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Dirac operators on Lorentzian manifolds
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Volker Braun
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Joined: 26 Apr 2005
Posts: 6

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

I just want to point out one detail. Lets take the usual 9+1 (or 3+1 for
historical reasons Smile dimensions, and split the tangent bundle TM=V+W
into time and space directions. If the "time-direction" line bundle W is
not trivial, then we are in such deep troubles that closed timelike curves
seem harmless in comparison. So for any physical spacetime, w_1(W) better
be zero.

On Sat, 23 Apr 2005 06:11:10 -0400, Thomas Mautsch wrote:

Quote:
While in Riemannian geometry the condition on a manifold
to carry a spin structure is that the second Stiefel-Whitney class
w_2(TM) be zero, in non-Riemannian geometry
the condition is connected to the splitting of M in space- and time-like
directions, TM = V + W, and it becomes:

w_2(TM) = w_1(V) u w_1(W).

And for any physically sensible M, that just reduces to w_2(TM)=0.
Volker
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Urs Schreiber
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Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

"Thomas Mautsch" <mautsch@math.ethz.ch> schrieb im Newsbeitrag
news:42694f6a@news1.ethz.ch...
Quote:
In news:<3cnkceF6pqrqiU1@news.dfncis.de> schrieb Urs Schreiber:
"Jack Tremarco" <jacktremarco@yahoo.com> schrieb im Newsbeitrag
news:1114017473.127196.51460@l41g2000cwc.googlegroups.com...
Urs Schreiber wrote:
"Kasper J. Larsen" <kjlarsen@hotmail.com> schrieb im Newsbeitrag
news:63ea8295.0504191455.50f6716e@posting.google.com...

Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold?

Yes.

This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.

Jack, could you be a little bit more specific about what you mean here?

The only global condition that you have is the obvious one, that your
manifold admits spinors at all, globally, which means that it admits a
spin bundle which means that it is spin. This same condition is there for
Riemannian signature. So the answer is indeed Yes.

I am sorry, Urs, but this statement looks like word-juggling to me. - If
you don't specify how you define these terms, how can we believe you that
the whole construction is so "obvious"?


My apologies if what I wrote looked like word-juggling. I am not sure why it
did to you, but that was certainly not my intention. I also don't see a
disagreement between what I wrote and what you write. But let's see. We'll
sort that out.

First of all let's distinguish between various flavors of the question that
is being discussed. Originally there was the question under what conditions
we can have a Dirac operator on a Lorentzian manifold and how it is defined.

Due to certain applications that people have in mind, this question will
immediately make them think of the question, when we can have a Dirac
operator and in addition have this operator satisfy some desired list of
properties.

I replied to the first question that you construct a Dirac operator on a
Lorentzian manifold in precisely the same way as you do on a Riemannian
manifold. In order for that to be meaningful we need to have a manifold
which admits a spin bundle. The Dirac operator acts on spinors (sections of
a spin bundle) and hence it is "obvious" (and please apologize if you find
this usage of the word obvious too vague) that we need the condition that
there are spinors in the first place on our manifold. If that is the case,
we can define the Dirac operator.

You don't seem to disagree with this stament. What you do point out is what
the conditions of having

1) Lorenttian manifold
2) with spin structure

mean in detail.

I would like to point out that once we have this we can write down
expressions like

\bar\psi D \psi

for \psi a spinor field (a section of the spin bundle) and construct an
action principle for this. For instance we could consider the field theory
describing those spinors coupled to general relativity, which would read

\int vol ( R + \bar \psi D\psi ) .

Or we could write down an action for some flavor of supergravity, for
instance. People do this all the time.

You point out that we will in general not have the additional properties
that there is a Hilbert space of spinors with a positive semi-definite inner
product (=scalar product) with respect to which the Dirac operator is
self-adjoint.

These problems are related to the fact that relativistic wave equations are
hard and under certain conditions impossible to be interpretable as
describing single particle dynamics.

For illustration purposes, let me mention some aspects of this for the case
where the Dirac operator under consideration is that on the exterior bundle,
i.e. the Dirac-Kaehler operator

D = d + del

where d is the exterior derivative for our manifold and del = \pm *d* is its
(formal, possibly) adjoint with respect to the Hodge inner product

<a|b> = \int a /\ * b

on differential p-forms a,b.

