Instantons in type IIA theory and correlators

[hines]

Hi,
in type IIA theory it is possible to compute correlation functions by infinite sums over instanton areas $$\sum e^{-Area}$$. Wouldn't that mean that it is possible to get two-point correlators? Imagine a torus with one brane on each of the two lattice vectors. They bound a parallelogram which is identical to the torus covering space. Is there anything that prevents us from summing up these areas to a valid correlator coming from just two branes?
Thank you,
hines

 

[suprised]

Yes there are configurations like that but this has nothing to do with 2-point correlators unless you insert vertex operators on the torus and/or on the branes. The two branes are not fields in a correlator but define the chosen background configuration. You may leave them out as well, then you have a torus around which the "unique" (up to multicovers) genus one closed string instanton can wrap. If you add branes then there are many more possibilities for open string instantons, and all need to be summed up to get the complete result.

 

[hines]

thanks for the reply. okay, so clearly it's not a 2-point function, my bad, but could there be a 4-point function with two different field insertions only like <\Psi_1 \Psi_2 \Psi_1 \Psi_2>? From your reply I understand that these two branes alone do not bound an open string instanton and therefore this correlator would vanish, correct?

 

[suprised]

From your reply I understand that these two branes alone do not bound an open string instanton and therefore this correlator would vanish, correct?

Degenerate instantons can be quite subtle. For example, it is known that open string instantons can fuse their boundaries so start to look like closed string instantons. However, typically there will be a divergence coming from the coinciding branes and therefore a correlator must be regularized in order to make it well-defined. The configuration you consider is however not the limiting case where four branes coincide pairwise, but only two branes, so I don't think it constitutes a well-defined open string correlator.

 

[hines]

Thank you~