# If QFT is a sum over 1-D topologies and String Theory over 2-D topologies, what is the corresponding theory for N-D topologies?

My understanding is that perturbative QFT can essentially be described as a weighted sum over 1-D topologies (ie Feynman graphs), and String theory is essentially the generalization to a sum over 2-D topologies. Why do we stop here? Is there a name for the theory defined as the sum over 3-D, or N-D topologies?

• Essentially a duplicate of physics.stackexchange.com/q/55431/2451 , physics.stackexchange.com/q/66948/2451 and links therein. Jun 13, 2014 at 16:54
• Ah, thanks, I hadn't found those before. They do basically ask the same question, although I would complain that their answers are too technical (if I was a string theorist I would know the answer already!) Jun 13, 2014 at 17:03

## 1 Answer

There is. They're all included in the name "string theory".

While it originally began as a theory of 1-dimensional strings, today it describe a quantum theory of many other $p$-dimensional (D$p$-branes, membranes, etc.). We just didn't bother finding another name for the theory. The key difference is that while a fundamental string is perturbative, the other objects are not (since they are very heavy)

• But I thought that p-branes emerged within string theory (which I'm defining is a theory of 1-dimensional strings and no larger), ie that there are dualities between stacks of strings and higher dimensional branes, and that therefore it isn't clear that these things represent the next logical step in the progression described in my post. Does the 1-D theory perturbatively "contain" through dualities the physics of all higher dimensional objects? Or are these branes really a separate theory altogether of higher dimensional objects? What connects the different dimensional theories? Jun 13, 2014 at 16:53
• Existence of D-branes can be perturbatively seen in string theory, essentially as objects that open strings can end on (with Dirichlet boundary conditions). Their dynamics can be seen more clearly sometimes in the context of T-duality (at the perturbative level) Of course, D-branes become immensely more interesting when treated non-perturbatively. Jun 13, 2014 at 20:56
• Compactifying and T-dualizing is often enough to take within various dimensional D-branes. S-duality takes from D-branes to other branes. For example, under an S-duality on IIB, D1 becomes F1 and D5 becomes NS5, etc. Jun 13, 2014 at 20:56