This case features most of the issues which we discussed by avoiding some
less essential technicalities.

First of all, this Dirac operator exists on every Riemannian or
pseudo-Riemannian manifold, no matter what.

We also always have an inner product <|>, which in the Lorentzian case is
indefinite. In order to really define it we need to restrict to sections of
the exterior bundle which are square integrable. Depending on the context
that one is dealing with this can be done in several ways.

Next one might ask under which conditions we can get from D a 'spatial'
Dirac operator D_s together with a true Hilbert space (H,[,]) for sections
of the exterior bundle restricted to some spatial hyperslice of our
manifold, such that the original
Dirac equation D psi = 0 becomes a Schroedinger-Dirac equation

i d_t psi = D_s psi

and hence can be (more or less) interpreted as a 1-particle theory.

Once necessary condition for this to work is that our spacetime is
stationary, i.e. that it has a timelike Killing vector.

Calling this Killing vector v we may be able to use the Clifford element
c(v) as an operator that turns our indefinite inner product space into a
Krein space and construct a true Hilbert space for D_s from it.

And so on and forth.

Agreed?
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Thomas Mautsch
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Joined: 06 May 2005
Posts: 96

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

In news:<3cnkceF6pqrqiU1@news.dfncis.de> schrieb Urs Schreiber:
Quote:
"Jack Tremarco" <jacktremarco@yahoo.com> schrieb im Newsbeitrag
news:1114017473.127196.51460@l41g2000cwc.googlegroups.com...
Urs Schreiber wrote:
"Kasper J. Larsen" <kjlarsen@hotmail.com> schrieb im Newsbeitrag
news:63ea8295.0504191455.50f6716e@posting.google.com...

Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold?

Yes.

This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.

Jack, could you be a little bit more specific about what you mean here?

Quote:
The only global condition that you have is the obvious one, that your
manifold admits spinors at all, globally, which means that it admits a spin
bundle which means that it is spin. This same condition is there for
Riemannian signature. So the answer is indeed Yes.

I am sorry, Urs, but this statement looks like word-juggling to me. -
If you don't specify how you define these terms,
how can we believe you that the whole construction is so "obvious"?

I would agree with Jack, that there are certain difficulties
with the definitions of spin structures, spinors, and Dirac operators
in pseudo-Riemannian spaces.

I had to look this up in the book:

"Spin-Strukturen und Dirac-Operatoren
ueber pseudoriemannschen Mannigfaltigkeiten"
by Helga Baum (Teubner-Verlag, Leipzig, 1981),

The results of this book have also appeared in English
without proofs in:

* Dlubek, Helga
Spinor-structures and Dirac-operators on pseudo-Riemannian manifolds.
Proceedings of the Conference on Differential Geometry and
its Applications (Nove Mesto na Morave, 1980), pp. 17--23,
Univ. Karlova, Prague, 1982.
* Baum, Helga
Spinor structures and Dirac operators on pseudo-Riemannian
manifolds. Bull. Polish Acad. Sci. Math. 33 (1985), no. 3-4, 165--171

Here are some facts from the book above:

First of all, the existence of pseudo-Riemannian structures
on manifolds is topologically restricting. -
For the existence of a Lorentz structure on a manifold M,
there has to exist a nowhere-vanishing line field on M.
Similar for pseudo-Riemannian metrics
of index (k,n-k) on an n-dimensional manifold,
there has to exist a splitting of the tangent bundle TM
into the direct sum
of a k-dimensional subbundle V and an (n-k)-dimensional subbundle W.
The first Stiefel-Whitney classes w_1 of these bundles,
which describe their orientability,
are invariants of the underlying pseudo-Riemannian structure.

Now, a metric of index (k,n-k) on M
gives us the bundle of orthonormal frames over M,
a principal bundle with structure group O(k,n-k).

The construction of Clifford algebra and Pin group Pin(k,n-k),
which is a natural double cover of O(k,n-k),
is very much like in the Riemannian case,
and there is plenty of literature about it, like, e.g.:

* Lawson, Michelsohn
Spin geometry. Princeton University Press, 1989.
Chapter I.
* Harvey, Spinors and calibrations. Academic Press, Boston, MA, 1990.

