The String Coffee Table

SLE is an approach to conformal field theory which emphasises CFT’s roots in statistical physics. 2-dimensional systems at their critical points exhibit scale invariance in that clusters of phases (e.g. clusters with all spins up or down in the Ising model) form fractal patterns. The main object of interest in the SLE approach to CFT are the wiggly boundary lines between these domains and their statistics.

The key tool for analyzing these domain walls is known as the Loewner differential equation – which gives SLE its name. This equation shows up as follows.

Consider a CFT with boundary. Fix a point on the boundary where the boundary conditions change (i.e. where a boundary condition changing operator is inserted). It’s helpful to keep a simple model in mind, so let’s imagine the Ising model such that all spins on the boundary left to our point are oriented upwards and all those to the right are oriented downwards.

For definiteness, map the CFT to the upper half plane $\mathbb{H}$ and let the special point be the origin.

Imagine a given configuration of the CFT, like an assignment of a spin value to each point, respecting the boundary conditions. Then there is a curve starting at the origin such that it has states of one sort (spin up, say) to its left and states of the other sort to its right.

It makes a difference if this curve is assumed to return to the real axis (this case is called chordal SLE) or not (called radial SLE), but that shall not concern me here.

Assume this curve is parameterized by a parameter $t \in [0,T)$. The heuristic idea is that, since crossing this curve is similar to crossing the origin where the boundary conditions change, we might want to perform a conformal mapping from the UHP without the curve to the full UHP such that the curve actually becomes part of the boundary.

Think of the curve as a zipper in the upper half plane. Now unzip this zipper.

So imagine cutting the upper half plane along that curve and then deforming it such that the point at parameter value $t$ ends up on the real axis with the states originally left of the curve now assembled to the left along the real axis and those formerly to the right aligned to the right along real axis.

By the Riemann mapping theorem there is indeed a conformal map $f_t$ accomplishing this. If a suitable normalization condition is imposed this map is even unique. Hence the conformal maps $f_t$ uniquely characterize the domain wall curves we started with. Instead of studying these curves, SLE studies the conformal maps that unzip them.

The fact that the domain wall curves exhibit what is called local growth with respect to the parameter $t$ is expressed by a linear differential equation satisfied by the corresponding unzipping map $f_t$. This equation is the Loewner differential equation and it reads

for some real function $\xi : \mathbb{R} \to \mathbb{R}$ determined by our curve. This real function gives the position on the real axis to which the point at parameter value $t$ of the domain wall curve is mapped.

Schramm showed that if the zipper curves are distributed according to a measure which is conformally invariant (which is the case when these curves are domain walls in a CFT) then $\xi_t$ is distributed like Brownian motion on the real line!

Somehow Brownian motion in one dimension knows about conformal invariance in two dimensions. The diffusion constant $\kappa$ of this Brownian motion depends on the conformal charge of the CFT. This can be seen as follows.

Let $B_t$ be Brownian motion with unit diffusion constant, then

by Schramm’s theorem.

Introduce the abbreviation

for the denominator appearing in the stochastic Loewner equation. In terms of stochastic differential calculus (where $dt dt = 0 = dt d\xi = d\xi dt$ but $d\xi d\xi = \kappa\, dt$) one finds that

The stochastic differential of a function $F$ on the upper half plane which is composed with $k_t$ is hence

Staring at this long enough (or, alternatively, looking at page 3 of hep-th/0309080) one finally sees the Virasoro generators appearing here in their $c=0$ incarnation $L_n = -z^{n+1}\partial_z$. Namely the operator $G_{k_t}$ inducing the map defined by $k_t$ satisfies

Hence for any state $|\omega\rangle$, the state $L_{-1}|\omega\rangle$ is a measure for the diffusion of $|\omega\rangle$ under the stochastic Loewner evolution induced by unzipping a domain wall curve, while $(-2 L_{-1} + \frac{\kappa}{2}L_{-1}^2)|\omega\rangle$ measures the drift.

It is of interest to consider those states that don’t drift at all under the Loewner evolution. This are those annihilated by $-2 L_{-1} + \frac{\kappa}{2}L_{-1}^2$. This again is true for highest weight states in a heighest weight representation with central charge

and conformal weight

This means the following: If the change in boundary conditions that we considered is induced by a primary operator of this special weight $h_\kappa$, then this operator does not change on average (since it has vanishing drift) under the unzipping of the domain wall and hence any correlation functions we might be computing with this operator inserted don’t change either.

That shall suffice for now. If you believe I have screwed up something in the above, check the details in hep-th/0309080.