8
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Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension: $$ [L_m,L_n]=(m-n)L_{m+n}+\frac{C}{12}(m^3-m)\delta_{m+n,0}, $$ It's called the Virasoro algebra.

Its unitary highest weight representations are completely given by the pair $(c,h)$ such that :

  • $C \Omega = c \Omega$
  • $L_{0} \Omega = h \Omega$

The number $c$ is called the central charge and $\Omega$ is the vacuum vector.
If $0 \le c < 1$, then the FQS criterion says that $c = c_{m} = 1 - \frac{6}{m(m+1)}$ with $m = 2,3,4,5,...$

Observation :
Forget that $0 \le c < 1$ and just consider the equation : $c=c(x) = 1 - \frac{6}{x(x+1)}$
Then : $(c-1)x(x+1)+6=0$
And so : $(c-1)x^{2} + (c-1)x + 6 = 0$
But the discriminant $\Delta = (c-1)^{2}-24(c-1) = (c-1)(c-25)$

Conclusion : $\Delta = 0$ iff $c=1$ or $25$

The $N$-supersymmetric extensions ($\mathfrak{Vir}$ corresponds to $N=0$) :
$\mathfrak{Vir}$ admits two $N=1$ supersymmetric extensions : the Neveu-Schwarz and Ramond algebras.
In this case, the discrete series central charge is given by : $c = c_{m} = \frac{3}{2}(1 - \frac{8}{m(m+2)})$

Observation :
Let the equation $c=c(x) = \frac{3}{2}(1 - \frac{8}{x(x+2)})$
Then : $(\frac{2}{3}c-1)x^{2} + 2(\frac{2}{3}c-1)x + 8 = 0$
And: $\Delta = 4(\frac{2}{3}c-1)^{2}-32(\frac{2}{3}c-1) = 4(\frac{2}{3}c-1)(\frac{2}{3}c-9)$

Conclusion : $\Delta=0$ iff $\frac{2}{3}c= 1$ or $9$

Let $d_{N}$ be the (allowed) space-time dimension of a $N$-superstring theory, then :

  • $d_{0}=25+1$
  • $d_{1}=9+1$

Questions:

  • Is there an explanation for the link with the observations above (or just a coincidence) ?

  • What are $d_{2}$, $d_{3}$ and $d_{4}$ ?

  • Is there a formula for $d_{N}$ ?

Remark for N>1 : if for all $N$, $\Delta$ is quadratic in $c$, then $\Delta = 0$ admits two solutions in $c$.
Then $c$ could be renormalized as $c'=k.c$ such that one of the two solutions is $c'=1$ for the time-dimension, and the other, the space-dimension.
This process runs for $N=0,1$. Now for $N=2$, $c_{m} = \frac{3m}{m+2}$ :
Let the equation $c=c(x)=\frac{3x}{x+2}$, then $(c-3)x+2c=0$ and $\Delta = (c-3)^{2}$.
Let $c'=c/3$ (the renormalization), then we "would" obtain $d_{2} = 1+1$.

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  • $\begingroup$ $\Delta = \beta^{2}-4\alpha \gamma$ is the discriminant of the polynomial $\alpha x^{2}+\beta x+\gamma$. $\endgroup$ – Sebastien Palcoux Oct 13 '13 at 13:46
  • $\begingroup$ I've asked the MO mods if they want it. I'll get back to you. $\endgroup$ – dmckee Nov 8 '13 at 16:55
  • $\begingroup$ (I said this was probably alright for MO, although a slightly more conceptual presentation might make some of the MO denizens happier. Let's give it a go.) $\endgroup$ – Todd Trimble Nov 8 '13 at 18:33
  • $\begingroup$ @ToddTrimble : I've improved the presentation, I hope it's ok. $\endgroup$ – Sebastien Palcoux Nov 8 '13 at 20:38
  • $\begingroup$ $N=2$ strings have spacetime dimension 4 (see e.g. arxiv.org/abs/hep-th/9211059). $\endgroup$ – Pavel Safronov Nov 9 '13 at 1:11

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