Weakness of Gravity and Transverse Spreading of Gravitational Flux. Drunken Risibility.


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

αU ∼ gI/(r˜MI)6-n

M2Planck ∼ rn6-nM8I/g2

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, MI the type-I string scale and gI 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 MPlanck, αand MI 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, MI, in particular is a free parameter, even if one insists that αU 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 MI to an intemediate scale or even all the way down to its experimentally-allowed limit of order the TeV. Keeping for instance gI, α and r˜MI fixed and of order one, leads to the condition

rn ∼ M2Planck/M2+nI

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.