## November 15, 2005

### CFT and SLE

#### Posted by Urs Schreiber

Just heard a talk by Roland Friedrich on stochastic Loewner evolution (SLE) and its relation to CFT. This is the same topic that John Cardy talked about recently under the title a new way of thinking about conformal field theory. Robert had reported some key ideas mentioned in Cardy’s talk here. I want to better understand this stuff. Here are some elementary notes.

The slides of John Cardy’s talk are available online. A useful recent review is

D. Bernard
Conformal Field Theories in Random Domains and Stochastic Loewner Evolutions
hep-th/0309080

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 $ℍ$ 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 $t\in \left[0,T\right)$. 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

(1)$\frac{\partial }{\partial t}{f}_{t}\left(z\right)=\frac{2}{{f}_{t}\left(z\right)-{\xi }_{t}}$

for some real function $\xi :ℝ\to ℝ$ 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

(2)${\xi }_{t}=\sqrt{\kappa }{B}_{t}$

by Schramm’s theorem.

Introduce the abbreviation

(3)${k}_{t}\left(z\right):={f}_{t}\left(z\right)-{\xi }_{t}$

for the denominator appearing in the stochastic Loewner equation. In terms of stochastic differential calculus (where $\mathrm{dt}\mathrm{dt}=0=\mathrm{dt}d\xi =d\xi \mathrm{dt}$ but $d\xi d\xi =\kappa \phantom{\rule{thinmathspace}{0ex}}\mathrm{dt}$) one finds that

(4)$d{k}_{t}=\frac{2}{{k}_{t}}\mathrm{dt}-d{\xi }_{t}\phantom{\rule{thinmathspace}{0ex}}.$

The stochastic differential of a function $F$ on the upper half plane which is composed with ${k}_{t}$ is hence

(5)$\begin{array}{ccc}dF\left({k}_{t}\right)& =& F\prime \left({k}_{t}\right){\mathrm{dk}}_{t}+\frac{1}{2}F″\left({k}_{t}\right){\mathrm{dk}}_{t}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dk}}_{t}\\ & =& F\prime \left({k}_{t}\right)\left(\frac{2}{{k}_{t}}\mathrm{dt}-d{\xi }_{t}\right)+\frac{\kappa }{2}F″\left({k}_{t}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dt}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$

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

(6)${G}_{{k}_{t}}^{-1}{\mathrm{dG}}_{{k}_{t}}=\left(-2{L}_{-1}+\frac{\kappa }{2}{L}_{-1}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{dt}+{L}_{-1}\phantom{\rule{thinmathspace}{0ex}}d{\xi }_{t}\phantom{\rule{thinmathspace}{0ex}}.$

Hence for any state $\mid \omega ⟩$, the state ${L}_{-1}\mid \omega ⟩$ is a measure for the diffusion of $\mid \omega ⟩$ under the stochastic Loewner evolution induced by unzipping a domain wall curve, while $\left(-2{L}_{-1}+\frac{\kappa }{2}{L}_{-1}^{2}\right)\mid \omega ⟩$ 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

(7)${c}_{\kappa }=\frac{\left(6-\kappa \right)\left(3\kappa -8\right)}{2\kappa }$

and conformal weight

(8)${h}_{\kappa }=\frac{6-\kappa }{2\kappa }\phantom{\rule{thinmathspace}{0ex}}.$

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.

Posted at November 15, 2005 6:45 PM UTC

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### Re: CFT and SLE

*cough*

Do you remember a certain chapter in our paper? Do you remember what it was that initially got you remotely interested in discrete geometry? :)

Posted by: Eric on November 17, 2005 1:35 AM | Permalink | Reply to this

### stochastic differential calculus

*cough*

Hi Eric,

yes, I remember. Everything under the sun is mutually related…

For those not knowing what you are refrring to:

Eric is alluding to the fact that there are graphs (binary trees, in fact) such that the algebra of discrete differential forms on them (which is a deformation of the ordinary algebra of differential forms in the continuum) automatically obeys those funny rules of stochastic differential calculus.

This, together with an application to mathematical finance Eric describes here.

Posted by: Urs on November 17, 2005 1:18 PM | Permalink | Reply to this

### Re: stochastic differential calculus

Hi Urs :)

I didn’t point out the relation to our paper merely to note that the Ito formula appears in both. I’ve got nothing more than a gut feeling, but at least that is something, that by trying to formulate something like CFT or SLE in a fundamentally discrete framework might tell you things even prior to taking the continuum limit. Is it possible that the stuff Thomas is referring to may be explored using discrete loop (polymer) space geometry?

It seems to me that you should be able to formulate the important questions completely rigorously within a discrete context. Then the continuum limit becomes a curiosity at best because you’ve already learn most of what you could learn in the discrete world.

Just another random thought…

Best regards,
Eric

Posted by: Eric on November 22, 2005 4:01 PM | Permalink | Reply to this

### Re: CFT and SLE

My first reaction when I heard about SLE (which most people except Oded Schramm pronounce Schramm-Loewner evolution) some years ago, was that it must be equivalent to CFT, something that has been investigated and verified by several people in recent years. After all, CFT says essentially everything worth knowing about 2D phase transitions, and so does SLE in a geometrical language. The two approaches also have the same major limitation, namely that they do not readily generalize to the physically more interesting 3D case, since they depend crucially on the infinite conformal group in 2D.

The idea that one can study statistical models in terms of domain walls is not really new: I believe that one reason why Polyakov invented his action in 1980 was that he wanted to apply it to domain walls in the 3D Ising model. However, nothing has come out of that in terms of exact information about 3D Ising. The important thing about SLE is that it can be effectively used for computations.

### Re: CFT and SLE

The important thing about SLE is that it can be effectively used for computations.

I am being told that some statements are easier to prove (rigorously) using SLE than standard CFT (and vice versa).

It seems (but please correct me if I am wrong) that the main thing about SLE currently is that it is fun to see how CFT can be addressed using stochastic and functional analytic tools.

But I am also being told that the hope is that current work being done on a generalization of SLE to more than one single domain wall line would yield a measure on the space of ‘cluster configurations’ (of the Ising model, say) that promises to provide a way to rigorously define the path integral for these models.

Apparently, handling conformal 2D sigma models (instead of just things like Ising or Potts) the SLE way is possible but requires more technology, involving bundles over the worldsheet. Somehow. Don’t know the details.

Posted by: Urs on November 17, 2005 1:02 PM | Permalink | Reply to this

### Re: CFT and SLE

I am being told that some statements are easier to prove (rigorously) using SLE than standard CFT (and vice versa).

I don’t disagree with this. What I meant is that you can compute most things of interest in 2D, using either CFT or SLE. Either formalism has it strengths and weaknesses, but the overlap is so great that they must be equivalent, more or less. In contrast, nothing is known exactly in 3D, although approximate results are of course abundant. One could study fluctuating domain walls in 3D as well, but you cannot map them onto some simple geometry using conformal transformations.

However, I have always believed that there is some algebraic structure in higher dimensions which has the same predictive power as CFT has in 2D. The reason is universality. CFT nicely explains universality in 2D, by identifying universality classes with representations of chiral algebras. But universality is not limited to 2D, and one may believe that higher-dimensional universality has a similar explanation as 2D universality.