gauge fixing

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[edit] Unit gauge

In order to fix the gauge g_{ab} = \delta_{ab}\, (after Wick rotation), we need to make use of the Faddeev-Popov procedure. Our gauge condition is F[g] = g_{ab} - \delta_{ab} = 0\,. Here a gauge transformation is to be understood as some combination of diffeomorphism and Weyl rescaling:

Infinitesimally, g_{ab} \to e^{2\omega} g_{ab}\, becomes

\delta g_{ab} = 2 g_{ab} \delta\omega\,,

while under an infinitesimal diffeomorphism \sigma^a \to \sigma^a + \delta \sigma^a(\sigma)\,, the metric changes as follows:

\delta g_{ab} = -\nabla_a \delta \sigma_b -\nabla_b \delta \sigma_a\,,

so that generally

\delta g_{ab} = 2 g_{ab} \delta\omega -\nabla_a \delta \sigma_b -\nabla_b \delta \sigma_a\,.

We can break this symmetric tensor into a part containing the trace and a part that is traceless:

\delta g_{ab} = (2 \delta\omega - \nabla_c \delta \sigma^c) g_{ab} - 2 (P_1 \delta\sigma)_{ab}\,,

where the traceless part is (P_1 \delta\sigma)_{ab} \equiv \frac{1}{2}(\nabla_a \delta \sigma_b +\nabla_b \delta \sigma_a - \nabla_c \delta\sigma^c g_{ab})\,.

Denoting the three infinitesimal gauge parameters by \delta\xi^\alpha = (\delta\sigma^a, \delta\omega)\,, the Faddeev-Popov procedure instructs us to compute the Faddeev-Popov determinant

\Delta_{FP} = \left.\operatorname{det}\frac{\delta F_{ab}[g^{\xi^\alpha}]}{\delta {\xi^\alpha}}\right|_{\xi = 0}\,.

Up to irrelevant normalization, this is given as a path integral over Grassmann scalars, i.e. Faddeev-Popov ghosts b_{ab}\, and c^{\alpha}\,:

\Delta_{FP} = \int\! \mathcal{D} b_{ab} \mathcal{D} c^{\alpha} e^{-S_{GH}}\,,

where the ghost action is

S_{GH} = \int\!d^2\sigma \,d^2\sigma' \, \sqrt{g} b^{ab}(\sigma) \left.\frac{\delta F_{ab}[g^{\xi^\alpha}](\sigma)}{\delta {\xi^\alpha}(\sigma')}\right|_{\xi = 0}c^{\alpha}(\sigma')\,.

Thus the components of the symmetric tensor b^{ab}\, correspond to different gauge conditions (one for each component of g_{ab}\,), while the components of c^\alpha = (c^a, c)\, correspond to the gauge parameters \xi^\alpha\,. If the gauge fixing is complete, then there should be as many parameters as components. Note that the 3 terms in \xi^\alpha\, (1 Weyl + 2 diffeomorphism) match up to the 3 independent components of the symmetric tensor g_{ab}\,. Now, \int\!d^2\sigma' \, \frac{\delta F_{ab}[g^{\xi^\alpha}](\sigma)}{\delta {\xi^\alpha}(\sigma')}c^{\alpha}(\sigma')\, is essentially an infinitesimal gauge transformation with c^{\alpha}\, as gauge parameter, as the expression replaces d\xi^\alpha\, with c^\alpha\,.

For example,

\int\!d^2\sigma' \, \frac{\delta F_{ab}[g^{\xi^\alpha}](\sigma)}{\delta \sigma^d(\sigma')} c^d(\sigma')\, = \int\!d^2\sigma' \, \left( - \delta_{bd} \nabla_a - \delta_{ad} \nabla_b \right) \delta^{(2)}(\sigma-\sigma') c^d(\sigma')\,,
= \left( - \delta_{bd} \nabla_a - \delta_{ad} \nabla_b \right) c^d(\sigma)\,.

