Notes on string theory #4: Polyakov action

In our ongoing read through of Polchinski’s textbook, we left off on page 12 having studied the first principle Nambu-Goto action {S_{NG}} for the string. We have glimpsed early on why string theory is a generalisation – or, one could also say, deformation – of point particle theory. The generalisation from point particles to strings will become more apparent in the quantum theory and, certainly, in the study of the string theoretic implications for geometry. In the short-term, however, our first major task will be to develop the path integral quantisation for a closed bosonic string in worldsheet Euclidean signature. But there is still a lot of stuff we must cover before we get there.

Recall that in constructing the Nambu-Goto action we observed, as was very much the case in point particle theory (wherein the square root function in the integrand was non-linear), the presence of a square root; quantising this action is therefore difficult because it is highly non-linear. So, before we start looking at the quantum physics of strings, we should like for an action of a more suitable form. The question is, what form may that be?

1.1. Polyakov action

Recall for ease of reference that the final version of the Nambu-Goto action may be written as,

\displaystyle  S_{NG} = - \frac{1}{2 \pi \alpha^{\prime}} \int_{\Sigma} d\tau d\sigma \ (- \det h_{\alpha \beta})^{1/2}. \ \ \ \ \ (1)

In the analogous case of the point particle, we modified the action by introducing an auxiliary field, which then enabled the elimination of the pesky square root. In the case of the string we’re going to follow the exact same strategy. In order to modify {S_{NG}}, we define an auxiliary field {\gamma_{a b} (\tau, \sigma)} with

\displaystyle  \gamma = \det \gamma_{a b} \ \ \ \ \ (2)


\displaystyle  \gamma^{\alpha \beta} = (\gamma^{-1})_{\alpha \beta}. \ \ \ \ \ (3)

The resulting string sigma model, or Polyakov action, is then given by

\displaystyle  S_{P}[X, \gamma] = - \frac{1}{4\pi \alpha^{\prime}} \int_{\Sigma} d\tau d\sigma \sqrt{\gamma} \gamma^{ab} \partial_a X^{\mu} \partial_b X^{\nu} \eta_{\mu \nu}, \ \ \ \ \ (4)

which describes the worldsheet dynamics. Here, {\eta_{\mu \nu}} is the induced metric, while the auxiliary field {\gamma_{ab}}, which is a symmetric 2-tensor, plays the role of an intrinsic metric on the string worldsheet with signature {(-,+)}. This intrinsic metric {\gamma_{ab}} is completely independent of the pullback metric {G_{ab} = \partial_a X^{\mu} \partial_b X^{\nu} \eta_{\mu \nu}}. This means that {\gamma_{ab}} will be a dynamic variable in the action and, inasmuch that one should think of it as an auxiliary field, a notable implication is that will lead to its own field equations.

The fundamental object of the theory is the string worldsheet. From the perspective of string theory spacetime is very much a derived concept. We start to see this upon the introduction of (4) by noting that, in its current form, the scalar fields {X^{\mu}(\tau, \sigma)}, with {\mu = 0,..., D-1}, are scalars on the worldsheet but vectors on the spacetime (evident due to the {\mu} index). Moreover, the fields {X^{\mu}(\tau, \sigma)} aid in describing both the properties of the spacetime in which the string propagates as well as the vibrational modes of the string (i.e., the internal degrees of freedom). As we’ll see later on, in the Polyakov formalism the worldsheet geometry is endowed with the metric {\gamma_{ab}} together with a set of matter fields, and the scalar fields {X^{\mu}} can be described by a general sigma model which encodes the embedding of the string in the D non-compact spacetime dimensions.

It is furthermore useful to note that the metric {\gamma_{ab}} is in general curved and furnishes the worldsheet with the Levi-Civita connection. The associated covariant derivative {\nabla_a} on the worldsheet is defined with respect to {\Gamma^a_{bc} = \frac{1}{2}h^{ad}(\partial_b h_{cd} + \partial_c h_{bd} - \partial_d h_{ab})}. For scalar fields, also recall that {\nabla_a X^{\mu} = \partial_a X^{\mu}}.

