The Nambu-Goto Action

String Theory without the Baggage, Episode 22

Does anyone else miss the warm embrace of Summer? Whatever. The coming winter is a great opportunity to stay in and do some mathematics.

Previously we’ve seen that the stress-energy tensor for the string is

\(T^{ab} =\frac{\pi}{2} \left[ \left(\partial^{a}X^{\mu}\partial^{b} X_{\mu}\right) - \frac{1}{2}\gamma^{ab}\left( \partial^{m}X^{\mu}\partial_{m}X_{\mu} + \Lambda \right) \right].\)

This was derived from the Polyakov action,

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

Following GSW, let’s go ahead and define the matrix G

\(G_{ab} = \partial_{a}X^{\mu}\partial_{b}X_{\mu},\)

so that

\(S = \frac{\kappa}{2} \int d^{n}\sigma \, \sqrt{-\gamma} \;\gamma^{ab}G_{ab}.\)

The constraint equations follow from the vanishing of the stress-energy tensor. In terms of G these are

\(G_{ab} = \frac{1}{2}\gamma_{ab}\gamma^{mn}G_{mn}.\)

We can then compute det G,

\(\det G = \frac{(\gamma^{mn}G_{mn})^{2}}{4}\det\gamma.\)

We can take the square root,

\(\sqrt{- G} =\frac{1}{2} \sqrt{-\gamma} \;\gamma^{ab}G_{ab}\)

But this is just the action for the string. Therefore, we have found the nonlinear representation of the string action,

\(S = -\frac{\kappa}{2} \int d^{2}\sigma \,\sqrt{-G}.\)

The determinant of the matrix G_ab is of course

\(\det G = G_{00}G_{11} - G_{01}G_{10}.\)

Let’s try to compute this thing explicitly.

The Conformal Gauge and the Nambu-Goto Action

Let τ, σ be the timeline and spacelike coordinates on the worldsheet, respectively. In the conformal gauge, where we use the various local symmetries of S to force the worldsheet metric to be flat,

Diffeomorphisms of the Worldsheet

we find that det G can be represented¹ as

\(\det G = \dot{X}^{\mu}\dot{X}_{\mu}X^{\prime\nu}X_{\nu}^{\prime} - (\dot{X}^{\mu}X^{\prime}_{\mu})^{2}.\)

Here dots and primes represent derivatives with respect to τ and σ, respectively. The nonlinear from of this action, in the conformal gauge, is therefore

\(S = \kappa \int d\tau\, d\sigma \, \sqrt{\dot{X}^{\mu}\dot{X}_{\mu}X^{\prime\nu}X_{\nu}^{\prime} - (\dot{X}^{\mu}X^{\prime}_{\mu})^{2}}.\)

This is the infamous Nambu-Goto action, which directly shows that the action is proportional the area swept out by the worldsheet of the string.

One Last Important Point

As we saw with the the relativistic point particle action, studied previously

What we talk about when we talk about Manifest Lorentz Covariance

the manifestly covariant phase space is different from the explicit representation. Importantly, it can² lead to different Hamiltonians.

A Different Hamiltonian

So too is the case for the Nambu-Goto string versus the Polyakov string. While it has - implicitly - the two reparametrization invariances³ of the Polyakov action, the Nambu-Goto string has no concept of Weyl invariance. It has no auxiliary metric to cary that scaling symmetry.

Weyl invariance is something specific to the Polyakov form of the action, and while central to the study of strings, it sometimes breaks down in the quantum theory. That is to say, the Polyakov string picks up an anomaly in its Weyl symmetry, unless certain conditions are met. Famously, those conditions include a constraint on the apparent⁴ dimension of spacetime.

The conditions surrounding the cancellation of this anomaly apply equally to quantization of the Nambu-Goto action, although it is much less clear how to do this. The Polyakov action provides us a geometric frame from which to understand the problem.

Broadly speaking, the nonlinear form of the action is incredibly awkward to use in practice. This was true with the point particle. To convince yourself of this for the string, derive the canonical momentum for X from the Nambu-Goto action.

1. The astute reader will note that, in the conformal gauge, the latter term vanishes via the constraint equations. This is where things start to get sticky. det G here is defined independent of any constraint choice, although the constraints were used to show that Polyakov action was equivalent to the nonlinear one whose form takes the square root of det G. We must be mindful about context as the symplectic structure should be derived prior to imposing constraints. Hence, an the canonical momentum for the string action in the Polyakov form will be a priori different for its nonlinear form. Keeping constraints and gauge choices clear in your head is one of the many challenges of this sort of work.

2. Although remember this is a choice

3. That is, a diffeomorphism invariance of the worldsheet.

4. Although again, remember our philosophy here. We’re going to remain agnostic about what that statement means, observationally.