A spin structure over the manifold M can be defined as reduction
of the principal O(k,n-k)-bundle of orthonormal frames
to a principal Pin(k,n-k)-bundle
which is compatible with the natural projection from Pin to O.

Already here appears the first slight complication:
While in Riemannian geometry the condition on a manifold
to carry a spin structure is that the second Stiefel-Whitney class
w_2(TM) be zero, in non-Riemannian geometry
the condition is connected to the splitting of M in space- and time-like
directions, TM = V + W, and it becomes:

w_2(TM) = w_1(V) u w_1(W).

This looks like a first reason to restrict oneself to
manifolds that carry space- and time-orientations,
and we have not yet started to define the spin group...

O.K., given a spin structure,
we can define the associated spinor bundle,
lift the Levi-Civita connection
to a compatible connection on the spinor bundle,
and define the Dirac operator.

Problem is, that all we can do in this setting is "geometric"
(we can e.g. define parallel spinors and harmonic spinors,
spinors annihilated by the Dirac operator)
but nothing "analytic", because for analytical considerations
we need the Dirac operator to act on a (complex!) *Hilbert space*. ...

Quote:
It would be strange if otherwise, given that we do live in a Lorentzian
spacetime and we do observe spinors.
Note that the question "Can one define a Dirac operator D on a spin bundle
over M" is different from the question "Can we make sense of quantum field
theory of fermions on M." QFT on curved spaces is hard.

.... But the problem does not start with QFT:
Even to do simple spectral theory for the Dirac operator,
we need a Hilbert space for it to act on.

Now Harvey goes in his book a great length
to prove that there exists a natural inner product on (real) spinors
that is compatible with the action of the Clifford algebra,
so that the Dirac operator would be *formally* self-adjoint
with respect to this product.
*BUT* this product may take values in R, C, or H,
and it may be symmetric or anti-symmetric,
and after complexification of the spinor bundle
(which we *must* inevitably do
to get a *complex* Hilbert space of spinor sections),
the inner product we obtain will - in the non-Riemannian case -
not be positive definite.

Now, in the later chapters of Helga Baum's book
there is a description for the definition of a "natural"
positive-definite hermitean product on the complex spinor bundle,
but to construct this product,
one has to fix a time- and a space-orientation on M,
i.e. one has to break the non-compact symmetry group
to the compact group SO(k)xSO(n-k).

But even then, the Dirac operator will *not* be essentially self-adjoint
in the Hilbert space defined by this scalar product,
only its "real and imaginary part" will be essentially self-adjoint;
and this makes that the spectrum of the Dirac operator
will not only consist of pure eigenvalues. -
There will also be essential spectrum and even rest spectrum.

Well, a Dirac operator with rest spectrum might
from the point of view of physics sound like nonsense,
but who can say, where we went wrong??
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

"Urs Schreiber" <Urs.Schreiber@uni-essen.de> schrieb im Newsbeitrag
news:3c4r7cF6kammeU1@news.dfncis.de...

Quote:
One would have to work out the conditions on b_i^j_k(x) for this to give a
well-defined continuum limit. This has not been discussed in the
literature as far as I am aware and I also don't see that Akhmedov
addresses this point.

Today he has a new preprint on this issue:

http://golem.ph.utexas.edu/string/archives/000558.html
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kneemo
science forum beginner


Joined: 26 Apr 2005
Posts: 2

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Bubbling AdS space and 1/2 BPS geometries Reply with quote

jcgonsowski@yahoo.com Wrote:
Quote:
Smith has bosonic strings so the M-theory is 24D + 3D\ntreated
differently but Lee Smolin at least used to look at starting\nhere to
include the bosonic side of heterotic strings.

I recall Smith using the complexified exceptional Jordan algebra for
his model. This is a (complex) 27-dimensional algebra, over the
bioctonions. Now, is it Smith's intention to use the (complex) 11
dimensions of the complexified Jordan algebra, or use the 11 (real)
dimensions of the self-adjoint part (the exceptional Jordan algebra)
for space-time. This would seem to make a difference, as we would be
working with 3 complex dimensions, along with a bioctonionic space, in
the 11-dimensional complex case.

Would space-time as 11=8+3 complex dimensions be consistent? Or must
space-time dimensions be strictly real?

