## May 3, 2004

### Talk: N=2 NCG, fields and strings

#### Posted by urs

Tomorrow I’ll travel to Hamburg, where on Wednesday I’ll give a talk at the theory seminar of University of Hamburg, on behalf of a kind invitation by Thorsten Pruestel. We had first met at the last DPG spring conference, where I learned of the approach by Pruestel’s group concerning gauge theories with nonunitary parallel transport, which is an attempt to describe (possibly discretized) gravity by means of a special non-unitary component of a gauge connection. Prof. Fredenhagen is also interested in noncommutative field theories, and hence my talk will be on the stuff that Eric Forgy and myself developed a while ago

Eric Forgy & Urs Schreiber: Discrete Differential Geometry on $n$-Diamond Complexes (2004)

as well as its applications to field theory and string theory.

Here I’d like to give a first sketch of what I am going to say in that talk, mostly in the hope that Eric Forgy will spot the major omissions. :-)

(I am having problems with my internet connection, that’s why the following is not fully properly formatted. I am hoping to improve this entire entry tomorrow.)

So this is the sketch of what I am going to say:

======================== N=2 NCG, fields and strings ========================

two mottos:

a) “geometry is supersymmetric quantum theory”

b) “lattice effects are a feature, not a bug”

- contents -

0) introduction

1) general formalism

2) application to mathematics

3) applications to field theory

4) application to string theory

5) conclusion

——

1) general formalism —————————————–

Dimakis & Mü Hoissen formalism:

N=2 generalized spectral ‘tuple’:

exterior calculus $\Omega \left(𝒜,d\right)$ over some algebra $𝒜$

now turn $\mathrm{Omega}\left(𝒜\right)$ into inner product space in order to get generalized spectral triple $\left(𝒜,d,〈\cdot \mid \cdot 〉\right)$

‘generalized’ because it violates some of the axioms of Connes’s NCG, but can be shown by explicit (and relatively simple!) example to give perfectly reasonable NCGs

direct interpretation:

$\Omega \left(𝒜,d\right)$ is (one) NCG generalization of exterior algebra

$〈\alpha \mid \cdot \beta 〉$ is just the NCG version of the Hodge inner product $\int \alpha \wedge \star \beta$

(1)$〈\cdot \mid \cdot 〉\to 〈\cdot \mid \stackrel{̂}{g}\mid \cdot 〉$

$\stackrel{̂}{g}$ is the metric operator which turns ONB forms into coordinate differentials (maps from chart to manifold)

–> Minkowski metric is special in that it is the unique metric constructible fromt the gluing 1-form, i.e. without introducing “further structure”

${d}^{†}$ is exterior coderivative

$\left\{d,v\to \right\}$ is Lie derivative, etc.

Clifford algebra and spinors constructible in Dirac-Kaehler formalism

2) application to mathematics ——————————————————

get NCG version of Hodge star operator:

* A |0> = A^\dagger|vol>

works whenever a |vol> can be defined, i.e. as general as possible!

easy to check that this Hodge has correct continuum limit

3) application to field theory ——————————————

metric formalism allows concise formulation of lattice gauge theory on curved space:

d may equivalently be replaced by “gluin 1-form” rho:

but rho is nothing but the trivial holonomy 1-form

so introduce non-trivial holonomies H

YM field strength is simply H^2 - lattice corrections are a feature, not a bug!!

so YM action on curved space simply

spinors and fermion doubling:

d+d^dagger Dirac-Kaehler operator, always available:

self-adjoint, no fermion doubling slightly non-local

gauge covariant Dirac operator “d + d^dagger –> H + H^\dagger”

4) applications to string theory: ————————————————–

other deformations of the scalar product are possible, what do they correspond to?

answer: further background fields, like Kalb-Ramond fields

take A to be continuous loop space, then super-Virasoro algebra gives a generalized spectral triple as above all backgrounds can be obtained by deforming the inner product, can be made explicit for all massless NS backgrounds, T-duality (S-duality?)

string field theory: ————————–

BRST operator is exterior derivative on gauge group:

Hata shows analogue of holonomy 1-form for OSFT - relation to gluing 1-form??

