More Superstring Questions
Squark
Hello dear group.
Some ppl might have noticed I didn't reply anything on the "Anti D-Branes"
and "D-Branes and K-theory" threads even though they were in the middle of a
very interesting discussion. This happened because my replies (apparently) were
consumed by some kind of internet demons. In this post I gather the key
superstring questions which arised in my mind during those discussions or
elsewhen. Everyone are welcome to tackle some or all of these.
1) Is there a good reference on the topological string theories (the A and
the B models) available online? Where do these models stand with respect to the
full fledged superstring theory?
2) Is there a good summary of everything that is known on superstring vacua?
What is involved in classifying these vacua? Basically, if we consider
asymptotically flat string theory compactified on a manifold X, we have to
more or less compute all of the classical solutions of the equations of motion
for the massless fields arising from the string theory that have the form R^n x X
with n-dimensional Poincare symmetry. What is the main difficulty here? The
higher order corrections to the equations of motion?
Is the 11d vacuum unique? Are the 10d vacua include the usual 5 only?
3) In Calabi-Yau compactifications, a complex structure somehow magically
arises on the compactified space. Why does that happen? My best guess so
far was that we get a symplectic structure* from the RR 2-form. That,
however, doesn't work for type IIA. Moreover, I see no reason the RR 2-form has to
be non-degenerate.
4) Branes wrapped on a submanifold which has only Spin_c rather than Spin
structure carry "twisted" rather than ordinary gauge bundles. "Twisted"
gauge bundles also arise on the whole spacetime in the presence of a non-zero B
field. Does it mean an effective B-field arises on those submanifolds in
some odd sense?
* Since we're talking Kaehler manifolds and we already have a metric, we can
equivalently speak of adding a complex or a symplectic structure, as long as
the compatibility conditions are satisfied.
Best regards,
Squark
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Lubos Motl
I think that your latest questions are extremely good, and all of us will
probably have to work hard to answer all your questions.
Sergei Gukov gave us an introductory talk about topological string theory
at Harvard on Wednesday; unfortunately you were not there. ;-) I can only
repeat your question: Does someone know a good "textbook" on topological
string theory? I might need one, too. :-)
Topological string theory is otherwise a modification of string theory on
the worldsheet - similar to type II strings - that assigns different
worldsheet spin to various worldsheet fields (the process of "twisting").
As a result, the theory defined on spacetime is topological in nature:
open topological strings generate various Chern-Simons theories, while the
closed strings describe a Kahler gravity (A-model) or the Kodaira-Spencer
theory (B-model).
As the previous sentence indicates, there are two models, that differ much
like type IIA from type IIB - namely the A-model and the B-model. One
usually defines these models on a Calabi-Yau target space. The observables
calculated in the A-model only depend on the Kahler moduli of the
Calabi-Yau space, while the observables of the B-model only depend on the
complex structure. The latter case turns out to be much easier: the
B-model is much more calculable - even if one adds D-brane instantons like
Witten did - and the full power of complex analysis can be used.
Finally, however, one can see that the A-model is the mirror dual of the
B-model, so in principle we should be able to apply this mirror duality to
solve the questions of the A-model and the B-model with the same degree of
difficulty.
Topological string theories on regular spaces that are known so far can't
be equivalent to the full string theory. Usually they are thought of as a
theory generating the observables that are guaranteed to be holomorphic -
the so-called F-terms. A partition sum of topological string theory is
typically the pre-potential in the full physical string theory. In other
words, sometimes they can be reinterpreted as very special amplitudes in a
corresponding model of the full string theory. Witten's recent article is
special, because it suggests a sort of duality - topological string theory
should be dual to the gauge theory (a description of D3-branes), at least
in the planar limit. The full power and relevance of topological string
theory within the whole framework of string theory is not understood yet,
and so far it is still legitimate to think of topological string theory as
an unphysical toy-model for real string theory. This situation might
change in the future.
> 2) Is there a good summary of everything that is known on superstring
> vacua? What is involved in classifying these vacua?
I think it is a too broad question. Once we understand the vacuum, and
everything that can fluctuate in it :-), in some sense, we understand
everything, and therefore 50 percent of string theory papers might be
described as papers "that study superstring vacua".
> Basically, if we consider asymptotically flat string theory
> compactified on a manifold X, we have to more or less compute all of
> the classical solutions of the equations of motion for the massless
> fields arising from the string theory that have the form R^n x X with
> n-dimensional Poincare symmetry. What is the main difficulty here?
You formulated the situation very well. If you focus on spaces of the form
R^n x X - the classical candidates for the real world - you will be able
to derive that X must be Ricci-flat. If you want some unbroken
supersymmetry, X must be a Calabi-Yau space (see below). From each
Calabi-Yau, one can consider a plenty of other modified solutions with
various fluxes (e.g. the gauge fields) living on the Calabi-Yau space
which deform the geometry a little bit. Usually one finds that the fluxes
generate potential/masses for some scalar fields, and it is pretty
complicated.
Even the question of the vacua with unbroken supersymmetry is a pretty
complex one. There are tens of thousands of topologies of Calabi-Yau
spaces X known (when Andy Strominger first found that Calabi-Yau spaces
were relevant for string theory, he went to the library and found one
Calabi-Yau space - the quintic - and therefore he thought he had the
unique theory of everything), and each of them has a finite number of
moduli (between a few and a few hundreds), continuous parameters
describing the shape. These moduli (scalar fields) probably choose a
minimum once supersymmetry is spontaneously broken, but the way how to
calculate the potential is known only partially. Moreover, it is known
that string theory allows a smooth process of topology change, and the
topological changing transitions can essentially connect all (or almost
all) possible topologies X. The structure of the moduli space - the space
of vacua - is pretty complicated, especially once we add the fluxes and
branes and other possibilities, and it is just not known whether the full
dynamics forces us to fall into one of several possible vacua, or whether
the "landscape" (let me use this Susskind's term) has very many different
minima where intelligent observers can happily live. There are several
convincing arguments that the number of minima is large indeed, but there
are certainly no rational arguments that we should give up our attempts to
identify the correct vacuum precisely - or more precisely than we have
been able so far.
These Cartesian products are just a special subclass of stringy vacua, and
we have many more candidates today whose qualitative features resemble the
Standard Model (or a GUT or some extension of them) plus gravity: they
involve M-theory on G2 manifolds, F-theory on Calabi-Yau four-folds,
intersecting D-brane models, Horava-Witten models on Calabi-Yau
three-folds, and all of them can have fluxes etc. Some of these models are
known to be dual to one another, but the complete picture is not yet
clear.
> The higher order corrections to the equations of motion? Is the 11d
> vacuum unique? Are the 10d vacua include the usual 5 only?
Yes, according to everything we know, there is a unique SO(10,1) invariant
vacuum, which is what we really called M-theory, and five SO(9,1)
invariant ten-dimensional (stringy) vacua. Once you allow a flux or
cosmological constant (or massive type IIA supergravity or
nonsupersymmetric type 0 strings etc. that also imply that the space can't
be flat once the coupling is nonzero), the spectrum of possibilities
increases. The number of possibilities and the richness of physics also
increases once you compactify more dimensions: well that's not surprising
because what you call "4-dimensional vacuum" corresponds to non-trivial
non-vacuum dynamics of the extra dimensions - in some sense, a
4-dimensional vacuum is about the things that you make to the
10/11-dimensional vacuum to make it realistic: adding fluxes, branes,
warping and/or compactification.
> 3) In Calabi-Yau compactifications, a complex structure somehow magically
> arises on the compactified space. Why does that happen? My best guess so...
The origin of the complex structure might be magical for a physicist, but
certainly not for a mathematician because they defined the Calabi-Yau
space to be a Kahler manifold (which automatically implies that it must
have a complex structure) with a vanishing first Chern class (imagine that
it has no nontrivial holomorphic 1-cycles). Eugenio Calabi conjectured in
1957 that each such a manifold can be given a Ricci-flat metric of SU(3)
holonomy (which means that by parallel transport, you can't rotate a
vector by an arbitrary rotation in SO(6), but only by a rotation in the
SU(3) subgroup of SO(6)). The Kahler manifolds must have a U(3) holonomy,
and the Ricci-flatness implies that the holonomy must be inside the SU(3)
subgroup. This conjecture about the existence of the Ricci-flat metric was
proved in 1977 by Shing-Tung Yau, and therefore the manifolds are called
Calabi-Yau manifolds.
The condition of Ricci-flatness is derived from the Einstein-like
equations of motion (that follow from conformal symmetry of the nonlinear
sigma model that describes the propagation of strings on the curved target
space), and the related, but independent condition of SU(3) holonomy is
equivalent to the unbroken 1/4 of the original supersymmetry (note that
SU(4)=SO(6) is the most general holonomy, and if we reduce it to SU(3), 1
component out of the 4 spinor components that generate the SU(4) is
constant, and therefore it can defined a covariantly constant spinor -
which gives rise to unbroken supersymmetry).
> far was that we get a symplectic structure* from the RR 2-form.
No, the complex structure is only related to the metric field. It is the
metric that defines the holonomy (infinitesimal loops of parallel
transport are determined by the Riemann tensor), and the SU(3) holonomy
tells you how the 6 coordinates are combined into 3 complex coordinates.
If supersymmetry is unbroken, various other fields - such as the B-field -
are combining with some of the metric excitations to give you
supermultiplets. But the complex and Kahler structure are directly
associated with the metric field only, not with the antisymmetric tensor
fields.
> That, however, doesn't work for type IIA. Moreover, I see no reason
> the RR 2-form has to be non-degenerate.
Whom should it be degenerate with?
> 4) Branes wrapped on a submanifold which has only Spin_c rather than
> Spin structure carry "twisted" rather than ordinary gauge bundles.
> "Twisted" gauge bundles also arise on the whole spacetime in the
> presence of a non-zero B field. Does it mean an effective B-field
> arises on those submanifolds in some odd sense?
There might be some sense in which it is true, but you are too fast for
me. On D-branes, B-field can be replaced by the internal electromagnetic
field inside the brane, only B-F is gauge invariant. Nontrivial bundles of
this B-F is what defines the lower-dimensional D-branes via K-theory. It
is useful to realize what the spin structure is good for physically, and
it is good both for the gauge fields as well as fermionic fields, and
whether or not your brane preserves some supersymmetry. There are no
problems to define the spinors/fermions on a brane that wraps anything,
and you may imagine that you are using the spin structure from the
spacetime.
> * Since we're talking Kaehler manifolds and we already have a metric,
> we can equivalently speak of adding a complex or a symplectic
> structure, as long as the compatibility conditions are satisfied.
Sorry, but Kahler manifolds *always* have a complex structure (that you
can use as a symplectic structure, too). See e.g.
http://mathworld.wolfram.com/KaehlerManifold.html
You might be confusing these terms because "complex structure" and "Kahler
moduli" are two independent pieces of information about a Calabi-Yau
manifold. They are independent indeed, but it only makes sense to talk
about the Kahler moduli if you have *some* complex structure, i.e. only if
you have a Kahler manifold. ;-)
Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Squark
"Lubos Motl" <mo...@feynman.harvard.edu> wrote in message
news:Pine.LNX.4.31.03122...@feynman.harvard.edu...
> If you focus on spaces of the form
> R^n x X - the classical candidates for the real world
I'm not sure those are "the classical candidates for the real world",
in particular since one usually takes n = 4, whereas I think it more
interesting to take n = 1.
> These moduli (scalar fields) probably choose a
> minimum once supersymmetry is spontaneously broken, but the way how to
> calculate the potential is known only partially.
I.e you can prove there is a non-vanishing quantum potential?
Where can I find investigation of this potential?
> Moreover, it is known
> that string theory allows a smooth process of topology change, and the
> topological changing transitions can essentially connect all (or almost
> all) possible topologies X.
Locally. I expect a topology change between R^n x X and R^n x Y
to cost an infinite amount of action, in some sense, at least when n > 2,
the same as analogous behavior in QFT.
> There are several
> convincing arguments that the number of minima is large indeed,
And where can I find those?
> M-theory on G2 manifolds,
G2 manifolds as in manifolds with G2 symmetry? Where do they
come from?
> intersecting D-brane models
Hmm, sounds interesting. With large extra dimensions, a la
Rundal-Sundrum?
