Notes on string theory: Symmetries of the Polyakov action (additional comments)

Let’s engage in further discussion about the symmetries of the Polyakov action. As it is an important topic, it will serve as a useful exercise to go through things a bit more deeply.

As I cited in a previous post, Polchinski describes the symmetries of the S_{P} action early on by noting on p.13,

  1. D-dimensional invariance under the Poincaré group.

        \[X^{\prime \mu} (\tau, \sigma) = \Lambda^{\mu}_{\nu} X^{\nu}(\tau, \sigma) + a^{\mu} \]

        \[\gamma^{\prime}_{\alpha, \beta} (\tau, \sigma) = \gamma_{\alpha \beta} (\tau, \sigma) \]

  2. Diffeomorphism invariance.

        \[X^{\prime \mu} (\tau^{\prime}, \sigma^{\prime}) = X^{\mu} (\tau, \sigma) \]

        \[ \frac{\partial \sigma^{\prime c}}{\partial \sigma^a} \frac{\partial \sigma^{\prime d}}{\partial \sigma^b} \gamma^{\prime}_{cd} (\tau^{\prime}, \sigma^{\prime}) = \gamma_{\alpha \beta} (\tau, \sigma) \]

  3. Two-dimensional Weyl invariance.

        \[ X^{\prime \mu} (\tau, \sigma) = X^{\mu} (\tau, \sigma) \]

        \[\gamma^{\prime}_{\alpha \beta} (\tau, \sigma) = exp (2 \omega (\tau, \sigma)) \gamma_{\alpha \beta} (\tau, \sigma) \]

I think it is useful to add some elaborating comments. There is also a nice point of discussion that broaches the introduction of conformal Killing vectors as related to the metric transformation under local Weyl rescaling and diffeomorphism. From a purely pedagogical point of view, a few comments can also be made directly relating the above symmetries with conserved quantities via Noether’s theorem.

Local and global symmetries

To begin, let’s return our attention to the symmetries of the Polyakov action. I think it is important to highlight that S_{P} has both local and global symmetries. Moreover, S_{P} has symmetries of the worldsheet and of the background spacetime. Furthermore, it is important to reemphasise that the local and global symmetries are considered and discussed from the perspective of the theory of the 2-dim worldsheet.  Thus, in terms of spacetime symmetries, we know S_{P} is manifestly Poincaré invariant due to the very nature of its construction. This is a benefit of how S_{P} was derived, and it can be said that the the Poincaré group is a global internal symmetry from the perspective of 2-dim field theory. As a point of emphasis, however, one should note that X^{\mu} can take on the interpretation as either spacetime coordinates or dynamical fields on the WS. This means that when considering Lorentz transformations, for example, while local spacetime symmetries, they are in fact global symmetries of the WS. In other words, the Poincaré transformations are global symmetries; but reparameterisations and Weyl transformations are local symmetries, with the latter used when making a decision on a gauge (Becker, Becker and Schwarz, p.30).

Proof of d-dimensional invariance under the Poincaré group can be found by noting,

    \[X^{\prime \mu} (\tau, \sigma) = \Lambda^{\mu}_{\nu} X^{\nu}(\tau, \sigma) + a^{\mu} \]

Where, \Lambda_{\mu \nu} = - \Lambda_{\nu \mu} and a^{\mu} is a constant. It follows,

    \[ S^{\prime} [\tau, \sigma] = \frac{T}{2} \int d^2 \sigma \sqrt{- \gamma} \gamma^{\alpha \beta} g_{\mu \nu} \partial_{a}(X^{\mu} + \Lambda^{\mu}_{\delta}X^{\delta}) \partial_{b}(X^{\nu} + \Lambda^{\nu}_{\delta}X^{\delta}) \]

    \[= S + \frac{T}{2} \int d^2 \sigma \sqrt{- \gamma} \gamma^{\alpha \beta}(\Lambda_{\mu \delta} \partial_{a} X^{\mu}\partial_{b}X^{\delta} + \Lambda_{\nu \delta} \partial_{a}X^{\delta} \partial_{b}X^{\nu}) + \mathcal{O}(\Lambda^2) \]

    \[ = S + \frac{T}{2} \int d^2 \sigma \sqrt{- \gamma} \gamma^{\alpha \beta} (\Lambda_{\mu \delta} + \Lambda_{\delta \mu})\partial_{a}X^{\mu}\partial_{b}X^{\delta} + \mathcal{O}(\Lambda^2) = S + \mathcal{O}(\Lambda^2) \]