comparison with lattice CS theory: int (H-d)HH

5) conclusion —————————–

one step away from D&MH: a small step in the formalism, a large step in terms of results

close algebraic connection to continuum theory, allows relatively powerful and general translation of continuum concepts to NCG

long standing problem of NCG Hodge star finds elegant solution

based on Connes central insight: geometry = supersymmetric quantum theory

violates some of Connes’ axioms but can explicitly be shown to suffer no problems due to that

works nicely on noncompact and semi-Riemannian spaces: “problems” are dealt with just as in continuum theory (e.g. solutions to KG/Dirac equation must not be integrated in time - gauge fixing)

applicable not only to discrete spaces - insights even to continuum theory (string backgrounds)

Posted at May 3, 2004 11:36 PM UTC

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### Re: Talk: N=2 NCG, fields and strings

Good morning! :)

You’ve given quite a lot of info here. I’ll give as much feedback as I can :)

two mottos:

a) “geometry is supersymmetric quantum theory”

b) “lattice effects are a feature, not a bug”

I think one of the main themes for your talk should be to try to make a clear distinction between what we have done and what others have done.

Point a.) is obvious to those who are familiar with Witten’s work (among others, notably Fröhlich) and point b.) is obvious to those who work in lattice theories. But the combination of the two ideas is really what sets this work apart I think.

1) general formalism:

Dimakis & Müeller Hoissen formalism:

N=2 generalized spectral ‘tuple’

exterior calculus $\Omega \left(𝒜,d\right)$ over some algebra $𝒜$

Except for the part about the “tuple”, this almost sounds like standard discrete geometry, but it is a little more than that. We are now thinking of the algebra of operators (as in spectral “tuple”). This is a departure from the usual discrete approaches.

now turn $\Omega \left(𝒜\right)$ into inner product space in order to get generalized spectral triple $\left(𝒜,d,〈\cdot \mid \cdot 〉\right)$

This is really the “holy grail” of the entire enterprise. Because you came in and we solved the problem within a matter of weeks, I still don’t think you fully realize what a great thing this is :) Personally, I spent 6 years banging my head against a wall trying to come up with a working theory. Others before me have spent even longer.

Looking back, it is obvious, but that doesn’t mean it is any less significant :) The big highlights for our contribution involve a unification of points a.) and b.).

1.) Thinking of discrete forms as operators (motivated by SQM)

2.) Postulating the existence of adjoint operators and discovering their required properties, e.g.

- 0-forms are only self adjoint for flat spaces, i.e.

0-forms self-adjoint $↔$ space is flat

- adjoint edges can be thought of as pointing in the opposite direction (rough analogy to Feynman diagrams)

$\stackrel{̂}{g}$ is the metric operator which turns ONB forms into coordinate differentials (maps from chart to manifold)

This is pretty neat, but may be confusing (I know it is for me). I usually think of the metric a providing a means to map a 1-form to a unique vector field and vice versa. The issue of primary versus dual spaces seems a little confused in our stuff. This “metric operator” maps a $p$-form to a $p$-form.

$\to$ Minkowski metric is special in that it is the unique metric constructible fromt the gluing 1-form, i.e. without introducing “further structure”

This is pretty neat and is probably related to Sorkin’s work with “posets”. You can obtain the Minkowski metric (up to a conformal factor) by simply defining the causal structure of a space. Our gluing 1-form essentially defines the causal structure as well, so it is not too surprising to see Minkowski space emerge.

By the way, did I tell you why I call it the “gluing 1-form”? The gluing 1-form is dual to the “gluing 1-chain” :) I called it the gluing 1-chain because it gives rise to a discrete version of the “join” of two chains. If you think of forms and chains as a bialgebra system, then the coboundary is a derivation, i.e. satsifies graded Leibniz, with respect to the product of forms (cochains). However, the boundary map is not a derivation with respect to the dual product of chains. However, if you define a $\rho$ augmented product of chains

(1)$a*b=a\rho b,$

where $\rho$ is a 1-chain, then the boundary map is a derivation. The $\rho$-augmented chain product is what is called the “join” in algebraic topology. The 1-chain that gives rise to the join operation is the gluing 1-chain :)

2) application to mathematics ——————-

get NCG version of Hodge star operator:

$\star A\mid 0〉={A}^{†}\mid \mathrm{vol}〉$

works whenever a $\mid \mathrm{vol}〉$ can be defined, i.e. as general as possible!

The Hodge is more than just mathematics. Without the Hodge, there is no physics either :)

easy to check that this Hodge has correct continuum limit

Nice :)

self-adjoint, no fermion doubling slightly non-local

Be ready to back up this claim. Some people might be sensitive to this because it is a long outstanding problem.