> Yes, according to everything we know, there is a unique SO(10,1) invariant
> vacuum, which is what we really called M-theory, and five SO(9,1)
> invariant ten-dimensional (stringy) vacua.
How do you show this (or at least show some evidence)?
> No, the complex structure is only related to the metric field. It is the
> metric that defines the holonomy (infinitesimal loops of parallel
> transport are determined by the Riemann tensor), and the SU(3) holonomy
> tells you how the 6 coordinates are combined into 3 complex coordinates.
This is interesting. What you're saying is that a subgroup of SO(6)
isomorphic
to SU(3) defines a complex structure w.r.t which the subgroup is ineed SU.
This works only for manifolds for which the subgroup is indeed SU(3) rather
than a proper subgroup thereof.
> > * Since we're talking Kaehler manifolds and we already have a metric,
> > we can equivalently speak of adding a complex or a symplectic
> > structure, as long as the compatibility conditions are satisfied.
>
> Sorry, but Kahler manifolds *always* have a complex structure (that you
> can use as a symplectic structure, too). See e.g.
I know that, of course. What I meant is that you may regard the defining
information of a Kahler manifold as three objects: metric, complex
structure and symplectic structure, any two of which define the third.
This trick works already on the vector (tangent) space level. What you're
saying is that in many cases the metric only is sufficient the other two,
using
its holonomy.
Robert C. Helling
> 1) Is there a good reference on the topological string theories (the A and
> the B models) available online? Where do these models stand with respect to the
> full fledged superstring theory?
I don't really know, but I would look up some old papers of Witten
(hint: have a look at the references of the twistor paper).
> 3) In Calabi-Yau compactifications, a complex structure somehow magically
> arises on the compactified space. Why does that happen? My best guess so
> far was that we get a symplectic structure* from the RR 2-form. That,
> however, doesn't work for type IIA. Moreover, I see no reason the RR 2-form has to
> be non-degenerate.
No, the complex structure comes with the killing spinor: The CY
has a covanriantly constant spinor psi and from that you can compute
psi-bar Gamma^ij psi and psi-bar Gamma^ijk psi. The first is the
complex structure J and the second is the holomorphic three form Omega.
Robert
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Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
> I'm not sure those are "the classical candidates for the real world",
> in particular since one usually takes n = 4, whereas I think it more
> interesting to take n = 1.
Well, it is usually assumed that cosmological issues can be separated, and
one can study local physics, at distances much shorter than the
cosmological scales, using something that looks like R^4 x X or a warped
version of it. Normally we would say that our Universe has 3, not 0, large
spatial dimensions, right? ;-)
> I.e you can prove there is a non-vanishing quantum potential?
> Where can I find investigation of this potential?
There is no general proof because the configuration space has many
branches that behave very differently. To achieve exactly flat directions
with N=1 supersymmetry is unlikely. See e.g. Gukov Vafa Witten paper about
the superpotentials that are generated by fluxes,
http://arxiv.org/abs/hep-th/9906070
> Locally. I expect a topology change between R^n x X and R^n x Y
> to cost an infinite amount of action, in some sense, at least when n > 2,
> the same as analogous behavior in QFT.
In the theories that are understood most thoroughly, i.e. those with 8
supercharges, it costs no action at all (or an arbitrarily small action)
because the process is completely smooth. If you studied less
supersymmetric contexts where you needed to go off-shell, it would cost a
nonzero action, but for the creation of a finite bubble inside your
Universe with a different topology, it would again cost a finite action.
Of course, if you want to change the topology off-shell in the whole
Universe simultaneously, the necessary action will be proportional to the
volume of the Universe. But I guess that you agree that ice can melt, even
though classification of H2O molecules as ice only makes sense for an
infinite amount of molecules, and it costs infinite energy to melt
everything. ;-)
> > There are several
> > convincing arguments that the number of minima is large indeed,
>
> And where can I find those?
The classical construction that "almost" proves the existence of a large
number of these vacua - and the main technical result supporting the
current claims that the anthropic principle should be taken seriously - is
the KKLT paper
http://arxiv.org/abs/hep-th/0301240
Of course, there are related papers, but you will find them.
> G2 manifolds as in manifolds with G2 symmetry? Where do they
> come from?
Nope. G2 manifolds have a G2 holonomy, not a G2 symmetry. They are
7-dimensional manifolds such that a general parallel transport induces a
transformation on the tangential vector that always belongs to the G2
subgroup of SO(7), although a generic manifold would certainly allow you
to achieve an arbitrary element of SO(7) by a parallel transport around an
appropriate curve.
Once again, the set of all possible transformations of a tangential vector
that you can induce by parallel transports around all possible curves is
called *holonomy*. See e.g. page 7 of Aspinwall's K3 lectures
http://arxiv.org/abs/hep-th/9611137
to see all possible holonomies. They are SO(d) for a generic d-dimensional
manifold, U(d/2) for a complex Kahler manifold (where you combine the
coordinates to complex ones, and the holonomy only mixes the holomorphic
ones), SU(d/2) which is a Ricci-flat Kahler manifold called "Calabi-Yau",
USp(d/2) for a hyperKahler manifold, which is again a subset of Kahler
manifolds, USp(d/2) x USp(2) for a quaternionic Kahler manifolds. Then you
have a few exceptional cases - namely G2 holonomy for 7-dimensional
manifolds and spin(7) holonomy for 8-dimensional manifolds. Then you can
have coset spaces G/H whose holonomy is typically G, and finally you can
have Cartesian products M1 x M2 whose holonomy is H1 x H2. A theorem shows
that there are no other possibilities unless I forgot an option.
G2 holonomy means that the transformations induced on a vector by a
parallel transport are in the G2 subgroup of SO(7) - the subgroup that
fixes that antisymmetric tensor of 3rd rank that defines the octonion's
multiplication table (G2 is also the automorphism group of the octonions).
Alternatively, you may define G2 as the subgroup of spin(7) that keeps
invariant a chosen spinor in the 8-dimensional spinor representation of
spin(7). This makes it clear that the G2 manifolds break the supersymmetry
to 1/8 of the original amount, which is exactly the right way to get the
phenomenologically interesting N=1 SUSY from N=8 SUSY of M-theory in flat
space.
> > intersecting D-brane models
>
> Hmm, sounds interesting. With large extra dimensions, a la
> Rundal-Sundrum?
First of all, Lisa Randall would not like that you call Randall-Sundrum
"large" extra dimension models because in their two-brane model, these
extra dimensions are short (like 30 Planck units), and still able to
generate the hierarchy. Randall-Sundrum are "warped" extra dimensions,
not necessarily "large".
Yes, intersecting brane models often want to have large dimensions, and a
fundamental scale at a couple of TeV, but people have tried other things,
too. To see examples of the intersecting brane models, look at e.g.
http://arxiv.org/abs/hep-th/0212177
http://arxiv.org/abs/hep-th/0303197
http://arxiv.org/abs/hep-th/0303208
http://arxiv.org/abs/hep-th/0309070
> > Yes, according to everything we know, there is a unique SO(10,1) invariant
> > vacuum, which is what we really called M-theory, and five SO(9,1)
> > invariant ten-dimensional (stringy) vacua.
>
> How do you show this (or at least show some evidence)?
We don't have a complete proof, and various no-go theorems often have
loopholes anyway, but let me try. Let us study the low-energy limit of a
theory, and assume that it is nontrivial. For more than 32 supercharges,
you would be guaranteed that the (massless) particles must have spin
greater than two. There are theorems showing that such high-spin fields
can't have interactions that are compatible with the gauge invariance -
the gauge invariance is necessary to decouple the negative-norm states. So
it means that 32 is the maximum number of supercharges in a physical
theory, and this number requires gravitons with spin 2. In 11 dimensions,
a spinor has 32 real components, and therefore 32 is also the minimal
nonzero number of supercharges. No supersymmetry in 11D would almost
certainly generate a large cosmological constant at the Planck scale,
which is incompatible with the SO(10,1) symmetry. Therefore you need 32
supercharges. You can then prove that the SUSY multiplets must have
2^{32/4} = 256 states by quantizing the unbroken half of the
supersymmetry. There must be a spin 2 particle, it must be the graviton
which has 9.10/2.1-1 = 44 physical polarizations (transverse traceless
tensor) in d=11. The remaining 84 bosonic fields are then guaranteed to
come from the 3-form potential C_{MNP}, and the 128 fermionic partners
must be gravitinos, and the low-energy action for this field content is
completely determined by SUSY. Once you have 11-dimensional supergravity,
you conjecture that there is a unique ultraviolet completion, and because
you learn a lot of things about it and everything seems to be unique for
many years, you use the Occam's razor to conclude that the SO(10,1)
invariant physics has probably a unique background.
> This is interesting. What you're saying is that a subgroup of SO(6)
> isomorphic to SU(3) defines a complex structure w.r.t which the
> subgroup is ineed SU. This works only for manifolds for which the
> subgroup is indeed SU(3) rather than a proper subgroup thereof.
Which proper subgroup do you exactly mean? If the dimension of the proper
subgroup is smaller than the dimension of SU(3) - now I talk about the
"continuous part" of the group only, which is connected to the identity
(because the discrete groups are even less important), then you get
something much more special and constrained than a Calabi-Yau manifold.
For example, if the holonomy is within the SU(2) subgroup, then you get
locally a K3 x T^2 where K3 is a unique hyperKahler d=4 manifold with the
SU(2) holonomy. A K3 x T2 is certainly not what we mean by a Calabi-Yau
manifold - for example, it has nontrivial first homology. Just be sure
that all interesting things that can be called Calabi-Yau three-folds have
*exactly* SU(3) holonomy and not a smaller one, and they always allow you
to decide what is the complex structure as a function of the holonomy.
> I know that, of course. What I meant is that you may regard the defining
> information of a Kahler manifold as three objects: metric, complex
> structure and symplectic structure, any two of which define the third.
If you knew a metric of a Calabi-Yau space, you would know everything
about it, including the complex and Kahler structure. And vice versa, you
need the complex structure and the Kahler 2-form to know the full metric.
I suppose that "symplectic structure" was meant to be a Kahler 2-form,
because the symplectic structure in this case is otherwise the same
information as the complex structure.
> This trick works already on the vector (tangent) space level. What
> you're saying is that in many cases the metric only is sufficient the
> other two, using its holonomy.
It is true for all spaces that can be called Calabi-Yaus i.e. whose
holonomy is exactly SU(d/2) where d>4 (the d=4, K3 case is special because
you can choose many different complex structures that are related by SU(2)
transformations).
All the best
Aaron Bergman
<Pine.LNX.4.31.031224...@feynman.harvard.edu>, Lubos
Motl wrote:
> The classical construction that "almost" proves the existence of a large
> number of these vacua - and the main technical result supporting the
> current claims that the anthropic principle should be taken seriously - is
> the KKLT paper
So, let me first confess that I haven't read the paper yet, but
from the second had stuff I've heard about it, I'd read those
quotes around 'almost' as scare-quotes.
>> Hmm, sounds interesting. With large extra dimensions, a la
>> Rundal-Sundrum?
>
> First of all, Lisa Randall would not like that you call Randall-Sundrum
> "large" extra dimension models because in their two-brane model, these
> extra dimensions are short (like 30 Planck units), and still able to
> generate the hierarchy. Randall-Sundrum are "warped" extra dimensions,
> not necessarily "large".
IIRC, that's RS1, not RS2. They get rid of the brane in RS2,
right, to get AdS space with infinite extent but finite volume or
something like that.
Aaron
--
Aaron Bergman
<http://zippy.ph.utexas.edu/~abergman/>
Squark
"Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote in message
news:bs6co1$a5tal$2...@ID-40416.news.uni-berlin.de...
> No, the complex structure comes with the killing spinor: The CY
> has a covanriantly constant spinor psi
I.e. its total covariant derivative vanishes identically? What is the
general definition of Killing spinor? I guess it doesn't have to be
covariantly constant, since a Killing vector isn't always covariantly
constant?
Squark
> Well, it is usually assumed that cosmological issues can be separated, and
> one can study local physics, at distances much shorter than the
> cosmological scales, using something that looks like R^4 x X or a warped
> version of it. Normally we would say that our Universe has 3, not 0, large
> spatial dimensions, right? ;-)
Yes, the keyword being "large", i.e. it is assume the universe is of the
form R^4 x X the (asymptotic) size of the "X" part being of the order of the
Planck length. Btw, if radiative corrections indeed generate a potential for
the moduli which leaves only a finite number of vacua, I suppose one of
the prominent possibilities is that correspond to some self-dual X (w.r.t.
