Notes on String Theory: Conformal Group in 2-dimensions

We have so far considered the conformal group in d-dimensions. However, our interest is in 2-dimensions. So it will be of benefit to study the 2-dim conformal group in particular. To do so, we take d = 2. We will now also be working with a 2-dim Euclidean metric, such that g_{\mu \nu} = \delta_{\mu \nu}.

With this set up, the first task is to construct the generators. To do so we recall the equation we previously found,

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

When d = 2, this equation becomes (substituting also for the Euclidean metric),

(2)   \begin{equation*} \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} = (\partial \cdot \epsilon)\delta_{\mu \nu} \end{equation*}

When we take the coordinates (x^1, x^2) and as we calculate for different values of \mu and \nu, the above equation reduces rather nicely.

For \mu = \nu = 1, we arrive at 2\partial_{1}\epsilon_{1} = \partial_{1}\epsilon_{1} + \partial_{2}\epsilon_{2} \implies \partial_{1}\epsilon_{1} = \partial_{2}\epsilon_{2}.

For \mu = \nu = 2, we arrive reversely at 2\partial_{2}\epsilon_{2} = \partial_{1}\epsilon_{1} + \partial_{2}\epsilon_{2} \implies \partial_{2}\epsilon_{2} = \partial_{1}\epsilon_{1}.

Now, for the symmetric case where \mu = 1 and \nu = 2 (and, equivalently by symmetry, \mu = 2 and \nu = 1), we arrive \partial_{1}\epsilon_{2} + \partial_{2}\epsilon_{1} = 0. It follows, \partial_{1}\epsilon_{2} = -\partial_{2}\epsilon_{1}.

Notice, from these results, we have two distinguishable equations:

(3)   \begin{equation*} \partial_{1}\epsilon_{1} = \partial_{2}\epsilon_{2} \end{equation*}

(4)   \begin{equation*} \partial_{1}\epsilon_{2} = -\partial_{2}\epsilon_{1} \end{equation*}

The engaged reader may notice something interesting here. These equations are nothing other than the Cauchy-Riemann equations.
What this means, moreover, is that the conformal Killing equations in 2-dim reduce to the Cauchy-Riemann equations. Secondly, we have found that in 2-dim the infinitesimal conformal transformations that are of primary focus obey these equations.

Why is this noteworthy? Well, we know that in the theory of complex variables we’re working with analytic functions. As Polchinski explicitly communicates (p.34), the advantage here is that in working with analytical functions we can employ the coordinate convention (z, \bar{z}). This means, firstly, our conformal transformations can be written

(5)   \begin{equation*} z \rightarrow f(z) \ \ \ \bar{z} \rightarrow \bar{f}(\bar{z}) \end{equation*}

Where z = x + ix^2 and \bar{z} = x - ix^2. And so, in terms of infinitesimal conformal transformations, we may write

(6)   \begin{equation*} \epsilon = \epsilon^1 + i\epsilon^2 \ \ \bar{\epsilon} = \epsilon^1 - i\epsilon^2 \end{equation*}

The implication, also, is that \bar{\partial}f = \partial \bar{f} = 0.

Now, recall that our goal stated at the outset is to obtain the generators. To do so, we need to review the appropriate coordinate transformations. But before this, note that there are infinite coordinate transformation in 2-dim that corresponds to the fact that the conformal algebra is infinite dimensional. What we want to do is consider infinitesimal coordinate transformations of the particular form,

(7)   \begin{equation*} z \rightarrow z^{\prime} = z + \epsilon(z) \end{equation*}

(8)   \begin{equation*} \bar{z} \rightarrow \bar{z}^{\prime} = \bar{z}^{\prime} + \bar{\epsilon}(\bar{z}) \end{equation*}

These transformed coordinates may also be written in the following way,

(9)   \begin{equation*} z \rightarrow z^{\prime} = z - \epsilon_{n}(z)^{n+1} \end{equation*}

(10)   \begin{equation*} \bar{z} \rightarrow \bar{z}^{\prime} = \bar{z}^{\prime} - \bar{\epsilon}_{n}(\bar{z}^{n+1}) \end{equation*}

Where we have included consideration of the basis functions derived by expanding \epsilon(z) and \bar{\epsilon}(\bar{z}).

In the near future, we will spend some time constructing more of the notation, so hopefully this discussion will also clarify. But what should be emphasised in the meantime is that, for the generators, we take the derivative for the transformed coordinates. This gives us our expansion with which we want to focus on the infinitesimal terms. So, for example, when we take the derivative of z^{\prime} (and, likewise, \bar{z}^{\prime}) we arrive at,

    \[ = \frac{\partial}{\partial z}(z - \epsilon_{n}z^{n+1}) = 1 - \epsilon_{n}(n+1)z^n - z^{n+1}\partial_z\epsilon_n \]

From which we find the infinitesimal generators that we have been looking for,

(11)   \begin{equation*} l_n = -z^{n+1}\partial_z \end{equation*}

(12)   \begin{equation*} \bar{l}_n = -\bar{z}^{n+1}\partial_{\bar{z}} \end{equation*}

These are the infinitesimal generators that generate the symmetries we’ve been exploring. Classically, (11) and (12) generators satisfy the Virasoro algebra. Furthermore, notice that the set {l_{n}, \bar{l}_{m}} for integer values of m and n form an algebra with the following commutation relations,

(13)   \begin{equation*} [l_m, l_n] = (m - n)l_{m + n} \end{equation*}

(14)   \begin{equation*} [\bar{l}_{m}, \bar{l}_{n}] = (m - n)\bar{l}_{m + n} \end{equation*}

(15)   \begin{equation*} [l_m, \bar{l}_{n}] = 0 \end{equation*}

With this algebraic structure in mind, there are a couple of things that are important to note. Firstly, from the commutation relations it can be seen that the conformal algebra in 2-dim is isomorphic to the Witt algebra. Secondly, we made reference to the classical theory, but in the quantum theory adjustments need to be made to the commutation relations. This is where the central extension is included, with a term proportional to the central charge. It is also the case, as we will see, a new algebra is formed that is isomorphic with the Virasoro algebra. Thirdly, not all of the generators are clearly defined globally, and so this algebra may be viewed are primarily local (see Wray, p.85 for details).

Finally, when we work out and consider the global conformal group, the generators (11) and (12) come into contact with the Möbius group. To show this, we may explore the special case in which l_{0, \pm 1} and \bar{l}_{0, \pm 1}. For instance, consider an infinitesimal coordinate transformation, and note that we find in the following cases:

* l_{-1} = -\partial_z generates rigid translations of the form z^{\prime} = z - \epsilon;

* l_{0} = z -\epsilon z generates dilatations;

* l_{-1} = z - \epsilon z^2 generates special conformal transformations.

When we collect these transformation we can describe globally defined conformal diffeomorphisms, which gives the Möbius transformation:

    \[ z \rightarrow \frac{az + b}{cz + d} \]

Where ad - bc = 1. Note also that the collection of above transformations and the 2-dim conformal group are isomorphic with the SL(2, \mathbb{C}) / \mathbb{Z}_{2} group. This is the group of projective conformal transformations. A list of other particulars and constraints can be reviewed in sections 3.1 and 3.2 of Joshua D. Qualls “Lectures on Conformal Field Theory”.

References

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

Joshua D. Qualls. (2016). “Lectures on Conformal Field Theory” [lecture notes].

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

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