# Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras?

Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by $$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$ $$[c,L_n]=0.$$

Similarly, following the standard conventions in the literature, the commutation relations of the Neveu-Schwarz super Lie algebra are given by $$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{8}(m^3-m)c,$$ $$[J_\alpha,J_\beta]_ +=2L_{\alpha+\beta}+\delta_{\alpha,-\beta}\frac12(\alpha^2-\frac14)c,$$ $$[L_m,J_\alpha]=(\frac12m-\alpha)J_{m+\alpha},$$ $$[c,L_n]=0,\qquad [c,J_\alpha]=0.$$

The Virasoro algebra is a subalgebra of the Neveu-Schwarz algebra by $L_n \mapsto L_n$ and... $c \mapsto \frac32c$. Clearly, one could change the normalizations in the second set of formulas to avoid the annoyance of having to send $c$ to $\frac32c$.

So whyyy do people do it that way?
Why do physicists take their standard formulas not consistent with each other?
There must be a reason.

• You're being a little too harsh with the physicists. In fact, the original literature does not use the same notation for the two central charges in the Virasoro and super-Virasoro algebras, hence there is no notational inconsistency. The central charge has physical meaning and it's not just an artifact of normalisaiton. – José Figueroa-O'Farrill Aug 1 '12 at 9:54
• "The original literature does not use the same notation for the two central charges": ok, I did not know that; what is the standard (or original) notation? "The central charge has physical meaning and it's not just an artifact of normalisaiton": are you referring to what Scott Carnahan wrote in his answer? If not, could you elaborate? – André Henriques Aug 1 '12 at 16:24

As far as I can tell, a $\sigma$-model with $d$ dimensional target space will have Virasoro central charge $d$ with bosonic strings, and $3d/2$ with supersymmetric strings. I believe the normalizations were chosen so that the constant $c$ reflects the dimension of spacetime in which the strings are propagating (even if the specific model you are considering does not involve a spacetime manifold).