As for the WS symmetries, it is noted above that we have local diffeomorphism invariance and 2-dim Weyl invariance. Regarding the former,  if from the perspective of the WS we have a 2-dim field theory X^{\mu} coupled to 2-dim gravity, it has been noted in a past entry that S_{P} is invariant under any WS reparameterisation. For any diffeomorphism, X^{\mu} \rightarrow X^{\prime^{\mu}} the scalar field transforms more completely as,

    \[ X^{\mu} (\sigma) \rightarrow X^{\prime^{mu}} (\sigma^{prime})= X^{\mu} (\sigma(\sigma^{\prime})) = X^{\mu}(\sigma) + \epsilon^{c}\partial_{c}X^{\mu}(\sigma) \]

    \[ \equiv X^{\mu} + \delta X^{\mu} + \mathcal{O}(\epsilon^2) \]

    \[ \implies \delta X^{\mu} = \epsilon^{c}\partial_{c} X^{\mu} \]

It follows that the metric \gamma_{\alpha \beta} defined in a previous post transforms as,

    \[ \gamma_{\alpha \beta} (\sigma) \rightarrow \tilde{\gamma_{\alpha \beta}} \]

Which is commonly expanded and written  in the form (although, as with our entire discussion, notation may vary superficially),

    \[ \frac{\partial \sigma^{\prime c}}{\partial \sigma^{a}} \frac{\partial \sigma^{\prime d}}{\partial \sigma^{b}} \gamma_{c d}^{\prime} (\sigma^{\prime}) = \gamma_{\alpha \beta} + \delta \gamma_{\alpha \beta} + \mathcal{O}(\epsilon^2) \]

    \[ \delta \gamma_{\alpha \beta} = \epsilon^{c} \partial_{c} \gamma_{\alpha \beta} + (\partial_{a} \epsilon^{c}) \gamma_{c b} + (\partial_b \epsilon^c) \gamma_{ac} \]

    \[ = \nabla_a \epsilon_b + \nabla_b \epsilon_a \]

 

We also have Weyl invariance – or local conformal invariance. 

    \[ \delta X^{\mu} = 0 \]

    \[ \gamma_{\alpha \beta} \rightarrow exp(2 \Lambda(\sigma)) \gamma_{\alpha \beta} = \gamma_{\alpha \beta} + \delta \gamma_{\alpha \beta} + \mathcal{O}(\Lambda^2) \]

    \[ \delta \gamma_{\alpha \beta} = 2 \Lambda(\sigma) \gamma_{\alpha \beta} \]

This is an extra symmetry that we’ve been bestowed due to the simple fact that, as frequently mentioned, what we’re working with now is a 2-dim WS theory. Moreover, “the appearance of Weyl invariance for 2-dimensional worldsheets identifies String Theory as a very special generalisation of the point particle theory” (Weigand, p. 17). This special quality of ST will also become explicate when we broach the topic of the quantisation of the string and the lovely marriage that takes place with the path integral formalism. Indeed, there is a lovely approach to deriving the Nambu-Goto and Polyakov actions by way of the path integral formalism that reveals the many symmetries we’ve been discussing (I will perhaps dedicate a separate post to this study). This is a point worth emphasising, I think, because as we will see in the future the stringy generalisation is absolutely lovely in this regard.

Metric transformation and conformal Killing vectors

Following Weigand’s approach to bosonic string theory in his lecture notes, I think it is good to supplement reading Polchinski by highlighting how with the combination of diffeomorphism and Weyl rescaling, the metric undergoes the following transformation:

    \[ \delta \gamma_{\alpha \beta} = \nabla_{\alpha} \epsilon_{\beta} +\nabla_{\beta} \epsilon_{\alpha} + 2\Lambda \gamma_{\alpha \beta} \]

    \[ =  \nabla_{\alpha} \epsilon_{\beta} + \nabla_{\beta} \epsilon_{\alpha} - \nabla^{c} \epsilon_c \gamma_{\alpha \beta} + 2(\Lambda + \frac{1}{2} \nabla^{\alpha} \epsilon_{\alpha}) \gamma_{\alpha \beta} \]

    \[ \implies \delta \gamma_{\alpha \beta} \equiv (P \cdot \epsilon)_{\alpha \beta} 2 \Lambda^{\prime}\gamma_{\alpha \beta} \]

