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The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\gamma_{\alpha\beta}, $$ where $\gamma_{\alpha\beta}$ is called the induced metric and $g_{\alpha\beta}$ the dynamical metric on the world sheet. I have difficulties understanding the differences between these two metrics. I know that the latter is introduced in order to be able to remove the square root in the Nambu-Goto action, but I don't know what it means. The space in which the string propagates has just the Minkowski metric $\eta_{\mu\nu}$, if I am not mistaken. Furthermore, I think that the induced metric is derived by demanding

$ds^2$(whole space) = $\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ = $ds^2$(world sheet) = $\gamma_{\alpha\beta}d\sigma^{\alpha}d\sigma^{\beta}$

Is this correct? I am really confused by all these different metrics.

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Related: physics.stackexchange.com/q/17349/2451 –  Qmechanic Feb 28 '13 at 20:05
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up vote 4 down vote accepted

There are two manifolds that are involved in string propagation.

  1. The spacetime in which the string propagates.

  2. The worldsheet of the string itself.

The fields $X^\mu$ are embedding coordinates of the worldsheet in the spacetime manifold. This means that for each point $(\sigma^1, \sigma^1)$ on the worldsheet, $X^\mu(\sigma^1, \sigma^2)$ gives the coordinates of that point in the spacetime manifold.

In the case you are considering, the spacetime is taken to be Minkowski, so the metric is $\eta_{\mu\nu}$. Now we could ask

"Given that the worldsheet is a two dimensional embedded submanifold of Minkowski space, is there some way that this manifold inherits its metric from the metric on the ambient spacetime?"

This question is analogous to

"Given that the sphere $S^2$ is some two-dimensional embedded submanifold of Euclidean space $\mathbb R^3$, is there some natural sense in which it inherits its metric from $\mathbb R^3$?

The answer to both of these question is yes, and the metric on the submanifold that does this is precisely the induced metric. The formula expression the induced metric for a two-dimensional submanifold of some ambient manifold with metric $g_{\mu\nu}$ (not necessarily flat) in terms of embedding coordinates is $$ \gamma_{ab}(\sigma) = g_{\mu\nu}(X(\sigma))\partial_aX^\mu(\sigma)\partial_b X^\nu(\sigma), \qquad \sigma = (\sigma^2, \sigma^2) $$ You are right about the derivation of the induced metric, it comes from demanding that the distance measured between points on the embedded submanifold is calculated to be the same number whether you use the ambient metric, or the induced metric. To see that the above expression for the induced metric does this, simply note that the infinitesimal distance between any two points on the embedded submanifold can be written in terms of the ambient metric and the embedding coordinates as \begin{align} g_{\mu\nu}(X(\sigma))d(X^\mu(\sigma))d(X^\nu(\sigma)) &= g_{\mu\nu}(X(\sigma))\partial_a X^\mu(\sigma)\partial_bX^\nu(\sigma)d\sigma^ad\sigma^b \\ &= \gamma_{\mu\nu}(\sigma)d\sigma^ad\sigma^b \end{align} To get some intuition for all of this, recall that expression for embedding coordinates of $S^2$ in $\mathbb R^3$ is \begin{align} X(\theta, \phi) &= \sin\theta\cos\phi\\ Y(\theta, \phi) &= \sin\theta\sin\phi\\ Z(\theta, \phi) &= \cos\theta \end{align} and using these embeddings you should be able to show that the metric on the sphere is simply $$ \gamma_{ab}(\theta, \phi) = \mathrm{diag}(1, \sin^2\theta) $$

Let me know if that's unclear or if you need more detail!

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Thanks for this thorough explanation, the induced metric is totally clear to me now. Could you maybe elaborate a bit more on the dynamical metric $g_{\alpha\beta}$, which is also defined on the string world sheet? –  Funzies Mar 1 '13 at 9:23
    
Hmm, well what sort of elaboration are you looking for? It's true that the dynamical metric is defined on the worldsheet since it's defined on the entire ambient space... –  joshphysics Mar 1 '13 at 16:51
    
Well, it is a dynamical metric, which has an equation of motion that can be calculated by varying the action. But a metric is really a way to measure distances in spacetime, so how can this be dynamical and have different values? –  Funzies Mar 4 '13 at 17:56
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I would like to add that the geometric picture and relationship between Nambu-Goto and Polyakov actions are only hints and heuristics. Specifically, string scattering amplitudes are computed in a Lorentzian space, but the worldsheets are Euclidean. One way to see it is that topology changes don't respect causality, so branching worldsheets are problematic for a Euclidean worldsheet. Would be great if a string theorist could elaborate.

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