octonionic del Pezzo surface?
Andy Neitzke
Hello!
From JB's postings and a little bit of research I understand that
the octonionic projective space OP^2 is an interesting "sporadic"
object. Does anyone know whether there is a natural definition of an
"octonionic blow-up" which could be used to construct an octonionic
del Pezzo surface?
I would be interested in any reference, but below is a little more
motivation:
I am not even sure what category such an object should inhabit --
but what I am hoping for is that it exists and that in the appropriate
category the exceptional symmetries which appear on the usual complex
del Pezzo surfaces are also visible on the octonionic ones. The
reason for the inquiry is that there have been attempts to identify
the exceptional symmetries of the usual complex del Pezzo surface with
the U-duality symmetries which occur in toroidal compactification of
M-theory. One problem with such an idea is that one can't find the
transverse Lorentz symmetry anywhere in the del Pezzo surface; but
because of the funny fact that F_4 / Spin(9) = OP^2, one might imagine
that one really should have started with F_4, with Spin(9) then
responsible for the light-cone Lorentz symmetry and OP^2 responsible
for the U-duality (via its blow-ups.)
--
Andy Neitzke
nei...@fas.harvard.edu
John Baez
Andy Neitzke <nei...@fas.harvard.edu> wrote:
>From JB's postings and a little bit of research I understand that
>the octonionic projective space OP^2 is an interesting "sporadic"
>object. Does anyone know whether there is a natural definition of an
>"octonionic blow-up" which could be used to construct an octonionic
>del Pezzo surface?
I know everything about the octonions - well, at least compared
to most people - but I have never heard of an octonionic version of
the "blow-up" construction in algebraic geometry.
>I am not even sure what category such an object should inhabit --
>but what I am hoping for is that it exists and that in the appropriate
>category the exceptional symmetries which appear on the usual complex
>del Pezzo surfaces are also visible on the octonionic ones.
The only thing I know about del Pezzo surfaces is that they are
surfaces named after a guy called "del Pezzo". So, I can't be
very helpful here. If you were looking for a category of
8-dimensional manifolds that are more special than Kaehler
manifolds or hyperKaehler manifolds, I'd suggest you try
octonionic manifolds:
http://math.ucr.edu/home/baez/week193.html
But if you want a natural category of 8n-dimensional manifolds
that includes OP^2, I can't help you!
Andy Neitzke
>>I am not even sure what category such an object should inhabit --
>>but what I am hoping for is that it exists and that in the appropriate
>>category the exceptional symmetries which appear on the usual complex
>>del Pezzo surfaces are also visible on the octonionic ones.
>
> The only thing I know about del Pezzo surfaces is that they are
> surfaces named after a guy called "del Pezzo". So, I can't be
Oh, this is a fun story! I assumed that you knew it since you know a lot
about the exceptional groups. The definition of a "del Pezzo surface" is
that it's a compact two-dimensional nonsingular variety over C with ample
anticanonical divisor (morally, "positive curvature.") If we wrote
"one-dimensional" instead of "two-dimensional" then that definition would
single out P^1 as the unique such object; in two dimensions there are 10
such surfaces instead of 1.
So what are these 10 special surfaces? You get nine of them by starting
with P^2 and blowing-up k<=8 points in general position (call the surface
obtained in this way B_k), and the tenth one is P^1 x P^1. (It turns out
that blowing up one point on P^1 x P^1 gives you B_2, not something new.)
Anyway, the surface B_k has a lot to do with the exceptional group E_k
(suitably defined for k<6). Most of the connections have to do with the
fact that blowing up a smooth point produces a curve of self-intersection
-1, so that the second homology (with its usual intersection product) is a
rectangular lattice of signature (1,k), and furthermore the orthocomplement
of the canonical divisor turns out to be exactly the root lattice of E_k.
Furthermore the Weyl group of E_k acts by diffeomorphisms. So all kinds of
geometrical objects on B_k (lines, rulings, curves of degree 4, whatever)
fall into representations of E_k, and there are natural algebraic E_k
bundles over B_k, and probably a lot more connections I don't know about.
