projection. The ground state tachyon state is filtered out, the first excited state is allowed, and every other excitation (if we only count the fermionic excitations) is kept. The overall, "diagonal" GSO projection is needed for the spacetime fields to obey the spin-statistics relation (integer spin states must respect the Bose-Einsteni statistics; half-integer spin states must respect the Fermi-Dirac statistics). Both projections are needed for the spacetime supersymmetry to exist.

There is also a world sheet consistency explanation why such projections exist. The reason is that we need R-R, R-NS, NS-R, and NS-NS sectors (the latter is the most natural one) in which the left-moving fermions (first label before the hyphen) and right-moving fermions (second label after the hyphen) may be both periodic or antiperiodic. To allow the world sheet to preserve the signs of the fermions or flip them (independently for left-movers and right-movers; in "type 0" theories, only the "synchronized" sign flips are allowed), you must declare the operator flipping the symmetry to be a gauge symmetry on the world sheet. But if it is a gauge symmetry, physical states have to be invariant under it, and that's where the GSO projections come from.

Amusingly enough, the GSO projections have reduced the number of basis vectors to one quarter because the two parities were random numbers from the set \(\{+1,-1\}\). This is beautifully compensated by the fact that we have four sectors, NS-NS and the three others. So in some rough counting, the combination of "many sectors" and the "corresponding GSO projections" doesn't change the number of states. This holds for any "orbifolds" and "compactifications producing winding numbers", too. The GSO projections may be understood as a special case of the orbifolding procedure.

Wrong signs

The spectrum of open-string excitations arising from open strings attached to an anti-D-brane by both end points is the same as the spectrum for a D-brane. After all, an anti-D-brane may be understood as a D-brane rotated by 180 degrees (although spacetime-filling D-branes don't leave you with transverse dimensions in which you could rotate them; and this picture of rotation also fails for D-instantons which are just points and can't be nontrivially rotated, either, because one needs at least one transverse and one longitudinal direction to perform such a rotation).



However, if you study open strings whose one endpoint is attached to a D-brane while the other endpoint is attached to an otherwise parallel anti-D-brane (in the intermediate cases, such an anti-D-brane may be described as an antiparallel D-brane with the opposite orientation than the first one), you will find out that the diagonal GSO projection is reverted upside down. For brane-antibrane systems, the states that were kept at brane-brane systems are killed, and the states that used to be killed and preserved.

In particular, the ground state tachyonic state \(\ket 0\) is preserved and it is a part of the physical spectrum. So if your D-brane and anti-D-brane are (anti)parallel and coincide, string theory predicts that the effective field theory will contain spinless fields that are tachyons arising from open strings stretched between a brane and an antibrane. (Well, the ground state is tachyonic only for small enough transverse separations. The transverse distance \(a\) adds \((Ta)^2\) to the squared mass \(m^2\); for high separation, the mass itself goes like \(Ta\). Relative angles and internal excitations along the open strings raise the squared mass, too.)

The fate of the tachyon

As discussed earlier, such a pathological, supersymmetry-breaking, tachyon-infected configuration of D-branes and anti-D-branes may occur in otherwise consistent type II string theories with otherwise stable, beautiful, and supersymmetric D-branes that happen to meet their antiobjects (yes, the "anti" is the same "anti" as the "anti" in "antimatter" – at least, it's the most straightforward generalization of antimatter from pointlike particles to extended objects – and it has almost nothing to do with "anti-Semites" and other things).

If we trust the perturbative string theory, and we surely should if the coupling constant is low, it seems obvious that there is a tachyon. Well, it's a complex one because it transforms as a bifundamental representation under \(U(M)\times U(N)\) if we deal with a stack of \(M\) D-branes and \(N\) coincident anti-D-branes. There is an instability. Ashoke Sen wasn't scared of such an instability. He wouldn't scream "Look at the instability, catastrophe, planetary or cosmic emergency: abandon all fossil fuels, don't look at this theory, it's scary" or "get used to it, natural catastrophes will be here constantly" as Michio Kaku, a co-father of string field theory.



