## January 26, 2004

### The search for discrete differential geometry

#### Posted by Urs Schreiber

At first it may look like a trivial problem whose solution should have been known for ages, being used all over the place for applications in mathematics, theoretical physics and engineering. But surprisingly, it has apparently not fully been understood yet. I am talking about the adaption of the full machinery of differential geometry to discrete spaces. In particular, I am being told that the metric aspects of such a theory still puzzle many researchers, the Hodge star operator, for instance, being notoriously hard to come by on general discrete spaces.

A while ago Eric Forgy has convinced me that it may be worthwhile to think about these issues, and after a very intesive collaboration we came up with what looks like an interesting approach to discrete metric differential geometry to us. Now we are trying to communicate our results with other researchers in the field.

Since most of this exchange is currently going on by e-mail and since this puts severe restrictions on the amount of true interaction and maybe cross-fertilization as soon as more than two people are involved, I was wondering if maybe we’d need some sort of discussion forum. This blog entry is supposed to be the entry point to such a discussion. To get started, I list some relevant liks to the current literature below. The list won’t be comprehensive at all at the moment, but I am planning to update it as we go along.

Ok, so here are some links to people and texts currently dealing with discrete differential geometry. This is just a very rough first approximation to a comprehensive link list. I am going to flesh this out as soon as I find the time.

First of all, Dimakis and Mueller-Hoissen have written many papers on dicrete differential geometry. For the moment I’ll just cite this list.

Jenny Harrison says she has been developing a new discrete theory of exterior calculus using chainlets. I don’t know if the new results are already published.

There is an interesting talk by Sullivan about his ideas on discrete geometry.

Alain Bossavit is an expert on discrete electromagnetism using finite elements and Whitney forms.

A. Sitarz has papers on Noncommutative Geometry apporaches to discrete spaces, for instance Noncommutative Geometry of Finite Groups.

In Berlin there is a ‘Sonderforschungsbereich’ concerned with Discrete Differential Geometry, Quantum Field Theory, and Statistical Mechanics. See their list of publications.

Our own notes (which are still in a draft stage) are

E. Forgy, U. Schreiber Discrete Differential Geometry on $n$-Diamonds.

More links, some of which need to be incorporated into this list, can be found here of course.

Posted at January 26, 2004 11:07 PM UTC

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### Re: The search for discrete differential geometry

Hi Urs! :)

The relationship between string theory and discrete differential geometry is not obvious to me, but a lot of the papers by Dimakis and Mueller-Hoissen mention the word now and then :)

I hope that the other hosts do not mind if we use this powerful forum to discuss our work out in the open. If anyone besides us ends up reading this, let me just quickly say that Urs and I typically end up writing several emails to each other each day. It is usually me coming up with some whacky idea and then he’ll turn it into something fairly intelligible. It’s great fun! :)

Urs is a graduate student studying string theory. I somehow managed to get him hooked on discrete differential geometry, and he is now set on applying it toward string theory. So this is not really as off topic as it might seem.

Until any of the other gracious hosts object, we’ll probably try this forum out as a way to communicate our ideas to one another and to anyone else who might find what we have to say interesting.

It should go without saying that we welcome any and all comments along the way. There is a great potential to have some really nice discussion here.

Best wishes,

Eric

Posted by: Eric on January 27, 2004 2:56 AM | Permalink | Reply to this

### Re: The search for discrete differential geometry

I will sketch some of the central ideas of the approach that Eric and I have been following. The basic idea is quite simple and involves essentially a straightforward merging of the concepts of Noncommutative Geometry (NCG) with that of abstract differential calculus. The result is indeed nothing but an NCG defined by an $N=2$ spectral triple, i.e. one which has not only one but two (anticommuting) Dirac operators.

So let $𝒜$ be an associative algebra of functions characterizing a topological space. In the present case this will be the group-valued algebra of functions on a denumerable set of points of a discrete space.

Over $𝒜$ one can consider differential calculi $\Omega \left(𝒜,d\right)$ whose elements are generated by elements $a$ of grade 0 and elements $da$ of grade 1 (with $a\in 𝒜$), where $d$ is a formal exterior derivative which satisfies ${d}^{2}=0$ as well as the graded Leibnitz rule. Elements of grade $p$ in this algebra are the analoga of differential $p$-forms.

In order to obtain a substitute for the Hodge inner product $〈\alpha \mid \beta 〉=\int \alpha \wedge \star \beta$ known from the continuum, define on $\Omega \left(𝒜,d\right)$ (regarded as a vector space) a non-degenerate (sesquilinear) inner product

(1)$〈\cdot \mid \cdot 〉:\Omega ×\Omega \to C\phantom{\rule{thinmathspace}{0ex}}.$

The elements $a\in 𝒜$ of the original algebra as well as $d$ are represented on the resulting inner product space $ℋ\left(\Omega ,〈\cdot \mid \cdot 〉\right)$ as operators in the obvious way. Taking adjoints of these operators with respect to $〈\cdot \mid \cdot 〉$ in particular yields ${d}^{†}$, the analogue of the exterior coderivative.

The information about the metric geometry of the discrete space is encoded in $〈\cdot \mid \cdot 〉$ and hence inherited by ${d}^{†}$. The connection to the familiar formulation of Noncommutative Geometry is obtained by noting that $d±{d}^{†}$ are two odd-graded Dirac operators on $ℋ$, so that $\left(𝒜,ℋ,d±{d}^{†}\right)$ is a spectral triple.

Now denote by $\star$ an operator on $ℋ$ (if it exists) which satisfies

(2)${d}^{†}=±\star d{\star }^{-1}\phantom{\rule{thinmathspace}{0ex}}.$

Define a volume-like form to be an element $\mid \mathrm{vol}〉$ of $\Omega \left(𝒜,d\right)$ that is of maximal grade and annihilated by ${d}^{†}$: ${d}^{†}\mid \mathrm{vol}〉=0$.

With these definitions the following fact holds: Given the inner product space $ℋ\left(\Omega ,〈\cdot \mid \cdot 〉\right)$ we have

(3)$\star \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exists}\phantom{\rule{thinmathspace}{0ex}}⇔\phantom{\rule{thinmathspace}{0ex}}\mid \mathrm{vol}〉\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{exists}$

and they are related by

(4)$\mid \mathrm{vol}〉=\star \mid 1〉$

and

(5)$\star \mid {a}_{0}\phantom{\rule{thinmathspace}{0ex}}d{a}_{1}\cdots d{a}_{p}〉={\left({a}_{0}\phantom{\rule{thinmathspace}{0ex}}d{a}_{1}\cdots d{a}_{p}\right)}^{†}\mid \mathrm{vol}〉\phantom{\rule{thinmathspace}{0ex}}.$

It follows in particular that $\star$ exists iff there is a solution $\mid \mathrm{vol}〉$ to the equation $〈d\alpha \mid \mathrm{vol}〉=0,\phantom{\rule{thinmathspace}{0ex}}\forall \phantom{\rule{thinmathspace}{0ex}}\alpha \in \Omega \left(𝒜,d\right).$ In the case of discrete geometry on topologically hypercubic graphs and complex valued $a\in 𝒜$ this condition can be solved explicitly for $\mid \mathrm{vol}〉$. The above equation then defines the Hodge star operator on these spaces.

This is spelled out in detail in section 4 of our notes.

Posted by: Urs Schreiber on January 27, 2004 6:54 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

It seems like there is quite some excitement about Thiemann’s paper. Sounds interesting.

One quick comment…

One neat consequence of what Urs outlined was that if elements of the algegra are self-adjoint, then the space is “flat”. This was surprising to both of us and explained one of the reasons I had been struggling over the years. I was trying to have both non-flat spaces and self-adjoint algebras. This was a crucial insight.

Gotta run. More later…

Eric

Posted by: Eric on January 27, 2004 8:13 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Eric,

I need help in finding the reference to the approach to discrete diff geometry by means of using Whitney forms, finite elements, interpolation, and so on. Thanks!

Posted by: Urs Schreiber on January 28, 2004 11:27 AM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Urs,

I don’t think there exists the biblical reference on discrete methods for which you are looking :)

I particularly like the “Japanese Papers” by Alain Bossavit, whose reference can be found on that link you provided to his web page

http://www.lgep.supelec.fr/mse/perso/ab/bossavit.html

I see his book is also available for download, which is a good reference.

This

http://www.math.unm.edu/~stanly/mimetic.html

is probably the best resource for finding “who’s who” in discrete methods. The list of references (in .bib format and everything! :)) there is pretty helpful.

You might, in particular, like to follow Tonti’s link (if it works, my proxy is complaining).

You’ll find all you ever wanted to know about (applied) discrete methods by browsing the list of people and their web pages.

Where does the time go…

Eric

Posted by: Eric on January 28, 2004 10:23 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Deformed differential geometry related to finite differences has been developed by Majid ten
years ago, as an example to his q-group approach. Possibly the right setting for finite
differences would be the tangent groupoid, as it includes the possibility of a continous limit.

Speaking of continous limit, Brouder hep-th/9904014 shows that Butcher trees, which
classify finer and finer Runge-Kutta methods, can be related to Connes-Kreimer trees, which
control the renormalization group (hep-th/9904044 etc). Besides, CK trees seem to be related to
the group of diffeomorfisms of a geometrical object.

As for lattices, let me point that commutative spectral triples over discrete lattices have
been classified by Alain Connes (example at page 15 of hep-th/9603053), by
T. Krajewsky (hep-th/9701081) and by M. Paschke and A. Sitarz (q-alg/9612029)

See also hep-th/0312276 and also the big series of papers and preprints by Jian Dai.

A non-go result for Poincare Duality on naive discrete dirac operators was given
by M. Gokeler and T. Schucker in hep-th/9805077. A workaround has been proposed
by A. Rivero ;-) at math-ph/0203024, hep-th/0204238.

I have not read “Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory”,
which seems closer to string theory or at least to the stringy view of NCG. To me, string theory is
about having distances and energy-momentum adscribed to the same physical entity, while field theory
(and very specifically fields in the discrete lattice) are about having distances and momentum adscribed to
different entities, sometimes called vacuum and particles respectively.

Posted by: alejandro rivero on February 6, 2004 3:04 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Alejandro,

Thanks for your post and the references. I wasn’t aware of those two most excellent workaround papers from last year :) I’ll have to check them out.

