Notes on string theory: Polyakov action – Symmetries, classical equivalence, equations of motion, and Virasoro constraints

We know from a past note that the Nambu-Goto action can be written as,

    \[ S_{NG}= - \frac{1}{2 \pi \alpha^{\prime}} \int d\tau d\sigma \sqrt{- h} \]

Where I’ve used slightly different notation than in the past in order to align with Polchinski, and so it should be restated h = det(h_{\alpha \beta}). This is just a superficial point. (I prefer to use \gamma here for the auxilliary worldsheet metric described below).

One more thing we might recall. It was mentioned in the last post that, as in the case of the point particle, when studying the action S_{NG} problems arise in the quantum theory that are associated primarily with the presence of the square root. So, for the study of the quantum physics of strings, we should like for an action of another form. What form may that be? Recall that in the case of the point particle, we modified the action by introducing an auxiliary field. This enabled the elimination of the pesky square root. Analogously, we can take the same approach for S_{NG}. In fact, in majority of textbooks and certainly in most lecture series I’ve reviewed, introduction to the Nambu-Goto action is followed almost immediately by the standard explanation of how S_{NG} can be modified with some sort of auxiliary field.

However, it is worth pointing out that this new auxiliary field, which we will define as \gamma_{\alpha \beta} (\tau, \sigma), is actually an “intrinsic” metric on the worldsheet (WS). In other words, it is a dynamic variable in the action and, inasmuch that one should think of it as an auxiliary WS metric, another notable implication is that it also leads to its own field equations. So, moving forward, \gamma_{\alpha \beta} will represent the WS metric and h_{\mu \nu} will represent the induced metric. It may also be noted,

    \[ \gamma = \det \ \gamma_{\alpha \beta} \]

    \[ \gamma^{\alpha \beta} = (\gamma^{-1})_{\alpha \beta} \]

From this, the Polyakov action, which we may also think of in terms of the string sigma model action, takes the form:

    \[ S_{p} [X, \gamma] =-\frac{1}{2}T \int_{\sum} d\tau d\sigma \sqrt{-\gamma} \gamma^{\alpha \beta} h_{\alpha \beta} \]

    \[ =-\frac{T}{2} \int_{\sum} d\tau d\sigma \sqrt{-\gamma} \gamma^{\alpha \beta} \partial _{\alpha} X^{\mu} \cdot \partial _{\beta} X^{\mu} \]

    \[ = -\frac{1}{4 \pi \alpha^{\prime}} \int_{\sum} d\tau d\sigma \sqrt{-\gamma} \gamma^{\alpha \beta} \partial _{\alpha} X^{\mu} \partial _{\beta} X^{\nu} g_{\mu \nu} \]

In that this is the Polyakov action, it was discovered independently by Brink, Di Vecchia, and Howe and by Deser and Zumino. Here, X(\tau, \sigma) are scalar fields. Another way to put this: the spacetime coordinates X^{\mu} are promoted to dynamical fields. These fields are the centre of focus when it comes to the 2-dim field theory on the WS. From the point of view of the WS, the action describes the way in which these fields are coupled to 2D gravity. The indices \mu and \nu correspond to the target space while \alpha and \beta correspond to the 2D WS. There are also several other points worth emphasising.

First, the Polyakov action indeed looks very much like a 2D sigma model. Compare, for example, with a 4D sigma model in QFT where g_{\mu \nu} is the metric and where we’re working on a general spacetime background:

    \[ S_{\sigma} = \int d^{4} x \sqrt{g} g^{\mu \nu} \partial_{\mu} \phi^{M} \partial_{\nu} \phi^{N} G_{MN} \]

This connection makes a lot of sense, given the relation between Quantum Field Theory and String Theory.

Second, S_{p} is classically equivalent to the NG action. We will prove this in just a moment. Additionally, what is also nice is how the variation of the action with respect to the metric \gamma_{\alpha \beta} gives the energy-momentum tensor (again, see below).

Third, in that one of the primary advantages of S_{P} is how it provides the correct quantum theory, my understanding here is that S_{P} has the advantage of bilinearity with respect to the X(\sigma) fields. As Polchinski writes, “Its virtues” are “especially for path integral quantization” (p.34) in that it enables or allows for direct access to quantum procedures in Fock space. Moreover, it is not that S_{NG} cannot be quantised. Indeed, there’s a lot that can be accomplished with it in general. It is just that things can and do become untidy with the S_{NG} form of the action – we end up with square roots everywhere and I think it is generally more difficult to derive the wave equations, momenta identities, and other things. It seems to me that S_{P} is easier to generalise when considering curved or arbitrary backgrounds. This advantage is emphasised as early as Chapter 2 in Polchinski, where we begin to learn how, when generalised, S_{P} takes the form of an interacting field theory. Hence, the treatment of the string sigma model as a synonym.

