List of Compact 7 dimensional Einstein manifolds

[arivero]

The most recent version of the theorem, as stated by Nikonorov in 2004

Let G a connected compact semisimple Lie group, which acts almost effectively on a seven-dimensional simply-connected homogeneous space $M^7=G/H$. If $(G/H,\rho)$ is a homogeneous Einstein manifold, then it is either a symmetric space or isometric to one of the manifolds in i.1 to i.6

These are the spaces having each exactly one Einstein metric, the symmetric one:

s.1 $$S^7$$ being diffeomorphic to $$SO(8)/SO(7)$$

s.2 [tex]S^5 \times S^2[/itex] being diff to $${SO(6)\over SO(5)} \times {SO(3)\over SO(2)}$$

s.3 $$S^4 \times S^3$$ diff to $${SO(5)\over SO(4)} \times {SU(2)\times SU(2) \over SU(2)}$$

s.4 $${SU(3)\over SO(3)} \times S^2$$ diff to $${SU(3)\over SO(3)} \times {SO(3)\over SO(2)}$$

s.5 $$S^7$$ diff to $$Spin(7)/G_2$$

(and note also $$SU(4)/SU(3)$$, which produces again the standard metric on S^7)

and besides we have: -two spaces with a single invariant Einstein metric:

i.1 $$Sp(2)/SU(2)$$

i.2 Stiefel $$V_{5,3} \equiv SO(5)/SO(3)$$

-two spaces with two metrics available:

i.3 again $$S^7$$ now as $$Sp(2)\over Sp(1)$$

i.5c $$T_1S^3 \times S^2$$, an special case (1,1,0) of the family 5.

-two families with one metric for each embedding:

i.4 The biparametric (a,b)=$$SU(3)\times SU(2) \over SU(2) \times U(1)$$

i.5a the triparametric (a,b,c)=$$SU(2) \times SU(2) \times SU(2) \over SO(2) \times SO(2)$$
which has the subfamily of special cases
i.5b (a,b,0) each with a pure factor space $\times S^2$ and then the special case i.5.c above

-and one family with two metrics for each embedding

i.6 the biparametric (a,b)= $$SU(3)\over SO(2)$$

Note I am abusing the notation of the parameters because the table in the theorem does not follow previous notations. This is a tradition in the field, it seems.

Family i.4 is sometime referred as "simply connected Witten spaces". Family i.6 is known to matematicians as Allof-Wallach spaces.

All the three families can be buit as principal $S^1$ fiber bundles:
i.4 over $CP^2 \times S^2$
i.5 over $S^2 \times S^2 \times S^2$
i.6 over $SU(3)/T^2$

All the three families produce groups of isometry greater than g by an U(1) factor.

 

[arivero]

Compare this theorem with Duff et al table 6 pg 64
s.1 is Round S7
s.2 is M(1,0), Group SU(4)xSU(2)
s.3 similarly listed. Ok.
s.4 has group SU(3)xSU(2)
s.5 is discussed in page 40
i.1 listed perhaps as a SO(5)/SO(3), see pg 42??
i.2 listed as V5,2, group SO(5)xU(1) ?? Or listed without name?
i.3 is discussed in page 40
i,4 with group SU(3)xSU(2)xU(1)
i.5 with group SU(2)xSU(2)xSU(2)xU(1)
i.6 with group SU(3)xU(1)
and some special cases listed in the families, basically where U(1) promotes to SU(2)

According the table, only the squashed S7 (i.3? or s.5?), the unknown i.1 and the family i.6 have G2 holonomy.

 

[arivero]

It is claimed in Nikonorov, without prrof (only reference to a PhD thesis) that all the spaces in i.5 are diffeomorfic to the product of spheres S3 x S2 x S2. I can believe that this result happens because they are asked to be simply connected (so the above referred fibre bundles are asked to have coprime winding numbers)

So the good news is that the only parametric families are the two that can be related to the standard model, in the low and high energy limits.