T-duality or mirror symmetry)? Btw, are there known mirror self-dual CYs?
Also, what do you called "a warped version of it"? AdS_4 x X?
On the level of local physics many or all of these vacua might merge. I.e.
if we consider compact space we might find out there a single superselection
sector and we need no anthropic principle or anything of the like.
Not that I think the anthropic principle can actually be of any help in
other cases.
> In the theories that are understood most thoroughly, i.e. those with 8
> supercharges, it costs no action at all (or an arbitrarily small action)
> because the process is completely smooth.
I don't see how it "no action at all" follows from "completely smooth".
In continuous symmetry breaking, the vacua are also connected on-shell
and we still get different superselection sectors - the action cost
comes from the kinetic part of the action. So we still get different vacua
as long as #(large spacetime dimensions) > 2.
> If you studied less
> supersymmetric contexts where you needed to go off-shell, it would cost a
> nonzero action, but for the creation of a finite bubble inside your
> Universe with a different topology, it would again cost a finite action.
Yes, of course. That's precisely what I said, you can do topology change but
only locally, not in the asymptotics.
> ...USp(d/2) for a hyperKahler manifold, which is again a subset of Kahler
> manifolds, USp(d/2) x USp(2) for a quaternionic Kahler manifolds.
I thought hyperKahler and quaternionic Kahler is the same thing :-( Indeed
it would seem so since USp(d/2) is the group of "quaternionic d/2 x d/2
unitary matrices". What is the USp(2) factor doing there?
> A theorem shows
> that there are no other possibilities unless I forgot an option.
Summarizing all said, the holonomy group can be any simple Lie group
except F4, E6, E7, E8? Or are those the options you forgot? :-)
> This makes it clear that the G2 manifolds break the supersymmetry
> to 1/8 of the original amount, which is exactly the right way to get the
> phenomenologically interesting N=1 SUSY from N=8 SUSY of M-theory in flat
> space.
Is a G2 manifold automatically Ricci flat?
> First of all, Lisa Randall would not like that you call Randall-Sundrum
> "large" extra dimension models because in their two-brane model, these
> extra dimensions are short (like 30 Planck units), and still able to
> generate the hierarchy. Randall-Sundrum are "warped" extra dimensions,
> not necessarily "large".
She would probably also dislike me calling her "Rundal" :-) OK. What is
so special / surprising bout their model w.r.t. other ways to compactify,
then?
> No supersymmetry in 11D would almost
> certainly generate a large cosmological constant at the Planck scale,
> which is incompatible with the SO(10,1) symmetry.
Hmm. Essentially you are saying the classical Minkowski spacetime
solution would have no quantum analogue?
> Once you have 11-dimensional supergravity,
> you conjecture that there is a unique ultraviolet completion, and because
> you learn a lot of things about it and everything seems to be unique for
> many years, you use the Occam's razor to conclude that the SO(10,1)
> invariant physics has probably a unique background.
So basically it's the good old method of studying the vacua classically,
but this time you have to allow for different classical Lagrangians
(different supergravity theories in 10d, for instance).
> Which proper subgroup do you exactly mean? If the dimension of the proper
> subgroup is smaller than the dimension of SU(3) - now I talk about the
> "continuous part" of the group only, which is connected to the identity
> (because the discrete groups are even less important), then you get
> something much more special and constrained than a Calabi-Yau manifold.
I realize that. Can we have a (closed) proper subgroup with dimension that
is not smaller??
> I suppose that "symplectic structure" was meant to be a Kahler 2-form,
> because the symplectic structure in this case is otherwise the same
> information as the complex structure.
First of all remind me what is the Kahler 2-form :-)
> It is true for all spaces that can be called Calabi-Yaus i.e. whose
> holonomy is exactly SU(d/2) where d>4 (the d=4, K3 case is special because
> you can choose many different complex structures that are related by SU(2)
> transformations).
Since SO(4) is (modulo global issues)
SU(2) x SU(2) and, if the holonomy group is one
of these SU(2), the other provides these
transformations?
Aaron Bergman
>
> I thought hyperKahler and quaternionic Kahler is the same thing :-( Indeed
> it would seem so since USp(d/2) is the group of "quaternionic d/2 x d/2
> unitary matrices". What is the USp(2) factor doing there?
It's just a different possibility for holonomy -- I forget the
origin of the names.
>> A theorem shows
>> that there are no other possibilities unless I forgot an option.
>
> Summarizing all said, the holonomy group can be any simple Lie group
> except F4, E6, E7, E8? Or are those the options you forgot? :-)
U(n) isn't simple. Neother is the QK holonomy group. None of the
others you list are possible.
>> This makes it clear that the G2 manifolds break the supersymmetry
>> to 1/8 of the original amount, which is exactly the right way to get the
>> phenomenologically interesting N=1 SUSY from N=8 SUSY of M-theory in flat
>> space.
>
> Is a G2 manifold automatically Ricci flat?
Yes. A G2 manifold has a covariantly constant spinor. Apply the
commutator of two covariant derivatives to that spinor to show
that it is Ricci flat.
> First of all remind me what is the Kahler 2-form :-)
g_{i\bar{j}} dz^i /\ d\bar{z}^\bar{j}
Urs Schreiber
news:3fec...@news.012.net.il...
>
> "Lubos Motl" <mo...@feynman.harvard.edu> wrote in message
> news:Pine.LNX.4.31.031224...@feynman.harvard.edu...
> > In the theories that are understood most thoroughly, i.e. those with 8
> > supercharges, it costs no action at all (or an arbitrarily small action)
> > because the process is completely smooth.
>
> I don't see how it "no action at all" follows from "completely smooth".
There has recently appeared an interesting analysis by Mohaupt's group which
is reviewed in
Jarv, Mohaupt, Saueressig, Singular compactifications and cosmology
http://xxx.uni-augsburg.de/abs/hep-th/0311016
and which refers to
http://xxx.uni-augsburg.de/abs/hep-th/0310174^
and
http://xxx.uni-augsburg.de/abs/hep-th/0310173 .
Using an effective action for the transition point the authors find a
cosmological model where the universe oscillates around the flop transition
point with respect to the CY factor.
Lubos Motl
> Yes, the keyword being "large", i.e. it is assume the universe is of the
> form R^4 x X the (asymptotic) size of the "X" part being of the order of the
> Planck length.
OK, I meant 4 "very large" dimensions. The "X" might be much bigger than
the 4-dimensional Planck length. Then those dimensions of "X" are called
"extra large dimensions", but they are still much smaller than the 4
dimensions that we know from everyday life.
> Btw, if radiative corrections indeed generate a potential for the
> moduli which leaves only a finite number of vacua, I suppose one of
> the prominent possibilities is that correspond to some self-dual X
> (w.r.t. T-duality or mirror symmetry)?
Yes, this has been a popular guess - a self-dual point of any kind of
duality that you may imagine is a natural place for a minimum (c.f. the
critical line of the zeta function), but a reliable answer requires us to
know more about these generic regions.
> Btw, are there known mirror self-dual CYs?
Very good question. I don't know, but there are articles that mention it:
http://www.google.com/search?q=%22mirror+self-dual%22+calabi-yau&num=100&filter=0
> Also, what do you called "a warped version of it"? AdS_4 x X?
A warped product X x Y is an X-fibration over Y where the overall size of
X is a function of Y. If the metric of the ordinary product "X x Y" is
ds^2 = dx^2 + dy^2,
where x and y collectively denote all coordinates of X and Y respectively,
then the metric of the warped product "X x Y" is
ds^2 = f(y)dx^2 + dy^2
A modern textbook version of a warped geometry are the Randall-Sundrum
models that assume that we live on a (3+1)-brane inside a 5-dimensional
spacetime that looks as AdS_5, and that can be represented as a warped
product R^4 x R.
ds^2 = exp(-2t) dx^mu dx_mu + dt^2, mu=0,1,2,3
> On the level of local physics many or all of these vacua might merge. I.e.
> if we consider compact space we might find out there a single superselection
> sector and we need no anthropic principle or anything of the like.
I hope so, but if you've solved the problem completely, then I
misunderstood it. ;-)
> Not that I think the anthropic principle can actually be of any help in
> other cases.
This is why some people try to use it. :-) Others think that even if it
can seem to be of any help, it is a virus that should be avoided.
> I don't see how it "no action at all" follows from "completely smooth".
I thought that by the action you meant the action of the instanton that
signals the instability - the action is a counterpart of the total "size"
(given by the exponent in the exponential decay of the wavefunction) of
the potential barrier for tunneling. A smooth transition corresponds to
the limit where this barrier disappears and its action goes to zero. If it
takes no effort to go through the tunnel - because there is no rock - the
action (=effort) is zero, is not it?
> In continuous symmetry breaking, the vacua are also connected on-shell
> and we still get different superselection sectors - the action cost
> comes from the kinetic part of the action.
Do you mean the Higgs mechanism and the "different" vacua at the bottom of
the Mexican hat potential left behind? They are not different at all! By
gauge symmetry, all these vacua are physically equivalent. It would be
pretty disastrous for the electroweak theory if it generated an infinite
degeneracy of physical vacua. The directions along the valley are the
Goldstone bosons, and if you started with a gauge symmetry, these
Goldstone bosons become the third polarizations of the massive gauge
bosons (they are "eaten"). But there is still one Lorentz-invariant vacuum
only!
> So we still get different vacua
> as long as #(large spacetime dimensions) > 2.
My thinking might be slower now, but I don't understand which different
vacua you mean.
> Yes, of course. That's precisely what I said, you can do topology change but
> only locally, not in the asymptotics.
Well, even if we don't talk about the topology change, it is pretty
unusual for humans to change something in the whole Universe. ;-)
> I thought hyperKahler and quaternionic Kahler is the same thing :-( Indeed
> it would seem so since USp(d/2) is the group of "quaternionic d/2 x d/2
> unitary matrices". What is the USp(2) factor doing there?
HyperKahler and quaternionic Kahler manifolds are similar, but they differ
by this extra USp(2) on the quaternionic Kahler side. The coordinates on
the tangent space of these manifolds can be organized into d/4 quaternions
(d is the real dimension) and the holonomy is a multiplication of the d/4
x 1 quaternionic vector (column) by a quaternionic matrix from
U(d/4,H)=USp(d/2) from the left. However recall that quaternions are
non-commutative. Consequently, you can also multiply the d/4 x 1
quaternionic column by a 1x1 quaternionic "matrix" that is now put on the
right, and you get a different result. In the complex (Kahler) case this
would be impossible because this number could be included in the matrix on
the left.
Note that for real 4-dimensional manifolds the notion of a quaternionic
Kahler manifold is redundant: USp(2) x USp(2) is (locally) isomorphic to
SO(4), and a quaternionic Kahler manifold is in fact a generic real
4-dimensional manifold in this case.
> Summarizing all said, the holonomy group can be any simple Lie group
> except F4, E6, E7, E8? Or are those the options you forgot? :-)
No, I did not forget them. In the 19th century people used to think that
F4 was possible, too, but it was later proved that no manifolds with F4
holonomy exist, and this is the case of the E_k symmetries, too. When you
summarized, you ignored the subtleties such as the quaternionic Kahler
manifolds - which have a non-simple holonomy group :-) - but you probably
thought that the extra "times USp(2)" was a typo of mine. No, it was not.
> Is a G2 manifold automatically Ricci flat?
Good question, and the answer is yes. It is also the case of the manifolds
with SU(d/2) holonomy - and of course USp(d/2) which is a subset - as well
as the spin(7) holonomy 8-dimensional manifolds. BTW the master thesis of
this Japanese guy below might be a helpful first text to understand
something about the G2 compactifications.
http://www2.yukawa.kyoto-u.ac.jp/~hamanaka/shimizu_t03.ps
> She would probably also dislike me calling her "Rundal" :-) OK.
We can ask her which of these two things is more annoying. ;-)
> What is so special / surprising bout their model w.r.t. other ways to
> compactify, then?
