The Polyakov Action

String Theory without the Baggage, Episode 18.

Hey Friends!

Alright. Dust off your geometry toolkits because we’re about to get into it. Here’s our first in a series of pieces about the basics of the bosonic string.

Sean

Here is a beautiful wall along Devil’s Canyon in Northwest Colorado, and it is precisely where you can direct any and all metric signature complaints for this series.

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Let’s consider an embedding of one manifold into another. Let Σ and 𝑀 be two Riemannian manifolds such that

\(\dim \Sigma = n,\quad \mathrm{and}\quad \dim M = N > n.\)

We’ll equip Σ with a metric 𝛾 and 𝑀 the metric 𝑔. Now let 𝑋 be a smooth injection:

\(X: \Sigma \rightarrow M.\)

We aim to make this a physical object, and study the dynamics of Σ as it moves through 𝑀. Following our study of the relativistic point particle, we want to consider the dynamics associated to this embedding by minimizing the action

\(S = \kappa \int d^{n}\sigma \, \sqrt{-\gamma} \left( \frac{1}{2}\gamma^{ab}g_{\mu\nu} \partial_{a}X^{\mu}\partial_{b}X^{\nu} + \Lambda + R^{(n)}(\gamma)\right).\)

The first of the three terms is clearly the kinetic sector for Σ. The second is the analogue of a cosmological constant term on Σ - consistent with the embedding map. The third is the curvature of the metric 𝛾 on Σ itself, also consistent with the embedding. At this point we have said nothing about the dynamics of 𝑔, the gravitational metric on 𝑀, and assume it to be fixed.

In short, we have included basically every term we could think of in 𝑆 that might affect the dynamics of Σ that is consistent with local reparametrization invariance on Σ and global invariance on 𝑀. We could of course include more matter fields, gauge fields on 𝑀, etc, and couple them to Σ, but as with our study of the relativistic point particle, we’ll ignore those extra details for now.

The Point Particle (n=1)

To make sure we’re not insane, let’s verify these ideas for the case where 𝑛 = 1. Here Σ equals the worldline 𝛾 of a relativistic point particle. In particular, the worldline metric 𝛾𝑎𝑏 has only one component¹:

\(\gamma_{ab} \rightsquigarrow \gamma_{11} = \gamma.\)

Note that this means that the inverse metric is

\(\gamma^{ab} \rightsquigarrow \gamma^{11} = \frac{1}{\gamma}.\)

In this case,

\(S = \kappa \int d\sigma \, \sqrt{\gamma} \left( \frac{1}{2\gamma}g_{\mu\nu} \partial_{1}X^{\mu}\partial_{1}X^{\nu} + \Lambda + R^{(1)}(\gamma)\right).\)

First note that 𝑅^(1) actually vanishes in one-dimension, as the Riemann curvature tensor reads²

\(R^{a}_{\,bcd} = \Gamma^{a}_{bd,c} - \Gamma^{a}_{bc,d} -\Gamma^{a}_{cf}\Gamma^{f}_{bd} + \Gamma^{a}_{bf}\Gamma^{f}_{cd},\)

so that

\(R^{1}_{\,111} = 0,\)

and hence

\(R^{(1)} = 0.\)

Simplifying,

\(S = \kappa \int d\sigma \, \frac{1}{2\sqrt{\gamma} }g_{\mu\nu} \partial_{1}X^{\mu}\partial_{1}X^{\nu} + \sqrt{\gamma} \Lambda .\)

If 𝑔_𝜇𝜈 is the flat metric 𝜂_𝜇𝜈, and we write the constant³ Λ as

\(\Lambda = -\frac{(mc)^{2}}{2},\)

we can take 𝜆 = √𝛾 and see that this is nothing more than the relativistic point particle action:

\(S = -\int d\sigma \, \frac{1}{2\lambda }\dot{X}^{\mu}\dot{X}_{\mu} +\frac{\lambda}{2}(mc)^{2}.\)

Great! Next time, we’ll consider n > 1.

1. In this case we don’t need to worry about a Minkowski signature difference here, since there is only one coordinate. This means that we won’t have to worry about any minus signs in the square root of the determinant, insofar as we take the “mostly minus” signature convention. As you’ve probably seen by now, this makes a ton of sense for the point particle. I’d apologize for the chaotic flipping of these sign conventions, but I think it’s better to use the one that makes the most sense for your model. Also switching between them frequently is a good way to make sure you have a strong handle on what, precisely you are doing.

2. Note here the relativist convention that a comma implies a partial derivative.

3. Again, note that the minus sign that appears here is related to the fact that - for the point particle - we've been taking the “mostly minus” sign convention formerly famous with particle physicists.