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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

"Jack Tremarco" <jacktremarco@yahoo.com> schrieb im Newsbeitrag
news:1114017473.127196.51460@l41g2000cwc.googlegroups.com...
Quote:
Urs Schreiber wrote:

"Kasper J. Larsen" <kjlarsen@hotmail.com> schrieb im Newsbeitrag
news:63ea8295.0504191455.50f6716e@posting.google.com...

Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold?

Yes.

This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.

The only global condition that you have is the obvious one, that your
manifold admits spinors at all, globally, which means that it admits a spin
bundle which means that it is spin. This same condition is there for
Riemannian signature. So the answer is indeed Yes.

It would be strange if otherwise, given that we do live in a Lorentzian
spacetime and we do observe spinors.

Note that the question "Can one define a Dirac operator D on a spin bundle
over M" is different from the question "Can we make sense of quantum field
theory of fermions on M." QFT on curved spaces is hard.
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Jack Tremarco
science forum beginner


Joined: 19 Jun 2005
Posts: 7

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

Urs Schreiber wrote:

Quote:
"Kasper J. Larsen" <kjlarsen@hotmail.com> schrieb im Newsbeitrag
news:63ea8295.0504191455.50f6716e@posting.google.com...

Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold?

Yes.

This is true if you ignore non-compactness issues, which can be quite
serious. In generic time-dependent backgrounds the honest answer is
closer to a "no", at least if you demand mathematical rigor.

[Moderator's note: It depends on what sort of rigor you want to have.
The spinors (and the Dirac equation) on a curved manifold is defined
with the help of the vielbein (tetrad) that allows one to treat
the curved space locally in exactly the same way as the Euclidean
or Minkowski space. It's easy to define the Dirac operator. LM]
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: weakened nonabelian bundle gerbes and 2-bundles Reply with quote

"DM Roberts" <droberts@physics.adelaide.edu.au> schrieb im Newsbeitrag
news:Pine.LNX.4.62.0504201035210.13801@feynman.harvard.edu...


Quote:
All the work I've seen so far on 2-bundles with connection

Is there any other work on that than hep-th/0412325 ?


Quote:
has been in terms of local data - Lie(G) 1-forms etc.

More precisely, in hep-th/0412325 this is given in terms of a local holonomy
2-functor which is then decoded to yield local p-form data.


Quote:
Could we instead define a
connection as in the 1-bundle case as a sort of "bundle of subspaces"?
(heuristic definition only) That is, a splitting into horizontal and
vertical parts T = H \oplus V. We have a concept of 2-vector space from
HDA VI (math.QA/0307263) and there is the concept of a sub-2-vector
space (I think one could work backward from the direct sum of two
2-vector spaces to get apropriate definitions, or else in terms of
"images" and "kernels" of "linear transformations").


I expect that this should work and should be equivalent to the existing
definition. But as far as I know so far nobody has tried to spell that out
in detail.


Quote:
Ah, I see where the nonabelian surface parallel transport rears its
ugly head - how can we generalise the proof as per bundles without a
decent definition of this?


There is in fact a decent definition of nonabelian surface parallel
transport in strict G-2-bundles. This is unfortunately only hinted at in
hep-th/0412325, but I have reported on more details here:

http://golem.ph.utexas.edu/string/archives/000503.html

and, upon request, have clarified the context here:

http://golem.ph.utexas.edu/string/archives/000547.html#c002194 .

A more detailed exposition is underway:

http://www-stud.uni-essen.de/~sb0264/2NCG.pdf .

What I haven't shown yet, though, is indepence of this construction on the
choice of cover. I expect the proof to be completely analogous to the well
known abelian case.


Quote:
We could work from a position of physical insight perhaps. But as the
"physics" of this (H-flux in string theory, say)


H-flux gives rise to _abelian_ gerbes coupled to F-strings. Holonomy for
abelian 2-gerbes is well understood, parallel transport has recently been
studied by Picken. This is a special case of the nonabelian surface
transport mentioned above.

The challenge is to identify the physics that gives rise to _non_abelian
gerbes/2-bundles. The ordinary F-string in 10D does not couple to any
nonabelian 2-form, so it must be something else.

Several people expect this to be related to theories on stacks of N
M5-branes, where we have end-strings of open membranes on the 5-branes. For
N>1 these should sort of carry Chan-Paton-like degrees of freedom and couple
to nonabelian 2-forms which are known to be part of the spectrom on these
branes.

