# Quantum expressions for the Virasoro constraints

I am trying to derive the quantum form of the Virasoro constraints. $$L_{m} = \frac{1}{2} \sum_{n} :\alpha_{m-n}.\alpha_{n}:$$ :...: meaans normal ordering. Using the common commutator between modes of string when $$m=0$$ $$[\alpha^{\mu}_{n}, \alpha^{\nu}_{-n}]= n \eta^{\mu\nu}$$ we easily get: $$L_{0}= \frac{1}{2}\sum_{n=-\infty}^{\infty} \alpha_{-n}.\alpha_{n}= \frac{1}{2} \alpha_{0}^2 + \sum_{n=1}^{\infty} \alpha_{-n}.\alpha_{n}\\ +\frac{D}{2} \sum_{n=1}^{\infty} n.$$ My problem is here when calculating the last term. We know that it is infinite, forgetting it for a moment because text says we can use regularization to calculate its finite value. The simple calculation comes first

$$\frac{d}{da}\sum_n e^{-na} ~=~\sum_n ne^{-na}~=~ \frac{d}{da} \frac{1}{1-e^{-a}} ~=~ \frac{e^{-a}}{(1-e^{-a})^{2}} ~=~ \frac{1}{a^2}-\frac{1}{12}.$$

First, I could not understand $$(\frac{1}{a^2}-\frac{1}{12})$$ part. Second, the text concludes that: $$\frac{D}{2} \sum_{n=1}^{\infty} n= -\frac{D}{24}$$ where I could not follow. So my clear question is "how the regularization works"? It is said that $$a= 2πε /\ell$$.

References:

1. David MacMahon, String theory demystified, p. 79-80.

$$\color{red}{-}~\frac{d}{da}\sum_{n=0}^{\infty} e^{-na} ~=~\sum_{n=0}^{\infty}ne^{-na}~=~ \color{red}{-}~\frac{d}{da} \frac{1}{1-e^{-a}} ~=~ \frac{e^{-a}}{(1-e^{-a})^{2}} ~=~ \frac{1}{a^2}-\frac{1}{12}\color{red}{+{\cal O}(a^2)}.$$ The main point is that the regular terms $${\cal O}(a^2)$$ vanish as we remove the regularization $$a\to 0^+$$, while the singular term $$\frac{1}{a^2}$$ is cancelled by local counterterms. See e.g. this Math.SE post, this & this Phys.SE posts, and links therein.