By “Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory”, are you referring to this

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

http://arxiv.org/abs/hep-th/0004147

By the way, Urs and I would definitely value any feedback you might have regarding our paper. I believe that although what we have can probably be consumed under the huge umbrella that Connes’ constructed, the specific model we have seems to be particularly nice for doing concrete applications. I haven’t seen any reference with the same emphasis that developed from our work (e.g. the need for n-diamond complexes).

Best wishes,

Eric

PS: I am fairly certain the Poincare duality is trivial on an n-diamond complex. I could be wrong.

Posted by: Eric on February 6, 2004 4:40 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi guys,

Mi experience is that poincare duality is never trivial with spectral triples; I will bet that my own work have some failure in it that my friends have been nice enought to no report to me, so I will not depress… Probably the same can be said about the enormous effort of the two australian students, Rennie and Lord, who took upon them the effort to reproduce completely the commutative continuous example.

Yep, the other paper was J. Ambjorn et al. I do not know what to think about all the NCG variants outside the -almost extinct- first mainstream.

I am printing your text to take a look during the weekend. On my own, I believe that any discretization should be measured in units of plank constant, so I do not believe in naive discrete geometry over the space, nor the space-time.

Posted by: alejandro rivero on February 6, 2004 7:22 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Alejandro -

many thanks for all these references! I haven’t yet had the time to look in any detail at all this information. But here are some questions:

Could we perhaps clarify what we mutually mean by Hodge duality versus Poincare duality?

Could you try to explain the basic assumptions and ideas behind this no-go theorem that you mentioned? Also, what are the assumptions and ideas behind the workarounds that you are proposing?

Posted by: Urs Schreiber on February 6, 2004 7:12 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Urs and Alejandro,

When I said Poincare duality was trivial with diamond complexes, what I really meant was that constructing a Poincare dual grid was trivial. This I guess isn’t really what is usually meant by Poincare duality. For that, we would need to define a product between elements of the primary grid with elements of the Poincare dual grid. We have not done this. We have talked about it, but it was not obvious how to proceed. In that sense, Poincare duality is not trivial in our formalism either. Then again, we may have a slightly unorthodox way to view Poincare duality on a grid. This is due to the fact that our Hodge star does not map p-forms to (n-p)-forms on a dual lattice. Rather, we get (n-p)-form on the SAME lattice. Therefore, we could probably define a meaningful notion of Poincare duality on n-diamond complexes although it might not be exactly what one might usually think of (which is a good thing in my opinion).

I should say that I am also very much against ad hocness and naive discretizations. That is the primary motivation for this entire effort in the first place. So although Planck’s constant has not “emerged” from our work from some primordial concept, I’d really hesitate to say that what we did is naive (not that that is what I think you were implying).

I’ll return the favor and print out your papers for the weekend. I look forward to discussing them here.

Best regards,
Eric

Posted by: Eric on February 6, 2004 7:37 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Well Poincare duality, as used in NCG and other mathematical field, maps –note super and sub-indices– from H^p to H_{n-p}; on the contrary, I understand that the usual Hodge operator maps from p-forms to n-p forms.

My better grasping of Poincare Duality comes from third volume of Dubrovin-Fomenko-Novikov. Poor Fomenko crossed the border of madness sometime, according a website around. But anyway, the point to grasp is the role of the intersection or cap product. It is the operation that let us to reduce the number of variables in a integral, ie to do separate integration of (along) a single coordinate. Thus a fixed volume form plus an intersection product give us the tools to define Poincare Duality.

Posted by: Alejandro Rivero on February 7, 2004 10:53 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

So Hodge duality plus a way to send p-forms to p-chains should give us also Poincare-duality, right?

Posted by: Urs Schreiber on February 9, 2004 11:30 AM | Permalink | Reply to this

### Re: The search for discrete differential geometry

I would expect so. In fact my main worry about Froelich et al approach to Hodge operator is that it is added to the spectral triple, while I expected at least one such operator to come canonically from the volume form and the intersection product. Note that in NCG axioms the intersection product is required, and it should be. It is a important point, to be able to do calculus.

Posted by: arivero on February 9, 2004 1:10 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Whether or not to include the Hodge star, if it exists, in the spectral ‘triple’ is just a matter of notation, I’d say.

But yes, as I have mentioned before the existence of a volume form is equivalent to the existence of a Hodge star (at least if by volume form we mean a top form which is annihilated by the adjoint of $d$).

Posted by: Urs Schreiber on February 9, 2004 1:55 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

And on other hand the existence of such volume form is already required in the axioms for spectral triples, so the Hodge should follow sort of canonical way, that was my doubt against Froelich’s two papers; they seem to work with an ad-hoc Hodge.

Posted by: alejandro rivero on February 9, 2004 2:22 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Alejandro -

when you say ‘volume form’ you are not referring to the same thing that I am! :-) Let’s try to get in sync. If you don’t mind, let’s start from the very beginning:

We are given an associative algebra $𝒜$, a Hilbert space $ℋ$ on which $𝒜$ is represented and which supports a grading operator $\gamma$ with ${\gamma }^{2}=1$, $\left[\gamma ,𝒜\right]=0$ and an operator $D$ with $\left\{D,\gamma \right\}=0$. When all this is collected in an ordered tuple $\left(𝒜,ℋ,D\right)$ we speak of a spectral triple.

Now what is it that you are calling a ‘volume form’ in this construction?

What I would like to refer to as a volume form is the following: I consider the special case where there is a second odd-graded operator ${D}_{2}$ on $ℋ$ such that ${D}^{2}={D}_{2}^{2}$ and $\left\{D,{D}_{2}\right\}=0$. Then I set

(1)$d:=\frac{1}{2}\left(D+i{D}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$
(2)${d}^{†}=\left(d{\right)}^{†}=\frac{1}{2}\left(D-i{D}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$

As discussed by Froehlich, the existence of the second ${D}_{2}$ refines the grading of the Hilbert space form ${Z}_{2}$ to $Z$. What I am calling a volume form is this: A non-vanishing element $\mid \psi 〉\in ℋ$ which is of maximal grade and satisfies

(3)${d}^{†}\mid \psi 〉=0\phantom{\rule{thinmathspace}{0ex}}.$

This use of the term ‘volume form’ in NCG is my own invention and if it collides with any established convention I need to change my nomenclature.

Posted by: Urs Schreiber on February 9, 2004 2:43 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

The point is that $\gamma$ contains already the info for oriented volumes, lackin only of the metric information coming from $D$. So it can be told that $\gamma$ is the volume form. It is not standard use, anyway, because really one needs not only the representative $\gamma$ but the Hochschild n-cycle $c$ such that $\pi \left(c\right)=\gamma$. The situation is explicitly stated, for the commutative case, in page 12 of one of the fundational papers, hep-th/9603053

Posted by: Alejandro Rivero on February 9, 2004 5:31 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Alejandro, many thanks for your help! Apparently it is important to say that what Eric and I have done is NCG-inspired rather than true NCG, because I am not sure that our construction satisfies all the axioms on pp. 13 of that paper.

I have a couple of further questions:

1) Assume in an even number of dimensions that a $\gamma$ with $\gamma ={\gamma }^{*}$, ${\gamma }^{2}=1$, $\left[\gamma ,a\right]=0$, $\left\{\gamma ,D\right\}=0$ exist on $ℋ$ but that it does not come from any Hochschild cycle. What construction would fail? I.e., what is it good for to know the Hochschild cycle that $\gamma$ comes from?

2) What is the purpose of axioms (2’) and (7’) on pages 13,14? What constructions would fail if (2’),(7’) were not satisfied?

This is probably crucial for our approach. I can currently not see how in a discrete differential calculus equipped with an inner product (2’) should hold.

Posted by: Urs Schreiber on February 9, 2004 6:42 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

I am sorry I am been already pushed into my limit. Lets see. As for (1), I would say that you need the cycle because of the same reasons you wanted a volume form defined from a operator d. Note for instance that page 8 uses that $\pi \left(c\right)=1$ (odd case, here) to get the equation actually defining the circle ${S}^{1}$. In fact all the plethora of papers on NCG spheres are extensions of this example.

As for question (2), it is not easy to get it if you have not read the redbook, last chapter. There He :-) used two different algebras to define all the duality plays. When joining them in a single selfdual algebra, this leads to a new operator, J, keeping memory of the two previuosly separated algebras so that. The axiom 2, which makes sure that D is first-order (I was told), must be modifyed also to keep its previous significance. The new concept of using $J$ appeared in the preprint “Non Commutative Geometry and Reality”. It is not at the arxiv, but it is scanned at KEK database.

Aside… did you enjoyed my planck length==>compton length charade? I am happy to see, even if naively, that there is some quantum mechanics going on inside quantum gravity.

Posted by: alejandro rivero on February 9, 2004 7:32 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

it is not easy to get it if you have not read the redbook, last chapter.

Thanks, I’ll see if that helps. So do I understand correctly that there was a certain evolution of Connes’ axioms over the years?

Aside… did you enjoyed my planck length==>compton length charade? I am happy to see, even if naively, that there is some quantum mechanics going on inside quantum gravity.

I assume that you are referring to this? A nice observation. I am not quite sure what to make of it, though. I think area quantization rests on shaky grounds, but of course the dimensional analysis you do is suggestive even without strict area quantization.

Posted by: Urs Schreiber on February 10, 2004 9:49 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Alejandro,

Well your paper math-ph/0203024 certainly looks interesting at first glance :) I am now collecting all of the papers but I like the abstract of hep-th/9805077

We are unable to formulate lattice gauge theories in the framework of Connes’
spectral triples.

I think we also might have a counterexample :)

Best regards,
Eric

Posted by: Eric on February 6, 2004 7:53 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi again

About your paper: the main problem, when glancing over it, is a sense of deja-vu. I mean, people looking at the question of relating discrete derivations and non commutative geometry has already heard this music, and you need some catching notes to convince them to put attention. Of course a sound counterexample to Goelecker-Shucker should be a good thing.

Another interesting point I see is the calculation of divergences. Here it seems you have gone a little step forward than another papers, and it could be interesting if you were able to propose a couple of problems (classical EM fields or something so, with Gauss theorem and all that) and solve them by discrete differential geometry.

The relationship with stochastic calculus has also been invoked in Dimakis-Tzanatkis work, q-alg/960601, math-ph/9912016 … Now here I believe your paper is in position to do deeper things,in considering both the osmotic and symmetric derivative or the ambiguity in combining left and right derivatives. One expects that a whole new set of differential equations could be defined if one gets to save this ambiguity in the continuous limit without increasing the order of the differential equation.

