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.
Q&A (4871)
