Thursday, 20 November 2008


Edward Witten appeared here at CERN about the time of the first LHC beam. The coincidence suggests that he might have been created in a particle collision. That is however unlikely, since the entropy of one Witten is huge, even larger than that of a dragon (not to mention the fact that there were never any collisions at the LHC). In view of that, a more plausible explanation is that Edward is spending his sabbatical at CERN. My allusion to dragons was not entirely off-topic though, because Edward's presence seems to provoke awe and fear among the local string folk. I haven't yet quite discoverd why, but the story goes that around lunchtime each day they lock their offices, close the shutters, and hide in fireproof drawers.

Anyway, people out there don't want to know how Witten is doing, but what he is doing, so back to work... Last Tuesday Edward gave a talk entitled M2-Branes With Half Supersymmetry. The topic is far beyond my expertise and you should not expect any insight from me. I will try to summarize the main points, although it feels like reciting Bhagavad Gita in original.

Edward considers the 11-dimensional M-theory in the background of AdS4 x S7/$Z_n x Z_m$, which can be obtained as the near horizon geometry of a stack of M2 branes. This background preserves half of the original supersymmetry which corresponds to N=4 supersymmetry in 4D. The two discrete orbifold symmetries act on separate SO(4) components of the SO(8) symmetry group of S7. There are two orbifold fixed points who are $A_{n-1}$ and $A_{m-1}$ singularities on which $SU(n)$ and $SU(m)$ gauge theories live. Sitting on the Zn fixed point we see the SU(n) gauge theory in AdS4 x S3/$Z_m$ and, analogously, in the Zm fixed point we see the SU(m) gauge theory in AdS4 x S3/$Z_n$.

Edward argues that there are several interesting facts about this set-up:

  • The theory has a huge landscape of vacua that can be parametrized by elements x of SU(n) satisfying the condition $x^m = 1$ (because of the orbifolding) and elements y of SU(m) satisfying $y^n = 1$. There are $(\stackrel{n+m-1}{n}) \cdot (\stackrel{n+m-1}{m})$ such elements, so that the number of vacua grows factorially with n and m. It is surprising that so many vacua are encountered in a set-up with such a large amount of supersymmetry.
  • One can view the M2 branes as SU(m) instantons on $R_4/Z_n$ or, equivalently, as SU(n) instantons on $R_4/Z_m$. For some reason, the former point of view is called the Higgs branch, while the latter is called the Coulomb branch.
  • String theorists have their ways to count the number of instantons via D-brane configurations sitting at an orbifold point and the effective description in terms of quiver theories. Here, the quiver diagram for the Higgs branch contains the chain $SU(m) -> SU(m_0) x ... x SU(m_{n-1})$ and $U(p) -> U(p_0) x ... x U(p_{n-1})$ linked by bi-fundamental matter, where $p$ is the number of SU(m) instantons. Similarly, the Coulomb branch has the quiver with $SU(n) -> SU(n_0) x ... x SU(n_{m-1})$ and $U(\bar p) -> U(\bar p_0) x ... x U(\bar p_{n-1})$.
  • The integers m and n have a clear M-theory interpretation but the numbers of instantons p and $\bar p$ do not. But Gaiotto and Witten recently demonstrated the existence of a mirror symmetry that relates n and p, and also m and $\bar p$. This mirror symmetry allows one to describe both Higgs and Coulomb branches of M-theory.

This is it. I did not attempt to explain the physics but just to give a flavor of what Edward is brooding on these days. And don't ask me about the applications. God knows.


Kea said...

...but the story goes that around lunchtime each day they lock their offices, close the shutters, and hide in fireproof drawers.

LOL! Hilarious! But interesting about Z(m) and Z(n). Hmmm. If only M theorists wouldn't insist on the SU(N) gauge theory stuff, and start looking at S7 from the point of view of quaternionic Hopf bundles and qudits.

Anonymous said...

You haven't considered the Pauli effect?

Daniel de França MTd2 said...

Hi Jester and everyone!

Do you pople have a good review on quiver diagram?

Luboš Motl said...

Dear Kea, the text below is not quite optimized to answer your question, but try these comments about deconstruction where quivers are also defined etc.

Jester said...

Daniel, you may look at the original paper of Douglas and Moore, hep-th/9603167. I'm sure there many reviews but I'm not familiar with any.

Daniel de França MTd2 said...

Hi Jester,

Thanks for pointing out that article. It had a link to John Baez weekly findings 230. There, he linked to reviews of quiver representations :