(Actually, time ago I conjectured that a try to save this ambiguity in second order partial differential equations and beyond should force the definition of a sort of cabbibo angles and mass matrices)

Posted by: alejandro rivero on February 9, 2004 12:09 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

I am now in NY for a conference visiting Dennis Sullivan, Jozef Dodziuk, and Jenny Harrison. It is painfully obvious how inferior my knowledge in mathematics is :)

Eric

Posted by: Eric on February 9, 2004 2:27 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hello,

Urs mentioned that it might be good to write a short summary of my experience at the conference at CUNY.

I went for two meetings. The first was a conference honoring the retirement of Professor Edgar Feldman.

Geometric Analysis and its Applications:
A Conference in Honor of Edgar Feldman

February 7-8, 2004
Science Center, Fourth Floor
365 Fifth Avenue, New York, NY

What drew me to the conference was the convergence of three people I admire very much: Professor Dennis Sullivan, Professor Jozef Dodziuk, and Professor Jenny Harrison.

For me, meeting and getting a chance to discuss research with Professor Sullivan was not far from what I might experience meeting someone like Einstein or Dirac. He’s clearly so much more knowledgeable than I am that it was almost as if we spoke two different languages. At one point he said, “Mathematics has its own grammar not unlike any other language. What you are saying to me is like a sentence without a verb. What is the statement?!” Yes, I felt like an idiot, but I’d much prefer having someone be mercilessly honest rather than politely nodding, while quietly thinking the same thing :) He certainly had a lot of patience because he kept at me until I (kicking and screaming) was finally able to make a coherent statement about the work Urs’ and I have been doing. This was frustrating because I really am happy with our work and wanted to communicate the ideas better. I wish Urs was there to help! :)

The conference itself for the most part blew right over my head except for two talks: one by Professor Mark Hillery on quantum walks on graphs, the other by Professor Sullivan. Professor Jenny Harrison was scheduled to talk (and was one of the reason I attended), but she couldn’t make it due to her son coming down with a cold.

Since I am technically an engineer, Professor Hillery’s work reminded me a LOT of transmission line theory. Motivated by the fact that random walks are of significance for many algorithms in classical computing, he is hoping that the quantum mechanical version of a random walk will be of significance for developing algorithms in quantum computing. In this approach, you begin with some directed graph where each edge of the graph is replaced by two oppositely directed edges and an edge from node $j$ to node $j+1$ is denoted $\mid j,j+1〉$. He then defined a unitary time evolution operator

(1)$U\mid j-1,j〉=t\mid j,j+1〉+r\mid j,j-1〉$
(2)$U\mid j+1,j〉={t}^{*}\mid j,j-1〉-{r}^{*}\mid j,j+1〉$

where $t$ is basically a transmission coefficient and $r$ is a reflection coefficient. I am about 99% sure that this is essentially casting standard transmission line theory in a different language.

Professor Sullivan’s talk was interesting, but if you had followed Professor Baez’s quantum gravity seminar notes there were quite a lot of similarities. He was discussing algebra and coalgebras via diagrams and the algebraic relations there can also be viewed as simple topological relations of diagrams. It was fun.

However, the exciting part of the trip for me and what made the whole thing worth it was the Einstein Chair Seminar on the following Tuesday. The Einstein Chair Seminar is usually a single talk where Professor Sullivan basically has free reign to grill the speaker for as long as he wishes (according to Professor Baez :)). They usually last from 4-6 hours (or until the speaker passes out). The speaker was to be Professor Jenny Harrison talking about her work on “Discrete Exterior Calculus.” This Seminar was a little special because Professor Harrison doesn’t like to speak for more than 45 minutes, so instead of the usual grilling, they had a handful of people talk for 45 minutes each. Kind of like a mini afternoon conference. The theme of the afternoon was

DISCRETE DIFFERENTIAL GEOMETRY

Woohoo! :)

The first speaker was Professor Jozef Dodziuk. His talk was essentially a review of his Ph.D. thesis, which was WAY ahead of its time as far as applied finite-element methods are concerned (although that probably wasn’t the intended use he had in mind)

Finite difference approach to the Hodge theory of harmonic forms
Amer. J. Math., 98 (1976), 79-104.

I like this paper a lot. In it, he defines an inner product

(3)$〈a,b〉={\int }_{M}\mathrm{Wa}\wedge \star \mathrm{Wb}$

of cochains via the inner product of forms and the Whitney map $W:C\left(K\right)\to \Omega \left(M\right)$ from cochains on a simplicial approximation $K$ to differential forms on a smooth manifold $M$. With this inner product he defines a discrete adjoint exterior derivative $\delta :{C}^{p}\left(K\right)\to {C}^{p-1}\left(K\right)$ and Hodge decomposition falls out naturally. This is one of my all time favorite papers and it was a treat to see him present it in person.

The next speaker was Scott Wilson who I think was a student of Professor Sullivan’s. If not a student, Professor Sullivan seemed very much aware of his work as if he were an advisor. This paper hit on some topics I tried to ask Professor Dodziuk about regarding algebraic properties of the Whitney map. For one thing, the Whitney map is not an algebra morphism, which I think is at the heart of some of the issues addressed throughout the day. His talk began with a

Definition:
We say $\cup :{C}^{p}\left(K\right)\otimes {C}^{q}\left(K\right)\to {C}^{p+q}\left(K\right)$ is an approximation of $\wedge$ if $\forall f,g\in \Omega \left(M\right)$

(4)$\mid \mid W\left(\mathrm{Rf}\cup \mathrm{Rg}\right)-f\wedge g\mid \mid \le O\left(\mathrm{mesh}\right)$

where $f,g$ are forms, $R:\Omega \left(M\right)\to C\left(K\right)$ is the de Rham map, and O(mesh) is some constant that depends on the largest cell in mesh.

He makes the statement, “If $\cup$ is associative, then $\cup$ is not an approximation of $\wedge$.” He doesn’t prove this, so I am not sure if it is true or not. I think both he and Sullivan are pretty convinced it is true. This is somewhat odd because Urs and I DO have an associative product that DOES approximate the wedge product. However, our definition of “approximate” hasn’t been stated quite as formally as they did. This made me think that if the statement was true, then it could be because the definition of “approximate” is too tight. I suggested the alternative

Definition:
We say $\cup :{C}^{p}\left(K\right)\otimes {C}^{q}\left(K\right)\to {C}^{p+q}\left(K\right)$ is an approximation of $\wedge$ if $\forall a,b\in C\left(K\right)$

(5)$\mid \mid W\left(a\cup b\right)-\mathrm{Wa}\wedge \mathrm{Wb}\mid \mid \le O\left(\mathrm{mesh}\right).$

For some reason, I don’t think Scott was very interested in listening to what I had to say. Oh well. This alternative definition is pleasant in the sense that it says that you want your product to be such that the Whitney map is an algebra morphism. I have to admit that it would be kind of fun to disprove their statement :)

The next talk was by a friend of mine, Professor Robert Kotiuga. This wasn’t so much about discrete methods as it was about applied electromagnetics and I’m running out of energy, so I’ll skip his talk although it was definitely interesting.

The next talk was Professor Harrison. Unfortunately, her talk was a bit over my head so I still have lots of questions about it. Professor Sullivan seemed to be very interested in it and asked her a ton of questions. If was great to observe the two of them going back and forth. If only I had a brain and could have understood what they were talking about :) Anyway, I will try to see if Professor Harrison can maybe explain a bit about what that was all about.

The final talk was by Professor Sullivan. It’s kind of fun to watch him give a talk. It is almost as if he is just thinking out loud about some problem. I don’t even know if he prepares before standing up there. Apparently he has been working with a physicist Jae Suk Park. Together they are attempting to define exactly what a quantum field theory is in such a way that a mathematician can understand it. The talk was essentially constructing some discrete model of Poincare duality and cell complexes that sort of fit into Jae Suk Park’s framework. One interesting thing is that I thing Urs and my framework actually does fit into Jae Suk Park’s work. I hope to get a preprint from him so we can compare notes. I gave him a copy of our paper.

That’s it for now! Time to go home!

Ciao,
Eric

Posted by: Eric on February 12, 2004 10:30 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Eric -

many thanks for this most interesting report. It is very interesting to see that the questions the mathematicians find most interesting are so radically remote from the questions that I, as a physicist, consider interesting.

For instance I think it is trivial to see that what we have done has the correct continuum limit. (What I find much more interesting is how we can define nice action principles in the discrete framework, for instance.) But of course we should try to make that formally explicit if we want to communicate this idea. In order to do so, we somehow need to adapt the mathematician’s language. You have already made an attempt in this direction by reformulating that definition about continuum limits. I surely think that your remark is legitimate. But before I can do anythink to help I need to resolve the following problem:

Both definitions, the one used by Sullivan and your’s, involve the Whitney map. This, again, is only defined for simplices. But we would rather formulate the continuum limit in terms of hypercubes instead of simplices. None of these two definitions can directly be checked in our framework, because we don’t even have a Whitney map. Right?

Posted by: Urs Schreiber on February 13, 2004 8:23 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Urs,

None of these two definitions can directly be checked in our framework, because we don’t even have a Whitney map. Right?

Well, the Whitney map is defined originally for simplices, but if you understand what the Whitney map really says you will see that you can construct it for topological n-cubes easily enough. Don’t worry :)

Here is a conjecture of mine that would be great if we could prove.

If a manifold $M$ can be triangulated, then $M×R$ can be diamonated.

:)

If this is true, which I think it is, then we are in business. We simply need to write down the Whitney map for n-cubes. If this hasn’t been done, then it will be original, but I think it has been :)

I stated this during Dodziuk’s talk and everyone jumped on me. Revenge will be sweet :)

Eric

Posted by: Eric on February 13, 2004 8:41 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Eric Forgy is continuing to discuss issues that have been raised here with Dr. Bossavit, who made the following important remark:

I’m not inclined to doubt what Wilson says (and that Sullivan told me nine years ago): There seems to be a strong no-go result that says that you can’t have a discrete triple $\left(\star ,d,\wedge \right)$ with all the desirable properties. How we cope with that is a serious issue.

Unfortunately I am not yet familiar with the assumptions that enter this result and the list of ‘all desirable properties’. Perhaps it all depends on the nature of our desires!

