# How is the Nambu Goto action proportional to the world sheet area?

Hi,

(I'm not sure if this belongs in the BSM forum...apologies to the moderator if it belongs there.)

I'm working through Polchinski's book on string theory (volume 1) and I came cross the definition of the Nambu Goto action. I want to understand why the Nambu Goto action is proportional to the area of the world sheet. This is probably a trivial question but I'll ask anyway.

Specifically

$$S_{NG} = -\int_{M} d\tau d\sigma\frac{1}{2\pi\alpha'}(-\det(h_{ab}))^{1/2}$$

where

$$h_{ab} = \partial_{a}X^\mu \partial_b X_\mu$$

Thanks!

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samalkhaiat
In general, an n-dimensional object sweeps out an (n+1)-dimensional manifold (world-volume) as it moves through space-time. So, the simplest action, which is invariant under re-parametrization of the world-volume is the “volume” of the world-volume.
Nambu and Goto were trying to generalize the action of relativistic point particle, which is a 0-dimensional object sweeping curves $x^{ \mu } ( \tau )$(1-dimensional manifold) in space-time with $\tau$-invariant action,
$$S[x^{ \mu }]= - m \int d s = - m \int d \tau \sqrt{ \eta_{ \mu \nu } \dot{ x }^{ \mu } \dot{ x }^{ \nu } } = - m ( \mbox{ length } ) ,$$
to the case of string $X^{ \mu } ( \sigma , \tau )$ (i.e., 1-dimensional object sweeping out 2-dimensional world-surface) and demanding reparametrization invariance of the world-sheet. The area of the world-sheet is given by
$$d ( \mbox{ area } ) \sim \sqrt{ \det | g _{ a b } | } \ d \sigma \ d \tau ,$$
where the metric $g_{ a b } = \partial_{ a } X_{ \mu } \ \partial_{ b } X^{ \mu }$ is obtained by contracting the tangent vectors
$$\frac{ \partial X^{ \mu } }{ \partial \tau } \ \ ; \ \frac{ \partial X^{ \mu } }{ \partial \sigma } .$$
Because the area of a surface is independent of the parametrization, the Namu-Goto action
$$S[ X ] = T \int d ( \mbox{ area } )$$
is indeed invariant under an arbitrary change of world-sheet coordinates.

• 1 person
mfb
Mentor
String theory is beyond the Standard Model, I moved the thread.

maverick: I strongly suggest you to use even the Zwiebach book while studying Polchinski. It's very clear and well written, even though it doesn't use the full mathematical formalism of Polchinski.

• 1 person
julian
Gold Member
Note the Nambu Goto action has an obvious geometric meaning but it is not the action that ends up getting quantized usually...The Polyakov action is introduced instead which is classically equivalent. Interestingly see:

http://uk.arxiv.org/pdf/hep-th/0401172

maverick: I strongly suggest you to use even the Zwiebach book while studying Polchinski. It's very clear and well written, even though it doesn't use the full mathematical formalism of Polchinski.
Thank you samalkhaiat and Rexcirus.

Why does Polchinski refer to $h_{ab}$ as the metric (and explicitly say that it isn't the induced metric) when Zwiebach says it is actually the induced metric?

To be perfectly clear, Polchinski uses $h_{ab}$ in equation 1.2.9b in defining the Nambu-Goto action, whereas the notation used by Zwiebach is $\gamma_{\alpha\beta}$ in equation 6.44. Zwiebach calls his $\gamma_{ab}$ an induced metric on the world-sheet (just above equaton 6.42), but Polchinski says his $h_{ab}$ is a metric (as opposed to an induced metric).

samalkhaiat
Thank you samalkhaiat and Rexcirus.

Why does Polchinski refer to $h_{ab}$ as the metric (and explicitly say that it isn't the induced metric) when Zwiebach says it is actually the induced metric?

To be perfectly clear, Polchinski uses $h_{ab}$ in equation 1.2.9b in defining the Nambu-Goto action, whereas the notation used by Zwiebach is $\gamma_{\alpha\beta}$ in equation 6.44. Zwiebach calls his $\gamma_{ab}$ an induced metric on the world-sheet (just above equaton 6.42), but Polchinski says his $h_{ab}$ is a metric (as opposed to an induced metric).
The “metric” in the Nambu-Goto action is the induced metric. The fact that the world-sheet lives in space-time means that we can measure distances on it using the space-time metric: mapping the world-sheet into space-time means that it picks up a metric, the so-called induced metric. To find an expression for the induced metric, start from the metric on space-time
$$d s^{ 2 } = G_{ \mu \nu } ( X ) \ d X^{ \mu } \ d X^{ \nu } = G_{ \mu \nu } ( X ) \frac{ \partial X^{ \mu } }{ \partial \sigma^{ a } } \frac{ \partial X^{ \nu } }{ \partial \sigma^{ b } } d \sigma^{ a } d \sigma^{ b } .$$
If the displacement is to stay on the surface, then we may write this in terms of the induced metric on the surface
$$d s^{ 2 } = g_{ a b } ( X ) d \sigma^{ a } d \sigma^{ b } .$$
Comparing the two expressions, we find the induced metric
$$g_{ a b } ( X ) = G_{ \mu \nu } ( X ) \frac{ \partial X^{ \mu } }{ \partial \sigma^{ a } } \frac{ \partial X^{ \nu } }{ \partial \sigma^{ b } } = \partial_{ a } X^{ \mu } \ \partial_{ b } X_{ \mu } . \ \ \ (1)$$
Using this, the Nambu-Goto action becomes
$$S[ X ] = - T \int d \tau \ d \sigma \ \sqrt{ - g } = - T \int d \tau \ d \sigma \ \sqrt{ ( \dot{ X }^{ \mu } \cdot X_{ \mu }^{ ' } )^{ 2 } - ( \dot{ X }^{ \mu } )^{ 2 } ( X_{ \mu }^{ ' } )^{ 2 } } .$$
Mathematically, this is a formula for the area of a sheet embedded in Minkowski space-time. Notice that the action is a functional of the string fields only, i.e., $g_{ a b } ( X )$, which is not a function of the world-sheet coordinates $( \tau , \sigma )$, is not an independent field variable. And this is exactly the difference between the above Nambu-Goto action and the following Polyakov action
$$S[ h_{ a b } , X^{ \mu } ] = - \frac{ T }{ 2 } \int d \tau \ d \sigma \sqrt{ - h } \ h^{ a b } ( \tau , \sigma ) \ \partial_{ a } X^{ \mu } \ \partial_{ b } X_{ \mu } .$$
Here, $h_{ a b } ( \tau , \sigma )$ is trated as a field variable independent of the string field $X^{ \mu } ( \tau , \sigma )$, and therefore, is the intrinsic metric on the world-sheet, and not the induced metric. The identification $h_{ a b } \equiv g_{ a b }$ works only as the solution, Eq(1), to the classical equation of motion for $h_{ a b }$
$$\frac{ \delta }{ \delta h_{ a b } } S[ h , X] = 0 .$$

As for the books you have mentioned, I can not say anything because I have not seen any of them.

Sam