The [Virasoro algebras](http://en.wikipedia.org/wiki/Virasoro_algebra) $Vir\_c$ 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 $\ge 0$).



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> Can someone explain what happens at $c=25$?
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**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).



What happens to Virasoro at c=25?
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](https://mathoverflow.net/questions/6923/complexes-of-representations-with-complementary-central-charges), 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](http://arxiv.org/abs/0708.3398).


