Stringy Things

Notes on string theory: Generators of conformal transformations

1. Infinitesimal Generators of the Conformal Group

In the last post, we considered a brief introduction to conformal field theory in string theory. We also began to study the d-dimensional conformal group, as described in equations (1) and (2). What we want to do now is study the infinitesimal generators of the d-dimensional conformal group, and in doing so we will refer back to these equations.

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. Returning to (2) in the last post, infinitesimal coordinate transformations may be considered generally in the form {x^{\mu} \rightarrow x^{\prime \mu} = x^{\mu} + \epsilon^{\mu}(x) + \mathcal{O}(\epsilon^{2})}. For the scaling factor {\Omega (x)} we have {\Omega (x) = e^{\omega(x)} = 1 + \omega(x) + [...]}.

Now, the question remains: in the case of an infinitesimal transformation, what happens to the metric? It turns out that the metric is left unchanged. To consider why this is the case, we may consider (1) from the linked discussion. Moreover, if, as above, we take an infinitesimal coordinate transformation then we have

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

\displaystyle = g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (3)

However, to satisfy the condition of a conformal transformation, (3) must be equal to (1). So we equate (3) and (1),

\displaystyle \omega(x)g_{\mu \nu}(x) = g_{\mu \nu} + \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (4)

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} which then gives,

\displaystyle (\omega(x) - 1)g_{\mu \nu} = \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} \ \ (5)

For this to make sense, we must find some expression for the scaling term {\omega(x) - 1}. One way to proceed is to multiply both sides of (5) by {g^{\mu \nu}}. As we are working in {d} spacetime dimensions, it follows {g_{\mu \nu}g^{\mu \nu} = d}. Hence,

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

\displaystyle (\omega(x) - 1)d = g^{\mu \nu}\partial_{\mu}\epsilon_{\nu} + g^{\mu \nu}\partial_{\nu}\epsilon_{\mu} \ \ (6)

The left-hand side of (6) is simple to manage. Focusing on the right-hand side, we raise indices and relabel. This gives us a usual factor of {2}. Hence, for the RHS of (6),

\displaystyle = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu}

\displaystyle = 2 \partial_{\mu}\epsilon^{\nu} \ \ (7)

Therefore, substituting (7) into the RHS of (6) we get,

\displaystyle (\omega(x) - 1)d = 2 \partial_{\mu}\epsilon^{\nu} \ \ (8)

Now, if we divide both sides by {d} and simplify, we end up with

\displaystyle (\omega(x) - 1) = \frac{2}{d} \partial_{\mu}\epsilon^{\nu} = \frac{2}{d} (\partial \cdot \epsilon) \ \ (9)

For which we may note the substitution,

\displaystyle \frac{2}{d}(\partial \cdot \epsilon) = \partial_{\mu}\epsilon^{\nu} + \partial_{\nu}\epsilon^{\mu} \ \ (10)

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. Let us now study these solutions.

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

LHS:

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

RHS:

\displaystyle \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 then rearranging terms we find,

\displaystyle \Box\epsilon_{\nu} + (1 - \frac{2}{d})\partial_{\nu}(\partial \cdot \epsilon) = 0 \ \ (11)

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

\displaystyle \Box\epsilon_{\nu} = 0 \ \ (12)

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 conformal group in d-dimensions. This group is isomorphic to SO(2,d).

To complete our discussion, we may note that generally 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 d-dimensional conformal group and its algebra. In the next post, we will study the conformal group in 2-dimensions.

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].

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