The special thing about their model of compactification is that it is not
a compactification. :-) The RS2 paper is called "An alternative to
compactification" because in this case, the extra dimension can be, in
fact, infinitely large - and still, we won't see it (because we are stuck
on the brane) and we will even see 4-dimensional law of gravity (Lisa and
Raman make it possible for gravity to "localize" near the brane - via the
"potential" coming from the warping - even though gravity's main feature
is that it feels geometry of the whole spacetime). Then it is hard to call
such a dimension "compact" and the process henceforth can't be called
"compactification".
The model RS1, on the other hand, has the fifth dimension truncated by
*two* branes - so it is a line interval. The virtue of this model is that
it can naturally generate hierarchies from the exponential warp factor
exp(-2t) that becomes huge even if the separation of two things in the
fifth dimension (t) is something like 30. This model (RS1) provided a new
possible answer - an answer independent of supersymmetry - of the
hierarchy problem i.e. of the question why is gravity so weak at the
electroweak scale.
http://arxiv.org/abs/hep-ph/9905221
http://arxiv.org/abs/hep-th/9906064
> Hmm. Essentially you are saying the classical Minkowski spacetime
> solution would have no quantum analogue?
This is my guess, but I can't prove it rigorously.
> So basically it's the good old method of studying the vacua classically,
> but this time you have to allow for different classical Lagrangians
> (different supergravity theories in 10d, for instance).
That's right. Even at the classical level, the spectrum of theories is not
too large. A richer - but simple enough - classification of
Lorentz-invariant vacua appears in 9+1 = 10 dimensions. Let me remind you.
Let's assume that we need at least some supersymmetry to keep the
spacetime exactly flat (i.e. to avoid the cosmological constant). Again,
32 supercharges is the maximal amount so that you avoid massless particles
with spins above 2 which are problematic. In 10 dimensions, there are two
inequivalent chiralities of spinors, each of them having 16 real
components. If you want 32 supercharges, you either construct them from
two spinors of the opposite chiralities, or two spinors of the same
chirality.
Normally we would call these two possibilities (1,1) and (2,0)
supersymmetry, but in 10 dimensions, we call them type IIA and type IIB
supergravity, respectively (in both cases the massless multiplet must
contain the spin 2 graviton). Both these theories are found in string
theory - type IIA/IIB supergravity is the low energy limit of type IIA/IIB
superstring theory which we know pretty well. Nonperturbatively we find
out that type IIA string theory is really the 11-dimensional supergravity
on a circle (a 11-dimensional spinor decomposes into 10-dimensional
spinors of both chiralities) while type IIB string theory has SL(2,Z)
self-duality acting on the scalar fields (dilaton and axion).
There are two possibilities with 32 supercharges. What about 16
supercharges? The only possible supersymmetry is (1,0), i.e. type I. It
allows a gravitational supermultiplet, and it also allows gauge
supermultiplets. The gravitational multiplet itself is anomalous - some
hexagon (6 = 10/2 + 1, in the same way we have anomalous triangles in 4D
because 3 = 4/2 + 1) diagrams with a gravitino loop and 6 gravitons
attached should be zero by symmetries, but the divergences in the
low-energy effective field theory can't be simultaneously regularized to
give us zero. The pure type I supergravity in 10 dimensions has
gravitational anomalies - the general covariance is broken by quantum
effects.
In a similar way, you find out that the gauge multiplet has gravitational
as well as mixed (some bosons that are attached are gravitons, some of
them are gauge bosons) anomalies. Taking the Green-Schwarz mechanism
properly into account, one can see that all these anomalies cancel if the
gauge group is U(1)^496, U(1)^248 x E_8 - I started with these cases that
are most likely uninteresting physically - or E_8 x E_8 or SO(32). Let me
now exclude the many-U(1) solutions. The type I supergravity coupled to
E_8 x E_8 super Yang-Mills is the low-energy limit of one of the heterotic
string theories - that seems as M-theory on a line-interval at strong
coupling - and the SO(32) case is the low-energy limit of the other
heterotic theory as well as type I string theory (an orientifold of type
IIB whose strings are unoriented, and there are also open strings with
SO(32) Chan-Paton factors at the ends). Type I and SO(32) heterotic
theories are S-dual to one another, and they have the same low-energy
approximation.
Regardless whether we will have to apply the anthropic reasoning in 4D or
not, and regardless of all the technical difficulties with understanding
lower-dimensional vacua, I think that the picture of these 10 and
11-dimensional vacua is as clear as the Holy Trinity ;-). Much like Jesus
Christ looks like humans at low energies, already these 10 and
11-dimensional vacua share many important properties with the real world,
and unlike Jesus Christ, they allow us to make quantitative predictions
about these Universes above us. The richness of compactified stringy vacua
is impressive, too, but for me it is especially the miraculous clarity
(and unity) of these simple vacua in the highest possible dimensions that
makes me believe that string/M-theory is an exceptional structure that
must be studied.
> I realize that. Can we have a (closed) proper subgroup with dimension that
> is not smaller??
Nope :-), I just wanted to reiterate the same condition twice.
> First of all remind me what is the Kahler 2-form :-)
The metric of Kahler manifolds has a Hermitean structure, so to say:
ds^2 = omega_{i jBAR} dz^i dzBAR^jBAR
Note that the metric contains no (dz)^2 and (dzBAR)^2 terms. The
coefficients omega_{i jBAR} define the Kahler 2-form - more precisely
(1,1) form if you count the holomorphic and antiholomorphic indices
separately. The information about the metric of a Kahler manifold is
divided into two levels: first you say what is the complex structure -
i.e. how you can divide the coordinates to z and zBAR. Then you know that
the metric must only have the "mixed" terms and because of the complex
structure you know what "mixed" means, and you can determine what they
exactly are by the Kahler 2-form.
> Since SO(4) is (modulo global issues) SU(2) x SU(2) and, if the
> holonomy group is one of these SU(2), the other provides these
> transformations?
I think that the answer is Exactly!
> I.e. its total covariant derivative vanishes identically? What is the
> general definition of Killing spinor? I guess it doesn't have to be
> covariantly constant, since a Killing vector isn't always covariantly
> constant?
A Killing vector defines an infinitesimal coordinate transformations, and
if it is a Killing vector, it should keep the metric unchanged, and the
metric variation is the symmetric part of nabla_m vector_n. In the case of
Killing spinors, we are talking about an infinitesimal supersymmetry
transformation whose parameter is this spinor (that should be considered
an anticommuting object). For such a supertransformation to be a symmetry,
the variation of all the fields must vanish. The variation of the
gravitino field is proportional to the full (nabla_m spinor^a) - note that
the gravitino has one vector index (m) and one spinor index (a). There is
no symmetrization going on here - the only subtlety that we would
encounter is the special treatment of the trace - you may contract
(nabla_m spinor^a) with gamma^m_{a bDOT} to obtain another spinor_{bDOT} -
which is the dilatino and whose variation should also vanish anyway.
Therefore for spinors, the Killing spinor and the covariantly constant
spinor is the same thing.
Squark
"Aaron Bergman" <aber...@physics.utexas.edu> wrote in message
news:slrnbuqd7p....@cardinal1.Stanford.EDU...
> It's just a different possibility for holonomy -- I forget the
> origin of the names.
Okay, but which of them is the quaternionic analogue of a
Kahler manifold? I always thought it's the USp, but now it
turns out there's something slightly different called
"quaternionic Kahler".
> > First of all remind me what is the Kahler 2-form :-)
>
> g_{i\bar{j}} dz^i /\ d\bar{z}^\bar{j}
Okay, you guys managed to confuse me, now I need serious
de-confused. What I used to know is that a hermitean structure
can be decomposed into a real part which a metric and an
imaginary part which is the symplectic structure. Now this form
comes in. Is it different from the later? It's too late for my mind
to struggle with this formula.
Jeffery
>
> > M-theory on G2 manifolds,
>
> G2 manifolds as in manifolds with G2 symmetry? Where do they
> come from?
He means Joyce manifolds or manifolds with G_2 holonomy. Where they
come from is that we choose to use them because they are
7-dimensional, so if you take M-theory which is 11-dimensional, and
compactify 7 of the dimensions on a G_2 manifold, you end up with 4D
spacetime.
>
> > intersecting D-brane models
>
> Hmm, sounds interesting. With large extra dimensions, a la
> Rundal-Sundrum?
Intersecting brane models often construct non-supersymmetric Standard
Models with essentially the right spectrum. These particles are
vibrating strings whose two ends are attached to different D-branes
that intersect. Various particles are therefore forced to be localized
at different intersections. These models can again be realistic, and
they naturally lead to hierarchy of fermionic masses.
Jeffery Winkler
Lubos Motl
> Okay, but which of them is the quaternionic analogue of a
> Kahler manifold?
In some sense, the analogues of "U(N) Kahler" and "SU(N) Calabi-Yau i.e.
Kahler Ricci-flat" are "USp(2) x USp(d/2) quaternionic Kahler" and
"USp(d/2) hyperKahler i.e. Ricci-flat", in this order. Note that in both
cases, quaternionic *and* complex, there are two possibilities, and only
the more special one is Ricci-flat.
More generally, I don't understand why you think that everything about
these different manifolds should follow the same rules. Complex numbers
are not quaternions, and therefore you can't expect that all the analogies
will work without any subtleties!
> Okay, you guys managed to confuse me, now I need serious
> de-confused.
I don't think so. It seems that you were very confused before, and we with
Aaron have been de-confusing you for some time.
> What I used to know is that a hermitean structure
> can be decomposed into a real part which a metric and an
> imaginary part which is the symplectic structure.
Yours is a very non-standard (and counterproductive, as argued below) way
of parameterizing the data about the manifold, and if you make an internet
search about the "Hermitean structure" of "complex manifolds", you will
find your own previous and confusing discussions on SPR (with another
confused participant of this newsgroup) and almost nothing else. In other
words, you became a victim of your own confusion in the past. "Hermitean
structure" is usually associated with (linear) Hilbert spaces.
But yes, let me follow this non-standard approach to clarify what you
probably mean. If you look at one point of your complex manifold and
define the tangent space, there is a way to establish a Hermitean
(sesqui-linear) scalar product on this space between two tangent vectors
z_A, z_B:
b ( dz_{A}, dz_{B} ) = omega_{i jBAR} dz_{A}^i dz*_{B}^j
where * denotes the complex conjugation. The scalar product of dz_{A} with
itself is nothing else than the metric, and therefore the "Hermitean
structure" - i.e. the definition of the scalar product - contains the
information about the metric (plus something else). But because one wants
to view the tangent space as a complex linear space (and we also need to
know how to find the complex-conjugate vectors), the (structure needed for
the) scalar product also contains the information about the thing that you
call the "symplectic structure".
In this picture, you study one point only, and the "metric" contains less
information than the "scalar product": you must combine the metric with
the "symplectic structure" that knows how the real coordinates should be
paired into complex ones - and the whole information about the tangent
space at this single point can be called "Hermitean structure" or whatever
you like.
Now: the approach above is different from the usual terminology - and
thinking - in geometry. Real geometers always study the metric and other
data at the *whole* manifold: otherwise they would not be talking about a
general (or Calabi-Yau) manifold, but rather a linear space only. And the
Ricci-flat metric at the whole Calabi-Yau manifold knows everything
(unlike the "metric" at a single point that knows less that "everything"
at a single point) about all the data that we mentioned.
The metric at the whole manifold tells you what is the metric at each
point, of course. But it tells you even more.
The metric at the whole Calabi-Yau manifold has SU(d/2) holonomy, and
therefore it also tells you how to pair the coordinates into complex
coordinates at *each* point (there is one way only that is compatible with
the holonomy; more precisely there are two ways, you can also switch the
convention what you mean by holomorphic and what you mean by
antiholomorphic) - this pairing into complex coordinates is what you
called "symplectic structure", but it is exactly what is usually called
"complex structure". By "complex structure" we always mean that the
pairing of the coordinates into complex ones is defined for *all* points
of your manifold.
It is exactly the complex structure that we usually start with if we want
to determine the shape of the manifold. (More precisely, we start with the
dimension, then we say what is the topology, and then we define the
complex structure.) We don't say anything about the distances at the
beginning; we simply define how the real coordinates on the tangent space
at each point should be paired into complex coordinates. There are many
ways how to define a metric, but once we have defined the complex
structure, we only want to study the metrics that are of the Kahler form,
i.e. those that only contain the mixed holomorphic-antiholomorphic terms.