Thus the ghost action becomes

S_{GH} = \frac{1}{4\pi\alpha'}\int\!d^2\sigma \, \sqrt{g}b^{ab}\left[(2 c - \nabla_c c^c) g_{ab} - 2 (P_1\vec{c})_{ab}\right]\,,

where \vec{c}^a = c^a\, and the ghosts have been rescaled to yield an overall factor. Finally, we may integrate over c\,, which yields the functional Dirac delta function \delta[ b^{ab} g_{ab} ]\,. Thus we may take b^{ab}\, to be traceless from now on:

\Delta_{FP} = \int \!\mathcal{D} \sqrt{g} b\, \mathcal{D} c \, e^{\frac{1}{2\pi\alpha'}\int\!d^2\sigma \, b^{ab}(P \vec{c})_{ab}}\,.

Now we may insert \Delta_{FP} \delta[ F[g] ]\, into the original path integral to yield:

Z = \int\!\mathcal{D}X\,\mathcal{D} b\, \mathcal{D} c\, e^{-\frac{1}{4\pi\alpha'}\int\!d^2\sigma\,\sqrt{g}\partial_a X^\mu \partial_a X_\mu + \frac{1}{2\pi\alpha'}\int\!d^2\sigma \, b^{ab}(P \vec{c})_{ab}}\,,

where now \nabla_a \to \partial_a\,.

[edit] Conformal gauge

Sometimes we would like to fix the gauge only up to Weyl transformations, i.e., g_{ab} = e^{2\omega}\delta_{ab}\,. A suitable gauge condition is F[g] = g_{ab} - g^{\frac{1}{d}} \delta_{ab} = 0\,. Again, under an infinitesimal diffeomorphism \sigma^a \to \sigma^a + \delta \sigma^a(\sigma)\,, the metric changes as follows:

\delta g_{ab} = -\nabla_a \delta \sigma_b -\nabla_b \delta \sigma_a\,,

while

\delta g^{\frac{1}{d}}\, = \operatorname{det}^{\frac{1}{d}}(g_{ab} -\nabla_a \delta \sigma_b -\nabla_b \delta \sigma_a) - \operatorname{det}^{\frac{1}{d}}(g_{ab})\,,
= \operatorname{det}^{\frac{1}{d}}\left[g_{ac} (\delta^c_b -\nabla^c \delta \sigma_b -\nabla_b \delta \sigma^c)\right] - \operatorname{det}^{\frac{1}{d}}(g_{ab})\,,
\approx g^{\frac{1}{d}} [1 - \operatorname{tr}(\nabla^c \delta \sigma_b +\nabla_b \delta \sigma^c)]^{\frac{1}{d}} - g^{\frac{1}{d}}\,,
\approx g^{\frac{1}{d}} \left(1 - \frac{2}{d}\nabla_a \delta \sigma_a \right) - g^{\frac{1}{d}}\,,
= - \frac{2}{d}\nabla_a \delta \sigma^a \,.

Thus,

\delta F_{ab}[g] = -\nabla_a \delta \sigma_b -\nabla_b \delta \sigma_a + \frac{2}{d}\nabla_c \delta \sigma^c \delta_{ab}\, (see Conformal Killing equation). Note that for d = 2\,,
\delta F_{ab}[g] = -\frac{1}{2}(P_1 \delta\sigma)_{ab}\,.

Again, the Faddeev-Popov procedure may be used, introducing an anticommuting, symmetric, traceless field b^{ab}\, and an anticommuting field c^a\,. Then

\Delta_{FP} = \int \!\mathcal{D} b\, \mathcal{D} c \, e^{\frac{1}{2\pi\alpha'}\int\!d^2\sigma \, \sqrt{g} b^{ab}(P \vec{c})_{ab}}\,,

so that

Z = \int\!\mathcal{D}\omega\mathcal{D}X\,\mathcal{D} b\, \mathcal{D} c\, e^{-\frac{1}{4\pi\alpha'}\int\!d^2\sigma\,\partial_a X^\mu \partial_a X_\mu + \frac{1}{2\pi\alpha'}\int\!d^2\sigma \, \sqrt{g} b^{ab}(P \vec{c})_{ab}}\,.

Again, both b^{ab}\, and c^a\, have two independent components. Although \nabla_a \neq \partial_a\, because we are in the conformal gauge, it is possible to decouple the ghosts from \omega\, by using lightcone coordinates or complex coordinates on the world-sheet.

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