4) applications to string theory: ————————

other deformations of the scalar product are possible, what do they correspond to?

answer: further background fields, like Kalb-Ramond fields

take A to be continuous loop space, then super-Virasoro algebra gives a generalized spectral triple as above all backgrounds can be obtained by deforming the inner product, can be made explicit for all massless NS backgrounds, T-duality (S-duality?)

I hope I had some influence on this. String theory is so far beyond me, but I like impose my limited knowledge now and then :) It seems natural to consider deformations of the inner product.

string field theory: ———————-

BRST operator is exterior derivative on gauge group:

Hata shows analogue of holonomy 1-form for OSFT - relation to gluing 1-form??

comparison with lattice CS theory: $\int \left(H-d\right)\mathrm{HH}$

You might think about a more aesthetic expression. Why not simply write

(2)${S}_{\mathrm{CS}}={\int }_{M}A{d}_{A}A$

?? Or find some other more elegant expression. In my experience, people are prejudiced again discrete theories so anything that makes them seem less elegant than the continuum will feed any prejudices they might have.

5) conclusion ———————

one step away from D&MH: a small step in the formalism, a large step in terms of results

Neil Armstrong? :)

applicable not only to discrete spaces - insights even to continuum theory (string backgrounds)

Kewl :) I wish I could be there! Let me know how it goes :)

Best of luck,
Eric

Posted by: Eric on May 4, 2004 4:55 PM | Permalink | Reply to this

### Re: Talk: N=2 NCG, fields and strings

I think what I said about fermion doubling is true, but the point is that there are several properties that one is concerned about, like ‘doublers’, locality, chirality, etc. The Dirac-Kaehler operator of course beaks locality and chirality is there only in a certain sense.

But I generally agree on what you said.

You can find the transparancies which I fainally used below.

I had a very nice and productive time at DESY/University of Hamburg, having lots of intensive conversation with Thorsten Pruestel, Michael Olschewsky and Falk Neugebohrn who are working in Prof. Mack’s group on what they call gauge theories with ‘non-unitary parallel transport’.

The key idea is that they descrize using D&MH formalism, but on a symmetric lattice, i.e. edges pointing in both directions.

Instead of worrying about the continuum interpretation of the additional p-forms which are obained this way, they just write down discrete YM actions (on a Euclidean background) using the assumption that

- ordinary holonomies along opposite edges are the inverse of the holonomies along the original edge (usual behaviour)

- but some components of the ‘gauge group’ (which is not really a symmetry grou, then) are ‘non-unitary’ in the sense that the holonomy along one arrow is not the inverse of the holonomy along the opposite arrow.

They attempt to express the Higgs field as well as gravity as such ‘non-unitary’ gauge degrees of freedom.

For the Higgs at least, where I looked at the mechanism in detail, this works quite nicely: Using a modified Kaluza-Klein mechanism where there is a 5th dimension of spacetime along which we have the non-unitary parallel tranport you readily see that the YM field strength now also has components

(1)${F}_{+5,-5}\ne 0$

which are non-vanishing, because the discrete 2-forms consisting of two opposite edges don’t vanish.

When you compute the YM action this way you get the usual expression in 3+1d plus the usual KK terms and, due to the symmetric lattice and non-unitarity, plus a further term which is quartic in the non-unitary component of the gauge field and looks exactly like the Higgs potential.

We cannot write down a Hodge star for the symmetric lattice (due to the fact that there is no volume form) but this has no bearing on this construction, because the inner product used to write down the YM action of course still exists. Mack’s group now also tries to get gravity in the Palatini formulation in a similar way as the Higgs field appeared above. This is apparently much more involved and also, I’ve been told, more awkward. I am wondering if it wouldn’t instead be interesting to use the above Higgs mechanism and put it on a curved spacetime using our metric formalism.

BTW, after my talk Prof. Fredenhagen and his student Jochen Zahn complained about my claim that for self-adjoint discrete 0-forms the inner product must be Euclidean. They are right that we need to emphasize that in deriving this claim we make the postulate that we want the result of acting with a form annihilator on a form creator to be an element of the algebra. A priori, the result could also include for instace shift operators. I think one can still argue that ours is the natural restriction, but it is an axiom/assumption which we we need to make explicit.

It turns out that our way to include background fields like gravity in terms of the inner product, which, as I tried to emphasize in my talk, is very nicely justified by related mechanisms in NCG and string (field) theory, is completely different from the approach by Mack’s group, where (discrete) gravity is part of the gauge group in some generalized sense. I am wondering what kind of inner product is used in this approach, and in fact while writing this I realize that I should have asked about this.

