In "M-theory on manifolds of $G_2$ holonomy: the first twenty years" by Duff, it is claimed (e.g. in section 8) that for compactification on singular 7-folds to be possible, we need to consider not the 11D supergravity (SUGRA) approximation to M-theory but "full M-theory". Such singular compactifications are desirable due to the absence of chiral matter in smooth 7-fold compactifications.

In contrast, many publications on M-theory compactified on 7-folds seem to just do Kaluza-Klein reduction of 11D SUGRA on the singular 7-folds, not considering "full M-theory" (as far as I am concerned, the M2- and M5-branes are part of 11D SUGRA as solitonic objects, maybe I'm wrong/non-standard with that view?). One example of this is "On gauge enhancemenet and singular limits in $G_2$ compactifications of M-theory" by Halverson and Morrison, where no "full" M-theory is in sight as far as I can see. There are many other such papers where the SUGRA approximation is the essential starting point for the Kaluza-Klein reductions.

So what, exactly, is meant by Duff's remark that singular compactifications are only possible for "full M-theory"? In what way does this compactification of "full M-theory" differ from a standard Kaluza-Klein reduction, and how does it allow for singular compactifications while 11D SUGRA only allows for smooth compactifications?

  • $\begingroup$ I guess you have to read the following review. iopscience.iop.org/article/10.1088/0264-9381/19/22/301/meta. It is mentioned that supergravity approximation is not valid near singularities for some reason because otherwise it would not yield chiral fermions. $\endgroup$
    – ved
    Dec 5, 2016 at 9:00
  • $\begingroup$ @ved Hm, the only thing I can see there would be indeed the wrapping of the M2-brane which would be "M-theory", but as I already said in the question, this brane also occurs as a solitonic object in the SUGRA theory, so I'm still confused what "full M-theory" means here. $\endgroup$
    – ACuriousMind
    Dec 5, 2016 at 17:53
  • 2
    $\begingroup$ There is an answer at physicsoverflow.org/37968 $\endgroup$ Dec 12, 2016 at 11:30

1 Answer 1


Although membranes are described as solitons in supergravity, their dynamics cannot be. Their fluctuation degrees of freedom are fundamental strings or other kinds of membranes that supergravity cannot fully catch.

The simplest example is a string wrapping or winding a circle of the critical radius, which enhances the gauge group to non-Abelian. The corresponding gauge boson cannot be obtained from the Kaluza-Klein reduction. Essentially the same happens.

It is well-known that a Dirichlet brane (D-brane) describes $U(1)$ gauge group: There is a massless open string whose both ends are ending on the same brane, giving a massless gauge boson. When two D-branes are coincident, then another string "stretched" between different branes gives a massless gauge boson charged under two different gauge groups. Gauge symmetry is enhanced.

In compactification, the singular geometry mostly means the singular limit of "good enough" curved space, like so-called resolved $A-D-E$ singularity. Its geometry is chain of pointwise intersecting cycles, or a series of spheres touching each other at points. Although they are curved, essentially the same happens if we replace (1) the above brane position as the intersection point of spheres and (2) the above string with membrane like D3-brane. As the sphere wrapped by D3-brane shrinks to zero volume, we have light gauge bosons charged under the gauge group at the intersection. Note that M-theory is required to describe the charged gauge bosons; the neutral bosons can be obtained by Kaluza-Klein reduction in conventional supergravity.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.