Fourth, the WS metric here has a lovely quality in that its components, in a sense, play the role of Lagrange multipliers. These Lagrange multipliers impose the Virasoro constraints, which prove incredibly important moving forward. This is something we will talk a lot about (for example, when we get into the quantum theory of strings among other topics). The Virasoro constraints can first be derived (a nice derivation, in fact) from the NG action.

Symmetries of the Polyakov action

Fifth, as Polchinski notes (p.13), a number of other advantages include the symmetries that come with S_{P}:

  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.

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

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

But to maintain a clear picture of what is being developed, we should be careful to delineate the origin of our symmetries. I will save the reader a detailed description of the transformations, instead offering only a few brief comments (perhaps I will write a separate post on these matters so as to limit the length of the current discussion). In terms of spacetime symmetries, S_{P} is manifestly Poincaré invariant. From the perspective of 2-dim field theory, this is a global internal symmetry. As for the WS symmetries, we have local diffeomorphism invariance under \xi^{a} \rightarrow \tilde{\xi}^{a} (\xi) = \xi^{a} - \epsilon^{a} (\xi). From this we can describe the manner in which the fields transform, such as in how a scalar field from the WS perspective transforms or how the metric \gamma_{\alpha \beta} transforms as a WS 2-tensor (Weigand, 2015/16).

We also have Weyl invariance – or local conformal invariance. On the introduction of Weyl invariance that follows the introduction of the auxiliary metric embedded on the WS, Polchinski writes, “The Weyl invariance, a local rescaling of the worldsheet metric, has no analog in the Nambu–Goto form” (p.13). This is an extra symmetry that we’ve been granted on the basis of the fact that what we’re working with now is a 2-dimensional WS theory. It also turns out that Weyl invariance will prove crucial, as we will see when we look to the quantise the string.

More can be said on these matters, but I should like to keep focused on the main aims of this post.

Equations of motion

    \[ S_{P} [X, \gamma] = - \frac{1}{4 \pi \alpha^{\prime}} \int_{\sum} d\tau d\sigma \sqrt{-\gamma} \gamma^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} \]

To simplify things, let:

    \[ \sigma^{0} = \tau \]

    \[ \sigma^{1} = \sigma \]

And set T = \frac{1}{2 \pi \alpha^{\prime}}. So,

    \[ S_{P} [X, \gamma] = - \frac{T}{2} \int_{\sum} d^2 \sigma  \ \sqrt{-\gamma} \gamma^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} \]

Varying the action such that \delta \gamma = \gamma + \delta \gamma,

    \[ \delta S_{P} [X, \gamma] = - \frac{T}{2} \int_{\sum} d^2 \sigma \ \delta (\sqrt{-\gamma} \gamma^{\alpha \beta}) \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} \]

    \[ = - \frac{T}{2} \int_{\sum} d^2 \sigma \sqrt{-\gamma} \delta \gamma^{\alpha \beta} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} - \int_{\sum} d^2 \sigma \gamma^{\alpha \beta} \ \delta \sqrt{- \gamma} \partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} \]

Now, two things to note before proceeding further. To vary the metric \gamma^{\alpha \beta}, notice that if A is some two-by-two matrix, and \delta A is its variation,

    \[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{pmatrix} \]

And,

    \[ \delta A = \begin{pmatrix} \delta a_{11} & \delta a_{12} \\ \delta a_{21} & \delta a_{22} \\ \end{pmatrix} \]

It should be known and understood that the variation of det A can be found,

    \[ \delta \det A = (\det A) Tr(A^{-1} \delta A) \]

Where A^{-1} is the inverse of A.

With that said, and with the previous working in mind, let’s now think of \delta \gamma^{\alpha \beta}. We can use the identity above to vary \gamma:

    \[ \delta \gamma = \delta \det(\gamma_{\alpha \beta}) = \gamma (\gamma^{\alpha \beta} \delta \gamma_{\beta \alpha}) \]

But we must write the variation in terms of \gamma^{\alpha \beta}, not \gamma_{\alpha \beta}. What to do? Note, \gamma^{\alpha \beta} \gamma_{\beta \alpha} = \delta_{\lambda}^{\alpha} and \gamma^{\alpha \beta} \gamma_{\alpha \beta} = 2.

