January 28, 2006

Ghost D-Branes and Renormalization

Catch a very interesting discussion over at Cosmic Variance of a paper by Evans, Morris and Rosten, relating Morris’s Exact Renormalization Group for large-$N$ $SU(N)$ Yang Mills to the “Ghost D-brane” proposal of Okuda and Takayanagi.

The claim is that, by embedding $SU(N)$ Yang-Mills in a larger (nonunitary) theory, whose gauge groups is the supergroup $SU(N|N)$, spontaneously broken to $SU(N)\times SU(N)$, one can produce a gauge-invariant Pauli-Villars regulator, with which to implement the Exact RG. The latter theory, in turn, is what Okuda and Takyanagi argue is the world-volume theory of a stack of D-branes and ghost D-branes.

When the gauge symmetry is unbroken, the $SU(N|M)$ theory is equivalent to $SU(N-M)$, as far as computing gauge-invariant observables. In particular, there is a perfect cancellation of diagrams for $N=M$.

Turning on a nonzero Higgs VEV (separating the D-branes from the ghost D-branes) provides a cutoff for the original $SU(N)$ theory. Above the scale of the Higgs VEV, you get zero; far below it, the “original” $SU(N)$ degrees of freedom decouple from the ghost $SU(N)$.

Evans et al propose and AdS/CFT geometry realization of this idea, with the hope of connecting, in a explicit way, the “holographic RG” (evolution in the radial coordinate of AdS) with the “exact RG” of Morris.

Anyway, Takuya Okuda is over there, fielding questions, so take advantage …

Posted by distler at January 28, 2006 1:29 AM

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Re: Ghost D-Branes and Renormalization

Hi Jacques or Takuya,

do you actually understand why is the regularization using the supergroup connected with the appearance of the holographic fifth dimension? I have no clue. The simple reason why I have no clue is that so far I did not consider the choice of a gauge-invariant regulator to be more than a minor technical challenge for gauge theory, while holography is a shocking and mysterious feature of gauge theories.

Thanks,
Lubos

Posted by: Lubos Motl on January 29, 2006 7:52 PM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

Well, a lot of papers have been written, arguing that the radial direction in AdS acts like a (gauge-invariant) regulator. The new claim is that this is related to to the Pauli-Villars-like regulator that comes from embedding $SU(N)$ in $SU(N|N)$ and then spontaneously-breaking $SU(N|N)$ to $SU(N)\times SU(N)$.

I think your question is about the first claim, rather than the second.

Posted by: Jacques Distler on January 29, 2006 8:12 PM | Permalink | PGP Sig | Reply to this

Re: Ghost D-Branes and Renormalization

Dear Jacques,

most of us would agree that the energy scale encodes the radial dimension. That’s hopefully not a difficult question. The main question is why the physics becomes local in the fifth dimension - why locality and (local) Lorentz invariance including the holographic dimension emerges. May this be answered?

Best
Lubos

Posted by: Lubos Motl on January 29, 2006 8:56 PM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

I’m sorry. I don’t see why that’s the question at all.

The kind of question I have is why does the $SO(6)$-symmetric distribution of ghost D-brane charge they propose in their paper correspond to Morris’s Higgsing of $SU(N|N)$? Indeed, can the latter even be done in an $SO(6)_R$-symmetric fashion at all?

Naively, I can, at best, imagine preserving $SO(5)_R$.

Posted by: Jacques Distler on January 29, 2006 10:08 PM | Permalink | PGP Sig | Reply to this

Re: Ghost D-Branes and Renormalization

Sorry, Jacques, but are you saying that the emergence of locality in the holographic dimension is not an important or interesting question, or not a question at all?

I think that this is the single most mysterious general question about the gauge/gravity duality.

I completely agree with you that Higgsing of the gauge group is done by Higgses, and Higgs bosons are also charged under R-symmetry, which is why they break it, too. Does someone in the papers propose that you can break one without the other?

Posted by: Lubos Motl on January 30, 2006 8:29 AM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

I’m not sure what sort of answer you are looking for.

Their proposal is to study the near-horizon geometry of a system of N D-branes and N ghost D-branes.

My complaint is that this cannot be done in an SO(6)-invariant fashion, contrary to Figure 1 of their paper. Various subsequent arguments rely on the assumption of spherical symmetry, so that is a problem for them.

