Boundary conditions in String Theory

[GargleBlast42]

I have this, probably quite simple, problem. In the RNS superstring, when varying the action, we obtain in general a term $$\int d\tau [X'_{\mu}\delta X^{\mu}|_{\sigma=\pi}-X'_{\mu}\delta X^{\mu}|_{\sigma=0} + (\psi_+ \delta \psi_+ - \psi_- \delta \psi_-)|_{\sigma=\pi}-(\psi_+ \delta \psi_+ - \psi_- \delta \psi_-)|_{\sigma=0}]$$, where the notation should be standard (as e.g. in Becker-Becker-Schwarz).

My question is the following: couldn't one also take other boundary conditions as those that one takes usually? For example, couldn't $$X^{\mu}$$ be anti-periodic (i.e. an antiperiodic closed string), or cuoldn't we take a boundary condition for the fermion in the form $$\delta \psi_+=\delta \psi_-=0$$? Can one show that there are no solutions to such boundary conditions (because nobody does that in a textbook)?

 

[GargleBlast42]

I'm sorry for bumping this, but I would at least like to know, if it is too difficult to answer (i.e. nobody has been considering such boundary conditions), or is it just that it's so obvious, that it's not worth replying to :-)? Thank you for any ideas.

 

[Naty1]

Much beyond my pay grade...boundary conditions can be rather esoteric...somebody may yet answer,

good luck...

 

[Ben Niehoff]

You cannot set $\delta \psi_+ = \delta \psi_- = 0$, because the entire point of varying the action is that the variations of the fields are generic (i.e., nonzero).

Also, $X^\mu$ cannot be antiperiodic, because it is just an ordinary number. Either the string is closed or it isn't. Remember that $X^\mu$ simply describes the string's position/configuration in spacetime.

 

[GargleBlast42]

You cannot set $\delta \psi_+ = \delta \psi_- = 0$, because the entire point of varying the action is that the variations of the fields are generic (i.e., nonzero).

Also, $X^\mu$ cannot be antiperiodic, because it is just an ordinary number. Either the string is closed or it isn't. Remember that $X^\mu$ simply describes the string's position/configuration in spacetime.

Sorry, I meant of course $\delta \psi_+|_{\sigma=0/\pi} = \delta \psi_-|_{\sigma=0/\pi} = 0$.

With the antiperiodicity - well yes, you couldn't interpret the X's as space-time dimensions, but from the point of view of the 2D Field Theory it would be okay, right?