In compactifications of string theory, to preserve supersymmetry often branes need to wrap subspaces that are specified by holomorphic equations in the compactification space $X$. Additionally there exist complex structure deformations of $X$, which are often massless unless lifted by extra effects. These complex structure deformations alter which subspaces can be specified holomorphically, 'shifting the sand' underneath the branes. Hence to continue to preserve supersymmetry under a complex structure deformation, we must shift the branes together with the deformation, to track holomorphic subspaces.

But is this always possible? That is, are there cases where a brane locus cannot be continuously deformed to remain holomorphic under a complex structure deformation? I have not been able to find such examples in the literature. I would be grateful for a reference, or to hear a reason why such examples cannot occur.

  • $\begingroup$ I'm far from comfortable with these topics, but it seems to me that under a complex structure deformation, the complex submanifolds (representing cycles that the branes are supposed to be wrapped around?) will also be deformed, certainly continuously but probably even more nicely. You might have some parts of the base where certain cycles vanish as one approaches them, though those are usually of (complex) codimension $\geq 1$, hence one can always avoid them. $\endgroup$ – Danu Jan 6 '18 at 12:03
  • $\begingroup$ I also have to admit that I find this question pretty unclear---perhaps you could try to reword it and formulate your question more precisely? $\endgroup$ – Danu Jan 6 '18 at 12:09
  • $\begingroup$ @Danu Thanks for your comments. Can I ask which aspects you find most unclear, so that I can edit those? $\endgroup$ – diracula Jan 6 '18 at 14:02

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