# A Twisted boundary conditions for 2d CFT entanglement entropy

1. Dec 24, 2016

### ShayanJ

I'm trying to read this paper. Right now my problem is with equations 3.16 and 3.17.
I understand that in equation 3.16 we're putting some boundary conditions on the fields, but I have two problems with these boundary conditions:

1) The fields depend on both $t_E$ and $x$, i.e. $\phi(t_E,x)$. But writing the fields as $\phi(w-u)=\phi(x+it_E-u)=\phi( \ (x-u)+it_E \ )$, means that its assumed that the dependence of the fields on $t_E$ and $x$ is only through the combination $x+it_E$. If a general dependence was assumed, the author should have used $\phi(w,\bar w)$. So what is it? Is it really an assumption about the form of dependence on $t_E$ and $x$? If yes, where does it come from? Or maybe its just the author being careless?

2) Also I understand that these boundary conditions just mean that like going to the next Riemann sheet, you should first go around a complete loop around the branch point. But I don't understand why this should be the boundary condition. We're not going around a loop with those path integrals, we're going from $t_E=-\infty$ to $t_E=\infty$. So how come a loop is what we consider for the boundary conditions?

My problem with equation 3.17, is that how exactly can we convert the effect of those twisted boundary condition to calculating the two-point function of the "twist operator"? What is that twist operator anyway? How is it defined?

I'd appreciate any comment.
Thanks

2. Dec 29, 2016