\[\begin{align}
[{\cal L}_\xi, {\cal L}_\eta] &= {\cal L}_{[\xi,\eta]}
+ {1\over{2\pi i}}\int dt\ \dot q^\rho(t)
\Big( c_1\ \partial_\rho\partial_\nu\xi^\mu(q(t))\ \partial_\mu \eta^\nu(q(t))\ +\\
& \qquad\qquad\qquad +\ c_2\ \partial_\rho \partial_\mu \xi^\mu(q(t))\ \partial_\nu \eta^\nu(q(t)) \Big),\\
[{\cal L}_\xi, q^\mu(t)] &= \xi^\mu(q(t)), \\
[q^\mu(t), q^\nu(t')] &= 0.
\end{align}\]
If you use Firefox with NoScript, as I do, you need to whitelist jsdelivr.net and polyfill.io, otherwise only the uncompiled LaTex code will show up.
If anybody wonders, this is the Virasoro-like extension of the diffeomorphism algebra in multiple dimensions. e.g. on the \(d\)-dimensional torus. When \(d=1\), both terms proportional to \(c_1\) and \(c_2\) reduce to the ordinary Virasoro algebra on the circle, a piece of mathematics that is popular e.g. in string theory and statistical physics. By the way, I discovered the \(c_2\) term myself a long time ago, which is something that I am still proud of (\(c_1\) was discovered by Rao and Moody).
When I started this blog almost a decade ago I had planned to write about this and how it in my opinion fits into physics, hence the name of the blog. I never got around to do that, but that might change in the future.