Then the (1,1) form that defines the metric is called the Kahler form. It
is a closed form, and locally it can be written as the derivative of a
Kahler potential (which implies that it is closed):
omega_{i jBAR} = del_{i} delBAR_{jBAR} K
The Kahler potential K is a scalar, a (0,0)-form. Let me emphasize again
that K is not globally defined.
Are you de-confused now?
Aaron Bergman
>
> Okay, you guys managed to confuse me, now I need serious
> de-confused. What I used to know is that a hermitean structure
> can be decomposed into a real part which a metric and an
> imaginary part which is the symplectic structure. Now this form
> comes in. Is it different from the later? It's too late for my mind
> to struggle with this formula.
The Kaehler form and the symplectic structure are the same thing.
Lubo? Motl
> The Kaehler form and the symplectic structure are the same thing.
My comment was also oversimplified, but this one is simplified, too.
Let me start with an N-dimensional complex vector spaces with a scalar
product - imagine that it is the tangent space at one point of the
manifold. The scalar product of u,v is a complex number. Its real part
is exactly what we would call the scalar product of two real
2N-dimensional vectors. The imaginary part of this scalar product is
exactly what defines the symplectic structure on this space.
The unitary group U(N) is the group that preserves the whole complex
scalar product. In other words, it must preserve both the real part,
as well the imaginary part. The real part is preserved by O(2N) while
the imaginary part is preserved by Sp(2N,R), and in this sense U(N) is
the intersection of O(2N) and Sp(2N,R): the Kahler form at one point -
the whole complex scalar product - is the union of the data in the
real metric AND the symplectic structure. I think that this is how
Squark is looking at it.
You see that the symplectic structure is a weakened form (the
imaginary part) of the Kahler form that defines the complex scalar
product. Once you know the real metric on the tangential space, you
still don't know how to pair the real coordinates into complex ones -
and this extra information can be provided both by the complex
structure as well as the symplectic structure or the Kahler form.
Once we study the complex manifold globally, I think that we agree
that there is no useful notion of "symplectic moduli", or is there
one? It only makes sense to talk about the Kahler moduli, or about the
complex structure moduli, but they can't be mixed in any useful way.
Happy New Year
Lubos
Squark
> Do you mean the Higgs mechanism and the "different" vacua at the bottom of
> the Mexican hat potential left behind? They are not different at all! By
> gauge symmetry, all these vacua are physically equivalent.
This is in case you have a gauge symmetry. Two CYs that can are "smoothly
connected" don't have to be related by gauge symmetry.
> ...When you
> summarized, you ignored the subtleties such as the quaternionic Kahler
> manifolds - which have a non-simple holonomy group :-)
Sorry, did I say "simple"? I meant semi-simple of course! Product
manifolds already give you non-simple stuff. Also, Aaron pointed out U(n)
is not simple: and it's in fact not even semi-simple. Okay, correction then:
the holonomy group is the product of a semi-simple one with copies of
U(1), s.t. there are no F4, Ex factors. That is only true when you ignore
global issues though. In short, life insists to be complicated...
> ...Taking the Green-Schwarz mechanism
> properly into account, one can see that all these anomalies cancel if the
> gauge group is U(1)^496, U(1)^248 x E_8 - I started with these cases that
> are most likely uninteresting physically - or E_8 x E_8 or SO(32). Let me
> now exclude the many-U(1) solutions.
Well, you don't exclude them you show they correspond to no string theories.
Is there a fundumental reason for that? Do we really know there are no such
string vacua?
> Much like Jesus
> Christ looks like humans at low energies, already these 10 and
> 11-dimensional vacua share many important properties with the real world
So "M" also stands for Mary now? I hope you won't try to convince me
M-theory is a virgin, it hardly looks innocent. Especially after all of what
was
done to it during the years...
> The metric of Kahler manifolds has a Hermitean structure, so to say:
>
> ds^2 = omega_{i jBAR} dz^i dzBAR^jBAR
>
> Note that the metric contains no (dz)^2 and (dzBAR)^2 terms. The
> coefficients omega_{i jBAR} define the Kahler 2-form
I'll just reiterate what I asked Aaron: and is this form not a symplectic
structure? Not the symplectic structure people talk about when
discussing Kahler manifolds?
Lubos Motl
On Fri, 26 Dec 2003, Aaron Bergman wrote:
> So, let me first confess that I haven't read the paper yet, but
> from the second had stuff I've heard about it, I'd read those
> quotes around 'almost' as scare-quotes. [KKLT "almost" proved...]
There is no full proof that a Calabi-Yau with the properties that they
need exists, and it has also not been proved that all conceivable decay
channels - that could make such states highly unstable - have been
checked. There are potentially many other similar states into which one
could decay, and all possible connections between these states must be
known before one can make the final verdict about the lifetime.
The KKLT result is a very suggestive argument, but it is far from being a
well-established result - as well-established as various facts that
appeared during the Duality Revolution. Also, I think that it is clear
that the KKLT paper was constructed in order to support the pre-determined
idea about the string landscape. If four smart physicists try to find as
a convincing argument as possible - and they can rely on our incomplete
knowledge of various things - it is not so surprising that they can
construct a rather convincing argument even if it turns out to be wrong.
> IIRC, that's RS1, not RS2. They get rid of the brane in RS2,
> right, to get AdS space with infinite extent but finite volume or
> something like that.
Yes, but no RS model can be really called "large extra dimensions": in the
RS2 case it is essentially "an infinite large dimension". Lisa remembers
that a few years ago, she was asked to talk about the "large extra
dimensions". It was surprising to her because she was supposed to talk
about all extra-dimensional models except for hers. ;-)
Lubos Motl
> This is in case you have a gauge symmetry. Two CYs that can are "smoothly
> connected" don't have to be related by gauge symmetry.
That's right (and I hope that I have not written the converse). But it
does not change the fact that if they're smoothly connected, there is no
exponential supression of the transition (no nontrivial action of an
instanton).
> Sorry, did I say "simple"? I meant semi-simple of course!
Which is also slightly incorrect, as you later realized; U(N) is not even
semisimple. ;-)
> ...is not simple: and it's in fact not even semi-simple. Okay,
> correction then: the holonomy group is the product of a semi-simple
> one with copies of U(1), s.t. there are no F4, Ex factors. That is
> only true when you ignore global issues though. In short, life insists
> to be complicated...
Your new description is still far from being the exact one (especially
because you can obtain the product holonomies simply if you consider
Cartesian products of the manifolds), and it might be a good point to stop
the attempts to simplify the list of possible holonomies. ;-)
> Well, you don't exclude them you show they correspond to no string theories.
> Is there a fundumental reason for that? Do we really know there are no such
> string vacua?
Good question. The answer is: Not really, but the appearance of U(1)^{496}
is unimpressive in the sense that U(1)^496 is a contraction of E8 x E8 or
SO(32). Because it has the same dimension, it cancels the gravitational
anomalies, and because it is Abelian (and everything comes in the
adjoint), there are no gauge anomalies either. We don't know of any string
theory in 10D that would lead to U(1)^496 or E8 x U(1)^248 gauge groups,
and with our current knowledge of string theory, these options look
bizarre, but it is of course very far from a proof that they don't exist
and similar "impossible" things have become possible, after some twists
and turns, many times in history.
> So "M" also stands for Mary now? I hope you won't try to convince me
> M-theory is a virgin, it hardly looks innocent. Especially after all
> of what was done to it during the years...
There is a secret 12-dimensional F(ather)-theory, so Mary is not so
innocent. A puzzling feature is that the Mother is Male while the Father
is Female. Their children are the type II string theories in 10
dimensions which are younger (10 < 11 < 12). The relation between these
theories is T-duality i.e. they are sibblings.
> I'll just reiterate what I asked Aaron: and is this form not a symplectic
> structure? Not the symplectic structure people talk about when
> discussing Kahler manifolds?
The symplectic structure is the imaginary part of the complex scalar
product (on the tangent space) that you can obtain from the Kahler form.
Best wishes
John Baez
Merry Christmas!
Lubos Motl <mo...@feynman.harvard.edu> wrote:
>Squark wrote:
>> 3) In Calabi-Yau compactifications, a complex structure somehow magically
>> arises on the compactified space. Why does that happen?
>The origin of the complex structure might be magical for a physicist, but
>certainly not for a mathematician because they defined the Calabi-Yau
>space to be a Kahler manifold (which automatically implies that it must
>have a complex structure) with a vanishing first Chern class (imagine that
>it has no nontrivial holomorphic 1-cycles). Eugenio Calabi conjectured in
>1957 that each such a manifold can be given a Ricci-flat metric of SU(3)
>holonomy (which means that by parallel transport, you can't rotate a
>vector by an arbitrary rotation in SO(6), but only by a rotation in the
>SU(3) subgroup of SO(6)). The Kahler manifolds must have a U(3) holonomy,
>and the Ricci-flatness implies that the holonomy must be inside the SU(3)
>subgroup. This conjecture about the existence of the Ricci-flat metric was
>proved in 1977 by Shing-Tung Yau, and therefore the manifolds are called
>Calabi-Yau manifolds.
>
>The condition of Ricci-flatness is derived from the Einstein-like
>equations of motion (that follow from conformal symmetry of the nonlinear
>sigma model that describes the propagation of strings on the curved target
>space), and the related, but independent condition of SU(3) holonomy is
>equivalent to the unbroken 1/4 of the original supersymmetry (note that
>SU(4)=SO(6) is the most general holonomy, and if we reduce it to SU(3), 1
>component out of the 4 spinor components that generate the SU(4) is
>constant, and therefore it can defined a covariantly constant spinor -
>which gives rise to unbroken supersymmetry).
You allude to some difficult stuff in the first paragraph of your
reply - difficult stuff like Yau's proof of Calabi's conjecture -
but I hope Squark realizes that the answer to his question is not so
difficult - it's lurking in the *second* paragraph of your reply.
Let me expand on it a bit:
How does a complex structure "magically" arise on a Calabi-Yau manifold?
The answer is that for parallel translation on a 6d Riemannian manifold
to preserve a nonzero spinor field is the same as for it to preserve a
complex structure! In general, parallel translation on a 6d Riemannian
manifold gives holonomies in SO(6). Parallel transport of spinors on a
6d Riemannian manifold with spin structure gives holonomies in the double
cover of SO(6), which is SU(4) - this should not be shocking, since spinors
in 6 dimensions live in C^4. For there to be a covariantly constant nonzero
spinor field, these holonomies must preserve a nonzero vector in C^4. So,
they can't be arbitrary elements of SU(4); they have to live in an SU(3)
subgroup. But, SU(3) is just the subgroup of SO(6) that preserves a
complex structure! After all, "preserving a complex structure" is just
another name for "being complex-linear", so we have to be in SU rather
just SO.
Here is a more drawn-out version of the tale, taken from "week190".
At the end you'll see that Squark and I were puzzled by this before,
and Marc Nardmann put our doubts to rest.
......................................................................
The key principle to keep in mind is that any type of structure you
can put on a real inner product space yields a type of Riemannian
manifold. Each tangent space of a Riemannian manifold is a real inner
product space, and there's a god-given way to parallel transport tangent
vectors on a Riemannian manifold. So, if X is some type of structure
you can put on a real inner product space, you can define an "X-manifold"
to be a Riemannian manifold where each tangent space has an X-structure...
in a way that's preserved by parallel transport!
For example, X could be a "Hermitian structure" - a way of making a
real inner product space into a *complex* inner product space. Then
an X-manifold is called a "Kaehler manifold".
When we parallel transport a vector around a loop in a n-dimensional
Riemannian manifold, it can be rotated or reflected. In more jargonesgue
jargon, the holonomy around a loop defines an element of the group O(n).
But when your manifold is a Kaehler manifold, each tangent space becomes
a complex inner product space of dimension n/2, in a way that's preserved
by parallel transport. So, the holonomy around any loop must lie in
the unitary group U(n/2).
There's a converse to this, as well! So a Kaehler manifold is just a
Riemannian manifold where the holonomies all lie in U(n/2).
And this is how it usually works - or *always*, if you take care
to include all the necessary fine print. Thus many sorts of
X-manifolds are called "manifolds with special holonomy". See:
8) Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford U.
Press, Oxford, 2000.
For example, suppose X is a "quaternionic structure" - a way of making
a real inner product space into a quaternionic inner product space.
