Dirichlet branes, or their dual heterotic fivebranes and Horava-Witten walls – can trap non-abelian gauge interactions in their worldvolumes. This has placed on a firmer basis an old idea, according to which we might be living on a brane embedded in a higher-dimensional world. The idea arises naturally in compactifications of type I theory, which typically involve collections of orientifold planes and D-branes. The ‘brane-world’ scenario admits a fully perturbative string description.

In type I string theory the graviton (a closed-string state) lives in the ten-dimensional bulk, while open-string vector bosons are in general localized on lower-dimensional D-branes. Furthermore while closed strings interact to leading order via the sphere diagram, open strings interact via the disk diagram which is of higher order in the genus expansion. The four-dimensional Planck mass and Yang-Mills couplings therefore take the form

α_{U} ∼ g_{I}/(r˜M_{I})^{6-n}

M^{2}_{Planck} ∼ r^{n}r˜^{6-n}M^{8}_{I}/g^{2}

where r is the typical radius of the n compact dimensions transverse to the brane, f the typical radius of the remaining (6-n) compact longitudinal dimensions, M_{I} the type-I string scale and g_{I} the string coupling constant. By appropriate T-dualities we can again ensure that both r and r˜ are greater than or equal to the fundamental string scale. T- dualities change n and may take us either to Ia or to Ib theory (also called I or I’, respectively).

It follows from these formulae that (a) there is no universal relation between M_{Planck}, α_{U }and M_{I} anymore, and (b) tree-level gauge couplings corresponding to different sets of D-branes have radius-dependent ratios and need not unify at all. Thus type-I string theory is much more flexible (and less predictive) than its heterotic counterpart. The fundamental string scale, M_{I}, in particular is a free parameter, even if one insists that α_{U} be kept fixed and of order one, and that the string theory be weakly coupled. This added flexibility can be used to ‘remove’ the order-of magnitude discrepancy between the apparent unification and string scales of the heterotic theory, to lower M_{I} to an intemediate scale or even all the way down to its experimentally-allowed limit of order the TeV. Keeping for instance g_{I}, α_{U } and r˜M_{I} fixed and of order one, leads to the condition

rn ∼ M^{2}_{Planck}/M^{2+n}_{I}

A TeV string scale would then require from n = 2 millimetric to n = 6 fermi-size dimensions transverse to our brane world. The relative weakness of gravity is in this picture attributed to the large transverse spreading of the gravitational flux.

I am just a layman and I remember reading about the Weak Gravity Conjecture, which… do I have this right… says that our current physics would become inconsistent if gravity became much stronger, so there is a reason why we find gravity so weak. Well that made the Braneworld less appealing to me, bc I liked that it ‘explains’ the weakness of gravity but now it doesn’t seem like it needs to. Does it make sense to ask if the WGC affects the Braneworld theories?

No, this question couldn’t be a layman’s question. Thanks for quite a probing one. I think, WGC affects branes. In what sense, do I say this? General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity “leaks” into the bulk, behaving in a truly higher-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting and potentially testable implications for high-energy cosmology. Consequently, Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. So, the importance cannot be undermined.