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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).

André Henriques
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  • Why is not $Vir_c$ a Lie algebra? It is the universal central extension of the Lie algebra of diffeomorphisms of the circle. Of course $c$ is not a number, but the central element, which does act by a number $c$ in any irreducible module. – José Figueroa-O'Farrill May 07 '13 at 10:28
  • It's been a while and I don't have the relevant papers with me, but I think that the story at $c=25$ is dual to the story at $c=1$. There is a duality between modules with central charge $c$ and central charge $26-c$, in that the embedding diagrams between Verma modules at central charge $26-c$ are obtained from those at central charge $c$ by reversing arrows. – José Figueroa-O'Farrill May 07 '13 at 10:32
  • The relevant paper is this one: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=1619&option_lang=eng, but the PDF there is in Russian. It should be possible to find the English translation. – José Figueroa-O'Farrill May 07 '13 at 10:35
  • Answer to Jose's first comment: The Lie algebra that people refer to as "the Virasoro algebra" does not depend on a choice of real number. Its representation category contains a full subcategory which is equivalent to the representation category of $Vir_c$, namely, those representations where the central element acts by $c\cdot 1$. Anyways, it's just a matter of definitions... – André Henriques May 07 '13 at 19:45
  • André, just to make sure I understand. What do you call the universal central extension of the Lie algebra of diffeomorphisms of the circle? Is that what you have termed "the Virasoro algebra" in your previous comment? or is it what you call $Vir_c$? I agree that, contrary to the oversimplification in my previous comment, there are representations of the universal central extension of $\mathfrak{diff}(S^1)$ where the centre does not act like a scalar multiple. – José Figueroa-O'Farrill May 07 '13 at 21:47
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    Andre wants to consider the quotient of the universal enveloping algebra of Virasoro by the relation $k-c\cdot 1$ ($k$ is the central element of the Lie algebra, $c$ a scalar, and $1$ the identity in the UAE). – Ben Webster May 08 '13 at 02:48
  • Thank you Ben... and I'm sorry that this confusion necessitated so many comments to be clarified. – André Henriques May 08 '13 at 04:40
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    Are these objects related to bosonic string theory with its $25+1$ space-time dimension (sorry, random pattern matching content free comment) :-) – Suvrit May 08 '13 at 04:54
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    Ben, thanks for the clarification. In that case, then at the top of page 236 in this paper of Feigin and Fuks (the English summary of their longer paper in one of my previous comments) http://link.springer.com/content/pdf/10.1007%2FBFb0099939.pdf you will find the statement that at the level of Verma modules, $(h,c)$ and $(-1-h, 26-c)$ are anti-equivalent. This is just a symmetry of the determinant formula of the Shapovalov form. – José Figueroa-O'Farrill May 08 '13 at 09:50
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    The answer to S. Sra's question, however random, is "yes". The state space of a bosonic string can be identified with the semi-infinite cohomology of $Vir_c$ relative its centre. The usual complex is obtained by tensoring a Virasoro module (not necessarily irreducible) with $c=26$ with the module of semi-infinite forms. In fact, the duality which answers André's question was first noticed by Feigin in the paper where he defined semi-infinite cohomology. – José Figueroa-O'Farrill May 08 '13 at 12:45
  • @José: wow, so random turned out to be not entirely random---though unfortunately I am quite far from understanding the content and background that you have provided. Thanks! – Suvrit May 09 '13 at 03:53

2 Answers2

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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.

Community
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S. Carnahan
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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 $d_{N}$ ?

Sebastien Palcoux
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  • This appears to be more appropriate as a question than as an answer. – S. Carnahan Oct 13 '13 at 01:03
  • @S.Carnahan : The observation that $\Delta = (c-1)(c-25)$ is an easy way to obtain $1$ and $25$, and it's coherent with your answer with $26-c$ : if I'm not mistaken, the negative highest weight discrete series have central charges $c_{m} = 25 + \frac{6}{m(m+1)}$, and the equations $c=c(x)=25 + \frac{6}{x(x+1)}$ gives the same $\Delta = (c-25)(c-1)$. Now, I'm agree that the questions about the space-time dimension of the $N$-superstring theory, deserve their own post. I will post it. – Sebastien Palcoux Oct 13 '13 at 10:35
  • What is $\Delta$? – André Henriques Oct 13 '13 at 12:52
  • @AndréHenriques : $\Delta = \beta^{2}-4\alpha \gamma$ is the discriminant of the polynomial $\alpha x^{2}+\beta x+\gamma$. To avoid confusion : here $\alpha = c-1$, $\beta=c-1$ and $\gamma = 6$. – Sebastien Palcoux Oct 13 '13 at 13:50
  • I have posted a development of the last question of my answer here – Sebastien Palcoux Jul 13 '14 at 16:54