What Wilson and Sullivan say that I do not find controversial (because it doesn’t contradict what Urs and I constructed) is that you can not have an algebra that is both associative and graded commutative that will approximate the continuum theory. This statement seems fairly clear to me. What I find controversial is their statement that you must have nonassociativity. Our work (as well as that of Alain Connes, since our work fits into his framework, sort of) is a counterexample to this. You can have an associative but non-graded commutative algebra that does approximate the continuum.

I fully agree with what Eric writes here. I would like to clarify this issue in detail. Even though there are about 80 pages in our notes, the crucial idea by which we get a triple $\left(\star ,d,\wedge \right)$ with some desirable properties can be sketched in just a few sentences. This is already attempted in the introduction to our notes on discrete differential geometry.

Let me recall the basic steps in just a few words:

- Consider an associative algebra $𝒜$ which characterizes a generalized manifold.

- Consider a differential calculus $\Omega \left(𝒜,d\right)$ over $𝒜$.

- Equip $\Omega \left(𝒜,d\right)$, regarded as a vector space, with an non-degenerate sesqui-linear inner product $〈\cdot \mid \cdot 〉$, thus promoting it to the $Z$-graded inner product space $ℋ:=\left(\Omega \left(𝒜,d\right),〈\cdot \mid \cdot 〉\right)$.

- The differential $d$ and the elements $a\in 𝒜$ are represented as operators $\stackrel{̂}{d}$ and $\stackrel{̂}{a}$ on $ℋ$ in the obvious way.

- Let ${\stackrel{̂}{d}}^{†}$ be the adjoint of $\stackrel{̂}{d}$ with respect to $〈\cdot \mid \cdot 〉$.

- Let $\mid \mathrm{vol}〉\in \Omega \left(𝒜,d\right)$ be an element of top grade that is in the kernel of ${\stackrel{̂}{d}}^{†}$.

- Denote by $\star$ an operator on $ℋ$ such that

(1)$\star \stackrel{̂}{d}=±{\stackrel{̂}{d}}^{†}\star \phantom{\rule{thinmathspace}{0ex}}.$

- Given all this it is very easy to check that the following is true:

a) $\star$ exists iff $\mid \mathrm{vol}〉$ exists.

b) If they exist they are related by $\mid \mathrm{vol}〉=\star \mid 1〉$ and

(2)$\star \mid {a}_{0}d{a}_{1}\phantom{\rule{thinmathspace}{0ex}}\cdots d{a}_{p}〉=\left({\stackrel{̂}{a}}_{0}\left[\stackrel{̂}{d},{\stackrel{̂}{a}}_{1}\right]\phantom{\rule{thinmathspace}{0ex}}\cdots \left[\stackrel{̂}{d},{\stackrel{̂}{a}}_{p}\right]{\right)}^{†}\mid \mathrm{vol}〉\phantom{\rule{thinmathspace}{0ex}}.$

If we identify $\wedge$ with the product in $\Omega \left(𝒜,d\right)$ the above gives a triple $\left(\star ,d,\wedge \right)$ for all cases where the above $\mid \mathrm{vol}〉$ exists. In our notes we show that it does exist in particular on topologically hypercubic graphs and that the resulting $\star$ approximates the usual continuum Hodge star operator.

What are the properties of the triple that we construct?

- The only property of $\star$ that is guaranteed is $\star \stackrel{̂}{d}=±{\stackrel{̂}{d}}^{†}\star$.

- The only properties of $\stackrel{̂}{d}$ that are guaranteed are nilpotency and the graded Leibnitz propery.

- The product $\wedge$ is associative but non-commutative.

These properties are good from our point of view because they are sufficient to apply the formalism all over the place in physics. The construction is very similar to Connes’s NCG which also features non-commutativity and preserves associativity, being modeled after quantum mechanics.

(As Alejandro Rivero has kindly pointed out, some NCG papers state a couple of auxiliary axioms which we currently see no need for.)

To my mind the construction of the Hodge star in our formalism is a mere afterthought. (What I find more interesting is what we have to say about deformations of the inner product $〈\cdot \mid \cdot 〉$ and how these can describe metric geometry on generalized spaces as well as other ‘backgrounds’.) But still, seeing that a lot of people have thought about the construction of discrete Hodge star operators, I would very much like to try to understand how the above construction fits in with other approaches. Probably it is missing properties that are ‘desirable’ elsewhere.

Posted by: Urs Schreiber on February 23, 2004 5:10 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hello,

Looking back at my notes from Scott’s talk, I see he has their product of cochains expressed in an alternative form via

(1)$a\stackrel{˜}{\mathrm{cup}}b:=R\left[\left(\mathrm{Wa}\right)\wedge \left(\mathrm{Wb}\right)\right],$

where $R$ is the de Rham map and $W$ is the Whitney map. This essentially verifies my suspicion that their algebra involves an antisymmetrization (coming from $\wedge$).

This gives a deformation $\stackrel{˜}{\mathrm{cup}}$ of the existing associative cup product of cochains (as defined in Munkres). In my opinion, this implies a prejudice toward the continuum theory and tries to construct a discrete theory from that.

On the other hand, we could be prejudiced toward the discrete theory and take the associative cup product of cochains as being fundamental and then construct a modified continuum product $\stackrel{˜}{\wedge }$ defined by

(2)$f\stackrel{˜}{\wedge }g:=W\left[\left(\mathrm{Rf}\right)\mathrm{cup}\left(\mathrm{Rg}\right)\right]$

Note that $\stackrel{˜}{\mathrm{cup}}$ is nonassociative, but commutative, whereas $\stackrel{˜}{\wedge }$ is associative, but noncommutative. Note also the symmetry between the two definitions. Kind of suggestive, isn’t it?

Hence, it is becoming more clear to me that Urs and I do have something quite distinct from Wilson and Sullivan, but each approach should agree in the continuum limit.

Personally, I think losing associativity is more of a sin than losing commutativity. In fact, losing noncommutativity turns out to be a virtue here because it draws close parallels to noncommutative geometry and quantum theory. Quantum theory is kind of dead without associativity in some form or another.

Another interesting aspect of $\stackrel{˜}{\wedge }$ is that the Whitney forms close as an algebra, i.e. if $f=\mathrm{Wa}$ is a Whitney $p$-form and $g=\mathrm{Wb}$ is a Whitney $q$-form, then $f\stackrel{˜}{\wedge }g$ is a Whitney $\left(p+q\right)$-form. Whitney forms do not constitute an algebra under the usual $\wedge$-product, i.e. the $\wedge$-product of a Whitney $p$-form and a Whitney $q$-form is NOT a Whitney $\left(p+q\right)$-form. Whitney forms do constitute an algebra under the $\stackrel{˜}{\wedge }$-product.

Eric

PS: This is probably obvious, but I will point it out that since $d$ commutes with both $R$ and $W$, we have

(3)$d\left(a\stackrel{˜}{\mathrm{cup}}b\right)=\left(\mathrm{da}\right)\stackrel{˜}{\mathrm{cup}}b+\left(-1{\right)}^{\mid a\mid }a\stackrel{˜}{\mathrm{cup}}\left(\mathrm{db}\right)$

and

(4)$d\left(f\stackrel{˜}{\wedge }g\right)=\left(\mathrm{df}\right)\stackrel{˜}{\wedge }g+\left(-1{\right)}^{\mid f\mid }f\stackrel{˜}{\wedge }\left(\mathrm{dg}\right)$

so that in each case we satisfy both the graded Leibniz rule and nilpotency of $d$.

Posted by: Eric on February 23, 2004 9:51 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Eric -

great, this looks like considerable progress in comparing the different approaches!

Let me see if I follow what you said:

It is obvious that $\stackrel{˜}{\mathrm{cup}}$ is graded commutative and that $\stackrel{˜}{\wedge }$ is not.

For associativity one needs to know to which degree $W$ is the inverse of $R$. I recall that you once said

(1)$\mathrm{RW}=1$
(2)$\mathrm{WR}\sim 1\phantom{\rule{thinmathspace}{0ex}}.$

I assume that the $\sim$ means ‘in the continuum limit’?

Ok, using this it is immediate that $\stackrel{˜}{\wedge }$ is associative and $\stackrel{˜}{\mathrm{cup}}$ is so only in the continuum limit.

Yes, this looks good. As far as the algebraic product in our approach can be identified with the $\mathrm{cup}$ product (which I think it can) $\stackrel{˜}{\wedge }$ is certainly the natural thing to consider.

Can you show that for your generalization of the Whitney map to hypercubes we still have $\mathrm{RW}=1$?

Posted by: Urs Schreiber on February 24, 2004 11:33 AM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Good morning Urs (and anyone else who may be reading this)!

What I wrote in that post wasn’t really anything new. I simply put down some thoughts that I had been aware of for a long time. As far as I know, $\stackrel{˜}{\wedge }$ is my invention, but it wouldn’t surprise me to find it in some tome from 100 years ago :) I think this does point out the differences between our work and their’s.

Can you show that for your generalization of the Whitney map to hypercubes we still have $\mathrm{RW}=1$?

I certainly hope so since this is basically the definition of a Whitney form :)

I suspect that there is even some theorem somewhere saying that if you have any map $W:C\left(K\right)\to \Omega \left(M\right)$ satisfying $\mathrm{RW}=1$, then you will have $\mathrm{WR}~1$ (in the continuum limit). Take this with the usual grain of salt though.

To make the analogy between $\stackrel{˜}{\mathrm{cup}}$ and $\stackrel{˜}{\wedge }$ complete, I won’t really be satisfied until we can make a concrete connection between our algebra and that of the usual cup product of cochains. Of course it will be a little different, but I suspect all the pieces should fit together nicely. I made some progress toward this last night (which is just a continuation of some of the stuff I’ve been doing over the last couple of weeks).

Last night I picked up an old book on introductory abstract algebra (Abtract Algebra, Kuczkowski) and reviewed the isomorphism theorems. It goes something like this:

Given a homomorphism $f:G\to H$ with kernel $K$, then the groups $f\left(G\right)$ and $G/K$ are isomorphic.

This essentially says that the image of any homomorphism is equivalent to a quotient space. This helps make precise some of the half baked ideas I’ve been discussing with Urs behind the scenes.

Let me try to fit all this together now (while my energy lasts).

