Timeline for WZW primary correlations in terms of current algebra?
Current License: CC BY-SA 4.0
18 events
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| Oct 29, 2023 at 3:45 | history | edited | Joe | CC BY-SA 4.0 |
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| Oct 29, 2023 at 3:30 | history | edited | Joe | CC BY-SA 4.0 |
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| S Oct 26, 2023 at 18:02 | history | bounty ended | CommunityBot | ||
| S Oct 26, 2023 at 18:02 | history | notice removed | CommunityBot | ||
| Oct 25, 2023 at 20:12 | comment | added | Joe | In the boson example above, the $e^{i \alpha \phi(0)}$ correspond to different vacua $| \alpha \rangle$. More generally, they act as intertwiners taking the $| \beta \rangle$-module $M_\beta$ to the $| \alpha + \beta \rangle$-module $M_{\alpha+\beta}$ for all $\beta$. In this case, with $\sum_i \alpha_i = 0$, the operator $e^{i \alpha_n \phi(z_n)} \cdots e^{i \alpha_1 \phi(z_1)}$ maps modules $M_0 \to M_{\alpha_1} \to M_{\alpha_1 + \alpha_2} \to \cdots \to M_{\alpha_1 + \cdots \alpha_n} = M_0$. And indeed we can write this in terms of currents like above. | |
| Oct 25, 2023 at 20:04 | comment | added | Joe | I understand now that simple currents keep one in same VOA module (essentially by definition). As above, the better question is about correlators of primary fields. I think my question translates as follows. Take primaries $\Phi_1(z_1), \cdots, \Phi_n(z_n)$ corresponding to punctures $z_i$ and modules $M_i$. How does one write the projection to the vacuum sector of the operator $\Phi_1(z_1) \cdots \Phi_n(z_n)$ in terms of the VOA? This is only nonzero if $M_1 \otimes \cdots \otimes M_n$ decomposes with an identity representation. | |
| Oct 25, 2023 at 1:04 | comment | added | Pulcinella | To reiterate Nikita's point, if you write down what you mean by "in terms of WZW currents" you see it's incorrect. What I think you're saying is that every integrable representation $M$ of the affine VOA $V(\mathfrak{g})$ is generated by a single vector $m\in M$, which is false. Since if $M$ is generated by a single vector there's a surjection $V(\mathfrak{g})\twoheadrightarrow M$, i.e. $M$ must be a quotient of $V(\mathfrak{g})$. So it's easy to create counterexamples: $M=V(\mathfrak{g})\oplus V(\mathfrak{g})$, or indeed most smooth representations of $\hat{\mathfrak{g}}$ (i.e. reps of V(g)). | |
| Oct 25, 2023 at 0:31 | history | edited | Joe |
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| Oct 18, 2023 at 21:58 | history | edited | Joe | CC BY-SA 4.0 |
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| S Oct 18, 2023 at 16:39 | history | bounty started | Joe | ||
| S Oct 18, 2023 at 16:39 | history | notice added | Joe | Draw attention | |
| Oct 17, 2023 at 23:02 | history | edited | YCor |
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| Oct 17, 2023 at 18:42 | history | edited | Joe | CC BY-SA 4.0 |
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| Oct 17, 2023 at 18:37 | history | edited | Joe | CC BY-SA 4.0 |
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| Oct 17, 2023 at 18:31 | history | edited | Joe | CC BY-SA 4.0 |
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| Oct 16, 2023 at 21:49 | answer | added | Nikita Grygoryev | timeline score: 3 | |
| Oct 15, 2023 at 21:00 | history | edited | Joe | CC BY-SA 4.0 |
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| Oct 15, 2023 at 20:37 | history | asked | Joe | CC BY-SA 4.0 |