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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Divergencies

Representations of the algebra of diffeomorphisms on the circle is essentially QFT on the circle. It would be natural to think that the same is true in higher dimensions, but this is not the case. If we try to repeat the recipe used in one dimension, we formally arrive at an infinite central charge, due to an unrestricted sum over transverse directions. This is of course a nonsense result, and it stopped physicists, myself included, from going further. Normal ordering only works if we start from a classical representation acting on finitely many functions of a single variable, because then there are no transverse directions that cause infinities. In later posts I will show how to do that within QJT.

First we review the construction of representations of the Virasoro algebra in one dimension. Let us start with the Heisenberg algebra:
\[\begin{align}
[\phi_m, \pi_n] &= i\delta_{m+n}, \\
[\phi_m, \phi_n] &= [\pi_m, \pi_n] = 0,
\end{align}\]
where \(m\) and \(n\) are integers. The algebra of diffeomorphisms on the circle can be embedded in this algebra in the following way.
\[
L_m = i \sum_{n=-\infty}^\infty (m-n) \phi_{m-n} \pi_n
\]
The brackets with the elements of the Heisenberg algebra read
\[\begin{align}
[L_m, \phi_n] &= (m+n) \phi_{m+n}, \\
[L_m, \pi_n] &= n \pi_{m+n},
\end{align}\]
which are the transformation laws of a scalar field and a scalar density, respectively.

Embedding the diffeomorphism algebra into the Heisenberg algebra means that a representation of the latter gives rise to a representation of the former. Introduce a vacuum \(|0>\) which annihilates all negative modes.
\[
\phi_{-m} |0> = \pi_{-m} |0> = 0,
\]
for all \(m > 0\).  This does not quite give us a representation of the diffeomorphism algebra, because the vacuum energy is infinite.
\[
L_0 |0> = -i\sum_{n=1}^\infty n \phi_{-n} \pi_n |0>
= \sum_{n=1}^\infty n |0>.
\]
To avoid this problem we redefine \(L_0\) so that \(L_0 |0> = 0\). 
\[
L_0 = i \sum_{n=1}^\infty n 
  \Big( \phi_{n} \pi_{-n} - \pi_{n} \phi_{-n}\Big)
\]
This is called normal ordering, because it is equivalent to moving the infinitely many negative modes to the right. However, this changes the algebra that the generators \(L_m\) satisfy, which can be seen by calculating the bracket
\[
[L_m, L_{-m}] |0> = -\sum_{n=1}^{m-1} n(m-n)|0> = -{(m^3-m)\over6}|0>
\]
Comparing with the definition of the Virasoro algebra in the previous post we see that normal ordering leads to a central charge \(c = 2\). The central charge is positive because the construction was made with bosonic fields. Repeating the analysis with fermionic fields would lead to  \(c = -2\) instead.

Now let us try to repeat the analysis in two dimensions. The indices now become 2D integer vectors \(m = (m_0, m_1)\), and the diffeomorphism generators can be embedded in a Heisenberg algebra like this:
\[
L_\mu(m_0,m_1) = i \sum_{n_0=-\infty}^\infty 
\sum_{n_1=-\infty}^\infty (m_\mu-n_\mu) \phi(m_0-n_0,m_1-n_1) \pi(n_0,n_1)
\]
To build a representation we must introduce a vacuum which is annihilated by half the modes, e.g. those with \(m_0 < 0\). Again this leads to an infinite vacuum energy, which can be removed by moving modes with \(m_0 < 0\) to the right. But this kind of normal ordering goes wrong, because we have to reorder infinitely many terms. E.g., when we compute \([L_0(m_0,m_1), L_0(-m_0,-m_1)] |0>\) we formally get the central charge
\[
\sum_{n_0=1}^{m_0} \sum_{n_1=-\infty}^\infty n_0(m_0-n_0) = K {(m_0^3-m_0)\over6},
\]
where
\[
K = \sum_{n_1=-\infty}^\infty 1 = \infty.
\]
So the would-be central charge is infinite, due to the unrestricted sum over transverse modes. An infinite central charge is of course a sign that something has gone very wrong. Also, the diffeomorphism algebra in \(d > 1\) dimension does not have a central extension, because it would not be compatible with the Jacobi identities.

The conclusion is that representations of the Virasoro algebra in \(d > 1\) dimensions can not be built in the same way as we do when \(d=1\). This is because the natural arena is not QFT but QJT.

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