1.2. Classical equivalence between {S_{NG}} and {S_{P}}

The Polyakov action {S_{P}} is classically equivalent to the Nambu-Goto action {S_{NG}}. To see this equivalence we vary {S_P} with respect to the metric {\gamma}.

To simplify things, let {\sigma^{0} = \tau} and {\sigma^{1} = \sigma}, and we set {T = \frac{1}{2 \pi \alpha^{\prime}}}. Varying {S_P} such that {\delta \gamma = \gamma + \delta \gamma} we have

\displaystyle  \delta_{\gamma} S_{P} = - \frac{T}{2} \int_{\sum} d^2 \sigma \ \delta (\sqrt{-\gamma} \gamma^{ab}) \partial_{a} X^{\mu} \partial_{b} X^{\nu} \eta_{\mu \nu}

\displaystyle  = - \frac{T}{2} \int_{\sum} d^2 \sigma \sqrt{-\gamma} \delta \gamma^{ab} \partial_{a} X^{\mu} \partial_{b} X^{\nu} \eta_{\mu \nu} - \int_{\sum} d^2 \sigma \gamma^{ab} \ \delta \sqrt{- \gamma} \partial_{a} X^{\mu} \partial_{b} X^{\nu} \eta_{\mu \nu}. \ \ \ \ \ (5)

To compute the variation, note that if {A} is some two-by-two matrix, and {\delta A} is its variation, we have

\displaystyle  A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{pmatrix} \ \ \ \ \ (6)

and therefore

\displaystyle  \delta A = \begin{pmatrix} \delta a_{11} & \delta a_{12} \\ \delta a_{21} & \delta a_{22} \\ \end{pmatrix}. \ \ \ \ \ (7)

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

\displaystyle  \delta \det A = (\det A) Tr(A^{-1} \delta A), \ \ \ \ \ (8)

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

Now, returning to {\delta \gamma^{\alpha \beta}}, we can use these facts to obtain

\displaystyle  \delta \gamma = \delta \det(\gamma_{ab}) = \gamma (\gamma^{ab} \delta \gamma_{ab}). \ \ \ \ \ (9)

But we must write the variation in terms of {\gamma^{\alpha \beta}}, so we note that as {\gamma = \det \gamma_{ab} = \frac{1}{n!} \epsilon^{a_1 ... a_n}\epsilon^{b_1 ... b_n}\gamma_{a_1 b_1} ... \gamma_{a_n b_n}} then

\displaystyle  \delta \gamma = \frac{1}{n!} \epsilon^{a_1 ... a_n}\epsilon^{b_1 ... b_n}\gamma_{a_1 b_1} ... \gamma_{a_n b_n} + \frac{1}{(n-1)!} \epsilon^{a_1 ... a_n}\epsilon^{b_1 ... b_n}\gamma_{a_2 b_2} ... \gamma_{a_n b_n} = \gamma^{a_1 b_1}\gamma, \ \ \ \ \ (10)

which implies

\displaystyle  \delta \gamma = \gamma^{ab} \delta \gamma_{ab} \gamma \ \ \ \ \ (11)

and therefore

\displaystyle  \delta \gamma = - \gamma_{ab}\delta \gamma^{ab}\gamma. \ \ \ \ \ (12)

If we put these two together we clearly see

\displaystyle  \gamma^{ab} \delta \gamma_{ab} \gamma = - \gamma_{ab}\delta \gamma^{ab}\gamma, \ \ \ \ \ (13)

which is precisely the expression given in eqn. (1.2.15) in Polchinski. For the second term in (10) we compute {\delta \sqrt{-\gamma}} as follows

