"Theta functions in affine Kac-Moody algebras"의 두 판 사이의 차이
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==introduction== | ==introduction== | ||
===notation=== | ===notation=== | ||
| − | + | * Let <math>M=Q^{\vee}</math>. This is also the <math>\mathbb{Z}</math>-span of <math>W\theta</math> where <math>\theta</math> is the highest root | |
| − | * Let | + | * for <math>\gamma\in M</math>, define <math>t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}</math> by |
| − | * for | + | :<math>t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta </math> |
| − | |||
| − | |||
| − | |||
;definition | ;definition | ||
| − | + | Let <math>k\in \mathbb{Z}_{\geq 1}</math> be the level of <math>\lambda</math>. Note <math>\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta</math>. and <math>|\lambda|^2=|\bar{\lambda}|^2+2k\xi</math>. The theta function is defined by | |
| + | :<math> | ||
\begin{align} | \begin{align} | ||
| − | \Theta_{k,\lambda} &=e^{-\frac{|\ | + | \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ |
| + | &=e^{-\frac{|\lambda|^2\delta}{2k}}e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{k(\gamma,\gamma)}{2}+(\gamma,\lambda)}\\ | ||
| + | &=e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{1}{2k}(k\gamma+\lambda,k\gamma+\lambda)} | ||
| + | \end{align} | ||
| + | </math> | ||
| + | We also have | ||
| + | :<math> | ||
| + | \begin{align} | ||
| + | \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ | ||
&=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k( \mu,\mu ) \delta}\\ | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k( \mu,\mu ) \delta}\\ | ||
&=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )} | &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )} | ||
\end{align} | \end{align} | ||
| − | + | </math> | |
| + | |||
| + | ===analytic function=== | ||
| + | * FMS 605p, (14.318) | ||
| + | * let <math>(\zeta;\tau;t)=\zeta+ \tau \Lambda_0+ t\delta\in \mathfrak{h}^{*}</math> | ||
| + | * once we evaluate them at <math>\xi=-2\pi i (\zeta;\tau;t)</math>, we get | ||
| + | :<math> | ||
| + | \Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2} | ||
| + | </math> | ||
| + | ;thm | ||
| + | :<math> | ||
| + | \Theta_{\lambda}^{(k)}(\frac{\zeta}{\tau},-\frac{1}{\tau},t+\frac{|\zeta|^2}{2\tau})=(-\frac{i \tau}{k})^{r/2}\frac{1}{\operatorname{vol}(Q^{\vee})}\sum_{\mu\in P/k Q^{\vee}}e^{-2\pi i (\mu,\nu)/k} \Theta_{\mu}^{(k)}(\zeta,\tau,t) | ||
| + | </math> | ||
| − | == | + | ==<math>A_1</math> example== |
| − | + | * let <math>z=e^{-\alpha_1}</math> | |
| − | * let | + | ===level k=1=== |
| − | + | * <math>\lambda=0</math> | |
| + | :<math> | ||
\Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots | \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots | ||
| − | + | </math> | |
| − | |||
==related items== | ==related items== | ||
| + | * [[Kac-Peterson modular S-matrix]] | ||
* [[Affine Weyl group]] | * [[Affine Weyl group]] | ||
* [[Theta functions]] | * [[Theta functions]] | ||
| + | * [[Cube root of the j-invariant and E8]] | ||
==computational resource== | ==computational resource== | ||
* https://docs.google.com/file/d/0B8XXo8Tve1cxbW9iUTgtaThCM2s/edit | * https://docs.google.com/file/d/0B8XXo8Tve1cxbW9iUTgtaThCM2s/edit | ||
| + | |||
| + | |||
| + | [[분류:math and physics]] | ||
| + | [[분류:Lie theory]] | ||
| + | [[분류:migrate]] | ||
2020년 11월 16일 (월) 05:34 기준 최신판
introduction
notation
- Let \(M=Q^{\vee}\). This is also the \(\mathbb{Z}\)-span of \(W\theta\) where \(\theta\) is the highest root
- for \(\gamma\in M\), define \(t_{\gamma} : \mathfrak{h}^{*}\to \mathfrak{h}^{*}\) by
\[t_{\gamma}(\lambda)=\lambda+\lambda(c)\gamma-\left(\frac{1}{2}\lambda(c)|\gamma|^2+(\gamma,\lambda)\right)\delta \]
- definition
Let \(k\in \mathbb{Z}_{\geq 1}\) be the level of \(\lambda\). Note \(\lambda=\bar{\lambda}+k\Lambda_0+\xi\delta\). and \(|\lambda|^2=|\bar{\lambda}|^2+2k\xi\). The theta function is defined by \[ \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{-\frac{|\lambda|^2\delta}{2k}}e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{k(\gamma,\gamma)}{2}+(\gamma,\lambda)}\\ &=e^{\lambda}\sum_{\gamma \in M}e^{k\gamma}q^{\frac{1}{2k}(k\gamma+\lambda,k\gamma+\lambda)} \end{align} \] We also have \[ \begin{align} \Theta_{k,\lambda} &=e^{-\frac{|\lambda|^2\delta}{2k}}\sum_{\gamma \in M}e^{t_{\gamma}(\lambda)} \\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu-\frac{1}{2}k( \mu,\mu ) \delta}\\ &=e^{k\Lambda_0}\sum_{\mu\in Q^{\vee}+\frac{\bar{\lambda}}{k}}e^{k\mu}q^{\frac{k}{2}( \mu,\mu )} \end{align} \]
analytic function
- FMS 605p, (14.318)
- let \((\zeta;\tau;t)=\zeta+ \tau \Lambda_0+ t\delta\in \mathfrak{h}^{*}\)
- once we evaluate them at \(\xi=-2\pi i (\zeta;\tau;t)\), we get
\[ \Theta_{\lambda}^{(k)}(\zeta,\tau,t)=e^{-2\pi i k t}\sum_{\alpha^{\vee}\in Q^{\vee}}e^{-2\pi i k (\alpha^{\vee}+\lambda/k,\zeta)}e^{i\pi k \tau |\alpha^{\vee}+\lambda/k|^2} \]
- thm
\[ \Theta_{\lambda}^{(k)}(\frac{\zeta}{\tau},-\frac{1}{\tau},t+\frac{|\zeta|^2}{2\tau})=(-\frac{i \tau}{k})^{r/2}\frac{1}{\operatorname{vol}(Q^{\vee})}\sum_{\mu\in P/k Q^{\vee}}e^{-2\pi i (\mu,\nu)/k} \Theta_{\mu}^{(k)}(\zeta,\tau,t) \]
\(A_1\) example
- let \(z=e^{-\alpha_1}\)
level k=1
- \(\lambda=0\)
\[ \Theta_{1,0}=1 + q (1/z + z) + q^4 (1/z^2 + z^2) + q^9 (1/z^3 + z^3)+\cdots \]