Ishibashi (as in without the k)

Consider the annulus.

You can view it as a segment rotated around to meet up with itself. Supposing boundary condition a and b on the outside and inside, what you are doing is calculating the effect of two boundary changing operators at 0 and at infinity.

If you calculate the partition function you get some linear combination of characters over all levels h showing up with some coefficients n.



where  is the

But you could also thing of this same situation as a closed loop moving from the a side to the b side in imaginary time . In this picture we need to calculate the matrix element  

But what a and b make sense to put there? We have the condition     in the weak sense on this space.
Look at n=0, this says that the state lives in  with no cross terms.

Keep looking at the others and you see an explicit expression



equal combinations over everything allowed with that h.

a and b have to be a linear combination of these.



Now let's get back to that matrix element.



That last two pieces is the matrix element between the Ishibashi states. But we see the argument of the character is the victim of an S transformation, so we can rewrite it in terms of the character we had in the first picture.



These are called the Cardy conditions. It is an positive integer condition so it is restrictive. The operator content at the boundary is constrained to fit this.