Then an X-manifold is called a "hyperKaehler manifold", and this
just one where the holonomies lie in the quaternionic unitary group Sp(n/4).
Or, suppose X is a Hermitian structure together with an n/2-form.
Then an X-manifold is called a "Calabi-Yau manifold". This concept
of Calabi-Yau manifold works in any even dimension, while before I was
just talking about 6-dimensional ones! For parallel transport around
a loop to preserve an n/2-form as well as a Hermitian structure,
the holonomy must lie in SU(n/2). So, a Calabi-Yau manifold is the
same as one where the holonomies lie in SU(n/2).
We can define G2-manifolds in a similar way. But to do this, and
to see how they're related to 6-dimensional Calabi-Yau manifolds,
we need a detour into the theory of spinors. The reason is that
"N = 1 supersymmetric theories" work nicely when you can pick a
spinor at each point of space in a way that's preserved by parallel
transport. We call such a thing a "covariantly constant spinor
field". Actually, this spinor field needs to be nonzero to be
of any use, but that's so obvious people often don't mention it.
Now, a nonzero spinor isn't exactly an extra structure you can put
on a real inner product space, since spinors are representations
not of O(n) or even SO(n) but of the double cover Spin(n).
However, if you start with a *spin* manifold, you can think of a
nonzero covariantly constant spinor field as some extra structure
that reduces the holonomy group from Spin(n) down to some subgroup.
So, let's see what this extra structure is like in some examples!
For the examples I'll talk about, the key is that spinors in 5-, 6-, 7-
and 8-dimensional space are all very related, and all very related to
the octonions. You can see this from looking at the even part of
the Clifford algebra, because spinors are defined to be irreducible
representations of this algebra. Here's what the even part of the
Clifford algebra looks like in various dimensions:
dimension 1: R
dimension 2: C
dimension 3: H
dimension 4: H + H
dimension 5: H(2)
dimension 6: C(4)
dimension 7: R(8)
dimension 8: R(8) + R(8)
Here K = R, C, H stands for the real numbers, complex numbers and
quaternions, while K(n) means n x n matrices with entries in K.
I'll always be interested in *real* spinors, which are the irreducible
*real* representations of these algebras. I won't even keep saying
the word "real" from now on. If you eyeball the above chart, you'll
see that in dimensions 4 and 8 we get two kinds of spinor - called
left- and right-handed spinors - while in the other dimensions there's
just one kind. The way these spinors work is sort of obvious:
dimension 1: R
dimension 2: C
dimension 3: H
dimension 4: left and right, both H
dimension 5: H^2
dimension 6: C^4
dimension 7: R^8
dimension 8: left and right, both R^8
Now the cool part is that H^2, C^4 and R^8 are all secretly the
same 8-dimensional real vector space equipped with various
amounts of extra structure - i.e. the structure of a 4-dimensional
complex vector space, or a 2-dimensional quaternionic vector space.
And you'll probably be more bored than shocked when I tell you that
this 8-dimensional real vector space is yearning to become the OCTONIONS.
Let's see how we can use this to study specially nice manifolds
in 8, 7, 6 and 5 dimensions. We'll start in dimension 8 and climb
our way down by a systematic process. In 7 dimensions we'll get G2
manifolds, while in 6 dimensions we'll get Calabi-Yau manifolds.
Okay:
In 8 dimensions there are three different 8-dimensional irreps of the
spin group (the double cover of the rotation group):
the vector rep V
the left-handed spinor rep S+
the right-handed spinor rep S-
You can build a vector from a left-handed spinor and a
right-handed spinor, so we have an intertwining operator:
S+ tensor S- -> V
The cool part is that this map tells us how to multiply octonions!
More precisely, suppose we pick a unit vector 1+ in S+ and a unit vector
1- in S-. It turns out that multiplying by 1+ defines an isomorphism
from S- to V. Similarly, multiplying by 1- gives an isomorphism from
S+ to V. This lets us think of all three spaces as the same: THE OCTONIONS,
with m as the octonion product and 1+ (or 1- if you prefer) as its unit.
In fact, there's nothing special about writing our operator as
S+ tensor S- -> V
since all three of these reps are their own dual. This lets us
permute these guys and work with
V tensor S+ -> S-
or whatever we like. So, picking unit vectors in any 2 out of these
3 spaces gives us a unit vector in the third and makes all 3 into an
algebra isomorphic to the octonions.
This instantly implies that if we have an 8-dimensional spin manifold M
with nonzero covariantly constant sections of 2 of these 3 bundles:
the left-handed spinor bundle
the right-handed spinor bundle
the tangent bundle
we get a way to make all 3 of these bundles into "octonion bundles" -
meaning that each fiber is an algebra in a covariantly constant way,
where this algebra is isomorphic to the octonions.
This in turn implies that the holonomy group of the metric on M must
be a subgroup of G2 - the automorphism group of the octonions.
Let's call a manifold like this M an "octonionic manifold".
How do we get manifolds like this?
The easiest way is to take a 7-dimensional spin manifold N and let
M = N x R. The special 8th direction in M gives us a nonzero
covariantly constant vector field on M. So, to get the above
"2 out of 3" trick to work, we just need a nonzero covariantly
constant section of either the left- or right-handed spinor bundle
of M.
But as we've seen, spinors in 7 dimensions are secretly the same
as either left- or right-handed spinors in 8 dimensions. So, it
suffices to have a nonzero covariantly constant spinor field on N.
Thus, when N is a 7-dimensional spin manifold with a nonzero covariantly
constant spinor field, its spinor bundle automatically becomes an octonion
bundle!
Its tangent bundle doesn't become an octonion bundle, because it's
just 7-dimensional. But if you think about what I've said, you'll
see the tangent bundle plus a trivial line bundle becomes an octonion
bundle. This trivial line bundle corresponds to the *real* octonions,
while the tangent bundle of N corresponds to the *imaginary* octonions.
The imaginary octonions are 7-dimensional, and they have a "dot product"
and "cross product" rather like those in 3 dimensions. Since you
can use these to recover the octonion product, the group of
transformations of the imaginary octonions preserving the dot product
and cross product is again G2.
So, the tangent bundle of N becomes an "imaginary octonion bundle",
meaning that each fiber gets a dot product and cross product in a
covariantly constant way, making it isomorphic to the imaginary octonions.
This in turn implies that the holonomy group of the metric on N must
be a subgroup of G2.
People call a manifold like this N a "G2 manifold".
How do we get manifolds like this?
The easiest way is to take a 6-dimensional spin manifold O and let
N = O x R. To make N into a G2 manifold, we need a nonzero covariantly
constant spinor field on N.
But as we've seen, spinors in 6 dimensions are secretly the same
as spinors in 7 dimensions. So, it suffices to have a nonzero
covariantly constant spinor field on O.
Thus, when O is a 6-dimensional spin manifold with a nonzero
covariantly constant spinor field, its spinor bundle automatically
becomes an octonion bundle!
Its tangent bundle doesn't become an imaginary octonion bundle,
because it's just 6-dimensional. But if you think about what I've
said, you'll see the tangent bundle plus a trivial line bundle
becomes an imaginary octonion bundle. This trivial line bundle
corresponds to a particular direction in the imaginary octonions.
This in turn implies that the holonomy group of O must lie in the
subgroup of G2 fixing a direction in the imaginary octonions.
This subgroup is SU(3), so the holonomy group of O must be a
subgroup of SU(3).
People call a manifold like this O a "Calabi-Yau manifold".
How do we get manifolds like this?
The easiest way is to take a 5-dimensional spin manifold P and let
O = P x R. To make O into a Calabi-Yau manifold, we need a nonzero
covariantly constant spinor field on O.
But as we've seen, spinors in 5 dimensions are secretly the same as
spinors in 6 dimensions. So, it will suffice to have a nonzero
covariantly constant spinor field on P.
Thus, when P is a 5-dimensional spin manifold with a nonzero
covariantly constant spinor field, its spinor bundle automatically
becomes an octonion bundle!
Its tangent bundle doesn't become an imaginary octonion bundle,
because it's just 5-dimensional. But if you think about what I've
said, you'll see the tangent bundle plus two trivial line bundles
becomes an imaginary octonion bundle. These trivial line bundles
correspond to two orthogonal directions in the imaginary octonions.
This in turn implies that the holonomy group of P must lie in the
subgroup of G2 fixing two orthogonal directions in the imaginary
octonions. This subgroup is SU(2).
I'll call a manifold like this P an "SU(2) manifold".
Does my prose style seem stuck in a loop? That's on purpose;
I'm trying to make a certain pattern very clear. But the loop stops
here, or at least changes flavor drastically, because spinors stop
being 8-dimensional when we get down to 4-dimensional space.
Summary:
A) When M is an 8-dimensional spin manifold with 2 out of these 3 things:
a nonzero covariantly constant vector field
a nonzero covariantly constant left-handed spinor field
a nonzero covariantly constant right-handed spinor field
it automatically gets all three - and its tangent bundle,
left-handed spinor bundle and right-handed spinor bundle all
become octonion bundles. We call M an octonionic manifold.
B) When N is a 7-dimensional spin manifold with a nonzero
covariantly constant spinor field, its spinor bundle becomes an
octonion bundle, while its tangent bundle becomes an imaginary
octonion bundle. We call N a G2 manifold.
C) When O is a 6-dimensional spin manifold with a nonzero
covariantly constant spinor field, its spinor bundle becomes an
octonion bundle, while its tangent bundle plus a trivial line bundle
becomes an imaginary octonion bundle. We call O a Calabi-Yau
manifold.
D) When P is a 5-dimensional spin manifold with a nonzero
covariantly constant spinor field, its spinor bundle becomes an
octonion bundle, while its tangent bundle plus two trivial line
bundles becomes an imaginary octonion bundle. We call O an
SU(2) manifold.
An SU(2) manifold times R is a Calabi-Yau manifold;
a Calabi-Yau manifold times R is a G2 manifold;
a G2 manifold times R is an octonionic manifold.
You may not like how item A) is different from the rest.
Don't worry; people also study 8-dimensional spin manifolds
that admit just a nonzero covariantly constant left-handed *or*
right-handed spinor field. The holonomy group of such a manifold
must like in Spin(7), and such a manifold is called a Spin(7) manifold.
You may wonder how I knew that the subgroup of G2 fixing
one direction in the imaginary octonions is SU(3). You may
also wonder how I knew that the subgroup of G2 fixing two
orthogonal directions in the imaginary octonions is SU(2).
This is very pretty! I mainly just used two facts we've already
seen: the even part of the Clifford algebra in 6 dimensions is C(4),
while in 5 dimensions it's H(2).
The first of these facts implies that so(6) must sit inside
the traceless skew-adjoint matrices in C(4). In other words,
so(6) sits inside su(4). But
dim(so(6)) = dim(su(4)) = 15
so in fact so(6) = su(4). Indeed, SU(4) is the double
cover of SO(6), and it acts on the space of spinors, C^4,
in the obvious way. The subgroup fixing a unit spinor is
thus SU(3).
The second of these facts implies that so(5) must sit inside
the traceless skew-adjoint matrices in H(2). In other words,
so(5) sits inside sp(2). But
dim(so(5)) = dim(sp(2)) = 10
so in fact so(5) = sp(2). Indeed, Sp(2) is the double
cover of SO(5), and it acts on the space of spinors, H^2,
in the obvious way. The subgroup fixing a unit spinor is
thus Sp(1)... which being the unit quaternions, is isomorphic to SU(2).
If you think about it a while, these results do the job.
-----------------------------------------------------------------------
Addendum: My definition of "Kaehler manifold" above was a bit
nonstandard. For a while, some of us on sci.physics.research started
worrying that it wasn't equivalent to the usual one! Luckily, it
turns out that it is. Here is some of our discussion of this issue.
John Baez wrote:
Squark wrote:
>John Baez wrote:
>> [Moderator's note: a Kaehler manifold has to be complex, not
>> just "almost complex". - jb]
>That's precisely my problem. You said that putting a Hermitian
>structure of the tangent space of a real manifold at each point
>(putting it on the tangent bundle, more accurately) makes it into
>a Kaehler manifold.
No, I did not say this! I'll remind you of what I actually said.
>However, there's the additional condition of the almost complex
>structure resulting on the manifold being an actual complex
>structure. This cannot be ensured on the "point level", i.e. it is
>not enough to speak of the kind of structure you put on the tangent
>space at each point, but it's important how those structures "glue
>together" (except the obvious smoothness part).
