Differential equations

Date: 2015-10-07; view: 476.

In this section, we discuss how the linear second order differential equations for various observables which arise from the stochastic aspect of SLE follow equivalently from the null state condition in CFT. In this context they are known as the BPZ equations [33]. As an example consider Schramm's formula (30) for the probability P that a point ζ lies to the right of γ, or equivalently the expectation value of the indicator function which is 1 if this is satisfied and zero otherwise. In SLE, this expectation value is with respect to the measure on curves which connect the point a₀ to infinity. In CFT, as explained above, we can only consider curves which intersect some -neighbourhood on the real axis. Therefore P should be written as a ratio of expectation values with respect to the CFT measure

(58)

We can derive differential equations for the correlators in the numerator and denominator by inserting into each of them a factor (1/(2πi)∫_Γα (z)T (z) dz + c.c., where α (z) = 2/(z−a₀), and Γ is a small semicircle surrounding a₀. This is equivalent to making the infinitesimal transformation z → z + 2 /(z − a₀). As before, the c.c. term is equivalent to extending the contour in the first term to a full circle. The effect of this insertion may be evaluated in two ways: by shrinking the contour onto a₀ and using the OPE between T and _2,1 we get

(59)

while wrapping it about ζ (in a clockwise sense) we get

(60)

The effect on _2,1 (r₂) vanishes in the limit r₂ → ∞. As a result we can ignore the variation of the denominator in this case. Equating Figs. (59) and (60) inside the correlation function in the numerator then leads to the differential equation (29) for P found in Section 4.1.