OK, so why did I want to know about the octonionic version? It's a crazy
idea, but if you're curious, here it is. There is a proposed "duality"
(hep-th/0111068) which relates some of the objects which appear in M-theory
(compactified on a k-dimensional torus) with curves lying on the del Pezzo
B_k. In particular, the exalted "M-theory in 11 flat dimensions" gets
identified with CP^2 in this duality. The outlier P^1 x P^1 gets
identified with the Type IIB string, which is also an outlier from the
M-theory point of view. Various numerological things work out well in this
duality, and there are enough coincidences to make you feel that it should
mean *something*, but only some aspects of M-theory are included so far --
some of the moduli are frozen to zero, and the Lorentz symmetry is hard to
see -- so it seems like to get the full story the del Pezzo surface needs
to be augmented in some way.
Switching to an octonionic surface seems like overkill at first, since then
(assuming everything works out like the complex case) you would roughly be
getting the exceptional symmetries *twice* -- once from the octonions and
once from the blow-ups -- but this could actually be right, because also in
M-theory one seems to find exceptional symmetries occurring in two
different ways: there is an E_k which appears on compactification as the
U-duality group, and there is another E_8 which appears already in 11
dimensions and is responsible e.g. for the E_8 x E_8 heterotic string. (It
could be that these are both somehow the same, but so far nobody has any
evidence for that, and actually there is an argument that they can't be the
same because the E_k appears in its non-compact form while the E_8 appears
in its compact form.) Another reason to think about the octonionic P^2 is
that I'm attracted to the idea that the embedding of Spin(9) in F_4 is
important for M-theory, because Ramond has discovered a way to get the
Lorentz content of M-theory naturally from this embedding (hep-th/0112261).
And anyway, these exceptional structures should be good for *something*,
right?
> But if you want a natural category of 8n-dimensional manifolds
> that includes OP^2, I can't help you!
OK, fair enough. I vaguely gather that I shouldn't expect such a thing at
least for n>2... and as for my dreamed-of octonionic del Pezzos, I suppose
I will have to do my own dirty work, if I can. Thanks for the info!
--
Andy Neitzke
nei...@fas.harvard.edu
Tony Smith
Andy Neitzke wrote:
"... a "del Pezzo surface" is ...[gotten]... by
starting with [complex C]P^2 and blowing up k<=8 points ...
as for octonionic del Pezzos, I suppose I will have to do my
own dirty work, if I can. ...".
As Amer Iqbal, Andy Neitzke, and Cumrun Vafa said in
http://xxx.lanl.gov/abs/hep-ph/0111068
"... M-theory on Tk corresponds to P2 blown up at k generic points;
Type IIB corresponds to P1 x P1.
The moduli of compactifications of M-theory on rectangular tori are
mapped to Kaehler moduli of del Pezzo surfaces.
The U-duality group of M-theory corresponds to a group of classical
symmetries of the del Pezzo represented by global diffeomorphisms.
The 1/2-BPS brane charges of M-theory correspond to spheres in the del Pezzo,
and
their tension to the exponentiated volume of the corresponding spheres.
The electric/magnetic pairing of branes is determined by the condition that
the union of the corresponding spheres
represent the anticanonical class of the del Pezzo.
The condition that a pair of 1/2-BPS states form a bound state is mapped
to a condition on the intersection of the corresponding spheres. ...".
In other words,
desingularization/blowup structures on the CP^2 complex projective plane
have interesting connections with string theory structures
and
maybe similar structures on an octonionic projective plane might have
more interesting connections with string theory structures.
A question is:
Which of the following do you want to be your generalized octonionic
projective plane:
(RxO)P^2 = OP^2 = F4 / Spin(9) (16-dim "real" octonionic projective plane)
(CxO)P^2 = E6 / Spin(10)xU(1) (its 32-dim "complexification")
(HxO)P^2 = E7 / Spin(12)xSp(1) (its 64-dim "quaternification")
(OxO)P^2 = E8 / Spin(16) (its 128-dim "octonification") ????
(For descriptions of those manifolds, see, for example,
Rosenfeld's book Geometry of Lie Groups.)
Being naive about all this stuff, I guess that you might
try to generalize what is said on page 321 of Einstein Manifolds
by (pseudonymous) Arthur L. Besse:
"... If M is a complex surface (m=2) ...[and]... c_1(M) is positive,
negative or zero, the Chern number c_1^2(M) is non-negative ...
