Reflection on this thought experiment shows that, however much we know about what has happened to my brain—we may know exactly what has happened to every atom in it—and to every other material part of me, we do not necessarily know what has happened to me. From that it follows that there must be more to me than the matter of which my body and brain are made, a further essential immaterial part whose continuing in existence makes the brain (and so the body) to which it is connected my brain (and body), and to this something I give the traditional name of ’soul’. I am my soul plus whatever brain (and body) it is connected to. Normally my soul goes where my brain goes, but in unusual circumstances (such as when my brain is split) it is uncertain where it goes.
I prefer Parfit's solution, which is to deny that there is any further fact here to know. The physical and psychological facts exhaust the facts. Once those are all specified, there is nothing left to know about the world. It would seem strange to posit two possibilities, alike in every objective and subjective respect, yet somehow differing in virtue of the "identity facts". For what would those consist in, and how could we ever grasp them? Our commonsense concept of identity tracks a familiar kind of continuity, but we have no reason to think there's anything further underlying it.