\displaystyle  \delta(\sqrt{-\gamma}) = - \frac{1}{2} \frac{\delta \gamma}{\sqrt{-\gamma}} = - \frac{1}{2} \frac{(-\gamma) \delta \gamma^{ab} \gamma_{ab}}{\sqrt{-\gamma}} = -\frac{1}{2} \sqrt{-\gamma} \delta \gamma^{ab} \gamma_{ab}. \ \ \ \ \ (14)

Therefore, putting everything together,

\displaystyle  \delta S_P = - \frac{T}{2} \int d^{2} \sigma \sqrt{- \gamma} \delta \ \gamma^{ab} (\partial_{a} X^{\mu} \partial_{b} X^{\nu} - \frac{1}{2} \gamma_{ab} (\gamma^{c d} \partial_{c} X^{\mu} \partial_{d} X^{\nu}))\eta_{\mu \nu} = 0, \ \ \ \ \ (15)

which gives the equations of motion

\displaystyle  \partial_{a} X \partial_{b} X - \frac{1}{2} \gamma_{a b} (\gamma^{c d} \partial_{c} X \partial_{d} X) = 0, \ \ \ \ \ (16)

which clearly gives

\displaystyle  \frac{1}{2} \gamma_{a b} (\gamma^{c d} \partial_{c} X \partial_{d} X) = \partial_{a} X \partial_{b} X, \ \ \ \ \ (17)

where it is recognised that {\partial_{a} X \partial_{b} X \equiv h_{ab}} as defined in our past discussion of the Nambu-Goto action. Hence, taking the square root of minus the determinant of both sides gives

\displaystyle  \frac{1}{2} \sqrt{-\gamma} \gamma^{ab} \partial_a X \partial_b X = \sqrt{-h_{ab}}. \ \ \ \ \ (18)

This shows that {S_p} and {S_{NG}} are classically equivalent, as given in eqn. (1.2.18) in Polchinski.

2. Symmetries

As Polchinski notes on p.12-13, the action (4) comes with many symmetries. These symmetries are both local and global, which is to say that {S_{P}} has symmetries of the worldsheet and of the background spacetime. But it is important to note that we are also speaking here from the perspective of the theory of the 2-dimensional string worldsheet.

As a point of emphasis, note that {X^{\mu}} can take on the interpretation as either spacetime coordinates or dynamical fields on the worldsheet. 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 choice of gauge.

2.1. Global symmetries

We start with global symmetries. In general, one can think of these as transformations in spacetime whose parameter(s) do not depend on where in the spacetime the transformation actually occurs. Just as in Einstein’s theory of gravity, invariance under global transformations will lead to conserved currents and charges via Noether’s theorem. We will look at this a little later.

\subsubsection{Poincaré transformations} Consider the background spacetime to be Minkowskian. Then the string theory (4) that lives in this space should have the same symmetries as Minkowski space and, in particular, our theory should be invariant under the Poincaré group.

Poincaré transformations are global transformations of the form

\displaystyle  X^{\prime \mu}(\tau, \sigma) = \Lambda^{\mu}_{\nu} (X^{\nu}(\tau, \sigma) + a^{\mu}), \ \ \ \ \ (19)

\displaystyle  \gamma^{\prime}_{ab}(\tau, \sigma) = \gamma_{ab}(\tau, \sigma), \ \ \ \ \ (20)

where {\lambda} corresponds to spatial rotations and boosts. Importantly, {\Lambda_{\mu \nu} = - \Lambda_{\nu \mu}}. The first term in (19) can therefore be seen as simply the spacetime Lorentz symmetry, which acts only on the fields {X^{\mu}} locally on the worldsheet. From the perspective of the 2-dimensional worldsheet, Poincaré invariance is in fact interpreted as a global internal symmetry.

