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The tachyonic string mode in perturbative bosonic string theory indicates that the "vacuum", flat Minkowski $\mathbb{R}^{25,1}$, is not really a vacuum. What is conjectured about tachyon condensation in this theory? Do we expect the theory to have a vacuum? Is there any way the condensate might generate fermions dynamically?

[Edit] Since Chris Gerig asked for more background: Tachyons, sadly for Star Trek writers, are typically an indication that the state you think is the vacuum of your system is, in fact, not the vacuum. This is because you can act on the ground state with a tachyon creation operator and get a new state with lower energy. For example, the Higgs potential is $$V(\phi) = \frac{1}{2}\lambda (|\phi|^2 - c^2)^2 = -\lambda c^2 \phi^2 + \frac{\lambda}{2} \phi^4 + const$$ If you do perturbation theory around $\phi_0 = 0$ instead of around one of the true minimum $\phi_0 = C$, with $|C| = \mu^2$ , you'll find the creation operators in $\phi_{pert} = \phi - \phi_0$ create tachyons, with negative mass-squared $m^2 = -\lambda c^2$. Create enough of these tachyons, and you'll turn the state $|\phi_0 = 0\rangle$ which you thought was the vacuum into one of the true vacua $|\phi= C\rangle$.

Bosonic string theory in 26d has tachons: the most basic closed and open string excitations, created by vertex operators with no derivatives. So it's a natural question to wonder about: what state do we get if we add tachyons to the false perturbative vacuum? Does this process converge? This is a pretty hard question to answer, since in bosonic string theory we don't have SUSY-protected quantities that we can compute to check our predictions. Which is why I asked what had been conjectured.

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Please give us more background to this question. –  Chris Gerig Jun 24 '12 at 18:16

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These are several rather different question.

First, bosonic string theory in $d=26$ has both open string tachyons and closed string tachyons. The open string excitations are attached to D-branes. In particular, open strings that can live everywhere are excitations of a spacetime-filling D25-brane. The list of these open string excitations includes a tachyon. This open string tachyon is a sign of instability of the D25-brane with respect to the complete annihilation of the D25-brane. It's a violent process but the released energy only goes like $1/g_{closed}$, proportionally to the tension of the affected D-brane, which is smaller – for a small $g_{closed}$ – than the energy densities of order $1/g_{closed}^2$ which occur in closed string processes.

The difference of potential energies before (local maximum of the potential) and after the tachyon condensation was conjectured by Ashoke Sen to coincide with the tension of the D25-brane. It has been verifified by various informal proofs as well as very sophisticated and mathematically rigorous proofs in open string field theory, especially cubic string field theory where the most complete steps towards the quantitative understanding of the tachyon condensation was achieved by Martin Schnabl.


The understanding of the closed string condensation is much less clear. Most likely, there doesn't exist any nearby local minimum and the potential for the closed string tachyon is unbounded from below in all directions, signalling a neverending instability that destroys the spacetime beyond repair.

Whether there are fermions in bosonic string theories is a different issue. One may mention type 0 string theory in $d=10$ – which are naively as purely bosonic as the $d=26$ string. However, it was proved by Shiraz Minwalla and pals that one may find fermionic solitons in those theories:


There are also papers claiming to find a dynamical process that interpolates between bosonic string theory and superstring theory. They require some time-dependent configurations, however. A time-dependent tachyon is a part of the picture, if I remember well. See e.g. this paper by Simeon Hellerman and Ian Swanson and other papers by the same authors (and followups and references):


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Thanks for the detailed answer, Lubos. –  user1504 Jun 24 '12 at 19:27
'...the potential for the closed string tachyon is unbounded from below in all directions...' Why is this most likely? It seems like it would be really hard to get a trustworthy description outside of a tiny neighborhood of the perturbative vacuum. –  user1504 Jun 24 '12 at 19:29
Dear user, for the open strings, we actually have a description that is valid "everywhere", at distances from the perturbative vacuum that are of the same order as the order at which totally new terms start to change the behavior (eg full D-brane annihilation). For the closed strings, we don't have any string field theory that fully works so it's harder. Still, there exists neither a good description where the local minimum could be nor what it could physically mean. Bosonic string theory is simple enough to conjecture that such special structures such as new stationary points are not there... –  Luboš Motl Jun 25 '12 at 9:34

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