# What is a good gentle introduction to the Virasoro algebra and its application in theoretical physics?

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I am looking for an as gentle and pedagogical as possible introduction that explains the Virasoro algebra and its applications in theoretical physics; finally I am interested in its application in string theory.

The short explanations in the physics books I am reading are not enough to make me feel comfortable about the Virasoro algebra so I need to read some more ...

I prefer shorter than whole book references which are optimally freely accessible, but if this does not work other things are welcome too.

I know a little bit about Lie algebras and conformal transformations, but the central extension issue confuses me. What I am looking for could be something at the level of these BRST lecture notes I could mostly follow.

retagged Apr 24, 2014

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A nice and friendly user at Quora has pointed out these lecture notes to me.

I have not jet fully read it, but is seems very useful to me and obviously explains some additional things I am missing.

answered Apr 12, 2013 by (5,440 points)
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The V. Algebra is a central extension of a Lie algebra of conformal complex transformations. As simple as you'd like it it is going to require full, or close to full undergraduate algebra (or even beyond), complex analysis and stuff....so odds are it is not going to be very gentle, and much even less if you want the applications to String Theory. Perhaps the best to do with this is to approach someone who masters the subject and learn directly from him.This post imported from StackExchange Mathematics at 2014-03-09 16:02 (UCT), posted by SE-user DonAntonio
answered Apr 3, 2013 by (0 points)
@DonAntonio I know a little bit about Lie algebras and conformal transformations, but exactly this central extension confuses me. What I am looking for could be something at the level of these BRST lecture notes I could mostly follow. I have not the possibility to aske somebody ind the "real world", and I can neither ask this at physics SE because there any reference / study material questions get immediately closed since some months ... :-/

This post imported from StackExchange Mathematics at 2014-03-09 16:02 (UCT), posted by SE-user Dilaton
@Dilaton Central extensions aren't really very hard to understand. They're a lot like direct products, but instead of keeping them completely separated, you allow isomorphic central subgroups of each group to overlap. You should search around for some explanations of central products - they are not difficult, just uncommon.

This post imported from StackExchange Mathematics at 2014-03-09 16:02 (UCT), posted by SE-user Alexander Gruber
Thanks for this hint @AlexanderGruber, so I will poke around a bit for central procuducts here.

This post imported from StackExchange Mathematics at 2014-03-09 16:02 (UCT), posted by SE-user Dilaton
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Some more sources (they are actually on string theory, but that clearly means they talk a lot about (super-)virasoro algebra and its applications in physics (specifically string theory) and as a quantisation of Witt Algebra, which is isomorphic to Conformal math.berkeley.edu/~kwray/papers/string_theory.pdf (An Introduction to String Theory), arxiv.org/pdf/hep-th/0207249v1.pdf (Introduction to String Theory), arxiv.org/pdf/hep-th/0207142v1.pdf (BUSTEPP Lectures on String Theory)

answered Jun 24, 2013 by (1,975 points)
edited Apr 24, 2014

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