Definition: A directed $n$-graph $G$ consists of $n+1$ countable sets ${G}_{0},...,{G}_{n}$ together with surjective source and target maps $s,t:{G}_{p}\to {G}_{p-1}$ satisfying $\mathrm{st}=\mathrm{ts}$ and $s{e}_{i}=t{e}_{i}=0$ for all ${e}_{i}\in {G}_{0}$ (0 is the emptyset).

The elements of ${G}_{p}$ are referred to as elementary $p$-paths and are often denoted by a $\left(p+1\right)$-tuple of indices, e.g. ${e}_{{i}_{0}...{i}_{p}}\in {G}_{p}.$ The source of an elementary $p$-path is then obtained by deleting the rightmost index, e.g. $s{e}_{{i}_{0}...{i}_{p}}={e}_{{i}_{0}...{i}_{p-1}}$ while the target of an elementary $p$-path is obtained by deleting the leftmost index, e.g. $t{e}_{{i}_{0}...{i}_{p}}={e}_{{i}_{1}...{i}_{p}}.$ Pictorially, an elementary $p$-path may be thought of a $p$-dimensional arrow extending from the source to the target, e.g. ${e}_{{i}_{0}...{i}_{p}}:{e}_{{i}_{0}...{i}_{p-1}}\to {e}_{{i}_{1}...{i}_{p}}.$ (This is supposed to remind you of $n$-categories.)

Now for another

Definition: The space of paths $P\left(G\right)$ on a directed $n$-graph $G$ is a free abelian group $P\left(G\right)=\underset{r=0}{\overset{n}{⨁}}{P}_{r}\left(G\right),$ where ${P}_{r}\left(G\right)$ is generated by the set of elementary $r$-paths ${G}_{r}$.

Having defined $P\left(G\right)$, we can now define the all-important boundary map $\partial :{P}_{p}\left(G\right)\to {P}_{p-1}\left(G\right)$ by defining how it acts on the bases, namely $\partial {e}_{{i}_{0}...{i}_{p}}:=\sum _{r=0}^{p}\left(-1{\right)}^{r}{e}_{{i}_{0}...\stackrel{̂}{{i}_{r}}...{i}_{p}}.$ This probably looks familiar to anyone who knows about simplices and simplicial complexes. It is basically the same expression as the boundary of a $p$-simplex. Let me point out the crucial difference. On a simplicial complex, every term on the right hand side is present, by definition. However, on a directed $n$-graph $G$, there may be terms on the right-hand side that are not present in $G$. We handle this by setting these terms to zero. This has the nontrivial affect of destroying the nilpotent property of the boundary map. For example consider the directed $2$-graph consisting of the six elementary paths $\left\{{e}_{\mathrm{ijk}},{e}_{\mathrm{ij}},{e}_{\mathrm{jk}},{e}_{i},{e}_{j},{e}_{k}\right\}.$ The boundary of ${e}_{\mathrm{ijk}}$ is then given by $\partial {e}_{\mathrm{ijk}}={e}_{\mathrm{jk}}-{e}_{\mathrm{ik}}+{e}_{\mathrm{ij}}={e}_{\mathrm{jk}}+{e}_{\mathrm{ij}}$ because ${e}_{\mathrm{ik}}=0$. The boundary of this is given by ${\partial }^{2}{e}_{\mathrm{ijk}}=\partial {e}_{\mathrm{jk}}+\partial {e}_{\mathrm{ij}}={e}_{k}-{e}_{i},$ which is not zero so that the boundary map is not nilpotent on the space of paths. Since one of my all-time favorite mottos is

“The boundary of a boundary is zero”

then I use this as kind of a guiding principle. Therefore, I refer to the space of paths suggestively as being “pre-geometric.” The next step is to look at “geometric” objects. These are the objects for which the boundary of a boundary is zero. More mathematically speaking, here is another

Definition: The kernel of ${\partial }^{2}$ is referred to as the space of chains and is denoted $C\left(G\right)=\underset{r=0}{\overset{n}{⨁}}{C}_{r}\left(G\right),$ where ${C}_{r}\left(G\right)$ is the subgroup of ${P}_{r}\left(G\right)$ for which ${\partial }^{2}=0$.

In other words, $c\in {C}_{p}\left(G\right)⇔{\partial }^{2}c=0.$ Now we are in business because the isomorphism theorem is within sight. Before getting to that, let me give another example that is less trivial (the space of 2-chains on that previous 6 element graph vanishes because there are no 2-paths for which ${\partial }^{2}=0$). Consider the directed 2-graph whose elementary paths are given by $\left\{{e}_{\mathrm{ijl}},{e}_{\mathrm{ikl}},{e}_{\mathrm{ij}},{e}_{\mathrm{jl}},{e}_{\mathrm{ik}},{e}_{\mathrm{kl}},{e}_{i},{e}_{j},{e}_{k},{e}_{l}\right\}.$ As before, we have ${\partial }^{2}{e}_{\mathrm{ijl}}={e}_{l}-{e}_{i}$ and ${\partial }^{2}{e}_{\mathrm{ikl}}={e}_{l}-{e}_{i}$ so that neither of these are 2-chains. However, the linear combination ${\partial }^{2}\left({e}_{\mathrm{ijl}}-{e}_{\mathrm{ikl}}\right)=0$ is a 2-chain. This element plays such a vital role in all of Urs and my work that we gave it a special name: the 2-diamond. The space of 2-chains on this directed 2-graph is then nontrival, i.e. it is generated by the 2-diamond ${e}_{\mathrm{ijl}}-{e}_{\mathrm{ikl}}$.

The next step is to define a projection map $A:P\left(G\right)\to C\left(G\right),$ which is the analog of antisymmetrization in the continuum. In other words, the space of paths $P\left(G\right)$ is analogous to the space of tensors and $C\left(G\right)$ is analogous to the space for differential forms (currents actually). Athough the map $A$ plays a role similar to antisymmetrization and I suggestively use the symbol “$A$”, it is important to realize that it is not antisymmetrization in the usual sense.

For example, with the previous directed 2-graph, there is only one nonvanishing 2-chain so we have $A\left({e}_{\mathrm{ijl}}\right)=1/2\left({e}_{\mathrm{ijl}}-{e}_{\mathrm{ikl}}\right)$ and $A\left({e}_{\mathrm{ikl}}\right)=-1/2\left({e}_{\mathrm{ijl}}-{e}_{\mathrm{ikl}}\right),$ where the factor of $1/2$ is required since we must have ${A}^{2}=A$. It is worth mentioning now that for any directed $n$-graph $G$, we always have ${C}_{1}\left(G\right)={P}_{1}\left(G\right)$ and ${C}_{0}\left(G\right)={P}_{0}\left(G\right)$ because ${\partial }^{2}$ vanishes on any 0- or 1-path so that $A\left({e}_{\mathrm{ij}}\right)={e}_{\mathrm{ij}},\phantom{\rule{1em}{0ex}}\forall {e}_{\mathrm{ij}}\in {P}_{1}\left(G\right)$ and $A\left({e}_{i}\right)={e}_{i},\phantom{\rule{1em}{0ex}}\forall {e}_{i}\in {P}_{0}\left(G\right).$ We can now bring in the isomorphism theorem. Since $C\left(G\right)$ is a subgroup of $P\left(G\right)$ and $A:P\left(G\right)\to C\left(G\right)$ is a homomorphism, that means that $C\left(G\right)\sim P\left(G\right)/\mathrm{ker}\left(A\right).$ In other words, we can write things like $A\left({e}_{\mathrm{ijk}}\right)\sim \left[{e}_{\mathrm{ijk}}\right]$ so that, for example, in that directed 2-graph, we’d have $\left[{e}_{\mathrm{ijl}}\right]=-\left[{e}_{\mathrm{ikl}}\right],$ which is what pops out of our notes, but now we have a more concrete connection to standard tools in algebraic topology. There is much more to say, for example, we can now define the dual spaces ${P}^{*}\left(G\right)$ and ${C}^{*}\left(G\right)$ with $〈{A}^{*}\left(\alpha \right),c〉:=〈\alpha ,A\left(c\right)〉,$ which would finally connect our algebra to the usual cup product of cochains, but I am out of energy and need to get some other things done first.

Ciao!
Eric

Posted by: Eric on February 24, 2004 4:47 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

This looks good.

I have always thought of the space of paths as the vector space over the set of elementary paths. But maybe you are right that it is better to define it as the free abelian group generated by elementary paths, which is like restricting the vector space to linear combinations of the basis elements with coefficients $±1$ (unless I misunderstood something). But, wait: In the antisymmetrization map you have a factor of 1/2! Hm… But $A$ need not be a projector, necessarily. We could replace that factor by unity, I think. $A$ would still be a homomorphism, wouldn’t it?

Ok, if this works as expected it should give a nice formulation of something we already had sort of understood. Am I right that what stopped us last time was the definition of the cup product on $C\left(G\right)$? Do you have that under control now?

Posted by: Urs Schreiber on February 24, 2004 6:52 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Urs!

I have always thought of the space of paths as the vector space over the set of elementary paths.

Me too!

But maybe you are right that it is better to define it as the free abelian group generated by elementary paths, which is like restricting the vector space to linear combinations of the basis elements with coefficients ± 1 (unless I misunderstood something).

Well, the coefficients are in Z for an abelian group, e.g. c+c+…+c = nc, but anyway, I switched viewpoints simply because it made the connection to the isomorphism theorem more explicit. I suppose that we might want to just go back to thinking of the space of paths as the free vector space generated by the set of elementary paths as opposed to the free abelian group. I don’t think it changes anything. Then again, you might be right and we should simply drop the requirement that $A$ be a projection and keep the integer coefficients. At least we are making progress! (I think) :)

But, wait: In the antisymmetrization map you have a factor of 1/2!

Oops! Good catch :)

Hm… But $A$ need not be a projector, necessarily. We could replace that factor by unity, I think. $A$ would still be a homomorphism, wouldn’t it?

Yes, you could be right. $A$ would certainly still be a homomorphism.

Ok, if this works as expected it should give a nice formulation of something we already had sort of understood.

Absolutely. In case someone is reading this besides us, the notes Urs and I have prepared are pretty much self contained and there are no problems that we are aware of yet. My attempt here is simply to relate what we did to what is already well known in algebraic topology (and to contrast with Wilson and Sullivan). This is motivated by my gut feeling that a lot of what we did is nothing but standard algebraic topology in disguise. I think we are close to making this connection explicit.

Am I right that what stopped us last time was the definition of the cup product on C (G) ? Do you have that under control now?