... blowing up a point ... by replacing a point p of M by the set
of (complex) tangent directions around the point, leaving unchanged
the remainder of M, ...[gives]... a new complex manifold M~
and a holomorphic mapping from M~ to M, bilholomorphic over M - {p},
and having a fiber isomorphic to CP^(m-1) over p ...
the fiber over p is called the exceptional divisor of the blowing up.
... the blowing up process decreases the Chern number c_1^2.
In particular, starting from the complex projective plane CP^2 ...
for which c_1^2 is equal to 9 ... by blowing up 9 points or more we
certainly obtain a complex surface whose first Chern class is indefinite.
... the complex surfaces S_r obtained from CP^2 by blowing up
r distinct points, 0 <= r <= 8, do have a positive first Chern class,
whenever those points are in general position ...
...[they]... are the only (compact) complex surfaces having positive
first Chern class, with CP^2 and CP^1 x CP^1 ...".
Another question is:
Is the property of holomorphy of the mapping M~ to M important
for your generalization, and if so, then how do you define
holomorphy for the octonionic structures?
(For example, would you generalize the Cauchy-Riemann equations
in the manner of section 4.2 of Nonassociative Algebras in Physics
by Lohmus, Paal, and Sorgsepp?)
Leaving aside the question of holomorphy,
do you generalize by picking a point p having a fiber isomorphic to
(RxO)P^1 = OP^1 (the 8-sphere?)
(CxO)P^1 (the 16-sphere?)
(HxO)P^1 (the 32-sphere?)
(OxO)P^1 (the 64-sphere?)
and then
looking at what the topological structure of the candidate generalizations,
to see what might be the corresponding structure to the Chern class of CP^2 ?
For instance, as stated in the paper by Iliev and Manivel at
http://xxx.lanl.gov/abs/math.AG/0306329
"... the real Cayley plane F4 / Spin(9) ... admit[s] a cell
decomposition R^0 u R^8 u R^16 and is topologically much simpler
...[than]...
the complex Cayley plane ... E6 / P6 [a subset of] PJ3(O) ...
... the restriction of J3(O) to the Levi part L =~ Spin(10)xC* of
the parabolic subgroup P6 of E6 ... give[s] 1 + 16 + 10 dimensions,
... the full decomposition [of J3(O)] .. In terms of Schubert cycles,
the Chern classes of the normal bundle to [complex]OP^2 [in] PJ3(O) are:
c_1(N) = 15 H ...".
As to what might be a natural exceptional divisor for the case of the
complexified octonionic projective plane (CxO)P^2 = E6 / Spin(10)xU(1),
perhaps the paper of Landsberg and Manivel at
http://xxx.lanl.gov/abs/math.AG/9908039
might be relevant, especially on pages 2, 24, and 25 where they say:
"... Let A denote ... the complexification of R, C, the quaternions H,
or the octonions O ...
... Desingularizations ... Orbits in P(J3(A)) ...
Let G / P = AP^2 [a subset of] PV be a Severi variety ...
There is a natural diagram
E = Q~^m --> AP^2
in in
PM -f-> sigma(AP^2)
pi\|/
AP^2*
where f is a desingularization of sigma(AP^2). The exceptional divisor
E of f is natrually identified with ... the set of points in (AP^2*)*
tangent to AP^2* along a quadric AP^1 =~ Q^m.
... we indicate
the nodes defining the space of F-points AP^2 with black dots *
and those defining the F-lines AP^2* =~ AP^2 with stars x.
The bundle on AP^2* ... and the quadric ... are below the diagram...
*--o--o--o--x
|
o
C^10
Q^8 ...".
Notation and details are described in the paper.
As of now, I don't know much about generalized del Pezzo surfaces
beyond what I have written above, but I hope that some day I will
find out more about the topological structure of the cases of
(HxO)P^2 = E7 / Spin(12)xSp(1) (64-dim "quaternification")
(OxO)P^2 = E8 / Spin(16) (128-dim "octonification").
Andy Neitzke also wrote:
"... And anyway, these exceptional structures should be
good for "something", right? ...".
I agree, but I will say that instead of working along the
lines of conventional string and M-theory, I am working
along the lines of my paper at
http://xxx.lanl.gov/abs/physics/0207095
which uses exceptional structures and Clifford algebras
in ways that are not widely accepted.
Tony Smith 1 July 2003