It is fairly obvious that the bosonic string action (4) that we’ve constructed is Poincaré invariant for the simple reason that, we’ve already seen that the Nambu-Goto action is invariant under Poincaré and we know that the Nambu-Goto action and Polyakov action are classically equivalent. But should one want to see this explicitly, return to (4). Consider a transformation of the coordinates, plugging in {X^{\prime} =\delta X^{\mu} = \Lambda^{\mu}_{k}X^{k} + a^{\mu}} so that

\displaystyle  \delta S_P = \frac{1}{2\pi \alpha^{\prime}} \int d^2 \sigma \sqrt{- \gamma} \gamma^{\alpha \beta} \eta_{\mu \nu} \partial_{a}(X^{\mu} + \Lambda^{\mu}_{\delta}X^{\delta}) \partial_{b}(X^{\nu} + \Lambda^{\nu}_{\delta}X^{\delta})

\displaystyle = S + \frac{1}{2\pi \alpha^{\prime}} \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)

\displaystyle  = S + \frac{1}{2\pi \alpha^{\prime}} \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), \ \ \ \ \ (21)

understanding that {\gamma^{ab}} is invariant such that {\delta \gamma^{ab} = 0}. We’ve also picked up a factor of 2 along the way and used the metric {\eta_{\mu \nu}} to lower the index on {\Lambda}.

What is also being implied here, recalling (18), is how if {\delta_{\gamma}S_{P} = 0} it follows the induced metric must take the form {h_{ab} = \frac{1}{2} \gamma_{ab} \gamma^{cd}h_{cd}}, which can be found eqn. (1.2.16) in Polchinski. If you divide this by the square root of minus its determinant we find that the intrinsic metric {\gamma} is proportional to the induced metric as expected

\displaystyle  h_{ab}(-h)^{1/2} = \gamma_{ab}(-\gamma)^{1/2}. \ (22)

2.2. Local symmetries

Let us now consider local symmetries and transformations on the worldsheet. If, previously, we considered global transformations as those that are dependent on spacetime coordinates, local symmetries are dependent on the worldsheet coordinates. The two local symmetries of the bosonic string are diffeomorphism (reparameterisation) invariance and Weyl invariance.

\subsubsection{Diffeomorphism invariance} If from the perspective of the worldsheet we have a 2-dimensional field theory {X^{\mu}} coupled to 2-dimensional gravity, we can see that the action {S_{P}} is invariant under any reparameterisation of the worldsheet coordinates as follows. Recalling the basic properties for the transformation of tensors and tensor densities, we find that for any diffeomorphism {X^{\mu} \rightarrow X^{\prime \mu}} the scalar field transforms completely

\displaystyle  X^{\mu} (\sigma) \rightarrow X^{\prime \mu} (\sigma^{\prime}) = X^{\mu} (\sigma) = X^{\mu}(\sigma^{\prime} + \epsilon(\sigma^{\prime}), \ \ \ \ \ (23)

using the infinitesimal parameterisation {X^{\mu}(\sigma) = X^{\prime \mu}(\sigma) + \epsilon^{\mu}(X^{\prime})}. Taylor expanding we end up with

\displaystyle  = X^{\mu}(\sigma^{\prime}) + \epsilon^{c}(\sigma^{\prime})\partial_{c}X^{\mu}(\sigma^{\prime}) + \mathcal{O}[\epsilon^2 (\sigma^{\prime})]. \ \ \ \ \ (24)

And then from this transformation law we see

\displaystyle  X^{\prime \mu}(\sigma) = X^{\mu}(\sigma) + \epsilon^{c}(\sigma)\partial_c X^{\mu}(\sigma) + \mathcal{O}[\epsilon^2(\sigma)]. \ \ \ \ \ (25)

Hence the variation {\delta X^{\mu}(\sigma)} of the scalar field {X^{\mu}(\sigma)} is given by the difference of the transformed and the original fields

\displaystyle  \delta X^{\mu}(\sigma) = X^{\prime \mu}(\sigma) + X^{\mu}(\sigma) = \epsilon^c(\sigma)\partial_c X^{\mu}(\sigma) + \mathcal{O}[\epsilon^2 (\sigma)], \ \ \ \ \ (26)

or, simply,

\displaystyle  \delta X^{\mu} = \epsilon^{c}\partial_{c} X^{\mu}. \ \ \ \ \ (27)

