The Spinorial Geometry Programme

Well it's been a quiet week on the seminar front. It also has seemed, at least to me, like it's been a quieter than usual last few months for new hep-th papers on the arxiv. There was one monday earlier this month, admittedly a holiday, when there was only one new paper, hep-th/0504234. It's not too easy to check at the hep-th level but you can look at the overall arxiv submission rate and see that since its beginning the submission rate has grown almost linearly. Unlike our exponential world population growth. Perhaps fewer scientists are being born! Also on the graph you can see that there is often a drop in submission rate going from December to January in consecutive years. Could there be a more objective reason than a desire to finish off work as the year ends, and then begin new research as the new year commences? There's also a frequent drop around July, when teaching slows down and summer is high, and presumably scientists travel to conferences, and other universities. Now if you put a straight edge along the graph the difference in month to month rate changes becomes most pronounced in the last few years, and perhaps, if anything, submission rate is beginning to level fall below the historical linear increase rate. We shall see in the next few years.

However submission rate is not in decline at KCL. That sole submission on the arxiv, that I referred to earlier, was written in part by Thomas Quella, a post-doc here at KCL. Also we are in the middle of a series of publications by George Papadopoulos et al. from KCL as part of his spinorial geometry programme aimed classifying all supersymmetric solutions, via his simplifying formalism.

The first spinorial geometry paper, "The spinorial geometry of supersymmetric backgrounds" by Joe Gillard, Ulf Gran and George Papadopoulos, appeared on the arxiv in October of last year. Since then there have been a series of three follow up papers, the latest one appearing on the arxiv this week:

"The spinorial geometry of supersymmetric IIB backgrounds" by U. Gran, J. Gutowski and G. Papadopoulos

"Systematics of M-theory spinorial geometry" by U. Gran, G. Papadopoulos and D. Roest

"The G_2 spinorial geometry of supersymmetric IIB backgrounds" by U. Gran, J. Gutowski and G. Papadopoulos

The motivating idea is that the killing spinor equations of any supergravity theory (set \lambda in the general killing spinor equation, given in the last link, to zero) can be solved for an arbitrary number of killing spinors in a systematic way by making use of the fact that the supercovariant derivative has a Spin group gauge symmetry (e.g. a Spin(1,10) gauge symmetry for the case of eleven dimensional supergravity - in passing we refresh our memory and recall that the Spin(t,s) group is the double cover of SO(t,s)). We won't get to the Killing equations in this post, instead we will focus on the stability group of spinors. Spinors can be classified by their stability subgroup, which is the subgroup of the spin group that leaves the spinor invariant. This offers a systematic way to study killing spinors.

At the start of the paper by Gillard et al. a recipe is given to write the spinors of supergravity, i.e. elements of a vector space which transform under Spin(1,10), as forms. The recipe commences by considering representations of Spin(10) instead of Spin(1,10). Complex/Dirac spinors in Spin(10) have 32-components and decompose into two 16-component irreducible representations. Mirroring this, the basis of our vector space, R^10 is split into two parts, each part spanning one copy of R^5. The two copies of R^5 are then related by some complex structure, identifying vectors in each R^5. So that the full vector space is determined from the orthonormal basis vectors of the "first" five coordinates, e.g. {e1,e2,e3,e4,e5}. Spinors are represented in the basis of forms, i.e. if S is a spinor, then it is expressed in the basis of 0-forms, 1-forms,...5-forms as S=lI+mA(eA)+[nAnB](eA^eB)+...[qAqBqCqDqE](eA^eB^eC^eD^eE). My apologies for the poor notation here, one day I will get mathplayer or some equivalent. This gives a canonical way to write Spin(10) spinors as forms. The action of the ten gamma matrices on the basis of forms is given in the paper and then gamma_0 is constructed (raising the field of play to Spin(1,10)) and the Majorana condition applied to the spinors.

Having written a spinor in this manner, use is made of a theorem concerning representations of SU(n) carried by the basis forms. I don't know the theorem used but it results in a statement similar to "SU(n) irreducible representations are carried by p-forms on C^n".

Additional comment: A theorem has that makes some inroads into understanding this step, albeit over a real vector space, is given in "Compact Manifolds with Special Holonomy" by Dominic Joyce, (Prop. 3.5.1) has the consequence that for a Lie subgroup, G, of GL(n,R), the bundle of p-forms over R^n splits into irreducible sub-bundles on which irreps of G act. [Thanks again Joe!]

This leads us not just to have irreps of SU(n), but also any Lie group. But this is the real case, so perhaps over the complex space, the reason for specialising to SU(n) irreps will become clear. Any explanations/corrections are welcome :)

If this holds then all the basis p-forms, that have been constructed are over C^5, and so carry an irrep of SU(5). The dimension of these irreps is (5 choose p), so that the zero-form and the five-form both carry one-dimensional irreps of SU(5). The dimension tells us how the irrep transforms under SU(5), in particular how many indices it transforms under: each index gives five degrees of freedom of the irrep. So that one dimensional irreps have zero indices and transform trivially as scalars under SU(5), five-dimensional irreps have one index and transform as a vector and ten-dimensional irreps have two indices and transform as skew-symmetric matrices (as the indices are still coupled to the form indices, so the symmetric part is zero). Both the zero-forms and the five-forms of the spinor basis transform trivially under SU(5), so to find spinors with stability group SU(5) we form our spinors out of the 1 and e1^e2^e3^e4^e5 basis elements. By imposing the Majorana condition on this general spinor we learn that there are only two linearly independent spinors with stability group SU(5). Which is rather neat.

I have to admit I had lots of difficulties even reading through the beginning of the first paper, but thanks to my good friend J. J. Gillard, who talked me through this stuff no less than five times, many of my difficulties have been overcome. If the increase rate of submissions to the archive is indeed slowing down perhaps it's because potentially prolific scientists everywhere are being distracted by people like me asking them to explain their work again, and again... :)