In the last post, we reviewed conformal invariance when d = 2. As a quick review, we saw how under coordinate change X^{\mu}(\sigma) \rightarrow X^{\prime \mu}(\sigma) the metric transforms as,

(1)   \begin{equation*} g_{\mu \nu}(\sigma) \rightarrow g_{\mu \nu}^{\prime}(\sigma^{\prime}) = \frac{\partial \sigma^{\alpha}}{\partial \sigma^{\prime \mu}} \frac{\partial \sigma^{\beta}}{\sigma^{\prime \nu}}g_{\alpha \beta}(f(\sigma^{\prime}))\end{equation*}

The conformal group is a subgroup of coordinate transformations that, up to some scale factor, leaves the metric unchanged. If we denote the scale factor as \Omega(x), then we may state this fact as follows

(2)   \begin{equation*}g_{\mu} \rightarrow g^{\prime}_{\mu \nu}(x^{\prime}) = \Omega(x)g_{\mu \nu}(x) \end{equation*}

The essence of a conformal transformation is defined in this way. Invariance under (2) is what gives definition to a conformal field theory (Wray, p.80). Further definition was offered in the previous post.

What we want to do now is study the infinitesimal generators of the conformal group. In other words, if we assume the background is flat, such that g_{\mu \nu} = \eta_{\mu \nu}, the essential point of interest here concerns the infinitesimal transformation of the coordinates. We may state this as,

(3)   \begin{equation*} x^{\mu} \rightarrow f^{\mu}x^{\nu} x^{\prime \mu} = x^{\mu} + \epsilon^{\mu}\end{equation*}

Now, the question remains: in the case of an infinitesimal transformation, what happens with the metric? It turns out that an infinitesimal transformation of the coordinates leaves the metric unchanged.

To further investigate the implications of a conformally invariant transformation, we may, in general terms, expand on (3). Moreover, if, as above, we take x^{\mu} \rightarrow f^{\mu}x^{\nu} x^{\prime \mu} = x^{\mu} + \epsilon^{\mu} then we have

    \[ g_{\mu \nu}^{\prime}(x^{\mu} + \epsilon^{\mu}) = g_{\mu \nu} + (\partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\nu})g_{\mu \nu} \]

(4)   \begin{equation*}= g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu}\end{equation*}

But it remains that, to satisfy the condition of a conformal transformation, (4) has to be equal to (1),

(5)   \begin{equation*} \omega(x)g_{\mu \nu}(x) = g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \end{equation*}

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Where, \omega(x) is just an arbitrary function denoting a very small deviation from identity. Thus, we may also write \omega(x) = \omega(x) - 1 such that

(6)   \begin{equation*} (\omega(x) - 1)g_{\mu \nu} = \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu}\end{equation*}

For this to make sense, we must find some expression for the scaling term \omega(x) - 1. To proceed, we multiply both sides of (4) by g^{\mu \nu}. As we are working in d spacetime dimensions, it follows g_{\mu \nu}g^{\mu \nu} = d. This gives,

    \[ (\omega(x) - 1)g_{\mu \nu}g^{\mu \nu} = (\partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu})g^{\mu \nu} \]

(7)   \begin{equation*} (\omega(x) - 1)d = g^{\mu \nu}\partial_{\mu}\epsilon_{\nu} + g^{\mu \nu}\partial_{\nu}\epsilon_{\mu}  \end{equation*}

The left-hand side is simple to manage. Focusing on the right-hand side, we raise indices and relabel. This gives us a usual factor of 2. Hence,

    \[ = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu} \]

(8)   \begin{equation*} = 2 \partial_{\mu}\epsilon^{\nu} \end{equation*}

Therefore, putting everything together,

(9)   \begin{equation*} (\omega(x) - 1)d = 2 \partial_{\mu}\epsilon^{\nu} \end{equation*}

Now, if we divide both sides by d and simplify, we get

(10)   \begin{equation*} (\omega(x) - 1) = \frac{2}{d} \partial_{\mu}\epsilon^{\nu} = \frac{2}{d} (\partial \cdot \epsilon) \end{equation*}

For which we may note the substitution,

(11)   \begin{equation*}  \frac{2}{d}(\partial \cdot \epsilon) = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu} \end{equation*}

Where we can see that the infinitesimal conformal transformation, \epsilon, obeys the above equation. What is significant about this equation? It is the conformal Killing equation. And, it turns out, solutions to the above correspond to infinitesimal conformal transformations (Wray, p.81).

From a certain moral standpoint, as we have come this far, it behooves us to study these solutions. This is our next task.

To simplify things, notice we can define \partial_{\mu}\epsilon^{\mu} = \Box. Taking the derivative of the left and right-hand sides of the conformal Killing equation,

LHS:

    \[ = \partial^{\mu}(\frac{2}{d}(\partial \cdot \epsilon))\]

RHS:

    \[ \partial^{\mu}\partial_{\mu}\epsilon_{\nu} + \partial^{\mu}\partial_{\nu}\epsilon_{\mu} = \Box\epsilon_{\nu} + \partial_{\nu}(\partial \cdot \epsilon) \]

Putting everything together, equating both sides and rearranging terms,

(12)   \begin{equation*} \Box\epsilon_{\nu} + (1 - \frac{2}{d})\partial_{\nu}(\partial \cdot \epsilon) = 0\end{equation*}

It is clear that when d = 2, our first equation may be written as

(13)   \begin{equation*} \Box\epsilon_{\nu} = 0 \end{equation*}

For d > 2, we arrive at the following commonly cited equations that one will find in most texts:

1) \epsilon^{\mu} = a^{\mu} which represents a translation (a^{\mu} is a constant).

2) e^{\mu} = \lambda x^{\mu} which represents a scale transformation. Note, this corresponds to an infinitesimal Poincaré transformation.

3) \epsilon^{\mu} = w^{\mu}_{\nu}x^{\nu} which represents a rotation, where w^{\mu}_{\nu}x^{\nu} is an antisymmetric tensor. Note, this antisymmetric tensor also acts as the generator of the Lorentz group. Also note, this corresponds to an infinitesimal Poincaré transformation.

4) \epsilon^{\mu} = b^{\mu}x^{2} - 2x^{\mu}(b \cdot x) which represents a special conformal transformation.

From these equations, and with the inclusion of the Poincaré group, we have the collection of transformations known as the \textit{conformal group} in d dimensions. This group is isomorphic to SO(2,d).

To complete our discussion, more generally, we note that we may also incorporate the following generators and thus the conformal group has the following representation:

1) P_{\mu} = -\partial_{\mu}, which generates translations and is from the Poincaré group.

2) D = -ix \cdot \partial, which generates scale transformations.

3) J_{\mu} = i(x_{\mu}\partial_{\nu} - x_{\nu}\partial_{\mu}), which generates rotations.

4) K_{\mu} = i(x^2\partial_{\mu} - 2x_{\mu}(x \cdot \partial)), which generates special conformal transformations.

This completes our review of the conformal generators. In the next post, we will study the conformal algebra in 2-dim.

References

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

Timo Weigand. (2015/16). “Introduction to String Theory” [lecture notes].

Kevin Wray. (2009). “An Introduction to String Theory” [lecture notes].