Decomposing NHEG algebra to a Kac-Moody algebra

AbstractNo Abstract Found

Let’s begin from the generators of the NHEG phase space: \[\chi[\epsilon(\vec{\varphi})]=-\vec{k}\cdot \vec{\partial}_\varphi \epsilon (\frac{1}{r}\partial_t+r\partial_r)+\epsilon \vec{k}\cdot\vec{\partial}_\varphi\] We can decompose it to a Kac-Moody algebra. To show it, let’s write the generators as: \[\chi[\epsilon(\vec{\varphi})]=\chi^t[\epsilon(\vec{\varphi})]+\chi^0 [\epsilon(\vec{\varphi})]\] in which: \[\begin{aligned} \chi^0 [\epsilon(\vec{\varphi})]&=-\vec{k}\cdot \vec{\partial}_\varphi \epsilon r\partial_r+\epsilon \vec{k}\cdot\vec{\partial}_\varphi\\ \chi^t[\epsilon(\vec{\varphi})]&=-\vec{k}\cdot \vec{\partial}_\varphi \epsilon \frac{1}{r}\partial_t\end{aligned}\] Interestingly, each one of the above generators constitute a closed algebra. The algebra of \(\chi^0\) is similar to the original NHEG algebra, while the algebra of \(\chi^t\) is an Abelian algebra: \[\begin{aligned} &\{\chi^{0} [\epsilon(\vec{\varphi})], \chi^{0} [\epsilon'(\vec{\varphi})]\}=\chi^{0} [\tilde{\epsilon} (\vec{\varphi})] \qquad \tilde{\epsilon} =\epsilon \vec{k}\cdot \vec{\partial}_\varphi \epsilon'-\epsilon' \vec{k}\cdot \vec{\partial}_\varphi \epsilon\\ &\{\chi^t [\epsilon(\vec{\varphi})], \chi^t [\epsilon'(\vec{\varphi})]\}=0\end{aligned}\] More interestingly, there is the following nice commutation relation between the two algerbas: \[\{\chi^0 [\epsilon(\vec{\varphi})], \chi^t [\epsilon'(\vec{\varphi})]\}=\chi^t [\tilde{\epsilon} (\vec{\varphi})] \qquad \tilde{\epsilon} =\vec{k}\cdot \vec{\partial}_\varphi (\epsilon \vec{k}\cdot \vec{\partial}_\varphi\epsilon')\] Using the Fourier expansion as our basis, the generators of the two algebras can be written as a linear combination of the following generators: \[\begin{aligned} &\chi^0_{\vec{n}}=e^{i\vec{n}\cdot\vec{\varphi}}(-i\vec{k}\cdot\vec{n}\,r\partial r +\vec{k}\cdot\vec{\partial}_\varphi)\\ &\chi^t_{\vec{m}}=e^{i\vec{m}\cdot\vec{\varphi}}(\frac{-i\vec{k}\cdot\vec{m}}{r}\partial_t) \end{aligned}\] Their commutation relations would be: \[\begin{aligned} &\{\chi^0_{\vec{m}},\chi^0_{\vec{n}}\}=-i\vec{k}\cdot(\vec{m}-\vec{n})\chi^0_{\vec{m}+\vec{n}}\\ &\{\chi^t_{\vec{m}},\chi^t_{\vec{n}}\}=0\\ &\{\chi^0_{\vec{m}},\chi^t_{\vec{n}}\}=-i\vec{k}\cdot\vec{n}\,\,\chi^t_{\vec{m}+\vec{n}}\end{aligned}\]

The central extension

Let’s fix the symplectic structure to be: \[\begin{aligned} \boldsymbol{\omega}^{LW}(\delta_{\chi_1}\Phi,\delta_{\chi_2}\Phi,\Phi)+d\Big(\delta_{\chi_1} \mathbf{Y}(\delta_{\chi_2}\Phi,\Phi)-\delta_{\chi_2} \mathbf{Y}(\delta_{\chi_1}\Phi,\Phi)\Big)\end{aligned}\] in which: \[\mathbf{Y}(\delta_{\chi}\Phi,\Phi)=-\eta_+\cdot \mathbf{\Theta}\,.\] Then the central extension for the algebra \(\chi^0\) can be calculated to be: \[\begin{aligned} C_{\vec{n},\vec{m}}&= -i(\vec{k}\cdot \vec{m})^3 \frac{S}{2\pi}\,\delta_{\vec{m}+\vec{n},0}\,.\end{aligned}\] As a result, we get rid of the unwanted factor \(2\).