As expected, the metric {\gamma_{\alpha \beta}} transforms as a 2-tensor

\displaystyle  \gamma_{\alpha \beta} (\sigma) \rightarrow \tilde{\gamma}_{\alpha \beta}, \ \ \ \ \ (28)

which can then be expanded

\displaystyle  \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)

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

\displaystyle  = \nabla_a \epsilon_b + \nabla_b \epsilon_a. \ \ \ \ \ (29)

Finally, the object {\sqrt{-\det \gamma}} transforms like a scalar density of weight 1, which can be seen by remembering from differential geometry that the determinant of a product of matrices is the product of determinants.

\displaystyle  \delta \sqrt{-\det \gamma} = \partial_c (\epsilon^c \sqrt{-\det \gamma}). \ \ \ \ \ (30)

As we have discussed here the topic diffeomorphisms on the worldsheet, there is also an interesting question about target space diffeomorphisms. I do not want to distract too much from Polchinski, so we’ll return to this question in a later note.

2.3. Weyl invariance

We also have Weyl invariance – or local conformal invariance. A Weyl transformation corresponds to a local rescaling of the worldsheet metric, leaving the other fields unaffected

\displaystyle  \delta X^{\mu} = 0,

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

\displaystyle   \delta \gamma_{\alpha \beta} = 2 \Lambda(\sigma) \gamma_{\alpha \beta}. \ \ \ \ \ (31)

Weyl invariance has no analogue in the Nambu-Goto form, which is to say that it is an extra symmetry that we’ve been bestowed in the Polyakov formalism. In fact, the presence of Weyl invariance for the 2-dimensional worldsheet theory aids in distinguishing why string theory is a unique generalisation of the point particle theory. To see how bosonic string theory is invariant under a Weyl transformation, we first show the transformation of {\sqrt{- \gamma}} as follows:

\displaystyle \sqrt{-\gamma^{\prime}} = \sqrt{- \det (\gamma^{\prime}_{ab})}

\displaystyle  = e^{2(2\omega(\sigma))/2} \sqrt{- \det (\gamma^{\prime}_{ab})}

\displaystyle  = e^{2\omega(\sigma)}\sqrt{- \gamma}, \ \ \ \ \ (32)

where {\omega(\sigma)} is for now considered to be an arbitrary transformation parameter.

Expanding (31) in \omega yields \gamma^{\prime ab} = e^{-2 \omega(\sigma)} \gamma^{ab} = 1 - 2\omega + ...)\gamma^{ab} , we see the infinitesimal variation of {\gamma^{ab}} has the form {\delta \gamma^{ab} = -2\omega \gamma^{ab}}. Therefore, considering {\sqrt{-\gamma}\gamma^{ab}} we find

\displaystyle  \sqrt{-\gamma^{\prime}}\gamma^{\prime ab} = \sqrt{-\gamma}e^{2\omega(\sigma)}e^{-2\omega(\sigma)}\gamma^{ab} = \sqrt{-\gamma}\gamma^{ab}. \ \ \ \ \ (33)

Hence, under a Weyl transformation {S_{P}} does not change, which shows that the variation of {S_P} under a Weyl transformation vanishes.

As Polchinski goes on to discuss on p.13, one of the most important implications of our theory (4) being Weyl invariant is that the stress-energy tensor associated with this theory is traceless.

