Notes on string theory: Fiducial metric


In Chapter 3 of Polchinski’s book, he references the fiducial metric. It is denoted as,

    \[ g_{ab} \rightarrow \hat{g}_{ab} \]

While this topic will be referenced in the future when exploring a study of the Polyakov path integral, and especially in a discussion about the symmetries used in the Faddeev-Popov method, we can at this point explore the notion of the fiducial metric by relating it to concepts already covered.

In short, the fiducial metric comes from the gauge freedom we’re afforded from the symmetries of the action, enabling us “to eliminate the integration over the metric, fixing it at some specific functional form” (p. 85). The important point to note is that we have our two gauge symmetries: diffeomorphisms and Weyl transformations. David Tong (p.109) puts it concisely, “We will schematically denote both of these by \zeta. The change of the metric under a general gauge transformation is g_{ab}(\sigma) \rightarrow g_{ab}^{\zeta}(\sigma^{\prime}).” The notation used here is shorthand for the following,

    \[ g_{ab}(\sigma) \rightarrow g_{ab}^{\zeta}(\sigma) = e^{2w(\sigma)} \frac{\partial \sigma^{\gamma}}{\partial \sigma^{\prime a}} \frac{\partial \sigma^{\lambda}}{\partial \sigma^{\prime b}}g_{\gamma \lambda}(\sigma) \]

Where, under the general gauge transformation g \rightarrow g^{\zeta} we have a change of metric.

Due to gauge freedom we can rewrite the metric in a simpler form, commonly as \hat{g}. This is the fiducial metric; it denotes our particular choice of gauge fixing. The common choice in the literature is the flat metric \hat{g}_{ab}(\sigma) = \delta_{ab}.

Again, while the topic is usually introduced at a more advanced stage, an additional now may prove beneficial for when we eventually indulge in conformal field theories. For instance, upon equipping the tools provided to us by utilising stringy CFTs, we may consider the WS as it relates to a cylinder (as the Euler characteristic vanishes).

For our present discussion, we shall maintain our usual (\tau, \sigma) coordinates and recall the metric \gamma. We also note the symmetric nature of this 2x2 metric with respect to its off-diagonal components.

    \[ \gamma_{\alpha \beta} = (\begin{bmatrix}\gamma_{00} & \gamma_{01} \\\gamma_{10} & \gamma_{11} \\\end{bmatrix}) \]

In that we have three independent components for the WS metric, we note that for the symmetry \gamma_{01} = \gamma_{10}. We can use the local symmetries already described to specify these three components. 

So, for reparameterisation invariance, we may perform a coordinate transformation which results in transforming the metric into a flat Minkowski metric.

    \[ \gamma_{\alpha \beta} = e^{\phi(\tau, \sigma)}\eta_{\alpha \beta} = e^{\phi(\tau, \sigma)}(\begin{bmatrix}-1 & 0\\0 & 1 \\\end{bmatrix})\]

As Polchinski notes, “One sometimes wishes to consider the effect of the diff group alone. In this case, one can bring an arbitrary metric to within a Weyl transformation of the unit form” (p.85). This is also known as the conformal group, which will be discussed again later. What is important for now, without jumping ahead, is that we can perform a Weyl transformation to remove the exponential factor. Thus, we are simply left with a transformation to a flat Minkowski metric. 

What does this mean? Well, for the S_{P} action that we’ve already grown familiar with, we already know that we can write is as,

    \[ S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^{2}\sigma \sqrt{- \gamma} \gamma^{\alpha \beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu \nu} \]

But now the determinate is just -1. As for the \gamma^{\alpha \beta} term, this is:

    \[ \gamma^{\alpha \beta} = \begin{bmatrix}-1 & 0 \\0 & 1 \\\end{bmatrix} \]

So, notice that we can compute for the remaining terms of the action,

    \[ \gamma^{\alpha \beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu \nu} = \gamma^{\tau \tau} \partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}g_{\mu \nu} + \gamma^{\sigma \sigma}\partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}g_{\mu \nu} \]

    \[= - \partial_{\tau}X^{\mu}\partial_{\tau}X^{\nu}g_{\mu \nu} + \partial_{\sigma}X^{\mu}\partial_{\sigma}X^{\nu}g_{\mu \nu} \]

    \[ - \partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu} \]

Here, we have the Lagrangian density for a set of massless free scalar fields. Substituting into S_{P}, as well as simplifying the notation by using the shorthand \frac{\partial X^{\mu}}{\partial \tau} = \dot{X}^{\mu} and \frac{\partial X^{\mu}}{\partial \sigma} = X^{\prime \mu},

    \[ S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^{2}\sigma (\partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu}) \]

    \[ = \frac{1}{4\pi \alpha^{\prime}} \int d^{2}\sigma (\dot{X}^{\mu 2} - X^{\prime 2}) \]

One may notice the expression in the integrand as a Virasoro condition. But the main focus here is a continued study of the EM tensor. In flat space \gamma_{\alpha \beta} = \eta_{\alpha \beta} as described. This means we can also write,

    \[ T_{\alpha \beta} = \partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu} - \frac{1}{2}\eta_{\alpha \beta}(\eta^{\lambda \rho} \partial_{\lambda}X^{\mu}\partial_{\rho}X_{\mu}) \]

And so, we may now look at each component of the EM tensor. We find,

    \[ T_{\tau \tau} = \partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} - \frac{1}{2}\eta_{\tau \tau}(\eta^{\tau \tau}\partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \eta^{\sigma \sigma} \partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu}) \]

    \[= \partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \frac{1}{2}(-\partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu}) \]

    \[= \frac{1}{2}(\partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} + \partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu}) \]

    \[ = \frac{1}{2}(\dot{X}^{\mu}\dot{X}_{\mu} + X^{\prime \mu}X_{\mu}^{\prime}) \]

Similarly, we can compute for T_{\sigma \sigma} which gives us the result,

    \[ T_{\sigma \sigma} = \frac{1}{2}(dot{X}^{\mu}\dot{X}_{\mu} + X^{\prime \mu}X_{\mu}^{\prime}) \]

For the off-diagonal terms,

    \[ T_{\tau \sigma} = \partial_{\tau}X^{\mu}\partial_{\sigma}X_{\mu} - \frac{1}{2}\eta_{\tau \sigma}(\eta^{\tau \tau}\partial_{\tau}X^{\mu}\partial_{\tau}X_{\mu} \eta^{\sigma \sigma}\partial_{\sigma}X^{\mu}\partial_{\sigma}X_{\mu}) \]

    \[= \partial_{\tau}X^{\mu}\partial_{\sigma}X_{\mu} = \dot{X}^{\mu}X_{\mu}^{\prime} \]

Similarly, we find,

    \[ T_{\sigma \tau} = X^{\prime \mu}\dot{X}_{\mu} \]

Therefore, the EM tensor takes the matrix form,

    \[ T_{\alpha \beta} = \begin{bmatrix}\frac{1}{2}(\dot{X}^{\mu}\dot{X}_{\mu} + X^{\prime \mu}X_{\mu}^{\prime}) & \dot{X}^{\mu}X_{\mu}^{\prime} \\X^{\prime \mu}\dot{X}_{\mu} & \frac{1}{2}(\dot{X}^{\mu}\dot{X}_{\mu} + X^{\prime \mu}X_{\mu}^{\prime}) \\\end{bmatrix} \]

We already know the EM tensor has zero trace. This of course can be computed. It also follows that the EM tensor appears in the equations of motion for S_{P} as before. But I leave it to the reader to explore these topics further.

References

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

David Tong. (2009). “String Theory” [lecture notes].