Edward Witten called the effective field theories for these branes once
tentatively "nonabelian gerbe theories":

http://www.maths.ox.ac.uk/notices/events/special/tgqfts/photos/witten/71.bmp
..

But I was being told that he has given up on making this precise. (?)

Hisham Sati is still arguing for this, e.g. in

I. Kriz and H. Sati
M-Theory, Type IIA Superstrings and Elliptic Cohomology
hep-th/0404013

H. Sati
M-theory and characteristic classes
hep-th/0501245

The most direct argument that this must be true that I know of is that in
section 5 of

P. Aschieri & B. Jurco,
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
hep-th/0409200 .

Recall that they argue as follows:

The M2 brane couples to the SUGRA 3-form. There seems to be no choice but
that this coupling is globally described by an abelian 2-gerbe/3-bundle,
just like in 1-dimension lower the coupling of the string to the KR 2-form
is globally described by an abelian 1-gerbe/2-bundle.

For the string we can derive from the fact alone that its bulk couples to an
abelian 1-gerbe the fact that its boundary couples to a nonabelian
0-gerbe/1-bundle, namely that living on the D-brane that the string ends on.

Schematically this works by noting that every abelian 1-gerbe G can be
written as a trivial gerbe G0 plus a lifting gerbe D(B) of a twisted
nonabelian 0-gerbe/1-bundle:

G = D(B) + G0.

B is the nonabelian 0-gerbe/bundle on the D-brane.

A similar relation holds for abelian 2-gerbes. They can be realized as a
lifting 2-gerbe of a twisted nonabelian 1-gerbe plus something else.

By analogy it is to be expected that this possibly twisted nonabelian
1-gerbe is that living on the 5-branes that the membrane ends on.

But what is interesting is that one can say more: The abelian 2-gerbe
coupled to the M2 brane is in fact a Chern-Simons 2-gerbe classified by the
Pontryagin class. These 2-gerbes are known to be the lifting 2-gerbes for
lifting an (Omega G)-gerbe to a \hat(Omega G)-gerbe, where Omega G is the
loop group of G and \hat(Omega G) its Kac-Moody central extension.

Incidentally, the \PG-2-bundles that we find in

Baez,Crans,Schreiber&Stevenson
Quote:
From Loop Groups to 2-Groups
math.QA/0504123


to be related to the group String(n) are known (not rigorously proven yet,
though) to be the same as these \hat(Omega G)-1-gerbes.

Combined with the argument by Aschieri&Jurco this would say that what lives
on a stack of M5-branes are these \PG-2-bundles. Since they also seem to be
related to elliptic cohomology (due to the appearance of String(n), for
one), this gives precisely the picture that Hisham Sati is arguing for in
the above papers.

But the details here still need to be written down.


Quote:
(the fault of the mathematicians, physicists or mathematical
physicists? Which came first, the chicken or the egg?) I don't know if
that will help.

Understanding the physical setups that give rise to nonabelian
gerbes/2-bundles would certainly help the general understanding.
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

On Thu, 14 Apr 2005, Thomas Larsson wrote:

Quote:
Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message
news:<3c4r7cF6kammeU1@news.dfncis.de>...

These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)

IOW, Akhmedov assumes the existence of a Killing metric,


Akhmedov's idea is based on 2-dimensional TFTs. These are in 1-1
correspondence with semisimple associative algebras, which, by definition,
have a nondegenerate Killing form (hep-th/9212154).

The fact that these algebras have to be associative and have a nondegenerate
Killing form is an algebraic reformulation of the invariance of the
corresponding TFT under re-triangulating moves.

Associativity describes the "fusion transformation", equation 3.1 and
figure 7 of hep-th/9212154. The bilinear form gives the
"bubble transformation", equation 3.2 and figure 8 in hep-th/9212154.

These "moves" are the 2D version of the Matveev moves, which are
equivalent to the Alexander moves or the bond-flip moves.


Quote:
Moreover, there is no problem to define 1-gauge
theories for gauge groups which do not admit a Killing metric, although
somewhat unusual.


The nondegenerate Killing form of the above mentioned algebras should not
be confused with any Killing form of any Lie algebra of any gauge group that might
appear once somebody manages to deforms these TFTs into something like a
surface holonomy.