Posted by: Jacques Distler on January 30, 2006 8:39 AM | Permalink | PGP Sig | Reply to this

Re: Ghost D-Branes and Renormalization

The ghost branes are smeared over the five-sphere, so the associated (unphysical) gauge group is broken. In Morris’ regularization scheme - as applied in a purely field theoretic context - this breaking is not required, but there is no reason why it can’t be introduced.

Posted by: Oliver Rosten on January 30, 2006 10:56 AM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

Hi Guys. Thanks for your interest in our paper on ghost branes and SU(N|N).

There are 6 adjoint scalars in N=4 SYM - give one a vev and you break U(N) to U(1)^N. If you give all 6 the same vev then SO(6) is left unbroken - this is what the 5-sphere distribution represents.

We don’t claim to understand why the RG flow equations of N=4 are the same as the field equations of strings on AdS - that’s the miracles after all. We have provided a field theoretic understanding of the radial direction in AdS - one does what one can. :-)

Posted by: Nick Evans on January 30, 2006 11:11 AM | Permalink | Reply to this

Rotational invariance

I’m sorry, but I must be extremely dense.

$\langle\phi^1\rangle=\langle\phi^2\rangle=\dots=\langle\phi^6\rangle=v$ is not an $SO(6)$-invariant configuration. The vector $(v,v,...,v)$ is just a particular vector in $\mathbb{R}^6$, and $SO(6)$ rotations act on it nontrivially (it’s preserved by an $SO(5)$ subgroup).

Physically, this is just the direction of separation of the stacks of D-branes and ghost D-branes in the transverse $\mathbb{R}^6$. Choosing such a direction necessarily breaks the transverse rotational invariance.

Even if $\langle\phi^1\rangle...\langle\phi^6\rangle$ lie in the same direction in the Lie algebra, I don’t see how you can avoid that they break the $R-symmetry$ down to (at least) $SO(5)$.

Posted by: Jacques Distler on January 30, 2006 11:49 AM | Permalink | PGP Sig | Reply to this

Re: Ghost D-Branes and Renormalization

Duality cascade = matrix-valued Hagedorn tachyon = off-shell black hole entanglement = multiwarped octonionic throat = infinite-order orbifold of the generalized conifold = Z_top = 0, right ?

Posted by: marcel steiner on January 30, 2006 5:46 AM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

Equal scalars brake S0(6) to SO(5). However, one can take diagonal scalars such that the (vector) eigenvalues point in different directions of R^6, have the same length and are uniformly distributed on S^5. Then, at large N, the eigenvalues become dense on the sphere and SO(6) is restored. Under rotation the eigenvalues are reshuffled but this is part of the gauge group. For large but finite N, SO(6) is only an approximate symmetry. A rotation can be followed by an S_N transformation appropriately chosen to bring the rotated configuration very close to the initial one. I believe that the above scalar configuration is the gauge theory dual to the brane configuration described in the paper of Evans, Morris and Rosten.

Posted by: Bogdan Morariu on January 30, 2006 1:15 PM | Permalink | Reply to this

Re: Ghost D-Branes and Renormalization

In the D-brane language, this is simply the prescription to distribute the ghost D-branes uniformly on a spherical shell of radius $v$. For finite $N$, as you say, this breaks $SO(6)$ to a finite group, but the full $SO(6)$ is restores in the $N\to\infty$ limit.

What this does, however, is Higgs $U(N|N)\to U(N)\times U(1)^N$, which is not quite what Tim Morris was doing in his previous papers (higgsing to $U(N)\times U(N)$ ).

Maybe this is good enough. But I’d want to see that spelled out.

Posted by: Jacques Distler on January 30, 2006 1:27 PM | Permalink | PGP Sig | Reply to this

Gauge symmetry breaking pattern

Nick Evans assures me that breaking to $U(N)\times U(1)^N$ is perfectly fine, from the point of view of carrying out Tim’s exact-RG program. (I confess that I have not studied Tim’s papers closely-enough, yet, but that sounds plausible.)

In which case, the ghost D-brane configuration, described above, does the job.

Posted by: Jacques Distler on January 30, 2006 3:19 PM | Permalink | PGP Sig | Reply to this

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