Here, as Weigand points out, this “linear operator P maps vectors symmetric traceless 2-tensors” (p. 17). And, what is nice is how, for any transformation \epsilon_{\alpha} where (P \cdot \epsilon)_{\alpha \beta} = 0, it can be said that the resultant effect can be undone by local Weyl rescaling. These \epsilon_{\alpha} terms are conformal Killing vectors (CKV).

Now, the presence of conformal Killing vectors in our discussion anticipates deeper engagement with conformal geometry, conformal symmetry and, one may have guessed it, Conformal Field Theory (CFT). 

But let us bracket these last few comments for a future post.

In the meantime, on the presence of conformal Killing vectors, let’s magnify in on this. To do so, as is common in most texts, I should first follow a discussion on the symmetries of the S_{P} action by noting that reparameterisation invariance implies the existence of conserved currents (the currents vanish on-shell, such that \nabla^{\alpha}T_{\alpha \beta} = 0 on-shell for X). We can write the conservation of the energy-momentum current in the following way,

    \[ \mathcal{P}_{\mu}^{a} = -T \sqrt{- \det \gamma} \gamma^{\alpha \beta} \partial_{\beta} X{\mu} \]

    \[\partial_{\alpha} \mathcal{P}_{\mu}^{a} = \nabla_{\alpha} \mathcal{P}_{\mu}^{a} = 0 \]

Moreover, we can denote the conserved currents as j_{b}^{f} = f^{a}(\sigma)T_{ab} for any function f^{a}(\sigma). It was noted in a past post that the energy-momentum tensor is traceless – that is to say it is zero – and this is a consequence of Weyl invariance, such that T_{a}^{a} = 0 as \delta S = 0. (For the reader comfortable with QFT, you will know or might recall that a local quantum field theory will have a conserved stress tensor). This can be shown by simply noting that, with conformal invariance of the action under reparameterisation, 

    \[ \gamma_{\alpha \beta} \rightarrow e^{\phi} \gamma_{\alpha \beta} \]

If we were to apply this in the context of an infinitesimal \phi notice,

    \[ \delta S = \frac{\delta S}{\delta \gamma^{\alpha \beta}} \]

Wherein,

    \[ \delta \gamma^{\alpha \beta} = \frac{\sqrt{-\gamma}}{4 \pi} T_{\alpha \beta}(e^{- \phi} - 1)\gamma^{\alpha \beta} \]

    \[ \implies \gamma^{\alpha \beta}T_{\alpha \beta} = 0 \]

Additionally, if reparameterisation implies conserved currents, we can determine that under global Poincaré transformations – this time invariance under global Lorentz transformation – we also find the conservation of angular momentum currents,

    \[ J_{\mu \nu}^{\alpha} = - T \sqrt{- \gamma} \gamma^{\alpha \beta}(X_{\mu}\partial_{\beta}X_{\nu} - X_{\nu}\partial_{\beta}X_{\mu}) \]

    \[ = X_{\mu}P_{\nu}^{\alpha} -  X_{\nu}P_{\mu}^{\alpha} \]

    \[ \nabla_{\alpha} J_{\mu \nu}^{\alpha} = 0 \]

The observations are, again, quite standard in the literature. Following Weigand (p.18) in particular, the following generalisation seems rather apt at this stage of our study. Returning to the brief mentioning of conformal Killing vectors, if every \epsilon_{alpha} which satisfies P \cdot \epsilon_{\alpha \beta} results in or produces a conserved current, we can say:

    \[ J_{\epsilon}^{\alpha} = T^{\alpha \beta} \epsilon_{\beta} \]

Where, as noted, \nabla_{\alpha} J_{\epsilon}^{\alpha} = 0. The proof for this is quite simple. You can find it on p. 18 in Weigand’s lecture notes, or, just recall from GR a study of Killing vector fields.

References

Katrin Becker, Melanie Becker, John H. Schwarz. (2006). “String Theory and M-Theory: A Modern Introduction“.

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].