Symmetries of a Free Massless Scalar in Two Dimensions

Question

On p. 49 of Polchinski's book, he says: "Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry -- much more than we will have occasion to mention."

Does anyone know what this is referring to? Is he just saying that there are a lot of different choices of stress tensors that generate the conformal symmetry?

Answer 1

I don't have the book here right now, so I am not sure what he is referring to exactly by that comment. But by modifying the energy-momentum tensor, you will change the central charge and actually get a whole new CFT. So I would not call that a symmetry of the free scalar theory.

However the theory does have a lot more symmetry, let me focus on one chiral sector only. As you know already, the special feature of two-dimensional CFT's is that the spin-two conserved current (EM-tensor) $\bar{\partial}T(z)=0$, automatically implies that there is an infinite number of conserved currents $\bar{\partial}(z^n\,T(z)) = 0$ in the theory. This is essentially the reason why the conformal algebra, extends to the infinite-dimensional Virasoro algebra i two-dimensions. Lets use the notation $W_2\equiv T$, where the index refers to the spin.

Free-field theories, however, contain an infinite tower of higher-spin conserved currents $W_s(z)$ where $s= 2, 3, \dots$ is the spin. Similar to the $W_2$ case, for any conserved current $W_s$, there is infinite number of associated conserved currents $\bar{\partial}(z^n\, W_s(z))=0$. Thereby each current $W_s$ extends the Virasoro algebra with infinite number of new generators, and there is by itself an infinite number of currents $W_s$. So this is a vast extension of the Virasoro algebra.

For example in the case of free complex scalar theory the currents are given by [A]

$W_s(z)= B(s)\sum_{k=1}^{s-1}(-1)^k\, A^s_k\, :\partial^k\phi\,\partial^{s-k}\bar{\phi}:(z)$,

where $B(s)$ and $A^s_k$ are constants. See in particular equations (2.11) and (2.18a)-(2.18e) in [A]. Here $W_2(z)$ is the usual energy-momentum tensor leading to $c=2$ and the Virasoro algebra. All the generators combined give rise to the so-called $w_\infty^{PRS}$-algebra, which contain the Virasoro algebra as a "small" subalgebra. Similar things can be done for other free-field theories, I myself used a similar construction for the free ghost system in a recent paper.

Higher-spin extensions of the Virasoro algebra are usually called $\mathcal W$-algebras and they do not lead to conventional Lie algebras, but certain types of non-linear algebras. See [B] for a review. The free field theories realize the rare type of $\mathcal W$-algebras which are usual (linear) Lie algebras.

In higher dimensions free-field theories also have an infinite tower of higher-spin conserved currents and thereby an infinite dimensional symmetry algebra. But each conserved current only lead to a finite number of generators in the algebra, unlike the the two-dimensional case.

Maybe Polchinski is referring to this vast number of higher-spin symmetries of the free-field theories?

[A]: Bakas and Kritsis - Bosonic realization of a universal $\mathcal W$-algebra and $Z_{\infty}$ parafermions [Nucl. Phys. B 343, 185 (1990)]

[B] Bouwknegt and Schoutens - $\mathcal W$ symmetry in conformal field theory [Phys.Rept. 223 (1993) 183-276]

Answer 2

I think he is referring to the Coulomb Gas formalism.

The 'usual' massless boson in 2D has the energy momentum tensor $T= -\frac{1}{\alpha}\partial X^\mu\partial X_\mu$ (normal ordered). This energy-momentum tensor is the generator of conformal transformations. The current $\partial X^\mu$ and vertex operator $e^{i\alpha X^\mu}$ are examples of fields which transform covariantly with respect to these conformal transformations. These fields have a certain conformal dimension (the current has $h=1$ and the vertex operators something like $h=\frac{1}{2\alpha^2}$, depending on your conventions) and the theory has a central charge of $c=1$.

However, we can modify the energy momentum tensor by adding a term:

$$T= -\frac{1}{\alpha}\partial X^\mu\partial X_\mu + V_\mu \partial^2 X^\mu$$

This energy-momentum tensor also generates conformal transformations, but a different kind. For instance, the field $\partial X^\mu$ is no longer primary (does not transform covariantly with respect to these transformations). The EM tensor in complex coordinates is: $$ T(z) =-\frac{1}{2} :~\partial X^\mu \partial X^\mu~: +i\sqrt{2}\alpha_\mu\partial^2X^\mu$$

where $\alpha_\mu$ is related to $V_\mu$. This extra term arises due to the addition of a linear, imaginary coupling to the theory.

In this new conformal theory the vertex operators are still primary, but with a shifted conformal dimension: $h=\alpha^2 - 2\alpha \alpha_0$. Another thing that can be checked is the central charge, which has been shifted to: $c=1-24\alpha_0^2$. A last interesting thing is that this new theory has a different neutrality condition.

Moral of the story: by the addition of an extra, linear coupling to theory we move away from the original massless boson. However, we do not lose the conformal symmetry, but instead uncover a whole zoo of conformal theories. The massless boson does not just realize the $c=1$ case, but can be used to construct many, many different central charges and different content of primary fields. For example, with this construction you can obtain all the minimal models.

See chapter 9 of the big yellow book by DiFrancesco, Mathieu and Senechal.

Answer 3

I haven't got the book, so not totally sure of the context, but he could just be referring to the fact that the conformal group in two spacetime dimensions is actually infinite dimensional. This is rather special, because the conformal groups of higher dimensional spacetime are finite dimensional - for example the conformal group of Minkowski space is 15 dimensional.

(The conformal group is the group of transformations of spacetime which induce transformations of the metric of the form $ g_{\mu\nu}(x) \rightarrow \Omega^2(x)g_{\mu\nu}(x) ). $