Instantons in type IIA theory and correlators

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Discussion Overview

The discussion revolves around the computation of correlation functions in type IIA string theory, specifically focusing on the role of instantons and branes in defining these correlators. Participants explore the possibility of deriving two-point and four-point correlators from configurations involving branes on a torus.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that correlation functions can be computed using sums over instanton areas, proposing the potential for two-point correlators with branes on a torus.
  • Another participant counters that while configurations exist, they do not directly lead to two-point correlators unless vertex operators are involved, emphasizing that the branes define the background rather than serve as fields in the correlator.
  • A participant acknowledges the misunderstanding regarding two-point functions and inquires about the possibility of a four-point function with two different field insertions, questioning if the correlator would vanish due to the absence of open string instantons bound by the two branes.
  • In response, it is noted that while degenerate instantons can exhibit complex behaviors, the specific configuration discussed does not yield a well-defined open string correlator due to the lack of necessary conditions for instanton contributions.

Areas of Agreement / Disagreement

Participants generally agree that the configurations discussed do not lead to straightforward correlators without additional conditions, but there is no consensus on the implications for four-point functions or the nature of instanton contributions.

Contextual Notes

There are limitations regarding the definitions of correlators and the conditions under which instantons contribute, as well as the need for regularization in certain configurations.

hines
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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
 
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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.
 
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?
 
hines said:
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.
 
Thank you~
 

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