Radial quantisation

Date: 2015-10-07; view: 448.

This is the most important concept in understanding the link between SLE and CFT. We introduce it in the context of boundary CFT. As before, suppose there is some set of fundamental fields {ψ (r)}, with a Gibbs measure e^−S[ψ][dψ]. Let Γ be a semicircle in the upper half plane, centered on the origin. The Hilbert space of the BCFT is the function space (with a suitable norm) of field configurations {ψ_Γ} on Γ.

The vacuum state is given by weighting each state by the (normalised) path integral restricted to the interior of Γ and conditioned to take the specified values on the boundary

(47)

Note that because of scale invariance different choices of the radius of Γ are equivalent, up to a normalisation factor.

Similarly, inserting a local operator (0) at the origin into the path integral defines a state | . This is called the operator-state correspondence of CFT. If we also insert (1/2πi)∫_C zⁿ⁺¹T (z) dz, where C lies inside Γ, we get a state L_n| . The L_n act linearly on the Hilbert space. From the OPE (45) we see that L_n| | ⁽ⁿ⁾ , and that, in particular, L₀| = h | . If is primary, L_n| = 0 for n 1. From the OPE (46) of T with itself follow the commutation relations for the L_n

(48)

which are known as the Virasoro algebra. The state | together with all its descendants, formed by acting on | an arbitrary number of times with the L_n with n −1, give a highest weight representation (where the weight is defined as the eigenvalue of −L₀).

There is another way of generating such a highest weight representation. Suppose the boundary conditions on the negative and positive real axes are both conformal, that is they satisfy , but they are different. The vacuum with these boundary conditions gives a highest weight state which it is sometimes useful to think of as corresponding to the insertion of a ‘boundary condition changing' (bcc) operator at the origin. An example is the continuum limit of an Ising model in which the spins on the negative real axis are −1, and those on the positive axis are +1.