Virasoro algebra

This post has been migrated from my old blog, the math-physics learning seminar.

Conformal Invariance in 2D

To begin, recall that in two dimensions, the conformal transformations are generated by holomorphic and anti-holomorphic transformations. At the infinitesimal level, let (\ell_n := -z^{n+1} \partial_z) be a basis of holomorphic vector fields. These satisfy the Witt algebra

[ [\ell_m, \ell_n] = (m-n)\ell_{m+n}. ]

Similarly, we can define (\bar{\ell}m = -\bar{z}^{n+1} \partial{\bar{z}}), and in addition to the Witt algebra these new generators satisfy ([\bar{\ell}_m, \ell_n]=0).

Now, we could try to define a 2D conformal quantum field theory to be a unitary representation of the Witt algebra (or rather, of two copies of the Witt algebra, since we have both holomorphic and anti-holomorphic vector fields–but nevermind that). But this is too naive.

Central Extensions

Recall that in quantum mechanics, states are represented by vectors in some Hilbert space (\mathcal{H}). However, the state (|\phi\rangle) and (\alpha|\phi\rangle) are physically equivalent for any non-zero complex number (\alpha). The reason, of course, is that the expectation value of an operator (\mathcal{O}) is defined to be (\langle \phi|\mathcal{O}|\phi\rangle / \langle \phi|\phi\rangle), and such expressions are invariant under rescaling in (\mathcal{H}).

Thus, a symmetry group (G) for a theory does not necessarily act via a map (G \to U(\mathcal{H})). It suffices to have a projective representation (G \to PU(\mathcal{H})). Let (\mathfrak{g}, \mathfrak{pu}) be the Lie algebras of (G) and (PU), respectively. A projective representation gives a map

[ \mathfrak{g} \to \mathfrak{pu}. ]

Since (PU) is a quotient of (U), we have a short exact sequence

[ 0 \to \mathbb{C} \to \mathfrak{u} \to \mathfrak{pu} \to 0. ]

Now let (\hat{\mathfrak{g}}) be defined as

[ \hat{\mathfrak{g}} = { (\xi, \eta) \in \mathfrak{u}\oplus\mathfrak{g} \ | \ \pi(\xi) = \rho(\eta) } ]

This comes with a natural projection (\hat{\mathfrak{g}} \to \mathfrak{g}). If we suppose that the projective representation (\rho) is faithful, then the kernel of this map is exactly (\mathbb{C}). Hence, a faithful projective representation of (\mathfrak{g}) yields a short exact sequence of Lie algebras

[ 0 \to \mathbb{C} \to \hat{\mathfrak{g}} \to \mathfrak{g} \to 0. ]

We have obtained a central extension of (\mathfrak{g}).

Virasoro Algebra

Finally, we can define the Virasoro algebra. It has generators (L_n) and (c), with defining relations

[ [L_m, L_n] = (m-n) L_{m+n} + \frac{c}{12}(m^3-m) \delta_{m+n,0}, [c, L_n] = 0. ]

The generator (c) acts as a scalar in any irreducible representation, and its value is called the central charge. The factor of (1/12) is entirely conventional. Now, the amazing fact is the following.

Theorem. Up to equivalence, the Virasoro algebra is the unique non-trivial central extension of the Witt algebra.

Proof sketch. This is essentially just a calculation. Any central extension has to be of the form

[ [L_m, L_n] = (m-n) L_{m+n} + A(m,n) c ]

for some function (A(m,n)). If we make the replacement (L_m \mapsto L_m + a_m c), then we have

[ [L_m, L_n] = (m-n) L_{m+n} + \left( A(m,n) + (m-n) a_{m+n} \right) c ]

Taking (n = 0), we have

[ [L_m, L_0] = m L_{m} + \left( A(m,0) + m a_{m} \right) c ]

Hence for (m\neq0) we can take (a_m = m^{-1} A(m,0)). Having done this, we are now free to assume that (A(m,0) = 0 ) for all (m). Then we may apply the Jacobi identity to deduce that (A(m,n)=0) except possibly for (m=-n), so that (A(m,n)) can be written in the form (A(m,n) = A_m \delta_{m+n, 0}). Finally, another application of the Jacobi identity yields a simple recurrence relation for the coefficients (A_m), and it is easily seen that every solution of this recurrence is proportional to (m^3-m).

Now we can take our (preliminary, and still too naive) definition of a quantum conformal field theory to be a unitary representation of the Virasoro algebra.

Stress-Energy Tensor and OPE

The operator (L_0) behaves like the Hamiltonian of the theory, and the Virasoro relations show that (L_n) for (n>0) act as lowering operators. Hence, in a physically sensible representation, the vacuum vector (|\Omega\rangle) will be annihilated by (L_n) for all (n > 0). Unitary requires (L_n^\dagger = L_{-n}), so additionally we have (\langle \Omega|L_n = 0) for (n < 0). Hence

[ \langle \Omega | L_m L_n | \Omega \rangle = 0 \ \textrm{unless}\ n \leq 0, m \geq 0 ]

Now define the stress-energy tensor to be the operator-valued formal power series

[ T(z) = \sum_n \frac{L_n}{z^{n+2}} ]

We can consider the vacuum expectation of the product (T(z) T(w)). By the above remarks, many terms in the expansion will vanish. In fact, it is a straightforward (but tedious!) exercise to check the following.

Theorem. The stress-energy tensor satisfies the operator product expansion

[ T(z) T(w) \sim \frac{c/2}{(z-w)^4} + \frac{2 T(w)}{(z-w)^2} + \frac{\partial_w T(w)}{z-w} ]

where (\sim) denotes that the left- and right-hand sides are equal up to the addition of terms with vanishing vev and/or regular as (z \to w).