Timeline for How to interpret the Sugawara construction from a physical or mathematical viewpoint?
Current License: CC BY-SA 4.0
16 events
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| May 19, 2022 at 8:24 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added doi and projecteuclid instead of the dead link
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| May 17, 2022 at 20:33 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed
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| Oct 21, 2019 at 12:42 | comment | added | soap | Let us continue this discussion in chat. | |
| Oct 21, 2019 at 11:08 | comment | added | José Figueroa-O'Farrill | @Soap I'm sorry to say that this may be my own imprecision. I tend to think of the loop algebra of a simple Lie algebra and the corresponding affine Kac-Moody algebra as roughly synonymous. | |
| Oct 20, 2019 at 22:01 | comment | added | soap | @JoséFigueroa-O'Farrill Just one more thing: you say that in the Sugawara construction one starts with a representation of a Kac-Moody algebra, but in the physics literature they seem to start with a central extension of a loop algebra instead (so they do not use the derivation extension). So I would expect them to use the (integrable) reps of central extension of a loop algebra, not the (integrable) reps of the affine Kac Moody algebra, to classify the WZW models... but they use the latter. | |
| Oct 20, 2019 at 17:55 | comment | added | José Figueroa-O'Farrill | @Soap In their paper String theory on group manifolds, Gepner and Witten show that any correlation function containing a highest-weight vector of a non-integrable representation vanishes. Hence one does not lose anything by restricting to integrable highest weight representations. | |
| Oct 20, 2019 at 17:22 | comment | added | soap | @JoséFigueroa-O'Farrill That I know. I should have been more explicit: why integrable representations? | |
| Oct 20, 2019 at 17:13 | comment | added | José Figueroa-O'Farrill | @Soap The mathematical condition of "highest weight" is equivalent to the physical condition of "minimal energy"; that is, representations where the energy is bounded below, which is a desirable physical property. | |
| Oct 20, 2019 at 16:11 | comment | added | soap | @JoséFigueroa-O'Farrill Why do we only use the integrable highest weight representations? | |
| Feb 26, 2010 at 15:09 | vote | accept | Xuexing Lu | ||
| Feb 26, 2010 at 15:09 | |||||
| Feb 25, 2010 at 22:19 | comment | added | José Figueroa-O'Farrill | If you define the quantisation of the WZW model in terms of the currents, so that the Hilbert space of the theory consists of all the integrable highest weight representations (up the relevant level) of the corresponding affine Kac-Moody algebra (I'm talking about the case of $\mathfrak{g}$ simple), then the Sugawara construction defines on those representations the structure of a Virasoro module. This, by definition, is an exact quantum conformal field theory. | |
| Feb 25, 2010 at 20:15 | comment | added | Konrad Waldorf | Hi José, how exactly does the Sugawara construction prove conformal invariance to all orders? In Witten's paper I can only see a calculation in one-loop-order. | |
| Feb 25, 2010 at 19:08 | history | edited | José Figueroa-O'Farrill | CC BY-SA 2.5 |
corrected a notational inconsistency: k and l both for the level
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| Feb 25, 2010 at 16:01 | vote | accept | Xuexing Lu | ||
| Feb 26, 2010 at 15:09 | |||||
| Feb 25, 2010 at 15:43 | history | edited | José Figueroa-O'Farrill | CC BY-SA 2.5 |
added 41 characters in body
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| Feb 25, 2010 at 15:37 | history | answered | José Figueroa-O'Farrill | CC BY-SA 2.5 |