We also talked about lots of other things, like LQG, Pohlmeyer invariants, discrete geometry and the relation to category theory, Connes-Lott models, Matrix models, which was all very interesting.

Posted by: Urs Schreiber on May 6, 2004 3:40 PM | Permalink | PGP Sig | Reply to this

### Re: Talk: N=2 NCG, fields and strings

Hello! :)

Just a quick note…

It sounds like you had another productive visit. That is great! :)

I don’t know if they would be interested, but I think it would be great if we could invite them to join us here at the coffee table to discuss things out in the open. It sounds like they are thinking of very similar things as we think about.

For example, the issue of directed versus symmetric graphs is an interesting thing to discuss. The thought has occured to me also that the extra dimension that arises from the opposite edges may be a feature and not a bug :) Then again, a directed graph giving a nice interpretation as a flow in time is nice too. How do they deal with time? Do they have some preprint available that discusses this material? My initial search must have missed it.

I’d also like to better understand their complaint about “0-forms self-adjoint $↔$ space is flat”. Do they have a non-flat inner product for which 0-forms are self-adjoint? That would be interesting. The fact that a Hodge star is not very natural on a symmetric graph causes me a little alarm, but nothing too serious. The important thing I’m concerned with is being able to write down a meaningful inner product and an adjoint exterior derivative. If we can do that, that is good enough for me (and Maxwell).

One outstanding question that I’m still slightly concerned with is more general manifolds. So far we only know for sure that our theory works on spaces globally like Minkowski space (and some simple cases that can be obtained by identitying points in Minkowski space, e.g. tori, etc). What about things like ${S}^{2}$? Can they handle arbitrary manifolds with ease? That would be interesting.

I think that all of our interests are different enough that we shouldn’t have any competing agendas. My ultimate goal is to write down Maxwell’s equations in the presence of antennas, buildings, and humans :) Your’s is primarily related to string theory and theirs seems to be on gauge theory and gravity. Sounds like it could make for some interesting exchanges of ideas to me :)

Gotta run for now…

Best wishes,
Eric

Posted by: Eric on May 6, 2004 4:06 PM | Permalink | Reply to this

### Re: Talk: N=2 NCG, fields and strings

Another quick thought, then I really must work on something or else I am dead meat :)

Remember when we were discussing ideas on how to construct finitary versions of arbitrary manifolds $M×R$? There, an “illusion” of having opposite edges, i.e. symmetric graphs, appeared but they were not really opposite because the end points were at different instances in time.

In this way, parallel transport could still be unitary, but there is no need for parallel transport from $i\to j$ to be the same as $j\to i$ because the nodes $i$, $j$ are actually at different time steps. In other words, to parallel transport a vector from node $i$ to node $j$ requires time to proceed by one step, so in fact $i\to j$ and $j\to i$ criss cross in space time even though they appear to be “opposite” in space. If I make this explicit by putting an index for time in parentheses, it would look like

(1)$i\left(n\right)\to j\left(n+1\right)$

versus

(2)$j\left(n\right)\to i\left(n+1\right).$

There is no reason to expect these parallel transports to be equivalent. I wonder if we can reinterpret their symmetric graph as a directed graph folded in the time direction?

I really gotta run :)

Cheers! :)

Eric

Posted by: Eric on May 6, 2004 4:24 PM | Permalink | Reply to this

### Re: Talk: N=2 NCG, fields and strings

How do they deal with time?

In case of the Higgs construction they are , at least in the simplest model, I think, using a hypercubic symmetric 5D graph where one of the directions of the edges is time, 3 are space and 1 is the extra KK dimension.

The paper dealing with this which I have looked at in a little more detail is

Michael Olschewsky: Higgsfelder als Verallgemeinerte Eichfelder (2002)

As you see, this is in German (being a diploma thesis, i.e. that which makes a student a graduate student in Germany) but probably you can get the idea by looking just at the formulas.

When dealing with gravity the approach is apparently to use a simplicial complex (as far as I understand this is motivated by the fact that one expects covariance to play a role, somehow) which makes D&MH formalism much more cumbersome, as we have discovered, too.

