# Issue with quantized branes?

JustinLevy
I was speaking with a gradstudent studying string theory the other day, and he mentioned that currently "branes" higher than just strings are currently treated classically. As he is still learning, he couldn't comment further.

My questions are this:
1) It is my understanding that the "prediction" of the number of dimensions comes from string theory only being finite/self-consistent for a certain set of dimensions, and they take the smallest of these as the prediction. Is it possible that when quantizing the higher dimensional branes, that this will restrict the set of "allowed" dimensions even more? Is it possible it will reduce this set to zero (ie. it can't be finite/self-consistent with quantized higher branes)? Or is there some mathematical theorem to protect it?

2) Are there technical issues preventing considering quantized branes? ...or is it technically straight-forward, but just very messy, so people focus elsewhere?

## Answers and Replies

Gold Member
One of the first calculations you do in learning string theory is to show that bosonic string theory is self-consistent only in 26 spacetime dimensions. However, another simple calculation shows that the bosonic string has a state of imaginary mass (tachyon), which means the whole thing blows up due to instability anyway.

Adding supersymmetry into the mix, a similar calculation shows that superstring theory is self-consistent only in 10 spacetime dimensions. And in this case, one can show that there are no tachyons, so the theory is at least stable. Trying to get it to make contact with our familiar 4-dimensional world is somewhat of a challenge.

I'm not aware of branes putting any further restriction on the number of spacetime dimensions...but I am still learning these things.

TFT
D-branes in string theory are defined as subspaces of spacetime on which open strings can end. The idea is that the open string spectrum can be described by fields living on the world volume of D-branes. Therefore, in string theory, D-brane actions are constructed using open string modes.

Generally, a p-brane is a p+1 dimensional subspace of spacetime, and the general p-brane world volume action cannot be quantized because it is nonlinear, or at least nobody knows how to quantize it.

Find a string theory textbook and read it seriously, if you really wanna understand these.

Generally, a p-brane is a p+1 dimensional subspace of spacetime, and the general p-brane world volume action cannot be quantized because it is nonlinear, or at least nobody knows how to quantize it.

Find a string theory textbook and read it seriously, if you really wanna understand these.
Interesting discussion. Unfortunately I have only seen claims that it doesn't wórk, but never a calculation which shows why it doesn't work.

Does it become inconsistent?
Are there obstacles to write down the action?
Are there obstacles to introduce a measure in the path integral?

Gold Member
Ah, TFT has it. The action itself is easy to write down; the problem is that it has a square root in it. And we only know how to quantize actions that are at most quadratic in the fields; if you expand the square root in a Taylor series, you get terms of all orders.

In the particular case of the string, one can re-write the theory using an auxiliary metric, forming the Polyakov action. This action is quadratic in the fields, so we can quantize it.

Is there an obstacle to doing the same procedure with p-brane actions? Sorry, I can't answer this right now...busy with some other stuff.

Gold Member
Formally, it is easy to quantize a (free) brane. Essentially, a quantized d-brane is a quantum field theory in d+1 dimensions. However, for d>1, the resulting quantum theory leads to divergences which nobody knows hot to remove. Only in the case d=1, i.e. for strings, there is a known way to remove the divergences. This is related to the mathematical fact that 2-dimensional conformal symmetry is much richer than a higher dimensional conformal symmetry.

how does the d>1 action look like?
is there a Polyakov-like formulation?

TFT
how does the d>1 action look like?
is there a Polyakov-like formulation?

For any free p-brane
$$S_p=-T_p\int \sqrt{-det(G_{ab})}d^{p+1}\sigma$$
where $$G_{ab}$$ is the induced metric on the world volume of the p-brane. The idea is that this action gives you the volume of the p-brane world volume. The p=1 case gives you the Nambu-Goto action for strings. Read GSW or Becker-Becker-Schwarz if you want the details.

OK, this is rather straighforward. As in the p=1 case there is a background metric which induces the 2-metric.

And what about a Polyakov-like formulation?

Homework Helper
It turns out that quantizing a higher dimensional brane is conceptually a bit different from quantizing a string. For example, people have been very interested in quantizing the M2 branes of M-theory. The quantization can be carried a long way along the lines of the Polyakov quantization of the string. However, one discovers a very important difference between the F1 string and the M2 brane, namely that the brane spectrum is continuous. The quantized M2 brane and related matrix models seem to describe multi-object states instead of a single isolated brane.

A simple way to understand why this might is to think about "spikes" on the brane. For a single string we may think of the energy as being given by the length times the tension. The string can't stretch a lot without costing a lot of energy. Now consider an M2 brane with energy given by area times brane tension. It is possible to form spikes of fixed or even small area that extend arbitrarily far from the brane for fixed or small energy cost. Thus the fluctuations of the brane seem more severe and non-local. In particular, one brane may actually look like many distant objects without a severe energy cost. Despite this conceptual difference, one can go a long way with the M2 brane theory including calculating all kinds of potentials, etc reproducing super gravity results. One also makes contact with the theory of multiple D0 branes and the BFSS conjecture.

Higher dimensional branes have other strange features such as a worldvolume action containing gauge fields. This leads to the AdS/CFT description of branes in terms of dual field theories. In some ways this is a better situation than the ordinary perturbative quantization of the string because we have something of a non-perturbative definition of branes in terms of various gauge theories.

• ... one discovers a very important difference between the F1 string and the M2 brane, namely that the brane spectrum is continuous. The quantized M2 brane and related matrix models seem to describe multi-object states instead of a single isolated brane.

Sounds rather strange as standard 2-dim. branes (with appropriate boundary conditions) always have a laplacian with disceret sprectrum.

Is there a reference where this spectrum issue is derived?