Geometrical Formulation

Previously we introduced the multi-dimensional Virasoro algebra in a Fourier basis on the \(d\)-dimensional torus, and showed that trying to build representations in the same way as in one dimension fails. Today I will describe a more geometrical formulation which will make the connection to QJT explicit.

Infinitesimal diffeomorphisms are generated by vector fields, which are locally of the form
\[
\xi = \xi^\mu\partial_\mu 
= \sum_{\mu=0}^{d-1} \xi^\mu(x) {\partial\over\partial x^\mu},
\]
where again we use Einstein's summation convention: two indices, one up and one down, are implicitly summed over. The bracket between two vector fields is
\[
[\xi, \eta] = \xi^\mu \partial_\mu \eta^\nu \partial_\nu -
\eta^\nu \partial_\nu \xi^\mu \partial_\mu.
\]
The diffeomorphism algebra, or algebra of vector fields, is the Lie algebra generated by the Lie derivatives \({\cal L}_\xi\):
\[
[{\cal L}_\xi, {\cal L}_\eta] = {\cal L}_{[\xi,\eta]}.
\]
In this formalism, the multi-dimensional Virasoro algebra takes the form in the blog banner.
\[\begin{align}
[{\cal L}_\xi, {\cal L}_\eta] &= {\cal L}_{[\xi,\eta]}
 + {1\over{2\pi i}}\int dt\ \dot q^\rho(t)
 \Big( c_1\ \partial_\rho\partial_\nu\xi^\mu(q(t))\ \partial_\mu \eta^\nu(q(t))\ +\\
& \qquad\qquad\qquad +\ c_2\ \partial_\rho \partial_\mu \xi^\mu(q(t))\ \partial_\nu \eta^\nu(q(t)) \Big),\\
[{\cal L}_\xi, q^\mu(t)] &= \xi^\mu(q(t)), \\
[q^\mu(t), q^\nu(t')] &= 0. 
\end{align}\] 
This is strictly speaking not a Lie algebra, because \(q\) appears non-linearly in the right-hand side. But we can extend these brackets to arbitrary functionals of \(q\), and then they define a proper Lie algebra.

To verify that this is a Lie algebra (anti-symmetry and Jacobi identities), we integrate by parts and throw away the boundary terms. Hence we must also add the condition that the integral of a total derivative vanishes.
\[
\int dt {dF(t)\over dt} \equiv 0.
\]
This is the case if the integral runs over a circle, or if we only consider vector fields which decrease sufficiently fast when \(t \to \pm\infty\). 

To connect this formulation with the previous one, we note that a basis for the vector fields on the \(d\)-torus is given by the plane waves \(i\exp(im\cdot x) \partial_\mu\). Define
\[\begin{align}
L_\mu(m) &= {\cal L}(i\exp(im\cdot x) \partial_\mu) \\
S^\mu(m) &= {1\over{2\pi}}\int dt\ \dot q^\mu(t) \exp(im\cdot q(t)) 
\end{align}\] 
These operators indeed satisfy the multi-dimensional Virasoro algebra in the Fourier basis. In particular, the vanishing of total derivatives leads to the condition
\[
m_\mu S^\mu = 
{-1\over{2\pi i}} \int dt\ {d\over dt}\Big(\exp(im\cdot q(t)) \Big) 
\equiv 0.
\]
We used the same letter \(q\) here as we did for the expansion point in the Taylor series. This is not a coincidence, and we will show later that they are the same. Hence the observer's position appears already in the very definition of the multi-dimensional Virasoro algebra, which explains why we need QJT to construct representations.

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