Quote:
IOW, you have several quantities for each plaquette, e.g. C_ijk, C^ijk,
C^i_jk etc. If you have a Killing metric, it is not important to distinguish
between them. Since I don't make that assumption, I need to be careful here.


Since you don't have that assumption you will have to do something else to
ensure that your construction is well defined (independent of details of
the latticization).


[...]
Quote:
However, tonight I woke up and realized that it is not.

V^n can be identified with the space of piecewise constant V-valued
functions, with n pieces. In the limit n -> infinity, this becomes the
space of functions from the boundary to V, LV.


This is what I wrote in the second post in this thread.


Quote:
One probably want some
continuity conditions in the limit. The gauge transformations in G^n
in G^n simply become the loop group LG, and acts on the functions in LV.


The concept of gauge group might require some care. For instance the
group of diffeomorphisms of the loop will play a role, too. I have
indicated in that previous post how we actually seem to get a coherent
2-group out of this (if it can indeed all be well defined).
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Thomas Larsson
science forum addict


Joined: 01 May 2005
Posts: 73

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Akhmedov: Nonabelian 2-holonomy using TFT Reply with quote

Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote in message news:<3c4r7cF6kammeU1@news.dfncis.de>...

Quote:
These TFT's that we are talking about can be shown to be in 1-1
correspondence with semisimple algebras with structure constants C_i^j_k.
The bilinear form has to be proportional to the "Killing form" g_ij =
C_i^r_s C_j^s_r of these, as found first in hep-th/9212154. (*)

IOW, Akhmedov assumes the existence of a Killing metric, which was exactly
my point. However, holonomies and inverses in 1-gauge theory are defined
without reference to this metric, so it does not naturally belong to the
domain of the problem. Moreover, there is no problem to define 1-gauge
theories for gauge groups which do not admit a Killing metric, although
somewhat unusual. This also works for 2-gauge theories in my sense, but
evidently not in Akhmedov's sense.

Quote:
In C_ijk, which of i,j,k are incoming and which are outgoing?


As discussed in the literature, e.g. the hep-th/9212154 and as I mentioned
in my original post (http://golem.ph.utexas.edu/string/archives/000542.html)
an upstairs index corresponds to ingoing and a downstairs index to outgoing
(or vice versa, depending on your conventions).

IOW, you have several quantities for each plaquette, e.g. C_ijk, C^ijk,
C^i_jk etc. If you have a Killing metric, it is not important to distinguish
between them. Since I don't make that assumption, I need to be careful here.

The only problem I saw with the formal continuum limit is that the
different surface holonomies take values in different spaces. If the
boundary consists of n links, the lattice surface holonomy belongs to V^n
(ignoring the difference between V and its dual) and the gauge
transformations belong to G^n, and it may be problematic to take the
limit of the n-fold tensor product. However, tonight I woke up and realized
that it is not.

V^n can be identified with the space of piecewise constant V-valued
functions, with n pieces. In the limit n -> infinity, this becomes the
space of functions from the boundary to V, LV. One probably want some
continuity conditions in the limit. The gauge transformations in G^n
in G^n simply become the loop group LG, and acts on the functions in LV.

There is something definitely right about this. As you know, I am a firm
believer in locality, and this is the reason why I don't like the 2-gauge
theories of Baez, Pfeiffer and others. If a finite-dimensional G acts on a
loop, in must be some non-local process which is smeared over the whole
loop at once. However, LG can act locally, with one copy of G at each
point of the loop. From this viewpoint it is encouraging that you and
Baez now seem forced into considering loop groups.

The generalization to higher gauge theory is now obvious: the p-holonomy
associated with a p-manifold M in p-gauge theory has a gauge symmetry
given by the manifold group living on the boundary dM, i.e. G^dM.
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Urs Schreiber
science forum Guru Wannabe


Joined: 04 May 2005
Posts: 127

PostPosted: Tue Apr 26, 2005 1:47 pm    Post subject: Re: Theories are increasingly theoretical Reply with quote

On Thu, 3 Feb 2005, Kea wrote:


Quote:
But I am a little confused as to why you think we need to stick with
String theoretic foundations. To my way of thinking N = 2
SUSY QM is not fundamental.