For the gravitational case the paper that I was pointed to is another diploma thesis:

Falk Neugebohrn: Allgemeine Relativitaetstheorie auf diskreten Mannigfaltigkeiten (2003)

Here the key idea is to formulate GR as a generalized gauge theory using Palatini formalism and those non-unitary holonomies and then to write down the discrete EH actions as

(1)$\int e\wedge e\wedge F\phantom{\rule{thinmathspace}{0ex}},$

where $F$ is the gravitational field strength, i.e. the Riemann tensor 2-form. Biggest problem is, I am being told, that the 4-form involved in this action is of course non-local in the discrete theory which makes gauge invariance of the above action problematic. This can be dealt with but at the cost that the resulting formulas tend to be awkward.

You should definitely have a look at Thorsten Pruestel’s PhD thesis:

Th. Prüstel, Gauge Theories with Nonunitary Parallel Transport (2003).

BTW, in my talk I was also asked how we would deal with the EH action. My canonical answer is that we would use just the spectral action principle by Connes (we could restrict to Riemannian metric and a toroidal compactification first in order to avoid technical problems), but of course I had to admit that we haven’t done it yet. Somebody should do this!! :-)

I’d also like to better understand their complaint about ‘0-forms self-adjoint $↔$ space is flat’.

The point is that we argue that acting with an elementary 1-form annihilator on an elementary 1 form must produce a 0-form. But that’s not the most general possibility. It could in principle produce anything which is of 0 grade. For instance it could produce a 0-form times a shift operator. Our argument only applies to the case where we demand (restrict to the special case) that the result is really an element of the algebra.

I wonder if we can reinterpret their symmetric graph as a directed graph folded in the time direction?

As far as I can see their construction really crucially uses 2-forms which return to the same point in spacetime, not just in space.

But I think there, at least from our point of view, really two independent assumptions in this ‘non-unitary holonomy’ concept. One is that the relation

(2)${g}^{†}={g}^{-1}$

for group elements, which we also have in our notes, is violated, the other is that opposite edges have not the inverse holonomy as the original edges. I think it would be interesting to go through the derivations in Michael’s thesis above and see what happens if we work on a diamond graph but with ${g}^{†}\ne {g}^{-1}$ or maybe the other way round.

Posted by: Urs Schreiber on May 6, 2004 4:57 PM | Permalink | PGP Sig | Reply to this

### Re: Talk: N=2 NCG, fields and strings

Hello,

Thanks for the link to Thorsten’s thesis! :) It is great! :)

I am beginning to wonder if the difference between their stuff and ours is not more than a reinterpretation and different notation? For example, I am looking at their $\star$-operation in Chapter 4. This looks very reminiscent of our $†$-operation. Could it be that these are really the same thing and any differences that follow are merely notational? We would think of a $†$-edge as having degree -1. Their $\star$-edge would have degree +1. I imagine there may be a map from one interpretation to the other. We should see if we can reproduce their Higgs stuff without opposite edges, but with adjoint edges taking their place.

Eric

PS: I’m still skimming the thesis so this issue may already be addressed and I haven’t seen it yet :)

Posted by: Eric on May 6, 2004 5:26 PM | Permalink | Reply to this

### Re: Talk: N=2 NCG, fields and strings

Hi Urs,

I have almost convinced myself (without really doing the work to justify it :)) that their work does not crucially depend on the existence of opposite edges in any way I can see. It does rely crucially on the existence of adjoint edges though. After all, I think the idea that parallel transport $i\to j$ must require an increment in time is a good rule of thumb. If they truly are tranporting back to the same point in spacetime, then that means they are tranporting back in time as well. The operator that should do this kind of thing is the adjoint operator, not an opposite operator. This really fits in with the idea of “non-unitary parallel transport”. It is not transporting along the opposite edge, it is transporting along the adjoint edge. Therefore, perhaps we’ve got

(1)${U}^{†}\left({C}^{†}\right)U\left(C\right)\ne 1$

rather than

(2)${U}^{†}\left(-C\right)U\left(C\right)\ne 1.$

See what I mean?

Of course, if anything I’m saying has any bases in truth, then we could really account for their “non-unitary transport” and probably the Higgs stuff by a simple deformation of our inner product.

This would fit in perfectly with the theme we’ve been developing :)

What do you think?

Eric

PS: I’m hoping some of the people you met at DESY can join us :)

Posted by: Eric on May 6, 2004 5:50 PM | Permalink | Reply to this

### Transparancies

Posted by: Urs Schreiber on May 6, 2004 3:34 PM | Permalink | PGP Sig | Reply to this

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