## April 9, 2005

### Lessons from low Dimensions

#### Posted by Urs Schreiber

I am currently visting a CFT workshop at University of Bonn,

Of the many talks, here I would like to mention aspects of two of them:

A. Zamolodchikov from Rutgers talked about

CFT with boundary interactions: Integrable ‘brane’ models

He discussed a linear 2-dimensional $\sigma$-model with target ${ℝ}^{2}$ and with boundary fixed on some closed curve in ${ℝ}^{2}$. Even though he loosely called this a ‘brane’ he did not require conformal invariance on the boundary with the consequence that there is a boundary RG flow which changes the shape of that curve. At least under certain conditions one can hence think of this model as ‘interpolating’ between two different conformal boundary conditions, one obtained when the curve shrinks to a point, the other one when it tends to a circle of infinite radius.

Next he wanted to find curves which give what is called integrable boundary conditions.

Any boundary condition on an open string can be represented (by a simple extension of the ‘image charge’ method known from electrostatics) by a certain off-shell state $\mid B〉$ of the closed string, called the boundary state.

Let $B$ be the operator which creates this state from the vacuum

(1)$B\mid 0〉=\mid B〉$

and which is purely left-moving (say). Then one calls this boundary state integrable if (like for an integrable Hamiltonian) there is an infinite set $\left\{{I}_{s}{\right\}}_{s\in ℕ}$ of (left-moving) operators on the string’s Hilbert space which all commute with $B$ and among themselves.

(2)$\left[{I}_{s},B\right]=0=\left[{I}_{s},{I}_{s\prime }\right]\phantom{\rule{thinmathspace}{0ex}}.$

This can be reformulated in a maybe more familiar looking way in the boundary state context by letting $\left\{{\overline{I}}_{s}\right\}$ be a family of right-moving operators such that

(3)${I}_{s}\mid B〉={\overline{I}}_{-s}\mid B〉\phantom{\rule{thinmathspace}{0ex}}.$

For the model under discussion, integrable boundary states can be obtained by combining ‘hairpin’ curves - which go one direction, make a sharp turn and return parallel to themselves - and straight lines. One simple thing you get from that is the result of gluing two hairpins. This looks like a stadium but is called the paper clip.

(I suspect the reason for that is just that in case this model ever turns out to play a role in string theory journalists may report that ‘String theorists have finally found their paperclip.’)

Since this boundary condition is integrable it naturally can be analyzed in some detail. It turns out that there is a big theorem which says that the partition function for the paperclip is proportional to the Wronskian of certain two solutions of a certain time-independent Schrödinger equation with a complicated but analytic potential. But the proof doesn’t tell you what the appearance of this Schrödinger equation here means.

If you find out, drop Prof. Zamolodchikov a note.

Then I feel like mentioning a talk by Ingo Runkel from the AEI about

Defects and dualities in conformal field theory

This is part of a huge body of work that has grown out of Witten’s observation that Wilson lines in Chern-Simons theory with boundary compute conformal blocks of a CFT on that boundary. Notably Christoph Schweigert has worked this out in a long series of long papers with a couple of other people (which I won’t try to list because I am sure to forget some of them, hep-th/0503194 for the latest), which are a treat to look at if you like diagrammatic reasoning (3-dimensional and in color!).

In this context one finds of interest a certain generalization of the ordinary D-brane concept, if you wish. So given a CFT with boundary we can impose the condition

(4)$T\left(x\right)-\overline{T}\left(x\right)=0$

for any point $x$ on the boundary. This gives a D-brane.

But a more general possibility is to attach a second CFT with boundary and demand that now

(5)${T}_{1}\left(x\right)-{\overline{T}}_{1}\left(x\right)={T}_{2}\left(x\right)-{\overline{T}}_{2}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$

This recovers the above as a special case for instance when the second CFT is trivial or when

(6)${T}_{2}\left(x\right)-{\overline{T}}_{2}\left(x\right)=0$

itself. This is called the reflective case and just produces two CFTs sitting on the same D-brane, in a sense.

The interesting stuff starts when both CFTs are allowed to mix, like when we solve the above condition by decreeing that

(7)${T}_{1}\left(x\right)={T}_{2}\left(x\right)$
(8)${\overline{T}}_{1}\left(x\right)={\overline{T}}_{2}\left(x\right)\phantom{\rule{thinmathspace}{0ex}}.$

This is called the transmissive case, since now the two ‘strings’ exchange their energy and momentum across the defect line.

It may happen that all correlators are independent of the exact location and shape of the defect line. Then it is called a topological conformal defect and useful for a host of cute magic.

This does actually arise in the Ising model, where it is related to the Kramers-Wannier duality that exchanges low and high temperature behaviour. Here the defect line correspond to lines of edges on which the ising coupling is inverted.

By the TFT-CFT relation all this lifts to analogous statement in three dimensions. By moving Wilson lines and defect Wilson lines around each other in various ways one can prove various nice statements about correlation functions with just a little diagrammatic reasoning.

Does anyone know if these defect lines have any physical interpretation for the case of the fundamental string?

Posted at April 9, 2005 5:32 PM UTC

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