I think so. At the moment all I have is a bunch of notes scribbled down. Part of the reason for explaining here is to force me to make it a little more concrete :)

One important comment is that what I had been calling $C\left(G\right)$ before is different than what I am calling $C\left(G\right)$ now because for a while I was using a modified boundary map. Now I am back to using the standard boundary map. The two agree on graphs without intermediate edges (see the notes for an explanation of what we mean by that). I believe things are correct now because the cooundary will now coincide with the one in our notes (whereas my modified coboundary didn’t, which is probably the source for the earlier difficulties relating the algebras).

Eric

Posted by: Eric on February 24, 2004 7:41 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

Hi Urs,

Seeing Scott Wilson’s thesis brought back memories of this discussion. You expressed some concern about how one would construct a Whitney map for $n$-cubes, which is key to proving statements about the continuum limit. Since you are now interested in synthetic differential geometry, which as you said, is kind of halfway between what we did and the continuum theory, I thought I would spell it out a little more just in case it might motivate some more thought.

The key to constructing a Whitney map for $n$-cubes is to reconsider how it works for $n$-simplices. For concreteness, lets consider $n$ = 3. Denote a 3-simplex via $\left[{i}_{0},{i}_{1},{i}_{2},{i}_{3}\right]$ and let $\mid \left[{i}_{0},{i}_{1},{i}_{2},{i}_{3}\right]\mid$ denote its volume. The barycentric (or area) coordinate ${\lambda }^{0}\left(p\right)$ is given by

(1)${\lambda }^{0}\left(p\right)=\frac{\mid \left[p,{i}_{1},{i}_{2},{i}_{3}\right]\mid }{\mid \left[{i}_{0},{i}_{1},{i}_{2},{i}_{3}\right]\mid }.$

The barcentric coordinate evaluated at the vertices is characterized by

(2)${\lambda }^{0}\left(p\right)=\left\{\begin{array}{c}1,\mathrm{if}p={i}_{0}\\ 0,\mathrm{if}p={i}_{1},{i}_{2},{i}_{3}.\end{array}$

To construct a Whitney map on $n$-cubes, we simply need a set of vertex functions satisfying the above. To do this, note that a point inside a 3-cube defines 8 sub-cubes that I will call “octants”. Therefore, we simply define

(3)${\lambda }^{0}\left(p\right)=\frac{\mathrm{Volume}\mathrm{of}\mathrm{octant}\mathrm{opposite}\mathrm{node}0}{\mathrm{Total}\mathrm{volume}\mathrm{of}n-\mathrm{cube}}.$

From these elementary Whitney 0-forms, we can construct higher degree forms as usual. For example,

(4)${\lambda }^{01}={\lambda }^{0}d{\lambda }^{1}-{\lambda }^{1}d{\lambda }^{0}.$

Before I hit the sack, let me also remind of the definition of convergence that I would like to use

Definition:
We say $⌣:{C}^{p}\left(K\right)\otimes {C}^{q}\left(K\right)\to {C}^{p+q}\left(K\right)$ is an approximation of $\wedge$ if $\forall a,b\in C\left(K\right)$

(5)$\mid \mid W\left(a⌣b\right)-\mathrm{Wa}\wedge \mathrm{Wb}\mid \mid \le O\left(\mathrm{mesh}\right).$
Posted by: Eric on October 19, 2005 8:53 AM | Permalink | Reply to this

### Re: The search for discrete differential geometry

A flash from the past!

Some discussions about Jenny’s material over on the n-Category Cafe caused me to reread this thread. This was a lot of fun! :)

I basically outlined here the definition of Whitney forms on $n$-cubes, but didn’t show that they actually work.

The properties we are after, and as far as I’m concerned the defining properties of Whitney forms, are that for an $n$-cube, Whitney 0-forms are defined as above and integrating them over the corresponding 0-cubes gives the Kronecker delta. That is easy to check.

Next, the Whitney 1-forms integrated over the 1-cubes should also be the Kronecker delta. That is also easy to check.

With a 2-cube whose nodes are labelled 0,1,2,3, the Whitney 1-form

(1)${\lambda }^{01}={\lambda }^{0}d{\lambda }^{1}-{\lambda }^{1}d{\lambda }^{0}$

vanishes on every edge other than [01] because ${\lambda }^{0}$ and ${\lambda }^{1}$ vanish on every other edge.

On the edge [01] (and only on this edge), we have

(2)${\lambda }^{0}+{\lambda }^{1}=1.$

Therefore,

(3)$d{\lambda }^{0}=-d{\lambda }^{1}$

and

(4)${{\lambda }^{01}\mid }_{\left[01\right]}=d{\lambda }^{1}=-d{\lambda }^{0}.$

Integrating this along [01] is obviously 1, i.e.

(5)${\int }_{\left[01\right]}{\lambda }^{01}={\int }_{\left[01\right]}d{\lambda }^{1}={{\lambda }^{1}\mid }_{\left[1\right]}-{{\lambda }^{1}\mid }_{\left[0\right]}=1.$

Taken together we have

(6)${\int }_{\left[\mathrm{ij}\right]}{\lambda }^{\mathrm{kl}}={\delta }_{\mathrm{ij}}^{\mathrm{kl}}.$

Whitney forms for higher $n$-cubes are also easy to check and all should satisfy

(7)${\int }_{\left[n-\mathrm{cube}i\right]}{\lambda }^{\left[n-\mathrm{cube}j}\right]={\delta }_{n-\mathrm{cube}i}^{n-\mathrm{cube}j}.$

Although it is obvious to us already, if we wanted to please the math camp (which is a noble goal) we can now prove convergence in the continuum limit with this Whitney map.

Posted by: Eric on June 1, 2008 3:02 PM | Permalink | Reply to this

### Spectral action principle and string theory

I am looking for work done on combining Connes’ spectral action principle with string theory.

I am thinking about something along the lines of the old papers by Ali Chamseddine, e.g. hep-th/9705153, hep-th/9701096, where it is shown that using the worldsheet supercharges as Dirac operators in the spectral action reproduces indeed parts of the superstring effective action.

But apparently this line of research has dies out? If that’s the case, does anyone know why? Just no interesting results or specific problems?

A related question: Is there anything known about calculation of entropy of black holes from within the NCG perspective using spectral actions?

Posted by: Urs Schreiber on February 10, 2004 9:15 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

Hmm yes, before the flood there were two marginal attemps to use Connes principle in string theory. This is, as a pure mathematical principle, unrelated to Connes-Lot model. One was from Chamseddine and another from the napolitan team, Fedele Lizzi and Giovanni Landi. Then it come all the stuff on Matrix Theory and all the other versions of noncommuativity in strings, and these essays were sweept into the big blue sea.

About your question in the axiomatic formalism, yes, in 1995 there was a displacement from a system of two algebras to a only, sort of self-dual, algebra plus a new operator, “reality”. For an example, in the old formalism the Connes-Lot model was about two algebras, one having U(1) times SU(2), the other having SU(3) and a spureus copy of U(1). Poincare Duality then was a kind of link between both algebras.

About black holes, I do not remember any, but I am getting old. Perhaps Chamseddine, again, could have some hint at gravity.

Posted by: alejandro rivero on February 11, 2004 1:03 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

Robert wrote in news:c0d6ee\$15u5r2\$1@ID-40416.news.uni-berlin.de

I don’t know anything, but let’s try to make things up: First there is an obstacle to get anything like this to work because the spectral action principle only works in euclidean signature (otherwise the spectrum of your Laplacian or Dirac operator that you use to encode the geometry) is badly behaved.

Can’t this problem be dealt with by using Krein space technology? The idea
goes back to

A.Strohmaier, On Noncommutative and Semi-Riemannian Geometry

and has a few more recent applications, i.e.

and

However, I am only vaguely familiar with this stuff and no semi-Riemannian
spectral action is mentioned explicitly in these papers.

Second, the spectral action is just a rewriting of the classical theory

Well, a rewriting of the action functional, really, which pertains to the quantum theory, too. Ok, this addresses a question I wanted to ask anyway: Somehow
Connes-Lott is supposed to be a “third road to quantum gravity”, but how much quantum is there in this approach?

Judging from papers like

I get the impression that the point is that by using appropriate spectral
triples (e.g. with algebras over discrete spaces) the path integral for the
spectral action, i.e. for some sort of quantum gravity, can actually be
evaluated sometimes.

I find it quite interesting how Chamseddine in hep-th/9701096 gets parts of
the the string effective action from using the worldsheet supercurrent, truncated to 0-modes of worldsheet fields, as the Dirac operator in the spectral action. I am wondering what would result if one used the full worldsheet supercurrent instead and computed to higher orders. Couldn’t this give some sort of closed superstring field theory?

To my mind this looks like an interesting question, Alejandro Rivero tells me that the advent of Matrix Theory apparently sort of scotched this approach.

I am aware that Connes himself has done research in Matrix Theory (e.g.
hep-th/9711162, thanks to Squark for this very valuable reference!) but apparently without seeing a relation to the spectral action principle.

Could there be one? Papers like hep-th/0310175 at least indicate a relation to random matrix theory.

Posted by: Urs Schreiber on February 12, 2004 3:47 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

>Somehow Connes-Lott is supposed to be a
>?third road to quantum gravity?, but how much
>quantum is there in this approach?

Good question. It seems it is the less travelled road. The idea, last time I looked upon it, was to built the NCG version of the group of diffeomorphisms of a manifold, in this case of a generic spectral triple. This was studied by Connes and Moscovici, and it was sort of success because renormalization group theory seems to emerge from this attack. Since then, I have seen some feints but no further progress :-(

We have a clue about first quantization in NCG contexts from a different point of view, the tangent groupoid of a manifold. But this way seems to follow his life separate from connes-lott- models.

I’ll take a look about Albuquerque etc.

Posted by: alejandro rivero on February 25, 2004 7:59 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

I did. Alburquerque is another “based on” model, it does not verify poincare duality across the limit I believe only a couple of australians have took an interest of keeping with this requeriment. It could be OK to forget about it if you have another suplementary axiom (as Hodge operator in your case). The only positive point of the paper is that they take seriusly the concept of dimension, but still I am at sure if it coincides with GAFA paper from Connes Moscovici.
So no, it is not the One.

Posted by: alejandro rivero on February 26, 2004 5:08 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

Wait, I am not following you! :-)

What do you mean by ‘not the One’?

