Spacetime Representation Theory: Setting the Scope of the ISE Method of Topological Redescription

Abstract: Spacetime dualities arise whenever two theories -- despite being structurally equivalent in some sense -- seemingly provide us with two radically different spatiotemporal descriptions of the world. This often involves radical differences in how the two theories topologically stage their states; Whereas one theory is about this type of particle/field on this smooth manifold, the other theory is about that type of particle/field arranged differently on that smooth manifold. For instance, the AdS-CFT correspondence relates a certain theory set in the bulk (our 3+1 dimensional spacetime) to another theory set on the boundary (a 2+1 dimensional spacetime). Another example (new in this paper) is the M\"{o}bius-Euclid duality: a theory about a certain type of particle floating around on the Euclidean plane can be topologically redescribed as instead being about a different type of particle living on a M\"{o}bius strip, and vice versa. The possibility of such alternative spacetime framings raises some significant questions about the epistemology and metaphysics of space and time. For instance, what are our topology selection criteria? Are they objective or conventional? Moreover, given that two spacetime theories are topological redescriptions of each other, what is the common core which they are equivalent descriptions of? As a step towards answering such questions, this paper develops a general framework (spacetime representation theory) for understanding our ability to topologically redescribe our spacetime theories. With this framework established, I will then discuss the ISE Equivalence Theorem which sets the scope of the recently developed ISE Method of topological redescription.