Right - in math jargon, we need some "integrability conditions" to
ensure that the complex structures on each tangent space fit
together to make each little patch of the manifold look like C^n.
Only then do we get a complex manifold. Otherwise we just have an
"almost complex manifold".
I didn't ignore this issue, but now you've got me worried
that I may not have handled it correctly. Here's what I wrote:
The key principle to keep in mind is that any type of structure you
can put on a real inner product space yields a type of Riemannian
manifold. Each tangent space of a Riemannian manifold is a real
inner product space, and there's a god-given way to parallel
transport tangent vectors on a Riemannian manifold. So, if X is
some type of structure you can put on a real inner product space,
you can define an "X-manifold" to be a Riemannian manifold where each
tangent space has an X-structure... in a way that's preserved by
parallel transport!
For example, X could be a "Hermitian structure" - a way of making a
real inner product space into a *complex* inner product space. Then
an X-manifold is called a "Kaehler manifold".
See?
I didn't say an X-manifold was a Riemannian manifold where
each tangent space is given a structure of type X.
I said it was a Riemannian manifold on which each tangent space is
given a structure of type X... IN A WAY THAT'S PRESERVED BY PARALLEL
TRANSPORT!
If I had left out that last clause, I'd obviously be in trouble.
This last clause is the only condition that relates what's going
on at different tangent spaces.
In particular, if X = "a Hermitian structure", an X-manifold is a
Riemannian manifold where each tangent space is equipped with a
complex structure J and a complex inner product h whose real part
is the original Riemannian inner product... such that h and J are
preserved by parallel transport.
I was hoping this definition is equivalent to the usual ones.
Now you've got me nervous... after all, before I can flame you
for misunderstanding me, I should be sure what I actually said
is right! :-)
My definition is conceptually simple, but it contains
some redundancy... let's squeeze that out and see what's left.
We start with an X-manifold where X = "a hermitian structure".
Each tangent space has a complex inner product h,
whose real part g is the original Riemannian metric,
and whose imaginary part we call w:
h = g + iw
Each tangent space also has a complex structure J on it.
We want all this stuff to be preserved by parallel transport.
So, at first it seems like we have 3 integrability conditions:
g, w, and J are preserved by parallel transport
But g is automatically preserved by parallel transport - that's
how the Levi-Civita connection is defined!
So, there are really just 2 integrability conditions:
w and J are preserved by parallel transport.
But we can always recover the imaginary part of the inner
product from its real part together with the complex structure:
w(u,v) = -g(u,Jv)
So, there is really just one integrability condition:
J is preserved by parallel transport.
Now, how does this compare to other definitions of Kaehler
manifold? Marc Nardmann wrote:
> I assume that you know
>
> (#) that every hermitian metric h on a complex manifold X has a
> decomposition h = g+iw, where g is a Riemannian metric on X_R
> (and X_R is the smooth manifold X without its complex structure),
> and w is a 2-form on X_R;
>
>(#) and that each of h,g,w determines the other two via the R-vector
> bundle morphism J: T(X_R) --> T(X_R) given by Jv = iv (where the
> holomorphic tangent bundle TX is canonically identified as a real
> vector bundle with T(X_R)). E.g. g(u,v) = w(u,Jv) up to a sign that
> depends on our definition of hermiticity.
>
> The hermitian metric h = g+iw is K"ahler
> if and only if w is closed,
> if and only if J, viewed as a real (1,1)-tensor field on X, is
> parallel with respect to the Levi-Civita connection of g.
It sounds like he's saying that a Kaehler manifold is a *complex*
manifold for which J is preserved by parallel transport. My proposed
definition is close, but it doesn't contain the crucial word *complex*.
Can we safely leave it out? I.e., is any almost complex Riemannian
manifold for which J is preserved by parallel transport automatically
complex???
I don't know. So, now I'm nervous.
I could try to show by a calculation that if J has vanishing
covariant derivative, it satisfies the integrability condition
that forces it to be a complex structure:
[Ju,Jv] - [u,v] - J[u,Jv] - J[Ju,v] = 0
However, I'm too lazy! I'm hoping Marc Nardmann or someone
will step in with either the necessary theorem, or a counterexample.
Btw, there is such a thing as an "almost Kaehler manifold",
which is an almost complex manifold where each tangent space
is equipped with a complex inner product h = g+iw such that the
imaginary part w is a closed 2-form. But, I don't see why
the existence of these things serves as a counterexample to
my hope.
Then Marc Nardmann confirmed my hope: any almost complex Riemannian
manifold for which J is preserved by parallel transport is
automatically complex, and thus a Kaehler manifold. He wrote
(in part):
John Baez wrote:
> It sounds like he's saying that a Kaehler manifold is a *complex*
> manifold for which J is preserved by parallel transport.
Yes. I forgot to discuss this issue in the post you're citing here. In
the stringy context, there's initially just the Riemannian metric, so it
is important to know how e.g. a holonomy condition implies the existence
of a complex structure, as opposed to a mere almost complex structure.
Let's see:
> My proposed definition
> is close, but it doesn't contain the crucial word *complex*.
>
> Can we safely leave it out? I.e., is any almost complex Riemannian
> manifold for which J is preserved by parallel transport automatically
> complex???
>
> I don't know. So, now I'm nervous.
>
> I could try to show by a calculation that if J has vanishing
> covariant derivative, it satisfies the integrability condition
> that forces it to be a complex structure:
>
> [Ju,Jv] - [u,v] - J[u,Jv] - J[Ju,v] = 0
>
> However, I'm too lazy!
It's very easy, so even laziness is no excuse :-). The hard part is
contained in the theorem you're citing here: that an almost complex
structure comes from a complex structure (which is then uniquely
determined) if (and only if) [Ju,Jv] - [u,v] - J[u,Jv] - J[Ju,v] = 0
for all vector fields u,v (in fact, the LHS of the equation is
tensorial, hence well-defined for *vectors*).
We need only the fact that the Levi-Civita connection is torsion-free:
[Ju,Jv] - [u,v] - J[u,Jv] - J[Ju,v]
= nabla_{Ju}Jv -nabla_{Jv}Ju -nabla_{u}v +nabla_{v}u
-J(nabla_{u}Jv -nabla_{Jv}u) -J(nabla_{Ju}v -nabla_{v}Ju)
= J(nabla_{Ju}v) -J(nabla_{Jv}u) -nabla_{u}v +nabla_{v}u
+nabla_{u}v +J(nabla_{Jv}u) -J(nabla_{Ju}v) -nabla_{v}u
= 0 .
Lubos Motl
I apologize but you are still confused. You summarized your own
presentation as follows:
> For there to be a covariantly constant nonzero spinor field, these
> holonomies must preserve a nonzero vector in C^4. So, they can't be
> arbitrary elements of SU(4); they have to live in an SU(3) subgroup.
> But, SU(3) is just the subgroup of SO(6) that preserves a complex
> structure! After all, "preserving a complex structure" is just
> another name for "being complex-linear", so we have to be in SU rather
> just SO.
No, this is not correct. The subgroup of SO(6) that preserves a complex
structure is U(3), not SU(3). If you pair six real coordinates into three
complex ones, you obtain a complex structure - which is associated with a
U(3) subgroup of SO(6). This complex structure does not have to preserve
any spinors. Indeed, the manifolds whose holonomy preserves the complex
structure are called the (complex) Kahler manifolds, their holonomy is
U(3) and most of them preserve *no* spinors.
Only a special subclass of Kahler manifolds - namely the Ricci-flat Kahler
manifolds - preserve a spinor. The condition of Ricci-flatness of a Kahler
manifold is equivalent to the holonomy being inside the SU(3) subgroup,
which is equivalent to an unbroken (one quarter of) supersymmetry (a
spinor is preserved). The manifolds of SU(3) holonomy are called the
Calabi-Yau manifolds.
It is true that the unbroken supersymmetry (the existence of a covariantly
constant spinor) - the material of my 2nd paragraph that you seem to like
- implies that the manifold has a complex structure, but it is certainly
not a necessary condition. The real answer to the question "where does the
complex structure of a Calabi-Yau come from" is in my 1st paragraph - in
fact, the Calabi-Yau spaces are *defined* as the complex Kahler manifolds
with a vanishing first Chern class. It is the first part of the definition
- complex Kahler manifolds - that guarantees the existence of a complex
structure. The second part of the definition is nice, and (assuming the
first one) it is equivalent to the SU(3) holonomy or to the unbroken
supersymmetry, but the second part has nothing to do with the existence of
the complex structure!
Even the correct direction of your (incorrect) equivalence - the statement
that a covariantly constant spinor implies the existence of the complex
structure of a Calabi-Yau three-fold - does not generalize to other
dimensions. The real 8-dimensional manifolds that preserve a spinor (in
the 8-dimensional Weyl spinor representation of spin(8)) are generically
the special manifolds of spin(7) holonomy that *don't* have any complex
structure. Yet, *all* Calabi-Yau manifolds in all dimensions have a
complex structure, because this is the first of the constraints that
define them!
All the best
R.X.
Lubos Motl <mo...@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.031228...@feynman.harvard.edu>...
> On 26 Dec 2003, Squark wrote:
>
>
>
> > Btw, are there known mirror self-dual CYs?
>
> Very good question. I don't know, but there are articles that mention it:
>
> http://www.google.com/search?q=%22mirror+self-dual%22+calabi-yau&num=100&filter=0
>
Hyperkaehler manifolds such as K3, or tori are trivially self-mirror.
But to my knowledge there is at least one good honest Calabi-Yau
(ie., with strict SU(3) holonomy) which is self-mirror, and it has
h21=h11=11. Correspondingly there are no world-sheet instanton
corrections to its moduli space, which is thus classical.
Aaron Bergman
In article <b034b730.03123...@posting.google.com>, Lubo? Motl wrote:
> Aaron Bergman <aber...@physics.utexas.edu> wrote:
>
>> The Kaehler form and the symplectic structure are the same thing.
>
> My comment was also oversimplified, but this one is simplified, too.
Kaehler manifolds are also symplectic manifolds. The Kaehler form
is a closed 2-form that is nondegenerate because it's just the
metric in disguise. That's all I meant.
> The unitary group U(N) is the group that preserves the whole complex
> scalar product. In other words, it must preserve both the real part,
> as well the imaginary part. The real part is preserved by O(2N) while
> the imaginary part is preserved by Sp(2N,R), and in this sense U(N) is
> the intersection of O(2N) and Sp(2N,R): the Kahler form at one point -
> the whole complex scalar product - is the union of the data in the
> real metric AND the symplectic structure. I think that this is how
> Squark is looking at it.
The definition that Squark is using is the same one from
Nakahara, I think. I don't have it on me, so I don't want to
quote from it.
[...]
> Once we study the complex manifold globally, I think that we agree
> that there is no useful notion of "symplectic moduli", or is there
> one? It only makes sense to talk about the Kahler moduli, or about the
> complex structure moduli, but they can't be mixed in any useful way.
If you just forget about everything else, there's certainly a
global nature of a symplectic manifold and a moduli space for
symplectic structures. For a Kaehler manifold, you have the
metric g(.,.), the Kaehler form w(.,.) and the complex structure
J(.) which are all tied together by, IIRC,
w(X,Y) = g(J(X),Y)
or something like that. So, you can pick any two to be your
independent structures, I think.
Serenus Zeitblom
> Dear John,
>
> I apologize but you are still confused. You summarized your own
> presentation as follows:
>
> > For there to be a covariantly constant nonzero spinor field, these
> > holonomies must preserve a nonzero vector in C^4. So, they can't be
> > arbitrary elements of SU(4); they have to live in an SU(3) subgroup.
> > But, SU(3) is just the subgroup of SO(6) that preserves a complex
> > structure! After all, "preserving a complex structure" is just
> > another name for "being complex-linear", so we have to be in SU rather
> > just SO.
What JB was saying here was simply that the correct answer
to squark's question: where does the complex structure
come from?" is " it comes from the parallel spinor". The
latter reduces the holonomy from SU(4) to SU(3), but SU(3)
is a subgroup of U(3), therefore the manifold is Kaehler.
