Timeline for What happens to Virasoro at c=25?
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18 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2014 at 7:59 | vote | accept | André Henriques | ||
| Oct 12, 2013 at 8:20 | answer | added | Sebastien Palcoux | timeline score: 0 | |
| May 14, 2013 at 3:13 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| May 13, 2013 at 19:34 | history | edited | André Henriques | CC BY-SA 3.0 |
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| May 9, 2013 at 3:53 | comment | added | Suvrit | @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! | |
| May 8, 2013 at 12:45 | comment | added | José Figueroa-O'Farrill | 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. | |
| May 8, 2013 at 9:50 | comment | added | José Figueroa-O'Farrill | 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) 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. | |
| May 8, 2013 at 4:54 | comment | added | Suvrit | Are these objects related to bosonic string theory with its $25+1$ space-time dimension (sorry, random pattern matching content free comment) :-) | |
| May 8, 2013 at 4:42 | history | edited | André Henriques | CC BY-SA 3.0 |
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| May 8, 2013 at 4:40 | comment | added | André Henriques | Thank you Ben... and I'm sorry that this confusion necessitated so many comments to be clarified. | |
| May 8, 2013 at 2:48 | comment | added | Ben Webster♦ | 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). | |
| May 7, 2013 at 21:47 | comment | added | José Figueroa-O'Farrill | 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. | |
| May 7, 2013 at 19:45 | comment | added | André Henriques | 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... | |
| May 7, 2013 at 10:35 | comment | added | José Figueroa-O'Farrill | The relevant paper is this one: mathnet.ru/php/…, but the PDF there is in Russian. It should be possible to find the English translation. | |
| May 7, 2013 at 10:32 | comment | added | José Figueroa-O'Farrill | 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. | |
| May 7, 2013 at 10:28 | comment | added | José Figueroa-O'Farrill | 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. | |
| May 7, 2013 at 10:01 | history | edited | André Henriques | CC BY-SA 3.0 |
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| May 7, 2013 at 8:58 | history | asked | André Henriques | CC BY-SA 3.0 |