Please note that I haven’t read Albuquerque in detail. I got the impression that they have a spectral action and manage to define and evaluate the path integral with respect to this action explicitly. Is that right? Do they have an explicitly evaluable path integral for an approximation of gravity?

What do you mean by ‘first quantization in NCG contexts’ in your previous message? Surely every spectral action yields a full fledged field theory?

You mentioned the ‘NCG version of the group of diffeomorphism of a manifold’. As far as I know the diffeomorphisms simply correspond to outer automorphism of the algebra $𝒜$. For instance on p. 16 of hep-th/9603053 is says

the group $\mathrm{Aut}\left(𝒜\right)$ of $*$ automorphisms of the algebra $𝒜$, which plays in general the role of the group $\mathrm{Diff}\left(M\right)$ of diffeomorphisms of the manifold $M$ […]

Ah, wait, I have just searched a huge pile of papers on my desk and found the one which I was really looking for. This is a most beautiful piece:

A. Chamseddine & A. Connes, The Spectral Action principle.

You surely know this one, but because it is so nice let me quote the enchanting koan that it contains (p.4):

(1)$1\to \mathrm{Int}\left(𝒜\right)\to \mathrm{Aut}\left(𝒜\right)\to \mathrm{Out}\left(𝒜\right)\to 1$
(2)$1\to 𝒰\to G\to \mathrm{Diff}\left(M\right)\to 1\phantom{\rule{thinmathspace}{0ex}}.$

This are two exact sequences of groups which are suggested to be in correspondence. Here $\mathrm{Aut}\left(𝒜\right)$ are the automorphism of $𝒜$, $\mathrm{Int}\left(𝒜\right)$ are the inner automorphisms and $\mathrm{Out}\left(𝒜\right)$ the outer ones. $𝒰$ is the internal gauge group of a physical model described by a spectral action, $G$ the symmetry group of the action and $\mathrm{Diff}\left(M\right)$ the diffeo group of the manifold.

Apparently this analogy/identification is enough to derive the algebra $𝒜$ (1.17) which gives rise to the standard model when inserted in the spectral action. Delightful.

Except that I don’t fully understand it. :-) Given any algebra $𝒜$, how do I determine its outer automorphisms? Help appreciated.

Posted by: Urs Schreiber on February 26, 2004 5:51 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

This is great :)

By the way, I’ve searched for the One for the past 7 years without success and I’m tempted to speculate that with Urs’ help, we found it. I think the rest is packaging, which is what I’m working on now.

So does you “One” mean the same as my “One”? What do you mean? I feel like we are knights searching for the holy grail :)

Eric

Posted by: Eric on February 26, 2004 6:03 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

hehe… I was playing uppercase upon the last phrase in your previous message “Could there be one?”, thus getting it out of context. (btw I think they expected the lowercase question could have had an affirmative answer).

Alburquerque has a Dirac operator but I doubt it has a spectral triple. Still, he is entitled to try the spectral action trick.

I am now unsure about if spectral action implies quantization. I thought it didnt, but it is true that it introduces a new scale. Hmm.

The koan, yes, is very very nice. I do not understand it today, but I think I understood it six or seven years ago, so it must be easy :-)

Sorry I am not of help this time… As for the Holy Grail, it is shown to the public in the Cathedral of Valencia, Spain (Actually I am born in lands where the french bards situated the action of the quest, so for us the epical tales do not sound very epic after all). Still, Eric, I guess that by One I mean a method to look the geometry of general relativity in a way such that the standard model emerges, or is constrained, by it.

Posted by: alejandro rivero on February 27, 2004 2:46 PM | Permalink | Reply to this

### Re: Spectral action principle and string theory

Let’s look at the koan again. I think it is pretty trivial:

Recall that

(1)$1\to 𝒰\to G\to \mathrm{Diff}\left(M\right)\to 1\phantom{\rule{thinmathspace}{0ex}}.$

Each arrow is a group homomorphism and any arrow maps the image of the previous arrow to the identity element, since this is supposed to be an exact sequence.

Since the homomorphism $1\to 𝒰$ simply sends the identity element of the trivial group on the left to the identity in $𝒰$ this implies that $𝒰\to G$ is injective which again says nothing but that $𝒰$ is a subgroup of $G$.

This entire subgroup is sent to the identity element by the next homomorphism $G\to \mathrm{Diff}\left(M\right)$. But this simply means that $\mathrm{Diff}\left(M\right)\subset G/𝒰$.

So all that is happening here is that the full symmetry group $G$ of the spectral action is divided into a subgroup $𝒰$ and the rest.

When we identify $G$ with the automorphisms $\mathrm{Aut}\left(𝒜\right)$ of some algebra $𝒜$ and the subgroup $𝒰$ (‘internal symmetry group’) with the inner automorphisms $\mathrm{Int}\left(𝒜\right)$ then the outer automorphisms $\mathrm{Out}\left(𝒜\right)$ (which are supposed to represent diffeomorphisms) are nothing but $\mathrm{Out}\left(𝒜\right)=\mathrm{Aut}\left(𝒜\right)/\mathrm{Int}\left(𝒜\right)$, i.e. essentially the automorphisms up to inner automorphisms, where inner automorphisms are just those automorphisms which come from unitary transformations $a\to ua{u}^{*}$ with $a,u\in 𝒜$.

Noting that for $𝒜={C}^{\infty }\left(M\right)$ the only unitary element is the unit function $f\left(x\right)=1$ one sees that in this case $\mathrm{Aut}\left(𝒜\right)=\mathrm{Out}\left(𝒜\right)$, i.e. every automorphism comes from a diffeomorphism.

Non-trivial inner automorphisms are obtained by adding addional structure. For instance by using complex functions on $M$ and hence the algebra $𝒜={C}^{\infty }\left(M\right)×C$ every function of the form $f\left(x\right)={e}^{i\varphi \left(x\right)}$ (with $\varphi \left(x\right)$ real )defines a unitary (multiplication) operator and hence an inner automorphism. This inner automorphism is still trivial, but acting on the Hilbert space $ℋ$ on which $𝒜$ is represented it induces phase transformations - and hence electromagnetism.

To make things a little more interesting take $𝒜={C}^{\infty }\left(M\right)×H$, where $H$ is the algebra of quaternions. Now the unitary elements of $𝒜$ are functions of the form $f\left(x\right)=\mathrm{exp}\left({\varphi }_{i}\left(x\right){\sigma }^{i}\right)$, where ${\sigma }^{i}$ are the ‘Pauli matrices’, i.e. the $\mathrm{SU}\left(2\right)$ generators. The inner automorphisms are now something like electroweak transformations.

By similar reasoning one should find the result in equation (1.17) of hep-th/9606001 which says that

(2)$𝒜={C}^{\infty }\left(M\right)\otimes \left(C\oplus H\oplus {M}_{3}\left(C\right)\right)$

is the unique algebra with outer automorphisms being the spacetime diffeomorphisms and inner automorphisms being the standard model gauge group. There are some subtleties, though, like equation (1.15), which I don’t understand yet.

Posted by: Urs Schreiber on March 1, 2004 12:02 PM | Permalink | Reply to this

### Alain’ bibliography

I just noticed that the following site is pretty up-to-date
http://www.alainconnes.org/bibliography.html

Besides some not-arxiv papers are available. for instance The local index formula in noncommutative geometry. Geom. Funct. Anal. 5 (1995), no. 2, 174-243. (with Henri Moscovici) [PS] .

Posted by: alejandro rivero on March 5, 2004 5:17 PM | Permalink | Reply to this

### Re: Alain’ bibliography

I see that the most recent publications deal with

- Hecke algebras

- Hopf algebras

- von Neumann algebras .

I haven’t looked at any of these recent papers by Connes. Are
they related? What is the general idea that is being studied
here?

Posted by: Urs Schreiber on March 5, 2004 7:42 PM | Permalink | PGP Sig | Reply to this

### Orthgonal Projectors and Discrete Calculus

It is fun to take new tools and apply them to old problems :)

In fact, sometimes I wish I could go back and relearn elementary physics knowing what I know now :)

After discussing the algebra of string field theory, a somewhat obvious observation occurred to me. For many years, I’ve been working on the development of a theory of discrete differential geometry. The basis of which is an graded (associative) algebra (with unit 1)

(1)$\Omega =\underset{p}{⨁}{\Omega }^{p}$

and a derivation $d:{\Omega }^{p}\to {\Omega }^{p+1}$ satisfying

(2)${d}^{2}=0$

and

(3)$d\left(\mathrm{ab}\right)=\left(\mathrm{da}\right)b+\left(-1{\right)}^{\mid a\mid }a\left(\mathrm{db}\right).$

Actually, the above is a general abstract differential geometry (including the usual one). The discrete part comes in upon define basis elements $\left\{{e}_{i}\right\}$ for the subalgebra ${\Omega }^{0}=𝒜$ satifying

(4)${e}_{i}{e}_{j}={\delta }_{\mathrm{ij}}{e}_{i}$

and

(5)$\sum _{i}{e}_{i}=1.$

It is completely obvious and I did know this before but the significance did not dawn on me until recently that these are the defining relations for a complete set of orthogonal projectors. Why this seems to be significant to me is that Urs and I developed a fairly complete working theory of discrete differential geometry, which I now appreciate as being based on a complete set of orthogonal projectors.

Recently, it has become known that there exists a complete set of orthogonal projectors for the string field algebra. Together with the BRST operator, we get a graded algebra of string fields. These two facts together mean that we can, in principle, define a discrete differential geometry of string fields. I don’t yet know what such a thing would mean, but it is my goal to figure it out :)

As a baby step in this direction, I decided to consider a simpler (yet non-trivial) complete set of orthogonal projectors. Namely, I decided to consider the Fourier modes for the discrete Fourier transform. This is also possibly of relevance for string theory because the terms in the 1d discrete Fourier transform are very much like the excitation modes of a string.

I’m hoping that an understanding of discrete differential geometry on these more abstract spaces might shed some light on things such as open string field theory.

Posted by: Eric on May 21, 2004 3:05 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

For those following this let me add that with respect to string field theory Eric is referring to our discussion here and with respect to the orthogonal projectors he is in particular thinking of the results presented in the paper

T. Kawano & K. Okuyama: Open String Fields as Matrices (2001),

where a method is discussed which seems to allow to map the space of all matter string fields (i.e. including everything but the ghosts) to that of $N×N$ matrices (with complex entries) for $N\to \infty$.

