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Bosonic string theory in arbitrary spacetime dimensions ?

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In the context of calculating the mass or potential energy squared of a bosonic unexcited string stretched between two D-branes

\(m^2 = V(r)^2 = (T_0r) - \frac{1}{\alpha'} \frac{D-2}{24}\)   

to model the potential energy of a quark-antiquark pair, I have heard that there exists a variant of the Nambu-Goto action with an infinite number of additional terms, which makes sense in an arbitrary number of spacetime dimensions, such as for example $D = 4$.

What does this "alternative Nambo-Goto action" look like, and what kind of "new physics" does it contain compared to the "ordinary" bosonic string theory?

asked Apr 27, 2014 in Theoretical Physics by Dilaton (4,295 points) [ no revision ]

Very interesting. Could you give an example of a source in which you heard about this variant of the Nambu-Goto action?  

Could it be that the extra terms in the action are not fixed, but change depending on the resulting dimension, and happen to zero off at 26? 

It is in the same book I used in my previous question, on p.530 in Chapter 23.3 ...

Hm, to go to zero for D=26 seems like a good idea to do for the additional terms. However I am still interested in what exactly they are doing for $D\ne 26$ ...

No, what I meant was whether the source told you that the new Nambu-Goto action is always the same, or there are different modifications at different dimensions so that there is always some different extra terms for each possible critical dimension? 

Unfurtunately he says absolutely nothing specific about the additional terms, apart from the fact that there are infinitely many of them ...

1 Answer

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One of the first papers on this topic is probably the one by Polchinski and Strominger (Phys.Rev.Lett. 67 (1991) 1681-1684). However, the first paper that springs to my mind when one talks of effective string theories is the one by Luscher and Weisz. (see also http://arxiv.org/abs/hep-lat/0207003) which I will discuss.

They begin with a classical string in $D$ dimensions (with coordinates $X^\mu$, $\mu=0,1,\ldots, D-1$) stretched between $X^1=0$ and $X^1=r$. Work in a static gauge where one identifies the worldsheet time $z^0=X^0$ and $z^1=X^1$, , the fluctuations about this string (or collective modes of the string) can be parameterized by a $(D-2)$ dimensional vector $\mathbf{h}$. The idea is to write an action in a derivative expansion with the fluctuations subject to Dirichlet boundary conditions at $z^1=0,r$. The initial term is  a Polyakov type action. Evaluating the value of the action for the string will lead to an expression for the quark-antiquark potential as a power series in $\tfrac1r$. The nice tweak to this story by Luscher and Weisz was the use of open-closed duality to constrain these coefficients. Their action takes the form

$S=   TL\  (r + \mu)  + S^{(2)}_0 + S^{(2)}_1 + \ldots $

where the superscript indicates the derivative order, $T$ is the tension, $L$ is the length of the compact time direction and $\mu$ is a constant.

$S^{(2)}_0 = \tfrac12 \int d^2z\ \partial_a \mathbf{h} \cdot \partial_a \mathbf{h} \quad,$

$S^{(2)}_1 =b \left(\tfrac12 \int dz^0\ (\partial_1 \mathbf{h} \cdot \partial_1 \mathbf{h})\ \Big|_{z^1=0} + \tfrac12 \int dz^0\ (\partial_1 \mathbf{h} \cdot \partial_1 \mathbf{h})\ \Big|_{z^1=r}\right)$

They were able to show using open-closed duality that $b=0$ -- a non-renormalization theorem. They also discuss the four-derivative terms (odd derivative terms are not present.) where they find two terms. Again, open-closed duality relates the two. The amazing part of this story is that up to four-derivative terms, the coefficients are precisely what one would get by expanding the Nambu-Goto term about the background. In other words, the Nambu-Goto action reproduces the effective quark-antiquark potential much better than one should expect.

Ofer Aharony and his students/post-docs/collaborators have been carrying out a systematic study of extending this work to higher derivatives, non-static gauges and so on. This paper was probably the first in that series. Here they show that there is only one unfixed coefficient (out of three) at sixth-order on imposing (I think) open-closed duality and for $D=3$ all coefficients get fixed. The introduction to this paper is probably a better summary of things.

I don't think quantizing these effective strings would be a way to get around the c=1 barrier. 

answered Apr 30, 2014 by suresh (1,535 points) [ revision history ]
edited Apr 30, 2014 by suresh

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