By the way, somebody once told me something that I don't
completely understand, but I will repeat it here anyway:
to go in the other direction, to deduce that there is a
parallel spinor, you just have to know [a] that the *algebra*
of the holonomy group is su(3) and [b] that your manifold
is 6D, compact and orientable. [I'm using small letters
for algebras] You don't need to know
the *group*. This is very non-trivial, because for example
you could never tell that a 4D manifold is spin just by
knowing that the algebra of its holonomy is su(2). Seems
that this is some very special property of 6D Calabi-Yaus.
Which brings us back to the title.
John Baez
Lubos Motl <mo...@feynman.harvard.edu> wrote:
>John Baez wrote:
>> For there to be a covariantly constant nonzero spinor field, these
>> holonomies must preserve a nonzero vector in C^4. So, they can't be
>> arbitrary elements of SU(4); they have to live in an SU(3) subgroup.
>> But, SU(3) is just the subgroup of SO(6) that preserves a complex
>> structure! After all, "preserving a complex structure" is just
>> another name for "being complex-linear", so we have to be in SU rather
>> just SO.
>No, this is not correct. The subgroup of SO(6) that preserves a complex
>structure is U(3), not SU(3).
You're right.
My mistake came from thinking that SU(3) was the intersection of U(3) and
SO(6), which in turn came from thinking that an n x n complex matrix that
has determinant 1 when viewed as a real 2n x 2n matrix, must automatically
have det = 1 when viewed as a *complex* matrix. But this is false, as the
example SO(2) = U(1) clearly shows!
>Only a special subclass of Kahler manifolds - namely the Ricci-flat Kahler
>manifolds - preserve a spinor. The condition of Ricci-flatness of a Kahler
>manifold is equivalent to the holonomy being inside the SU(3) subgroup [...]
Okay, so that's how we get from U(3) down to SU(3). I have known this at
times, like when I wrote the long article "week195" of which the above
lines were a summary. I just didn't know it when I wrote the above lines.
:-)
>Even the correct direction of your (incorrect) equivalence - the statement
>that a covariantly constant spinor implies the existence of the complex
>structure of a Calabi-Yau three-fold - does not generalize to other
>dimensions.
Right, it's just a lucky coincidence that SU(4) = Spin(6).
I like this because it's part of a system of related coincidences
about spin groups in dimensions 5,6,7 & 8, as discussed in "week195".
........................................................................
Puzzle #18 - Which would-be state was not allowed to join the United States,
and lasted only 4 years after its founding?
Hint: later, its capitol building mysteriously disappeared!
For the answer, see http://math.ucr.edu/home/baez/puzzles/18.html
Lubos Motl
> What JB was saying here was simply that the correct answer
> to squark's question: where does the complex structure
> come from?" is " it comes from the parallel spinor".
I think that John now agrees with me; the complex structure exists on
Calabi-Yau spaces because the Calabi-Yau spaces are *defined* as Kahler
manifolds that satisfy some additional conditions. John also knows that
one of his key sentences
> > > But, SU(3) is just the subgroup of SO(6) that preserves a complex
> > > structure!
emphasized by the exclamation mark is simply incorrect, because the group
that preserves the complex structure is U(3), not SU(3). In the case of
real six-dimensional manifolds, a covariantly constant spinor implies the
existence of a complex structure. But the reverse implication does not
hold, and in all other cases except for SU(2) and SU(3), the equivalence
is incorrect in both directions - for example for Calabi-Yau four-folds
whose holonomy must be SU(4)=spin(6) which is a subgroup of spin(7);
spin(7) also preserves a spinor, but not a complex structure. Spin(7) is
embedded into Spin(8) in the spinorial fashion. Also, G_2 holonomy
manifolds preserve a spinor, but they don't have a complex structure, and
so on.
> By the way, somebody once told me something that I don't completely
> understand, but I will repeat it here anyway: to go in the other
> direction, to deduce that there is a parallel spinor, you just have to
> know [a] that the *algebra* of the holonomy group is su(3) and [b]
> that your manifold is 6D, compact and orientable. [I'm using small
> letters for algebras] You don't need to know the *group*. This is very
> non-trivial, because for example you could never tell that a 4D
> manifold is spin just by knowing that the algebra of its holonomy is
> su(2). Seems that this is some very special property of 6D
> Calabi-Yaus. Which brings us back to the title.
Being a spin manifold is a rather global property of your manifold. The
holonomy is a global concept, too, but you can talk about the *local*
holonomy at each point - which only counts the parallel transport of the
contractible loops. For the local holonomy group, the local orientability
is automatic and the compactness is irrelevant, and you equivalence is
therefore simple to see because the holonomy algebra is *defined* from the
local holonomy group (around infinitesimal contours). I am not sure
whether your statement is correct for the global holonomy.
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
Volker Braun
obvious counterexample. First I'll rephrase the "theorem":
A compact 3-fold with vanishing Ricci curvature (equivalently, restricted
Holonomy in SU(3)) admits a constant spinor.
I think thats already wrong in 4 real dimensions, consider the Enriques
surface. It admits a Ricci-flat metric as the quotient of a free
involution on a K3, but has nonzero (torsion) canonical class. But a
parallel spinor would trivialize the canonical bundle.
In general, the question is "does a solution of the equations of motion
automatically imply Susy"? In general, the answer is of course to the
contrary.
-Volker
Serenus Zeitblom
> Since nobody contributed to Zeitblom's question, let me point out the
> obvious counterexample.
I must admit that I'm scratching my head at this point,
because your counterexample obviously isn't one! In fact,
the Enriques surface was exactly what I was thinking about
when I said that the theorem does *not* work in four dimensions!
First I'll rephrase the "theorem":
>
> A compact 3-fold with vanishing Ricci curvature (equivalently, restricted
> Holonomy in SU(3)) admits a constant spinor.
OK, I think I see where the problem is. This is *not* what I was saying!
I said that the holonomy algebra was *isomorphic* to su(3), NOT to
a subalgebra. In other words, your mistake lies in the tiny word "in".
I don't mean "in", I mean "equal to".
>
> I think thats already wrong in 4 real dimensions, consider the Enriques
> surface.
Exactly. The algebra of its holonomy group is su(2), yet it doesn't
have a parallel spinor --- it is not even a spin manifold! So I agree
that it does not work in 4 dimensions, and in fact that is what I
said in the original post.....you may be thinking that you can get
up to six dimensions by taking the product of an Enriques surface
with a torus, but that is precisely what is excluded by requiring
the holonomy algebra to be su(3) and *not* a subalgebra.
>
> In general, the question is "does a solution of the equations of motion
> automatically imply Susy"? In general, the answer is of course to the
> contrary.
I'm afraid that is a bit too vague for me to comment on.
But anyway the claim remains [not, I hasten to add, in any
way due to me!]. Let me state it in this way. In general,
the holonomy *group* is a complicated thing, because it requires
global information. The *algebra* is a local thing, easier to
understand because it is connected directly to the curvature.
For example, it is trivial that the holonomy algebra of a non-flat
2-dimensional surface is so(2). It is not so trivial to determine
whether the holonomy *group* is SO(2) or O(2) --- you need global
information about orientability. [Again note that so(2) is the
algebra, SO(2) is the group.] OK, so suppose that you have a
2n [real] dimensional compact manifold and you know that the
holonomy algebra is su(n). Is there a parallel spinor? In
general, the answer is no, as I said in the original post.
BUT if n=3 and you happen to know that the manifold is
orientable, then the answer is YES. For example, if you
already know that the 6-manifold is Kaehler, and Ricci-flat,
and *not* a local Riemannian product of lower-dimensional stuff,
then you can immediately conclude that there is a *global*
parallel spinor. As we agreed, this is not true if n=2
because an Enriques surface is Ricci-flat and Kaehler
but not spin. That in fact is the typical situation,
because you would not expect to get global information
[the holonomy group] out of local information [the holonomy
algebra]. Yet in SIX dimensions, miraculously it does work:
the algebra tells you the group! Which is pretty lucky for
string theorists!
I really should dig up my old lecture notes on this....
R.X.
> Volker Braun <volker...@physik.hu-berlin.de> wrote in message news:<pan.2004.01.22....@physik.hu-berlin.de>...
> > Since nobody contributed to Zeitblom's question, let me point out the
> > obvious counterexample.
>
> I must admit that I'm scratching my head at this point,
> because your counterexample obviously isn't one! In fact,
> the Enriques surface was exactly what I was thinking about
> when I said that the theorem does *not* work in four dimensions!
>
In fact in 6d there are known orbifold/free fermion examples, which
are not supersymmetric despite there exists a holomorphic 3-form,
so that the holonomy is SU(3).
The point is that usually when one talks about holonomy, one talks
about parallel transport of vectors; this is what is captured by
asking that c_1=0. But for SUSY one needs a covariantly constant
spinor, and this is a stronger requirement (roughly because a spinor
can be viewed as the square-root of a vector). This can be formulated
by requiring that 1/2 (c_1)=0, and in integral cohomology this does
not follow from c_1=0.
This is indeed a torsion phenomenon, and goes in the direction of
what Volker was saying. In CFT language, this boils down to having
an operator Omega surviving the GSO projection (which corresponde
to the holo 3-form), while on the other hand the space-time supercharge
Q (being the square-root of this operator) is projected out. In
terms of the more involved language of GSO projection in Gepner
models, preserving a supercharge Q amounts to requiring that all
physical operators have even U(1) charge and not just integral U(1)
charge.
Volker Braun
with the actual
Theorem: A compact 3-fold X with restricted Holonomy the whole
SU(3) [this implies vanishing Ricci curvature but is stronger] admits a
constant spinor [equivalently has Holonomy in SU(3)].
For a proof, we can assume X = Y/G is a quotient by a freely acting group
G, Y simply connected Calabi-Yau. Then the Hodge star of Y has zero's on
the boundary except for one's at the tips. So we find (I write h^ij for
the Hodge number $h^{i,j}$)
h^00(Y) = h^00(X) = 1
h^0i(Y) = h^0i(X) = 0 for 0<i<3
h^03(Y) = 1
Hence chi(O_Y) = 0 = chi(O_X). But this then implies h^03(X)=1. So there
exists a holomorphic 3-form which is [pulling back to Y] nowhere zero.
> algebra]. Yet in SIX dimensions, miraculously it does work:
> the algebra tells you the group! Which is pretty lucky for
> string theorists!
I think it actually works in all odd complex dimensions, i.e. real 6, 10,
14, .... So its more of a 50-50 thing, does not require too much luck :-)
Best,
Volker
Serenus Zeitblom
> In fact in 6d there are known orbifold/free fermion examples, which
> are not supersymmetric despite there exists a holomorphic 3-form,
> so that the holonomy is SU(3).
>
> The point is that usually when one talks about holonomy, one talks
> about parallel transport of vectors; this is what is captured by
> asking that c_1=0. But for SUSY one needs a covariantly constant
> spinor, and this is a stronger requirement (roughly because a spinor
> can be viewed as the square-root of a vector). This can be formulated
> by requiring that 1/2 (c_1)=0, and in integral cohomology this does
> not follow from c_1=0.
OK, that is very interesting. Any specific references?
The key word here of course is "orbifold". Nothing like that
can happen on a compact *manifold*. By the way, when I
asserted the existence of a global spinor, I guess I ought
to have said that the global spinor exists if you choose the
right spin structure.
R.X.
Phys.Lett.B205:471,1988
> The key word here of course is "orbifold". Nothing like that
> can happen on a compact *manifold*.
I actually meant even worse than orbifold. This phenomenon had been
originally observed in models built from free bosons or fermions,
which have a priori no geometric interpretation in terms of a
compactification manifold at all.
By the way, when I
> asserted the existence of a global spinor, I guess I ought
> to have said that the global spinor exists if you choose the
> right spin structure.
Yes, one typically simply defines the theory such that the supercharge exists,
though from the world-sheet point of view there is no particular
reason for that.
Volker Braun
Good point, since we are discussing non-simply connected manifolds we must
be careful to choose the supersymmetric spin structure. So whenever I say
"X admits a constant spinor" I really want to say "X admits a special
spin structure for which there exists a constant spinor".
Luckily, whenever the Holonomy is contained in SU(3), we can pick a
supersymmetric spin structure.
This is not quite a torsion phenomenon, it always can happen if H^1(X,Z_2)
is non-zero. The simplest example is the torus with antiperiodic
fermions in some direction - then a spinor has to have a zero somewhere,
so no constant spinors exist.
Best,
Volker