Twisted boundary conditions for 2d CFT entanglement entropy

[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

 

[AndyW]

for your post. I understand your confusion with equations 3.16 and 3.17. Let me try to clarify.

1) The dependence on $t_E$ and $x$ is only through the combination $x+it_E$ because the fields are being analytically continued to the complex plane. This is a common technique in physics, where we often extend our equations to the complex plane in order to better understand their behavior. The author is not being careless, but rather using a well-established technique.

2) The boundary conditions are chosen based on the physical behavior we are trying to model. In this case, the loop around the branch point represents the physical process of going from $t_E=-\infty$ to $t_E=\infty$. By requiring the fields to satisfy the boundary conditions, we are ensuring that our model accurately captures this physical process.

As for equation 3.17, the twist operator is a mathematical object that helps us understand the effects of the twisted boundary conditions on the two-point function. It is defined as a specific type of operator that acts on the fields in a particular way. You can think of it as a tool to help us solve the problem at hand.

I hope this helps clarify the concepts for you. If you have any further questions, please don't hesitate to ask.