Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
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

1 Answer 1

up vote 2 down vote accepted

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).

I don't have a good answer to your second question. Perhaps "it seemed like a good idea at the time".

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.