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Currently I am taking an introductory class in conformal field theory and ran into some confusion. Everywhere I run into Laurent expansions of several important quantities, and (mostly) all of them boil down to the following:

Suppose $\phi (z)$ is a primary field with conformal dimension $h$, then it may be Laurent expanded as:

$$\phi(z) = \sum_{n \in \mathbb{Z}} c_n z^{-n-h}$$

With $c_n$ the Laurent modes (constants). For a particular reference of this, see for example di Francesco's book eqn. (6.7) or Blumenhagen eqn. (2.42) in which it is claimed (without motivation beforehand):

$$T(z) = \sum_{n \in \mathbb{Z}} z^{-n-2} L_n$$

Is there any mathematical or physical reason why we include these conformal weights of the objects in the Laurent expansion (why not simply $\sum_{n \in \mathbb{Z}} z^{n} c_n$) ? Or is it simply convenient in calculations?

Bonus (related) question: in Blumenhagen's book (page 13) the following expression is stated:

$$z'=z + \epsilon(z) = z + \sum_{n \in \mathbb{Z}}\epsilon_n (-z^{n+1})$$

Where $\epsilon(z)$ is assumed to be meromorphic and $\epsilon_n$ constant. Is this particular expansion (so with $z^{n+1}$ instead of $z^n$) similarly done simply for convenience?

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  • $\begingroup$ It seems to me that this is a purely mathematical question although it might have an application in physics (as does most mathematics). It might be better on Mathematics SE. $\endgroup$ – sammy gerbil Mar 9 '17 at 0:10
  • $\begingroup$ How is it a mathematics question when it is asking about conformal weights? $\endgroup$ – bolbteppa Mar 9 '17 at 0:11
  • $\begingroup$ When you map between the plane and the cylinder, you are transforming your metric and so using 5.22 of DiFrancesco you pick up the extra $h$ terms, c.f. Ketov Ch. 1, 1.43. $\endgroup$ – bolbteppa Mar 9 '17 at 0:32
  • $\begingroup$ @bolbteppa Ah I see! Thanks for your reply, this indeed solves my question! $\endgroup$ – Wwhite Mar 9 '17 at 14:41
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Let me use the book by di Francesco et al. Including the conformal weight in the power of $z$ in the Laurent expansion, as in equation (6.8), allows you to have the nice property (6.10), namely $\phi_{-m} = \phi^{\dagger}_m$. For the energy momentum tensor, it gives $L_{-m} = L^{\dagger}_m$.

It is just a nice convention.

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