amazing…that viscosity measures(D brane analysis?) could have been hidden in all this talk?

I’ll let Olly try to convince you about regulators – as you’ve deduced it’s basically the same story as Pauli Villars though.

Flat space as a dual I can try to be helpful for you though. Stay in the pure AdS/CFT Correspondence – move the D3 onto a 5-sphere. The geometry is AdS outside the sphere and flat space inside. What does that mean? It means the theory is totally higgsed at the scale corresponding to the radius of the sphere and below that scale is a mass gap. The gravity description of this “nothing” is flat space. Our host, Clifford, played precisely this game in his enhancon papers removing repulsons and replacing with flat space…

cheers Nick

I can shed some light on the issue of unitarity at finite cutoff. As effectively said already, if the cutoff is sent to infinity, all fields in the spontaneously broken SU(N|N) theory become infintely massive, with the exception of the physical SU(N) field (and an unphysical copy which is decoupled). One of the crucial ingredients of our ERG approach – which is built in from the start – is that the partition function is invariant under the flow. Since we know that we are dealing with just SU(N) YM at the top end of the flow, we know that we must be dealing with the same theory everywhere along the flow. So, when computing physical quantities, the unphysical fields serve only to regularize the physical theory, and do not spoil unitarity. This works at finite N.

Oliver.

The normalization of the boundary state is fixed by demanding that the open string partition function is properly normalized in the presence of boundaries.

Perhaps it might be helpful to note that one can, technically, put either bosonic or fermionic Chan-Paton factors on the boundary; you get the same answer either way. Usually, this is thought of as an either/or proposition. The ghost D-brane proposal is to allow both bosonic and fermionic Chan-Paton factors in the same theory.

Shouldn’t I normalize this somehow with <B|B’>

No! Because <B|B’>=∞ !

The Boundary State is not a normalizable state in the Hilbert space.

<B|q^{H}|B’>

So, indeed, the normalization of the boundary state matters. That’s part of the data that goes into normalizing the path integral in the presence of boundaries.

I am surprised that by multiplying a boundary state |D> by (-1) you get a different state.
As you claim, a D-brane is turned into a ghost brane with opposite charges and tension.
I am a bit surprised by this since I would have thought that strictly speaking a state is not
given by a vector in a Hilbert space but by a ray in such a space. That is, only by a vector
up to normalisation by a non-zero complex number.

This would also follow from boundary CFT where the boundary state is defined by the condition
that certain operators (for example T, the left handed energy momentum tensor and its
right handed counterpart T-bar) agree on the boundary:

T|boundary>= T-bar|boundary>

But such equation again would characterise |boundary> at best up to a scalar multiple. So
I am quite surprised that a factor of (-1) has such drastic consequences. Maybe you
could comment on this?

Once we start transforming states by multiplication by (-1), nothing stops us from
considering exp(i phi)|D> for some phase exp(i phi). By a similar argument as yours,
I would conclude that this rotated state has a complex mass and charge since the
phase also multiplies amplitudes like

Is that correct?

I got a reply which I am not sure I am supposed to quote. It said something along the lines of the boundary state being a classical object from the target space perspective and thus a prefactor would matter. For example by multiplying by N I would get a state of N coincident branes.

I must say this does not really convince me. Could anybody else please say something about this?

But isn’t SU(n-m) when (n-m) is small dual to AdS_5xS^5 with small AdS length? If I am not missing something obvious then the same should go for SU(n|m) when n-m is small.

I agree the whole geometry is flat when there are equal numbers of D and ghost-branes, but don’t understand why the gauge theory is `dual’ to this flat geometry, as stated in the paper—probably I am missing something.

I don’t understand why it would be highly curved AdS in that limit. The sources of curvature seem to cancel each other out exactly (at least above a certain distance scale), and so flat space is pretty much the only option, I would say.

Cheers,

-cvj

I actually don’t fully understand why SU(N|N) is supposed to be dual to flat space in the ERG paper. I’d expect that, as N–>M, the SU(N|M) theory will be dual to increasingly highly-curved AdS_5xS^5. (Even if in the case N=M the near horizon limit isn’t really sensible).

Thanks. The reference to the TASI lectures looks promising.

Elliot