This depends a little on tastes and points of perspective, but
let me make some comments on how amazingly fundamental N=2 SUSY QM is from
a certain point of view:

To start with, in the "ordinary" case it is pretty much the same as deRham
theory on a manifold M. M is the configuration point of the particle, the
exterior bundle Omega(M) over M the corresponding superspace (every
exterior bundle is an N=2 superspace), the supercharges are the deRham
operators d, d^+, the Hilbert space H is that of suitable sections Gamma
in the exterior bundle and the inner product on that space is the Hodge
inner product

<a,b> = int a /\ * b

extended in the obvious way from a,b in Omega^p(M) to a,b in Omega(M).

Once you consider any manifolds at all this is about as fundamental as it
gets. See the beautiful work by Froehlich

hep-th/9612205
hep-th/9706132

for more.

In particular, there it is emphasized that the natural way to think about
this setup is as a certain spectral triple, namely (Gamma^0, H, d \pm
d^+).

So this let's us easily make the above yet more fundamental by decreeing
that with supersymmetric QM we want to mean in general just some spectral
triple (maybe not really any one but one having some basic properties,
if you like).

So if you like the point of view of that Cartier paper that you mentioned
this should be close to your heart. I think it has good chances to be
about as fundamental as it gets.

Froehlich in the last sections of the above mentioned papers makes some
attempts to lift this setup to the superstring, but this remained
tentative, as far as I am aware. A little more systematic attempt to do
something similar was published by Chamseddine in hep-th/9701096,
hep-th/9705153.

Alejandro Rivero once pointed out to me that one reason these attempts
were not further developed was because the rise of the BFSS matrix model and
interest in noncommutative field theories and open strings in
B-field backgrounds focused all stringy attention to the noncommutativity in
NCG, forgetting about the "spectral".

Be that as it may, after finding the results of hep-th/0401175 I fell in
love with the idea on looking at superstrings as susy QM on loop space.
With hindsight, that had to lead to the concept of categorification
eventually, which it did.

Using categories all over the place is enjoyable and useful, but
categorification is special.

I guess the point is that once you realize that category theory is the
language in which god wrote math it becomes clear that at the heart of it
one is dealing with omega-categories.

The step from set theory to category theory consists of realizing that
points are not enough, but that morphisms are important. The step from
category theory to 2-category theory replaces the points by morphisms once
again. Thinking this to the end the idea is that there are no points, but
just morphisms between morphism. Realizing this step by step is called
"categorification".

Phew, now I am getting on-topic for sci.philosophy.blah-blah. :-)

But maybe it is entertaining to note that "morphisms between morphisms"
rhymes with "worldsheets for worldsheets": It is well known that the string can
be thought of to be composed of strings itself:

Nucl Phys B293 (1987) 593

and

hep-th/9602049 .

And hence these consist again of strings, and so on.

As far as I understand from what Lubos told me
(http://golem.ph.utexas.edu/string/archives/000265.html#c000328)
this is at the heart of a big idea for a deeper understaning of M-theory:
hep-th/0111068 .

For these reasons I feel that categorifying spectral triples to learn
about strings is reasonably fundamental. All results that have shown up in
this approach so far also suggest that it is not completely on a wrong
track.


Quote:
Categorification isn't about categorifying
bundle structures piece by piece.


Well, yes, the "piece by piece" is a result of the insufficiency of the
human brain. :-)


Quote:
This is why (I think) Ross Street
says one should look at stack theory and leave gerbes alone.


You have to educate me here. Are you referring to stacks in the sense of
"fibered categories with certain properties"? In that case I don't
understand what tou mean because a gerbe is just a special case of a
stack.

And, by the way, a fibered category is just "half" the categorification of
a presheaf. 2-bundles know about string space, while gerbes do not. See

http://groups.google.de/groups?selm=ctbmgs%24b8s%241%40news.ks.uiuc.edu .


Quote:
In particular, recall that the notion of -point- becomes a geometric
morphism

\mathbf{Set} \rightarrow \mathbf{Sh}(M)

into the category of sheaves on a space M. This puts
geometry on a purely axiomatic footing.


(For those following this, Kea here is referring to the discussion on p.
400 on the paper by Cartier that he mentioned before.)