The potential energy curve for the complex tachyon is nothing else than the Mexican hat potential of a sort. The minima correspond to a complete destruction of the D-brane and the anti-D-brane.

Instead, he calmly analyzed what the outcome of this instability could be and he realized that the final outcome had to be completely peaceful. After all, the D-brane and anti-D-brane carry the opposite charges. They're antimatter to each other. So they may annihilate with each other and the tachyon is just a sign of this looming annihilation. When they annihilate, however, the potential energy for the tachyon rolls down lower. How much lower? Well, the decrease of the potential energy is exactly what you would expect from the complete destruction of the latent energy \(E=mc^2\) carried by the brane and the anti-brane. Because this energy is uniform and its density is given by the tension, Sen realized that it had to be the case that the energy densities obeyed\[

V(T=0) - V(T=T_{\rm min}) = 2\cdot {\rm Tension}_\text{D-brane}.

\] Because the point \(T=T_{\rm min}\) where the potential is minimized was rather far from \(T=0\), the difference above couldn't have been calculated in a straightforward fashion. Nevertheless, it was in principle possible to calculate the energy difference from string theory and Sen's formula above was therefore a conjecture about a particular result.

Various formal proofs of the conjecture were soon given – in boundary string field theory, by informal string arguments etc. – but the most explicit and indisputable proof, one that led to interesting new mathematical identities, was given in the framework of Witten's cubic string field theory by Martin Schnabl in 2005, in a paper that turned lots of assorted numerical data from numerical string field theory (which had already pushed the validity of the conjecture beyond all reasonable doubts) into the final indisputable analytic proof.

Generalization to other branes, bosonic string theory, SFT

Sen's insight meant one of those "unification moments" in which previously separated things were found to be two states in the same theory. Previously, one would either have D-branes, or no D-branes. There was no relation between such configurations. Sen realized that one could annihilate or recreated brane-antibrane pairs and this process could have been described as the evolution of some totally standard fields that follow from string theory.

This realization had many consequences. First of all, one could discuss not just brane-antibrane tachyons but also other tachyons that were arising on a single, non-supersymmetric D-brane. Such D-branes exist in string theory as well. After all, all D-branes of bosonic string theory are examples. (Additional examples of such non-supersymmetric branes are even-dimensional D-branes in type IIB and odd-dimensional branes in type IIA superstring theory, the "wrong parities".) It meant that open string tachyons lost their "catastrophe status". Whenever people encountered an open string tachyon before 1998, they would say "phew, horrible, catastrophe, let's go away from this mess". After 1998, they would say "nice, what an interesting instability, let's look where the instability ends". Of course, it's some mundane annihilation of objects in which most of the latent energy is released.

Needless to say, all the comments about string field theory above referred primarily to bosonic open string field theory. That's where the processes predicted by Sen may be studied most explicitly. The love is mutual: tachyon condensation became the main process that may be more naturally studied by string field theory than by other approaches to string theory. Various previous expectations that string field theory would be the answer to all open questions in string theory turned out to be wrong – an unjustified wishful thinking. It has pretty much all the limitations we see in any perturbative formulation of string theory. However, tachyon condensation is a discipline in which string field theory continues to shine.

Lower-dimensional branes as solitons

Aside from the complete annihilation of a brane-antibrane pair or a complete destruction of an unstable D-brane in various theories, one could study more subtle processes. The complex tachyon field \(T\) may have many values and in fact, only something like \(|T|^2\) has to have the right value for the potential energy to be minimized. The phase isn't determined and it may be nontrivial.

So one may actually start with a brane-antibrane system in which \(V(T)=V_{\rm min}\) almost everywhere but there is a localized topological obstruction in the configuration of the tachyon field (some kind of a kink, vortex, monopole, or instanton) that prevents the branes from disappearing completely. If that is the case, what is the annihilation going to end up with? Besides the light string radiation that carry the energy away, we are left with a lower-dimensional D-brane (or many such D-branes).