2.4. Energy-momentum tensor

In Quantum Field Theory, provided we are working in flat space, we have the following general formula for the energy-momentum tensor:

\displaystyle  T^{ab} = \frac{2}{\sqrt{- \gamma}} \frac{\delta S}{\delta \gamma_{ab}}. \ \ \ \ \ (34)

In string theory, when we vary {S_{P}} with respect to the background metric we arrive at the following

\displaystyle  T^{ab} = \frac{-4 \pi}{\sqrt{- \gamma}} \frac{\delta S_P}{\delta \gamma_{a b}}. \ \ \ \ \ (35)

To see this and to find eqn. (1.2.22) in Polchinski, we go back to considering a variation of {S_P} with respect to {\gamma}. Notice that, setting {\gamma^{ab} \partial_{a} X^{\mu} \partial_{b} X_{\mu} = \partial X^{2}} and remembering {\delta \gamma^{ab} = - \delta \gamma_{cd} \gamma^{ca} \gamma^{db}} and {\delta \gamma = \gamma \cdot \gamma^{ab} \delta \gamma_{ab}}, we can in fact write

\displaystyle  T^{ab} = \frac{4 \pi}{\sqrt{- \gamma}} \frac{1}{-4 \pi \alpha^{\prime}} \frac{\delta}{\delta \gamma_{ab}} \int [\frac{1}{2} (-\gamma)^{\frac{1}{2}} -\partial \gamma^{ab} \delta \gamma_{ab} (\partial X^{2}) - \sqrt{\gamma} \partial^{a} X^{\mu} \partial^{b} X_{\mu} \delta \gamma_{ab}] \ \ \ \ \ (36)

\displaystyle  \implies -\frac{1}{\alpha^{\prime}}[\partial^{a}X^{\mu}\partial^{b}X_{\mu} - \frac{1}{2} \gamma^{ab}\partial_{c}X^{\mu}\partial^{c}X_{\mu}]. \ \ \ \ \ (37)

When it comes to the constraints on {T^{ab}} for physical fluctuations, two important comments are necessary:

1. Diff invariance on the worldsheet implies {\nabla_{a} T^{ab} = 0}, and so the energy-momentum tensor is covariantly conserved;

2. Weyl invariance implies {T_{a}^{b} = \gamma^{ab}T_{ab} = 0}.

Indeed, we will find that the tracelessness of the energy-momentum tensor is a direct consequence of the symmetries imposed, such that when we compute the trace we find {T_{a}^{a} = 0}. Moreover, the tracelessness can easily be found

\displaystyle  T_{a}^{a} = -\frac{1}{\alpha^{\prime}} [ (\partial X)^{2} - \frac{1}{2} \gamma^{ab} \gamma_{ab} (\partial X)^2] = 0. \ \ \ \ \ (38)

This result reflects the fact that when we do a Weyl transformation, it does not effect the coordinates and matter fields of our theory, as already discussed.

In summary, we have found a very nice result

\displaystyle  \frac{\delta S}{\delta \gamma_{ab}} = T^{ab}= 0, \ \ \ \ \ (39)

which says that the equations of motion represent the vanishing of the energy-momentum tensor.

2.5. Klein-Gordon equation

Beneath eqn. (1.2.23), Polchinski makes the following important point. Note that what we are working with here is a 2-dimensional quantum field theory on the worldsheet. Essentially all of the machinery that we are constructing and defining focuses on perturbation theory of the 2-dimensional worldsheet theory. From the perspective of the worldsheet, diffeomorphism invariance (23) makes clear {X^{\mu}} are scalar fields and the spacetime index {\mu} is an internal index. So we see that the Polyakov action, from within the 2-dimensional point of view, describes Klein-Gordon scalars covariantly coupled to the metric {\gamma_{ab}}. To write this explicitly, we return to thinking about varying {S_P} with respect to {X^{\mu}}.