I think this is *one* way to look at a point. Seems to me that there are
many other concepts that we could "identify" with points. For instance in
NCG a point is a simple ideal in an algebra. Or is that secretly the same
as this Grothedieck's conception?
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richard miller
science forum addict


Joined: 24 Mar 2005
Posts: 95

PostPosted: Tue Apr 26, 2005 6:16 pm    Post subject: Re: Novice: Indivisibility of string Reply with quote

"John" <john@spam.is.evil.com> wrote in message
news:116qbkarhcd01aa@corp.supernews.com...

Quote:
I'm having a very hard time picturing what's vibrating. Vibration
seems to require parts of the string moving wrt other parts.
But doesn't that require there to be different parts, meaning
strings should be further divisible? Your example
of the rubber band doesn't help me, because it seems to me
that after enough divisions you're down to a string of 1 angstrom
diameter, and after that you've lost the rubber band...

Is this another one of those areas (like particle-wave duality
or 4 dimensional space-time) that just can't be pictured in terms
of our everyday models, or am I just having a hard time seeing
what should be an obvious point?


Sounds a nice question, we (one) have ascribed continuous laws over a length
of the Planck scale. do we have the justifiication for this continuity or
are all the action integrals etc. not validate nowadays (things have moved
on since the 80s?), at least as continuous functions? I don't know either.
Look forward to the answer.
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Robert C. Helling
science forum beginner


Joined: 30 Apr 2005
Posts: 22

PostPosted: Wed Apr 27, 2005 5:44 am    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

On Tue, 26 Apr 2005 11:47:35 -0400, Volker Braun <volker.braun@physik.hu-berlin.de> wrote:

Quote:
If the "time-direction" line bundle W is
not trivial, then we are in such deep troubles that closed timelike curves
seem harmless in comparison. So for any physical spacetime, w_1(W) better
be zero.

Why? Could you please expand this? What can you say about solution spaces
of you favourite wave equations?

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling School of Science and Engineering
International University Bremen
print "Just another Phone: +49 421-200 3574
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
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Volker Braun
science forum beginner


Joined: 26 Apr 2005
Posts: 6

PostPosted: Wed Apr 27, 2005 1:14 pm    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

I was not thinking of anything this complicated, certainly I did not
construct any solutions to the Dirac equation on such a manifold.

But if w_1(W) <> 0, then there is a closed loop in your manifold along
which the time-forward-direction flips. So how would you go about defining
past, future, and asymptotic states.

For example (ignoring spinors), a Moebius strip (S^1 x R^1)/Z_2. You take
local coordinates (t,x) where x runs along the circle and t is
perpendicular to it, pointing in the noncompact direction. The metric is
diag(-1,+1) such that x is space and t is time. Please define scattering
amplitudes :-)

Volker

On Wed, 27 Apr 2005 03:44:54 -0400, Robert C. Helling wrote:
Quote:
If the "time-direction" line bundle W is not trivial, then we are in
such deep troubles that closed timelike curves seem harmless in
comparison. So for any physical spacetime, w_1(W) better be zero.

Why? Could you please expand this? What can you say about solution
spaces of you favourite wave equations?
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Robert C. Helling
science forum beginner


Joined: 30 Apr 2005
Posts: 22

PostPosted: Thu Apr 28, 2005 10:15 am    Post subject: Re: Dirac operators on Lorentzian manifolds Reply with quote

On Wed, 27 Apr 2005 11:14:01 -0400, Volker Braun <volker.braun@physik.hu-berlin.de> wrote:
Quote:
I was not thinking of anything this complicated, certainly I did not
construct any solutions to the Dirac equation on such a manifold.

But if w_1(W) <> 0, then there is a closed loop in your manifold along
which the time-forward-direction flips. So how would you go about defining
past, future, and asymptotic states.

As you know, microscopic physics is invariant under time reversal
(well, most of it) so I don't see any immerdiate problems here.

Quote:

For example (ignoring spinors), a Moebius strip (S^1 x R^1)/Z_2. You take
local coordinates (t,x) where x runs along the circle and t is
perpendicular to it, pointing in the noncompact direction. The metric is
diag(-1,+1) such that x is space and t is time. Please define scattering
amplitudes Smile

But is that more complicated than scattering in other spaces that are
not asymptotically flat (or AdS or something similar with good
asymptotic regions)?

Robert

--
..oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling School of Science and Engineering
International University Bremen
print "Just another Phone: +49 421-200 3574
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
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