Fiducial metric
Stringy Things

Notes on string theory: Fiducial metric

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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 $2×2$ 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}{4pi 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}{4pi alpha^{prime}} int d^{2}sigma (partial_{tau}X^{mu}partial_{tau}X_{mu} + partial_{sigma}X^{mu}partial_{sigma}X_{mu}) ]

[ = frac{1}{4pi 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].

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