So, we can see how we obtain the correct expression if we vary \gamma^{\alpha \beta} \gamma_{\alpha \beta} = 2.

    \[ \delta \gamma^{\alpha \beta} \gamma_{\alpha \beta} +  \gamma^{\alpha \beta} \gamma_{\alpha \beta} = 0 \]

    \[ \implies \gamma^{\alpha \beta} \delta \gamma_{\beta \alpha} = -\delta \gamma^{\alpha \beta} \gamma_{\alpha \beta} \]

From this, we can determine that the variation of \gamma can be written more concisely as

    \[ \delta \gamma = - \gamma \ \delta \gamma^{\alpha \beta} \gamma_{\alpha \beta} \]

As an aside, one will find this exact expression in Polchinski (1.2.15), as we have invoked the general relation for the variation of the determinant (p.12). What is also being implied here, however little we have yet to truly explicate it, is how, if \delta_{\gamma}S_{P} = 0, it follows the induced metric h_{\alpha \beta} = \frac{1}{2} \gamma_{\alpha \beta} \gamma^{cd}h_{cd}. This, again, can be found on p.12 in Polchinski, as he sets himself the task of showing S_{P} and S_{NG} are classically equivalent. As we’re currently taking a different angle of attack, we’ll return to this point in a moment.

With that temporary departure from the main line of logic, let’s now ask on the basis of our previous result: what about the variation of \sqrt{-\gamma}? It remains to be said that we can show,

    \[ \delta(\sqrt{-\gamma}) = - \frac{1}{2} \frac{\delta \gamma}{\sqrt{-\gamma}} = - \frac{1}{2} \frac{(-\gamma) \delta \gamma^{\alpha \beta} \gamma_{\alpha \beta}}{\sqrt{-\gamma}} = -\frac{1}{2} \sqrt{-\gamma} \delta \gamma^{\alpha \beta} \gamma_{\alpha \beta} \]

Therefore, we can complete the variation of the action with respect to the metric, which was our goal. Carrying on from above,

    \[ \delta S_{P} [X, \gamma] = -\frac{T}{2} \int d^2 \sigma \ \sqrt{-\gamma} \delta \gamma^{\alpha \beta} \partial_{alpha} X^{\mu} \partial_{\beta} X^{\nu} g_{\mu \nu} - \frac{T}{2} \int d^2 \sigma - \frac{1}{2} \sqrt{-\gamma} \gamma_{\alpha \beta} \gamma^{c d} \partial_{c} X^{\mu} \partial_{d} X^{\nu} g_{\mu \nu} \ \delta \gamma^{\alpha \beta} \]

    \[ = \int d^{2} \sigma \sqrt{- \gamma} \delta \gamma^{\alpha \beta} (\partial_{\alpha} X^{\mu} \partial_{\beta} X^{\nu} - \frac{1}{2} \gamma_{\alpha \beta} (\gamma^{c d} \partial_{c} X^{\mu} \partial_{d} X^{\nu}))g_{\mu \nu} \]

This means that the EoM as a result of the variation \delta \gamma^{\alpha \beta} is found to be,

    \[ \partial_{\alpha} X \partial_{\beta} X - \frac{1}{2} \gamma_{\alpha \beta} (\gamma^{c d} \partial_{c} X \partial_{d} X) = 0 \]

Energy-momentum tensor of 2-dim field theory

One key point of emphasis is the relation to 2-dim scalar field theory, in that the WS is just the space where the field theory lives. On this note, recall a recent post introducing S_{P} by way of GR. It is a pertinent link. One reason I wrote that post is because I like the way in which there is a certain equivalence between the study of 2-dim gravity coupled to scalar fields and string theory (of the bosonic sort). Recall that in that post it was written,

    \[ T_{m n} : = -\frac{1}{T} \frac{1}{\sqrt{h}} \frac{\delta S_{P}}{\delta h^{m n}} \]

    \[ = \partial_{m} X^{\mu} \partial_{n} X_{\nu} - \frac{1}{2} h_{m n} h^{p q} \partial_{p} X^{\mu} \partial_{q} X_{\nu} = 0 \]

Notice, this is precisely what we have found when varying the action with respect to the metric. Indeed, when performing the above variation we arrive at the energy-momentum tensor. This is a nice connection, because we know all sorts of useful properties when it comes to the EM tensor. For example, we know it is traceless. And, indeed, this important property or feature of T_{mn} is something we’ll return to when get to stringy CFTs.

Classical equivalence

Earlier I mentioned S_{P} and S_{NG} are classically equivalent. I also referenced Polchinski’s engagement with the topic. We can now show this classical equivalence to be true.