\displaystyle  \delta S_P = - \frac{2}{4 \pi \alpha^{\prime}} \int d\tau d\sigma \sqrt{- \gamma} \gamma^{ab} \nabla_{a} X^{\mu} \cdot \nabla_{b} \delta X_{\mu}

Integrating by parts such that the covariant derivative {\nabla_{b}\delta X_{\mu}} acts on the rest of the action we obtain

\displaystyle  = \frac{1}{2 \pi \alpha^{\prime}} \int d\tau d\sigma \nabla_{b}[\sqrt{-\gamma}\gamma^{ab} \nabla_{a} X^{\mu}] \delta X_{\mu} + boundary \ terms

\displaystyle  \implies \nabla_{b}[\sqrt{-\gamma}\gamma^{ab}\nabla_{a}X^{\mu}] = 0

Notice, approached and written this way we see an explicit expression for the d’Alembertian

\displaystyle  \sqrt{-\gamma} \Box X^{\mu} = 0,

which makes sense, as our fields satisfy the massless Klein-Gordon equation in 2-dimensions on a curved background.

3. Boundary conditions

There are possible boundary conditions that preserve Poincaré invariance in D-dimensions. It is worth thinking about this. We start with,

\displaystyle  boundary \ term = - \frac{1}{2\pi \alpha^{\prime}} \int_{- \infty}^{\infty} d\tau \sqrt{-\gamma} \gamma^{\sigma \sigma} \partial_{a} X^{\mu} \delta X_{\mu} \rvert_{\sigma=0}^{\sigma=l},

\displaystyle  \implies \partial^{\sigma}X^{\mu}(\tau, \sigma) = 0, \ \ \ \partial^{\sigma} X^{\mu}(\tau, l) = 0. \ \ \ \ \ (40)

These are Neumann boundary conditions for an open string implying the existence of endpoints.

It follows {X^{\mu}(\tau, \sigma) = X^{\mu}(\tau, \sigma + l)}, where {X^{\mu}} must be periodic and where, for closed strings, as we go around the {\sigma} coordinates for all fields become identical {\sigma \approx \sigma + l}.

With regards to D-boundary conditions,

\displaystyle  \delta X_{\mu}(\tau, \sigma) = \delta X_{\mu}(\tau, l) = 0

\displaystyle  X^{\mu} (\tau, \sigma) = cont.

\displaystyle  X^{\mu}(\tau, l) = cont.

And what we find is that these D-boundary conditions break Poincaré invariance (we’ll return to this when we explore the topic of D-branes). For now, in covariant form, it is worth noting:

\displaystyle  boundary \ term = -\frac{1}{2 \pi \alpha^{\prime}} \int_{\sum} d\sigma \nabla_{b}[\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{\mu}\delta X_{\mu}

Motivated by the generalisation of divergence theorem, we use the following formula:

\displaystyle  \int_{\sum} \nabla_{a}v^{a} = \int_{\delta \sum} n_{a}v^{a} \ \ \ \ \ (41)


\displaystyle  \implies -\frac{1}{2 \pi \alpha^{\prime}} \int_{\partial \sum} n_b \partial^b X^{\mu}. \ \ \ \ \ (42)

And so, in covariant form,

\displaystyle  n_b \partial^b X^{\mu} = 0, \ \ \ \ \ (43)

which is, again, the Neumann boundary conditions for an open string.

4. Closing remarks

Returning to the opening parts of this note, I do not currently know of a forward derivation that works directly from the Nambu-Goto action to the Polyakov action. It would seem a non-trivial feat. Instead, we opted to follow the standard procedure by leveraging the point particle analogue and modifying our initial action through the inclusion of an auxiliary field. The result was that we obtained Polyakov action and the rest followed naturally. We also found that we can work backwards from the Polyakov action to the Nambu-Goto action, showing they are classically equivalent and, as we will start to review in the next section in Polchinski’s textbook, the former classical action indeed provides the correct quantum theory.

But one may not be entirely satisfied with this approach. We can instead introduce the Polyakov action in a slightly different way. We can take a more extended approach through General Relativity. So as we approach the end of Section 1.2 in Polchinski’s textbook, where a notable discussion unfolds regarding the importance of symmetries and the inclusion of the Einstein-Hilbert term to the action (4), we will take the opportunity in our review of this discussion to take a slight detour and look at the Polyakov action instead through the lens of Einstein gravity.