ok, i haven't looked at the problem in a long time, so all i've got is a general explanation (you can plug in the numbers for whatever circle you have).
(if you don't care about the math [which sorta makes it pointless] this is a very basic description: there is a lie algebra called the virasoro algebra; its almost just the lie algebra of the group of diffeomorphisms of the circle, but it's actually just one dimension bigger, being a "central extension" thereof; projective representations of the lie algebra of the group of diffeomorphisms of the circle correspond to honest representations of the virasoro algebra.)
i'm not gonna re-type my notes, but this post (from google groups) covers it pretty well:
"In 2D the conformal group is infinite-dimensional - it
consists of transformations which are analytic in z = x+iy and z* =
x-iy. If we concentrate on the former, we see that an infinitesimal
conformal transformation is generated by
L_m = z^{m+1} d/dz,
and they obey the algebra
[L_m, L_n] = (n-m) L_{m+n}.
However, it is easy to see that this is the algebra of vector fields
(= infinitesimal diffeomorphisms) on the circle, vect(1). If the
circle coordinate is x, the generators are
L_m = -i exp(imx) d/dx.
vect(1) has a central extension, known as the Virasoro algebra:
[L_m, L_n] = (n-m) L_{m+n} - c/12 (m^3 - m) delta_{m+n,0},
where c is a c-number known as the central charge or conformal
anomaly. This means that the Virasoro algebra is still a Lie
algebra - anti-symmetry and the Jacobi identities still hold. The
term linear in m is unimportant, because it can be removed by a
redefinition of L_0.
The m^3 term is a genuine quantum effect, which simply is there when
you quantize a string. When string theorists criticize Thiemann's
LQG string, they are basically complaining that he does not get
this term, which simply must be there.
There is some confusion about anomaly freedom here, because at the
end people want to eliminate the conformal symmetry. The nice way to
do this is to introduce a ghost pair b_m, c_n, satisfying fermionic
brackets
{ b_m, c_n } = delta_{m+n,0}.
One can now write down the BRST operator, which looks something
like (double dots denote normal ordering)
Q = sum_m :L_{-m} c_m: + 1/2 sum_{m,n} (m-n) :b_{-m-n} c_m c_n:
If the BRST operator is nilpotent, Q^2 = 0, we can identify
physical states with BRST cohomology. A state is physical if it
is BRST closed, Q|phys> = 0, and two states are equivalent if
they differ by a BRST exact term,
|phys> ~ |phys'> if Q( |phys> - |phys'> ) = 0.
It turns out that the ghost has central charge c = -26, so the BRST
operator is nilpotent if the L_m's have c = 26; this is where the 26
dimensions of they bosonic string comes from. However, the important
thing from my viewpoint is not that the end result is anomaly free,
but that an anomaly exists, even if only in intermediate
calculations. Thiemann does not have an anomaly even intermediately,
and from this string theorists (and myself) conclude that LQG is
wrong.
Let us now return to the math. A lowest-weight representation (LWR)
of the Virasoro algebra is characterized by a lowest-weight state
|h,c> satisfying
L_0 |h,c> = h |h,c>,
L_{-m} |h,c> = 0, for all -m < 0.
It is known that the only unitary LWR of vect(1) is the trivial one.
However, the Virasoro algebra has many unitary LWRs: the discrete
series with 0 <= c < 1, where
c = 1 - 6/m(m+1), m positive integer and
h = h_{p,q}(c) = (pm^2 - q(m+1)^2) / 4m(m+1)
(or something similar, I'm quoting from memory), and also all c > 1,
h > 0. Anyway, the important thing is that the only acceptable value
of (h,c) with c = 0 is h = 0 - this is the trivial representation.
That Thiemann obtains a non-trivial unitary representation of the
1D diffeomorphism group with c = 0 is thus very strange. It is
hard to see how that could be compatible with quantum theory.
The Virasoro algebra can be generalized to several dimensions -
an extension of the diffeomorphism algebra on the N-dimensional
torus, say. The generators are
L_k(m) = i exp(im.x) d/dx^k,
where m = (m_i) and m.x = m_i x^i and I use the summation
convention. The algebra depends on two parameters (abelian charges)
c_1 and c_2,
[L_i(m), L_j(n)] = n_i L_j(m+n) - m_j L_i(m+n)
+ (c_1 m_j n_i + c_2 m_i n_j) m_k S^k(m+n),
[L_i(m), S^j(n)] = n_i S^j(m+n) + delta^j_i m_k S^k(m+n),
[S^i(m), S^j(n)] = 0,
m_k S^k(m) = 0.
This is the Virasoro algebra in 1D because the last condition then
becomes m S(m) = 0, which only has the solution S(m) ~ delta(m).
The Virasoro extension is not central (does not commute with
everything) except in 1D. It is straightforward but somewhat tedious
to check that these relations indeed do define a Lie algebra.
Just as the Virasoro algebra is anomalous when c != 0, its higher-
dimensional sibling is anomalous unless both c_1 = c_2 = 0. And just
as for the Virasoro algebra, there is no non-trivial, unitary LWRs
unless the algebra is anomalous. The correct definition of
lowest-weight is more subtle in several dimensions. Let me just say
that the right definition does not introduce any anisotropy."
