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Now I mentioned before that I am chiefly interested in #GeneralizedComplex geometry underlying all this: All universes that exhibit #MirrorSymmetry are GC, and #branes are GC objects.
However, there is currently no general version for a "GC category of branes". Part of my current work is to define this in certain cases.

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The extremely simple example I showed earlier with the lines inside a circle is one extremely simple version of a #category of #branes: The disc with the circular boundary is the "universe" and the lines are the branes. The interaction between the branes is given by the cohomology associated to the intersection points. Because of its low dimension, this example is much less complex for higher dimensional universes like our own (which in string theory is 10-dim!).

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It turns out that #MirrorSymmetry can be described in terms of #branes, which I have described before: For each of the two different mirror geometries, we can define a #category of branes (again, this is very simplified and not exactly right), which is a mathematical structure encoding not only the branes themselves, but also in some sense their interactions. Two universes are mirror if the categories of branes are equivalent in a certain way.

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Several of my research projects involve #branes: This is also originally a string theory concept. Again, very simplified: A brane is something like a higher-dimensional version of a string (e.g. a plane or a 3d object inside a higher-dimensional universe), and it is also something strings can begin and end on.