** 1.1. Polyakov action **

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 , we define an auxiliary field with

and

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

which describes the worldsheet dynamics. Here, is the induced metric, while the auxiliary field , which is a symmetric 2-tensor, plays the role of an intrinsic metric on the string worldsheet with signature . This intrinsic metric is completely independent of the pullback metric . This means that 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 , with , are scalars on the worldsheet but vectors on the spacetime (evident due to the index). Moreover, the fields 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 together with a set of matter fields, and the scalar fields 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 is in general curved and furnishes the worldsheet with the Levi-Civita connection. The associated covariant derivative on the worldsheet is defined with respect to . For scalar fields, also recall that .** 1.2. Classical equivalence between and **

To compute the variation, note that if is some two-by-two matrix, and is its variation, we have

and therefore

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

where is the inverse of .

Now, returning to , we can use these facts to obtain

But we must write the variation in terms of , so we note that as then

which implies

and therefore

If we put these two together we clearly see

which is precisely the expression given in eqn. (1.2.15) in Polchinski. For the second term in (10) we compute as follows

Therefore, putting everything together,

which gives the equations of motion

which clearly gives

where it is recognised that as defined in our past discussion of the Nambu-Goto action. Hence, taking the square root of minus the determinant of both sides gives

This shows that and are classically equivalent, as given in eqn. (1.2.18) in Polchinski.

**2. Symmetries **

** 2.1. Global symmetries **

where corresponds to spatial rotations and boosts. Importantly, . The first term in (19) can therefore be seen as simply the spacetime Lorentz symmetry, which acts only on the fields 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 so that

understanding that is invariant such that . We’ve also picked up a factor of 2 along the way and used the metric to lower the index on .

What is also being implied here, recalling (18), is how if it follows the induced metric must take the form , 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 is proportional to the induced metric as expected** 2.2. Local symmetries **

using the infinitesimal parameterisation . Taylor expanding we end up with

And then from this transformation law we see

Hence the variation of the scalar field is given by the difference of the transformed and the original fields

or, simply,

As expected, the metric transforms as a 2-tensor

which can then be expanded

Finally, the object 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.

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 **

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 as follows:

where is for now considered to be an arbitrary transformation parameter.

Expanding (31) in yields , we see the infinitesimal variation of has the form . Therefore, considering we findHence, under a Weyl transformation does not change, which shows that the variation of 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 string theory, when we vary with respect to the background metric we arrive at the following

To see this and to find eqn. (1.2.22) in Polchinski, we go back to considering a variation of with respect to . Notice that, setting and remembering and , we can in fact write

When it comes to the constraints on for physical fluctuations, two important comments are necessary:

1. Diff invariance on the worldsheet implies , and so the energy-momentum tensor is covariantly conserved; 2. Weyl invariance implies . 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 . Moreover, the tracelessness can easily be foundThis 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 resultwhich says that the equations of motion represent the vanishing of the energy-momentum tensor.

** 2.5. Klein-Gordon equation **

Integrating by parts such that the covariant derivative acts on the rest of the action we obtain

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

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

** 3. Boundary conditions **

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

It follows , where must be periodic and where, for closed strings, as we go around the coordinates for all fields become identical . With regards to D-boundary conditions,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:

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

hence

And so, in covariant form,

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

** 4. Closing remarks **