Polyakov action

Question

I've started to read something about strings and I feel a little confused with the Polyakov action. The reason is because in this action you get two metrics, one of them is the induced metric over the world-sheet and the other is an arbitrary metric on every point of this world-sheet.

This is the definition I have of the Polyakov action: $\int d\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}h_{ab}$

Here $h_{ab}=\partial_{a}X^{\mu}\partial_{b}X^{nu}g_{\mu\nu}$ and is called induced metric, because you can get it from the metric of the spacetime. On the other hand, the second one $\gamma^{ab}$ is called metric as well, but is dynamical and arbitrary, because is as a field smeared on worldsheet, this new dynamical metric don't have any relation to $h_{ab}$, this $\gamma^{ab}$ get a dependence on $h^{ab}$ only when we start to work with the EOM with regard to this metric.

So now I'm confused. Which of this metrics $h^{ab}$ or $\gamma^{ab}$ I must use to rising and lowering indices in a tensor living in the world-sheet?

My answer would be $h^{ab}$, because it has a geometric meaning , but then why you give the name of metric to the other field $\gamma^{ab}$

Answer 1

The key here is to distinguish between two different manifolds. We have $M$ taken to be space-time, and within this manifold $M$, we imagine a string propagating which sweeps out a surface $\Sigma \subset M$.

The embedding functions of $\Sigma$ are $X^\mu(\tau,\sigma)$ which carry a $\mu$ index over space-time but are functions of coordinates defined for $\Sigma$ treated as a manifold in its own right.

The Polyakov action is,

$$S \sim \int d^2 \sigma \, \sqrt{-h}\, h^{ab} g_{\mu\nu}\partial_a X^\mu \partial_b X^\nu$$

where $h_{ab}$ is the induced metric on the surface $\Sigma$, and $g_{\mu\nu}$ is the metric of $M$, hence contracted with $X^\mu$ and $X^\nu$ which carry space-time indices.

Thus, if I have some object say $P^a$ which carries a worldsheet index $a$, to lower $a$ on $P^a$, one would use the metric on $\Sigma$ and thus $P_a = h_{ab}P^b$. Likewise, $M$ indices are raised and lowered with $g_{\mu\nu}$.

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A Subtlety

You may now think that if we're doing any operation that carries a space-time index, then we are doing it with respect to the manifold $M$, but this is not necessarily the case.

For example, suppose we have a sub-manifold $\Sigma \subset M$ embedded in $M$. We can define a covariant derivative, $\nabla_\mu$ which you would think means ordinary covariant differentiation w.r.t. $g_{\mu\nu}$.

However, this needn't be the case. In general, $\Sigma$ inherits two essentially equivalent metrics from $M$, the induced metric $\gamma_{ab}$ computed normally from the embedding and the first fundamental form,$^\dagger$

$$h_{\mu\nu} = g_{\mu\nu} \pm n_\mu n_\nu$$

which is a sort of projection of the metric on $M$; here $n_\mu$ is the normal vector. Thus, $\nabla_\mu$ could mean covariant differentiation on $\Sigma$ even though it carries a space-time index, if w.r.t. $h_{\mu\nu}$.

Usually, ambiguities like these are made clear in most sources, and a lot of the machinery of the differential geometry of sub-manifolds is not needed in introductory string theory texts.

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$\dagger$ Here $h_{\mu\nu}$ is not to be confused with $h_{ab}$. I picked $h$ simply because the first fundamental form is normally always named $h$. Note $\gamma_{ab}$ here is the same metric that makes an appearance in the Nambu-Goto action.

Answer 2

Indeed, there are two metrics, and there will be two sets of indices associated to tensors on the world sheet: one for the world sheet and one for the target space. For example, $X^{\mu}(\tau, \sigma)$ does not carry world sheet indices but it carries target space index $\mu$, so one should lower or raise $\mu$ by the target space metric. On the other hand, $\partial_aX^{\mu}$ carries both target space index $\mu$, and world sheet index $a$. So one should lower/raise $a$ by world sheet metric and lower/raise $\mu$ by target space metric .

One can also say this in the language of tensors being muti-linear maps, but I'm not sure it would be more helpful for this question.