Stringy Things

Notes on String Theory: Conformal Group in 2-dimensions

1. Introduction: Conformal Group in 2-dimensions

Following our previous study of the d-dimensional conformal group and the generators of conformal transformations, we now turn our attention to the study of the conformal group in 2-dimensions. Although we have taken some time to considered the d-dimensional conformal algebra, it should already be clear from past discussions that our interest is particularly in 2-dimensions. To begin our study of the 2-dimensional conformal algebra, where {d = 2}, note that we’re now employing a 2-dimensional Euclidean metric such that {g_{\mu \nu} = \delta_{\mu \nu}}. The first task is to construct the generators. Moreover, it can be found when studying the conserved currents on the WS (substituting for the Euclidean metric, see the last post),

\displaystyle \partial_{\mu}\epsilon_{\nu} + \partial_{\nu}\epsilon_{\mu} = (\partial \cdot \epsilon)\delta_{\mu \nu} \ \ (1)

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:

\displaystyle \partial_{1}\epsilon_{1} = \partial_{2}\epsilon_{2} \ \ (2)

\displaystyle \partial_{1}\epsilon_{2} = -\partial_{2}\epsilon_{1} \ \ (3)

If it is not obvious to the reader, it can be explicitly stated that these are nothing other than the Cauchy-Riemann equations. What this means, firstly, is that the conformal Killing equations reduce to the Cauchy-Riemann equations. Secondly, in 2-dimensions the infinitesimal conformal transformations that are of primary focus obey these equations.

Why is this notable? 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, that conformal transformations correspond with holomorphic and antiholomorphic coordinate transformations. These coordinate transformations are given by,

\displaystyle z \rightarrow f(z), \ \ \ \bar{z} \rightarrow \bar{f}(\bar{z}) \ \ (4)

Following Polchinski (pp.33-34), we are working with complex coordinates {z = \sigma + i\sigma^2} and {\bar{z} = \sigma - i\sigma^2}. It is also the case that {d^{2}x = dx^{0}dx^{1} = \frac{1}{2}dzd\bar{z}}. More will be said about this in the next section. Meanwhile, we may also define in the Euclidean signature and with complex variables,

\displaystyle \epsilon^{z} = \epsilon^0 + i\epsilon^1, \ \ \bar{\epsilon}^{\bar{z}} = \epsilon^0 - i\epsilon^1 \ \ (5)

In which {\epsilon} and {\bar{\epsilon}} are infinitesimal conformal transformations. This implies that {\partial_{z}\epsilon = 0} and {\partial_{z}\bar{\epsilon} = 0}.

And so, in terms of infinitesimal conformal transformations, we may write a change of holomorphic and antiholomorphic coordinates in an infinitesimal form,

\displaystyle z \rightarrow z^{\prime} = z^{\prime} + \epsilon(z), \ \ \ \bar{z} \rightarrow \bar{z}^{\prime} = \bar{z}^{\prime} + \bar{\epsilon}(\bar{z}) \ \ (6)

2. Generators of the 2-dimensional Conformal Group

What we want to do is obtain the basis of generators that produce the algebra of infinitesimal conformal transformations. To do so, we expand {\epsilon} and {\bar{\epsilon}} in a Laurent series obtaining the result,

\displaystyle \epsilon(z) = \sum_{n \in \mathbb{Z}} \epsilon_{n} z^{n+1} \ \ (7)

And,

\displaystyle \bar{\epsilon}(\bar{z}) = \sum_{n \in \mathbb{Z}} \epsilon_{n} \bar{z}^{n+1} \ \ (8)

With the basis of generators that generate the infinitesimal conformal transformations given by,

\displaystyle l_{n} = -z^{n+1}\partial_{z} \ \ (9)

And,

\displaystyle \bar{l}_{n} = -\bar{z}^{n+1}\partial_{\bar{z}} \ \ (10)

Classically, the above generators satisfy the Virasoro algebra. Moreover, it follows that these generators form the set {\{l_{n},\bar{l}_{n}\}} and this set becomes the algebra of infinitesimal conformal transformations for {n \in \mathbb{Z}}. The algebraic structure is given by the commutation relations,

\displaystyle [l_m, l_n] = (m - n)l_{m + n} \ \ (11)

\displaystyle [\bar{l_m}, \bar{l_n}] = (m - n)\bar{l}_{m + n} \ \ (12)

\displaystyle [l_m, \bar{l_n}] = 0 \ \ (13)

Importantly, the preceding generators obey the Witt algebra (Weigand, p.68). Also important, the generators that we’ve derived come into contact with the Möbius group (Weigand, p.69). To show this, we note the special case in which {l_{0, \pm 1}} and {\bar{l}_{0, \pm 1}}. Considering an infinitesimal coordinate transformation, 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 we can describe globally defined conformal diffeomorphisms as stated below, which give the Möbius transformation,

\displaystyle z \rightarrow \frac{az + b}{cz + d}

Where {ad - bc = 1}. A list of other constraints can be reviewed (Weigand, p.69).

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].

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