It can be shown that the topologically nontrivial vortex-like configurations of the tachyon fields (together with the natural gauge field that lowers the configuration as much as possible by making it "nearly pure gauge") defined on coincident D-branes and anti-D-branes carry the same Ramond-Ramond charges as lower-dimensional D-branes. Because those charges are preserved, the lower-dimensional D-branes must survive the annihilation process. They may be identified with topological defects of the tachyon fields!

This general insight became a whole mathematical subindustry of string theory in the case of D-branes wrapped on complicated Calabi-Yau-like manifolds. Because the lower-dimensional D-branes are given by possible "knots" on the tachyon field, one may describe in a completely new way how the branes may be wrapped on the manifold. Instead of "homology" which would be the previously believed mathematical structure classifying "how branes may be wrapped on cycles" (homology directly talks about topologically nontrivial "infinitely thin" submanifolds wrapping the big manifold), people started to talk about K-theory (the set of topologically inequivalent knots on the complex tachyon fields). Edward Witten would be the main industrialist promoting this new mathematical description of the wrapped D-branes.

K-theory sounds much like M-theory and F-theory but unlike M-theory or F-theory, it isn't any theory in a physics sense at all. The similarity in the names is misleading. Instead, it is a notion analogous to homology which is in many situations fully equivalent to homology but it may differ by some discrete subtleties, "torsion" etc., the information saying that twice or \(n\) times wrapped D-brane of some kind may unwind, and so on. K-theory (which naturally leads you to think that the lower-dimensional D-branes are "fat" and spread in the transverse dimensions a bit) began to be considered a more accurate description of the allowed D-brane charges than homology. After K-theory, certain heavily mathematical string theorists started to work with a more detailed mathematical structure knowing about the D-brane charges (plus other things), the so-called derived categories. As far as I understand it, it's the classification of the allowed D-brane charges together with some part of the dynamical information, something that allows the folks to study the lines of stability in new ways, too.

It's cool that those physics processes match some math definitions but I still think that if a physicist didn't know the concept of K-theory at the beginning, he could still be able to discuss the general processes first envisioned by Sen and he may decide to never invent the term "K-theory". In some sense, we're only getting what we're inserting. K-theory and its concise notions are important if you want to "mass produce" results about D-branes wrapped in general ways on complicated and general manifolds. But if you only know the rules of the game that involve the complexified tachyons and Sen's processes, you could be satisfied, after all. So I believe that K-theory – and derived categories – were almost exclusively promoted by folks trained as mathematicians who had been brainwashed to think that things like categories and K-theory were cool and important; and they were hardwired to look for their physics applications which they did find. But if they hadn't been brainwashed at the beginning, they could very well agree that those new mathematical concepts were not needed to understand the "laws of physics".

I think that a more conceptually new result was the research of tachyon condensation in non-commutative spacetimes i.e. in the presence of a nonzero \(B\)-field. Gopakumar, Minwalla, and Strominger had the great idea to study the "unusual but cool" limit of a very large non-commutativity – in some sense, the analogue of \(\hbar\to \infty\) limit. Well, Seiberg and Witten and others studied the same limit. But GMS looked at the tachyon condensation in that limit and realized that lower-dimensional branes become nothing else than individual cells in the quantum phase space identified with a subspace of the D-brane dimensions. The annihilating higher-dimensional D-brane is literally a "phase space composed of many cells" and one may annihilate them one by one, if you wish. At least that's what the mathematical analogy says (the interpretation of the D-brane is of course different than the interpretation of a phase space).

In the new unusual limit, the tachyon field may jump between the individual stationary points, whether they are maxima or minima of the potential, kind of "directly" and "seemingly discretely", without complicated questions how the interpolating curves exactly look like (which normally depends on the precise shape of the potential energy in between the stationary points, too).

