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In the context of the compactified free boson CFT ($c = 1$), we know that at special radii, additional vertex operators become dimension one and hence can be interpreted as conserved currents.

In other well-known cases (e.g., the stress-energy tensor or the $i\partial X$ current), we expand the current in Laurent modes and discover an associated algebra — the Virasoro algebra or the Heisenberg, respectively.

My question: I have not see any textbook doing similar mode expansion for the current generated by the aforementioned vertex operator. Why is that? Shouldn't we look at its modes to identify the algebra? Do we already know about this algebra from other easier methods, I guess?

My apologies if the question is too naive. I am just hoping that someone will emphasize in what respect is this current/algebra different than the other ones.

(I have only consulted Di Francesco et. al textbook, Ginsparg notes, and Blumenhagen textbook so far. If there is some other textbook that answers my question, please refer me to it.)

Qmechanic
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baba26
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1 Answers1

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If your extra currents have spin 1 then you're in the case Prahar mentions which is $R^2 = 1$ (in $\alpha' = 1$ units) and a chiral algebra of $\mathfrak{su}(2)_1$.

The more general case has the spin $k$ currents \begin{align} W^\pm(z) = e^{\pm 2i \sqrt{k} X(z)} \end{align} for some integer $k$. For the radius, we can have any $R^2 = p/q$ such that $k = pq$. Commutators of modes for this algebra are written in a recent paper and cited to Moore and Seiberg but both of these suppress certain terms. It would be a good exercise to work them out.

The general procedure here would use the correspondence between mode commutators and singular OPEs. Singular OPEs in the free boson are pretty simple so you can then use \begin{align} W^\pm(z) = \sum_{m \in \mathbb{Z}} W^\pm_m z^{-m-k}, \quad W^\pm_m = \oint_0 \frac{dz}{2\pi i} W^\pm(z) z^{m + k - 1} \end{align} in principle. This is discussed in a review on W-algebras.

Connor Behan
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