What happens to Virasoro at c=25?

Question

The Virasoro algebras $Vir_c$ are a family of infinite dimensional Lie *-algebras parametrized by a real number $c$, called the central charge.¹

I hear that there exist two critical values of the central charge $c$, where qualitative changes happen to the structure of $Vir_c$ and of its representations. Those are $c=1$, and $c=25$.

I understand what happens at $c=1$. For $c<1$, the invariant sesquilinear form on the Verma modules is not positive definite, and is positive semi-definite for a discrete set of parameters, while for $c\ge 1$, the sesquilinear form is always positive definite (if a certain parameter called $h$ is $\ge 0$).

Can someone explain what happens at $c=25$?

Edit: José Figueroa-O'Farrill's comments have been quite illuminating. I would still like to know if here is a property (such as unitarity) of $Vir_c$ that distinguishes the cases $c<25$, $c=25$, and $c>25$.

¹ Strictly speaking, $Vir_c$ is not a Lie algebra, but only an associative algbera as I want to impose the relation that identifies the central element with 1, and there is no notion of unit in a Lie algebra (see Ben Webster's comment for a better explanation).

Answer 1

I have an incomplete understanding of this, but I will try to say what I know.

For each $c \in \mathbb{C}$, we define $Verma_c$ to be the category whose objects are Verma modules $V_{h,c}$ of central charge $c$ and highest weight $h$ for some $h \in \mathbb{C}$, and whose maps are Virasoro-module maps. Feigin and Fuks worked out the morphisms in all of these categories (they are embeddings), and they found that the categories $Verma_c$ and $Verma_{26-c}$ are antiequivalent. In particular, when $V_{h,c}$ has an embedded Verma submodule $V_{h+x,c}$, the corresponding module $V_{-1-h,26-c}$ embeds into $V_{-1-h-x,26-c}$.

When you consider representations of Virasoro with central charge $0 < c < 1$ and positive $h$, the corresponding objects will have central charge $25 < 26-c < 26$, and highest weight that is negative. In particular, unitary representations don't seem to make an appearance anywhere in the complementary picture.

According to this MathOverflow question, Positselski has developed the antiequivalence into a derived category statement, where modules are resolved by complexes of Vermas. However, I have been unable to extract an explicit theorem about Virasoro from his monograph.

Answer 2

Here is an observation, I don't know yet if it has consequences :

N=0 case (Virasoro algebra) :
The unitary highest weight representations of the Virasoro algebra are completely given by the pair $(c,h)$. The number $c$ is called the central charge.
If $0 \le c < 1$, then the FQS criterion says that $c = c_{m} = 1 - \frac{6}{m(m+1)}$ with $m = 2,3,4,5,...$
Now, forget for a moment 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: $\Delta = (c-1)^{2}-24(c-1) = (c-1)(c-25)$

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

$\rightarrow$ I guess it's coherent with the answer of Scott on the negative highest weight representations.

N=1 case (Neveu-Schwarz and Ramond algebras) :
In this case, the discrete series central charge is given by : $c = c_{m} = \frac{3}{2}(1 - \frac{8}{m(m+2)})$
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 $9$

Remark : The space-time dimension $d_{N}$ in the superstring theory :

• $d_{0}=25+1$
• $d_{1}=9+1$

I guess it's not a coincidence.

If I'm not mistaken : $d_{2} = 2$, $d_{3} = 0$ and $d_{4} = -2$.

Is there a formula for $d_{N}$ ?