Twisted closed strings and closed string tachyons in general

The number of fantastic results that emerged thanked to Sen's visions was huge. One could perhaps say that the qualitative behavior of all open string tachyons in string theory has been understood.

This progress didn't immediately generalize to closed string tachyons. Why? Well, D-branes are objects whose tensions or masses scale like \(1/g\) where \(g\) is the closed-string coupling constant. However, the tree-level energy densities coming from closed strings – and therefore the tensions of more ordinary closed-string "solitons" – scale like \(1/g^2\) which is parameterically larger than \(1/g\) for a small \(g\). In this sense, D-branes are heavy because \(1/g\) diverges in the \(g\to 0\) limit but they're still infinitely lighter than what is needed for a significant change of the surrounding geometry.

If you study the gravitational acceleration around a D-brane, you know it is proportional to the mass which goes like \(1/g\) but also to Newton's constant which actually scales like \(g^2\). The latter wins so the "backreaction" (no relation to Sabine Hossenfelder) of a single D-brane is still negligible in the weakly coupled limit.

We may annihilate several D-branes and study the process as tachyon condensation. Enormous energies going like \(1/g\) for \(g\to 0\) are produced but in some sense, these are still minor processes. The energy densities needed to stabilize closed-string tachyons would have to scale like \(1/g^2\) and such huge changes affect the spacetime geometry by something of order 100 percent.

For this reason, closed string tachyons are still viewed almost in the same way as they were viewed before 1998, at least in practice. No canonical "stabilized minimum" of the potential energy curve is known and the closed string tachyon condensation is still a catastrophic process we would better avoid.

(Well, I was corrected by an expert that the previous paragraph is really no longer the case. Simeon Hellerman and Ian Swanson studied bulk closed-string tachyon condensation in various string theories, including supercritical ones, and found out it has an interesting and well-defined endpoint – in fact, these end points are apparently able to connect string vacua in different dimensionalities of spacetime in the same sense as the open string tachyon condensation merges configurations with and without D-branes. I won't go into these cool, relatively new, and advanced mechanisms in this introductory text.)

However, there's one class of exceptions pointed out and first studied by Adams, Polchinski, and Silverstein (APS, no relationship to the American Physical Society). They noticed that there can be closed-string tachyons in twisted sectors of various orbifold compactifications in string theory. Because states in the twisted sectors must be localized to the fixed points of the orbifold, the impact of these tachyons has to be spatially limited. In this respect, these particular closed-string tachyons have to resemble the open-string tachyons that had already been understood.

For example, if you study a simple \(\RR^2/\ZZ_n\) orbifold, the geometry looks like a cone with a deficit angle. Some tachyonic fields live at the tip of the cone and APS collected quite some evidence in favor of the proposition that the the condensation of the closed string tachyon is physically interpreted as the smoothening of the original sharp tip of the cone. The sharp iceberg is peacefully melting which means that there's no reason for fearmongering about the lethal twisted-sector tachyons. That's why APS named their paper Don't panic! Closed string tachyons in ALE spacetimes. I am sure that these three politically correct physicists were deeply sorry of this title a few years later, once the climate hysteria became a dominant symbol of their political allies and it turned the commandment "Do panic" into the most important insight of all of science. ;-)

OK, so open-string tachyons and closed-string tachyons in twisted sectors are more or less understood by now. Physicists no longer panic or abandon the theory when they see a tachyon. Instead, they calmly interpret these tachyons as sources of instabilities – instability that annihilate objects or whole chunks of spacetime and that may (but don't necessarily have to) lead to a new stable world with some interesting objects that may be left over.

For this reason, it was Ashoke Sen and his apprentices such as Edward Witten who unified nothingness and somethingness in physics – i.e. in string theory (because no other theory can achieve similar unifications) – and who discovered new perspectives on the somethingness in between and that's my explanation why he deserves his $3 million.

And that's the memo.