Hi Lubos

I agree with a lot of what you say. However, you appear not to
understand spin foam models at all.....but I won't waste time
defending them.

Let me ask you some questions:

1. What rigorous understanding do we have of confinement besides

Quark State Confinement as a Consequence of the Extension of the
Bose-Fermi Recoupling to SU(3) Colour; W. P. Joyce
http://arxiv.org/abs/hep-th/0306256

which fits into the topos quantisation picture (you can think of
Strings as categorified particles, like Urs Schreiber, if you like
- but this is NOT String Theory)

2. What is your physical interpretation of T-duality?

3. If twisted K-theory can't capture D-brane charges, what can? By
the way: category theory has an answer.

Regards
Kea

[Moderator's note: There seemed to be a problem with the quoting
hierarchy in the original submission. I have tried to improve the
formatting, hopefully in the way it was intended. -usc]

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It exchanges momentum with winding. Are you looking for anything else?

Can't it??

Sure, set theory has an answer, too. :-)

What? This is a derived result in the maths of strings. I do not
believe this is physics. It refers to naive internal degrees of
freedom. Even if one wants to believe the maths is physics, the
picture has to be more sophisticated than this.....see for instance

reference:
A MAD DAY’S WORK: FROM GROTHENDIECK TO CONNES AND
KONTSEVICH THE EVOLUTION OF CONCEPTS OF SPACE AND SYMMETRY, P.
Cartier, Bull. Amer. Math. Soc. 38, 4 (2001) 389-408
http://modular.fas.harvard.edu/sga/from_grothendieck.pdf

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Kea replies:

To me there is the physical interpretation of something and the
sophisticated description. You asked for the physical interpretation and it
is pretty obvious. But I agree that it would be nice to have a more
sophisticated description and it does exist and harmonizes with the more
lowbrow description.

.....see for instance

From skimming that text I don't see that the author mentiones it, but there
is in fact a very nice algebraic way to formulate the exchange of momentum
and winding and hence T-duality as an operation of a certain automorphism on
an NCG-like description of stringy spacetime. This is discussed for instance
in

hep-th/9511061
hep-th/9707202
hep-th/0401175 .

In fact, this is part of a long-term idea that lives in the back of my mind
and will hopefully one day evolve into a sufficiently coherent form:

Start with N=2 supersymmetric quantum mechanics. The sophisticated
formulation of that is an N=2 spectral triple in the language of Freohlich
and using the deformation by Witten

 (A, H, e^-W d e^W)

where A is some algebra of functions over some configuration space H, d is
the deRham operator on the exterior bundle over M, H is the Hilbert space of
sufficiently well behaved sections of that bundle equipped with the Hodge
scalar product and W is an operator on that Hilbert space. Hermitean W
encode background fields that the superparticle propagates in (notably
scalar potentials, gravity, torsion), anti-hermitean W encode gauge and
symmetry transformations on these background fields.

Now categorify this.

The configuration space M becomes a 2-space B, which we can think of as the
space of string configurations in M.

The exterior bundle over M becomes a 2-bundle over B.

This 2-bundle has a 2-sheaf of 2-sections. The operators H->H become
functors between such 2-sections. The arrow part of these must involve a
nilpotent graded operation. If we restrict to the reparametrization
invariant case this gives rise to the operator

  d + i K->

on the exterior bundle over loop space, where K-> is the inner product with
the loop space vector field that generates reparametrizations.

This operator is known to be the same as G + i\bar G, where G is the
supercharge of the RNS superstring.

The deformations e^W similarly categorify to functors between 2-sections.
One can find for all massless NSNS background fields and D-brane
configurations the corresponding e^W which turn the deRham Dirac operator
into that describing superstrings in these backgrounds. Anti-hermitean W
encode gauge and duality transformations of the background, in particular
T-duality.

I have large parts of that sketch worked out. But there are many details to
be made more precise.

Thank you for the references. I think you are one of few people thinking
seriously about where String theory might be going.

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. Categorification isn't about categorifying
bundle structures piece by piece. This is why (I think) Ross Street
says one should look at stack theory and leave gerbes alone.

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.

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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.

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

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 .

(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?

Urs Schreiber Wrote:            [Warning: text about category theory. LM]

Great! Do many String theorists think this way? I've quoted what
you said about category theory. It's on my door (a collection
point for interesting snippets).

If you'll allow me to refer to Ross Street's lectures here: Let
R be a commutative ring with unit. Spec(R) is
the space of prime ideals in R. It turns out that
Spec(R) is a sober space (every irreducible closed subset
is the closure of a unique singleton). Sober spaces are completely
recoverable from the category of elementary toposes.

For instance, when considering a Boolean algebra as a ring,
Spec(R) is the Stone space of the ring. Stone is a very
underrated historical figure. What are Stone spaces?

Recall that one takes the lattice structure
\mathcal{O}(X) of open sets of a space X as the
category underlying sheaves on X, which are contravariant.
Such lattices have a 0 (the empty set) and 1 (the set X).

In the category of topological spaces a point is specified by a
morphism from the one point space, which is an initial object. A
sufficiently general type of distributive lattice with 0 and 1 is
called a frame (see Mac Lane and Moerdijk). In the category of frames
the initial object is the 2 point lattice (0,1) so one defines a
'point' of a generalised space to be a morphism into
this object (remember the contravariance). But by this definition a
space might not have any points at all! A space is said to be
'geometric' if for any two objects of the lattice there exists a point
(morphism) p such that p^{-1} distinguishes
the objects.

Back to Stone. The category of sober spaces is equivalent to the
category of generalised spaces which are 'geometric'. This may be
viewed as a duality in which the two point space plays a special
'self-dual' role (it's called a schizophrenic object). Another example
of these so-called Stone dualities is Pontrjagin duality, for which
U(1) is the schizophrenic object.

So...what about NCG? Well, this is the question, isn't it? We need
2-toposes. This is my pet fundamental thing! To Ross Street a 2-topos
involves 2-stacks, which are, first of all, pseudofunctors from a site
C into Cat. The stack condition is a descent diagram, and
the inclusion of Stack(C) into
Ps(C^{\textrm{op}},Cat) is a nice biadjunction. Gerbes, as
you say, are related to this.

But the lattice theory is more fundamental. The logic of a topos
depends on it. Topos (1-stack) lattices are always distributive.
Quantum lattices are not. But quantum lattices are well understood, and
a proper understanding of 2-toposes means getting the lattice theory
right. I guess this is what I've been trying to say!

Regards Kea

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Right, this is getting off-topic. We have moved the discussion to

http://golem.ph.utexas.edu/string/archives/000509.html