To begin, we return to the EoM. Rearranging, we get:

    \[ \partial_{\alpha} X \partial_{\beta} X - \frac{1}{2} \gamma_{\alpha \beta} \gamma^{c d} \partial_{\alpha}X \partial_{\beta}X= 0 \]

Let’s write \partial_{\alpha}X \partial_{\beta}X as G_{\alpha \beta}. Then what we want to do is take the square root of the determinant of both sides.

    \[ G_{\alpha \beta} - \frac{1}{2} \gamma_{\alpha \beta} \gamma^{c d} G_{\alpha \beta} = 0 \]

    \[ G_{\alpha \beta} = \frac{1}{2} \gamma_{\alpha \beta} \gamma^{c d} G_{cd} \]

    \[ \sqrt{- \det(G_{\alpha \beta})} = \frac{1}{2} \sqrt{- gamma} \gamma^{c d} G_{c d} \]

    \[ \implies \sqrt{- \gamma} = \frac{2 \sqrt{-G}}{\gamma^{cd} G_{}cd} \]

Substitute for \sqrt{- \gamma} in S_{P}:

    \[ S_{P} = \frac{T}{2} \int d^2 \sigma \sqrt{- \gamma} \gamma^{\alpha \beta} G_{\alpha \beta} \]

    \[= \frac{T}{2} \int d^2 \sigma \frac{2 \sqrt{-G}}{\gamma^{cd} G_{}cd} \gamma^{\alpha \beta} G_{\alpha \beta} \]

    \[ = T \int d^2 \sigma \sqrt{-G} = S_{NG} \]

Wave equation

This time varying the S_{P} action with respect to X^{\mu}, such that X^{\mu} \rightarrow X^{\mu} + \delta X^{\mu}.

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

    \[ = \frac{T}{2} \int d^2 \sigma \delta X^{\mu} \partial_{\alpha}(\sqrt{- \gamma} \gamma^{\alpha \beta} \partial_{\beta} X^{\nu} g_{\mu \nu}) \]

Ignoring, once again, the boundary terms. The EoM is,

    \[ \partial_{\alpha}(\sqrt{- \gamma} \gamma^{\alpha \beta} \partial_{\beta} X^{\mu})=0 \]

The wave equation follows after a few important steps. We need to make a gauge choice. In particular, we shall work in the conformal gauge, such that \sqrt{- \gamma} \gamma^{\alpha \beta} \equiv \eta^{\alpha \beta}, where \eta^{\alpha \beta} is the 2-dim Minkowski metric. These manoeuvres are also principled on reparameterisation invariance for \gamma_{\alpha \beta} (Zwiebach, 2009, p. 476).

The EoM becomes,

    \[ \partial_{\alpha} (\eta^{\alpha \beta} \partial_{\beta} X^{\mu}) = \eta^{\alpha \beta} \partial_{\alpha}\partial_{\beta}X^{\mu} = 0 \]

This is the wave equation.

Virasoro constraints

To conclude this entry, I think it is worthwhile spending a few minutes thinking about the Virasoro constraints, simply because they are so important moving forward. In a future post on stringy CFTs, we will discuss the Virasoro algebra among other things. In the meantime, let us familiarise ourselves with the constraints.

For the Virasoro constraints, we once again invoke the conformal gauge. So, if we return to,

    \[ \partial_{\alpha} X \partial_{\beta} X - \frac{1}{2} \gamma_{\alpha \beta} (\gamma^{cd} \partial_{c} X \partial_{d} X) = 0 \]

The gauge condition \gamma_{\alpha \beta} = \rho^{2}(\xi) \eta_{\alpha \beta} gives,

    \[ \partial_{\alpha} X \partial_{\beta} X - \frac{1}{2} \eta_{\alpha \beta}(\eta^{cd} \partial_{c}X \partial_{d} X) = 0 \]

But we know that the expression in the parenthesis can be expanded. We’ve seen it before. So,

    \[ \partial_{\alpha} X \partial_{\beta} X - \frac{1}{2} \eta_{\alpha \beta} (- \dot{X}^2 + X^{\prime^2}) = 0 \]

Setting \alpha=\beta=1,

    \[ \dot{X}^2 + \frac{1}{2} (-\dot{X}^2 + X^{\prime^2}) = 0 \rightarrow  \dot{X}^2 + X^{\prime^2} = 0 \]

This gives one of the constraints. The second constraint can be found by setting alpha=1 and \beta=2,

    \[ \dot{X} \cdot X^{\prime} = 0 \]

The third constraint, then, comes from setting \alpha=\beta=2,

    \[ X^{\prime^2} - \frac{1}{2} (- \dot{X}^2 + X^{\prime^2})=0 \rightarrow \dot{X}^2 + X^{\prime^2} = 0 \]

However, we have already arrived at this constraint! So this last result is redundant. In the conformal gauge, we have now successfully derived the Virasoro constraints.