Since general experience with SFT seems to indicate that given any such algebra of the matter fields any odd graded, nilpotent operator $Q$ which satisfies Stokes’ law can be interpreted as the BRST operator associated with some background (classical solution of the theory), and because this means that SFT on this background is nothing but some differential calculus over the algebra $𝒜$ of $\infty ×\infty$ matrices, this suggests that one could gain isight into SFT by studying abstract differential calculus over such an algebra.

In general I think this is a good idea, but I also want to point out an important caveat:

The construction in the above paper is known to be a little too cavalier with some details of the SFT star product. The algebra constructed there relies heavily on the existence of the identity string field, for instance, but precisely this is known to be problematic, as discussed for instance in

Kishimorto & Ohmori: CFT description of identity string field (2002).

A more detailed discussion of possible problems is given in the discussion section by Kawano and Okuyama.

But despite of this I think it might be an entertaining and probably instructive undertaking to study all differential calculi over the algebr of $N×N$ matrices which admit a notion of ‘integral’ over generalized differential forms with respect to which Stokes’ law holds and which furthermore (as is the case in SFT) is supported on the space of $3$-forms.

Posted by: Urs Schreiber on May 21, 2004 3:25 PM | Permalink | PGP Sig | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

Ok. With that introduction out of the way, let me jump right into some fun stuff :)

Now that we know how discrete calculus works, let’s look at a very important discrete algorithm from this new perspective. Namely, the discrete Fourier transform (DFT). To begin let’s imagine $N$ nodes evenly spaced on a circle ${S}^{1}$. A function $f$ is expanded as

(1)$f=\sum _{m=0}^{N-1}f\left(m\right){e}_{m}.$

Now, the definition of the DFT is

(2)$f\left(m\right)=\sum _{n=0}^{N-1}\stackrel{˜}{f}\left(n\right)\mathrm{exp}\left(\frac{i2\pi \mathrm{mn}}{N}\right).$

Plugging this in above gives

(3)$f=\sum _{m=0}^{N-1}\sum _{n=0}^{N-1}\stackrel{˜}{f}\left(n\right)\mathrm{exp}\left(\frac{i2\pi \mathrm{mn}}{N}\right){e}_{m},$

which suggests the new basis

(4)${\stackrel{˜}{e}}_{n}=\sum _{m=0}^{N-1}\mathrm{exp}\left(\frac{i2\pi \mathrm{mn}}{N}\right){e}_{m}$

so that

(5)$f=\sum _{m}f\left(m\right){e}_{m}=\sum _{n}\stackrel{˜}{f}\left(n\right){\stackrel{˜}{e}}_{n}.$

(6)${\stackrel{˜}{e}}_{m}{\stackrel{˜}{e}}_{n}={\stackrel{˜}{e}}_{m+n}$

so that

(7)$fg=\sum _{m}f\left(m\right)g\left(m\right){e}_{m}=\sum _{m,n}\stackrel{˜}{f}\left(m\right)\stackrel{˜}{g}\left(n\right){\stackrel{˜}{e}}_{m+n},$

which demonstrates the convolution property :)

Anyway, this is just a new way to look at an old friend :)

A kind of neat thing to do from here is to consider states corresponding to the basis elements

(8)${e}_{m}\to \mid {e}_{m}〉$

so that

(9)$f\to \mid f〉=\sum _{m}f\left(m\right)\mid {e}_{m}〉.$

From here we can define dual states

(10)$〈{e}_{m}\mid$

satisfying

(11)$〈{e}_{m}\mid {e}_{n}〉={\delta }_{m,n}.$

In particular,

(12)$〈{e}_{m}\mid f〉=f\left(m\right)$

may be thought of as evaluating the function $f$ at the node $m$. Another suggestive way to interpret this would be as the integral of the 0-form $f$ over the point (0-chain) $m$, i.e.

(13)$〈{e}_{m}\mid f〉={\int }_{m}f=f\left(m\right).$

Now we can formally make the identification

(14)${e}_{m}=\mid {e}_{m}〉〈{e}_{m}\mid ,$

which makes the role of ${e}_{m}$ as a projector explicit.

Ok. We can now do the same trick for the Fourier bases ${\stackrel{˜}{e}}_{m}$ and first map them to states

(15)${\stackrel{˜}{e}}_{m}\to \mid {\stackrel{˜}{e}}_{m}〉=\sum _{n=0}^{N-1}\mathrm{exp}\left(\frac{i2\pi \mathrm{mn}}{N}\right)\mid {e}_{n}〉.$

The dual bases follow from the inverse DFT and can be written as

(16)$〈{\stackrel{˜}{e}}_{m}\mid =\frac{1}{N}\sum _{n=0}^{N-1}\mathrm{exp}\left(-\frac{i2\pi \mathrm{mn}}{N}\right)〈{e}_{n}\mid$

so that

(17)$\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid$

is also a projector and

(18)$\sum _{m=0}^{N-1}\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid =1.$

We should also note that, in contrast to ${e}_{m}$, we have

(19)${\stackrel{˜}{e}}_{m}\ne \mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid .$

unless I’ve made a mistake of course :)

However, we could now consider a new space where the $\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid$ were nodes and study the discrete calculus on this space. I don’t yet know what this would be like :)

For anyone who has taken quantum mechanics 101, everything here probably looks VERY familiar. I think that is pretty neat and highlights how closely the ideas of discrete calculus are engrained in the concepts of quantum mechanics.

If you switch to thinking in terms of QM, it is kind of neat to consider

(20)$\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid$

as an interaction and you can compute the probability of a particle at node ${e}_{p}$ making a transition to node ${e}_{q}$ under this interaction via

(21)$〈{e}_{p}\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid {e}_{q}〉=\frac{1}{N}\mathrm{exp}\left(\frac{i2\pi \left(p-q\right)m}{N}\right).$

It is kind of neat (maybe) :)

Then again, the complex probability probably means I’ve made some basic conceptual or algebraic mistake somewhere. It’s been eons since I looked at QM :)

That’s enough for now… (i’m tired :))

Eric

Posted by: Eric on May 21, 2004 4:17 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

That’s nice. (Though I don’t know if I see the relation to SFT yet!? :-)

Then again, the complex probability probably means I’ve made some basic conceptual or algebraic mistake somewhere.

Not necessarily. I haven’t tried to check all your signs and prefactors, but this result looks ok. There is no need for the right hand side to be real in general, because the left hand side is not necessarily real for $p\ne q$. Only for $p=q$ do we have $〈{e}_{q}\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid {e}_{q}〉=\mid 〈{e}_{q}\mid {\stackrel{˜}{e}}_{m}〉{\mid }^{2}\in R\phantom{\rule{thinmathspace}{0ex}}.$ And indeed, for $p=q$ the right hand side is real.

BTW, you don’t happen to know a nice trace-like thingy over some exterior calculus over the algebra of $N×N$ matrices which is supported on 3-forms, do you? ;-)

Posted by: Urs Schreiber on May 21, 2004 4:37 PM | Permalink | PGP Sig | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

BTW, you don’t happen to know a nice trace-like thingy over some exterior calculus over the algebra of $N×N$ matrices which is supported on 3-forms, do you? ;-)

Not off the top of my head, but I still don’t quite understand why you expect such a thing to be important. I would be tempted to put my effort into finding a $†$ operation on such forms and then study a Yang-Mills-like string field theory. Who knows? Unlike the sick Chern-Simon-like version, we might actually come up with something that works ;)

Eric

Posted by: Eric on May 21, 2004 4:51 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

Noncommutative algebras are discussed in

Deformations of classical geometries and integrable systems
A. Dimakis and F. Muller-Hoissen

A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical geometries. We also introduce harmonic maps on generalized Riemannian spaces into Hopf algebras and make contact with integrable models in two dimensions.

This paper points to Hopf algebras and some earlier (classic) work by Woronowicz.

Eric

Posted by: Eric on May 21, 2004 5:03 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

That’s nice. (Though I don’t know if I see the relation to SFT yet!? :-)

There might not BE any relation :)

Let me just clarify my motivation. I am beginning to suspect that the discrete calculus that we can construct in SFT might not be related to “points in physical space” the way things worked in our notes. I suspect they might be points in some abstract space that can be studied using the tools of discrete calculus. It is kind of a wild guess, but I had the hunch that if I could understand discrete calculus on some other simple, yet nontrivial, abstract space, then that would shed light on what we might end up building for SFT. The simple, yet nontrivial, example I decided to try was Fourier space, because there we also have a complete set of orthogonal projectors that are not related to dirac delta functions in spacetime.

The connection to SFT is merely wishful specualtion (even by my standards!) at the moment :)

Eric

Posted by: Eric on May 21, 2004 5:16 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

I see. Yes, the projector basis does not have to encode spatial position, and in the matrix basis used by Kawano&Okuyama it does not.

(This reminds me vaguely of Gavin Polhemus’s paper which we talked about here. He considered D-branes on a circle.)

Posted by: Urs Schreiber on May 21, 2004 5:30 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

I see. Yes, the projector basis does not have to encode spatial position

Exactly! :)

and in the matrix basis used by Kawano&Okuyama it does not.

I didn’t read K&0 closely enough to get that on my own, but it doesn’t surprise me too much. However, I am suggesting that any projector basis does encode position in some abstract space and we can study discrete calculus on this abstract space. My goal is to gain some understanding of this and maybe apply what we learn to SFT :)

One of the first things I’ve learned so far is probably pretty obvious. Since in the language of discrete calculus there really is no difference between a function defined in physical space and in Fourier space (it is just a change of bases), then the calculus you get will be identical. However, the expression in terms of bases will be different (of course).

If the bases used by K&0 do not represent spatial positions, then perhaps there is a different projector basis that does? That would be neat (and would give some credibility to some other ideas I’ve been thinking of).

Eric

Posted by: Eric on May 21, 2004 6:22 PM | Permalink | Reply to this

### Re: Orthgonal Projectors and Discrete Calculus

If you switch to thinking in terms of QM, it is kind of neat to consider

(1)$\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid$

as an interaction and you can compute the probability of a particle at node ${e}_{p}$ making a transition to node ${e}_{q}$ under this interaction

Oops! Although $\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid$ looks self adjoint to me, in fact, it is not. The interaction I was after should be something more like

(2)${H}_{\mathrm{int}}=\frac{1}{2}\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid +\frac{1}{2}{\left(\mid {\stackrel{˜}{e}}_{m}〉〈{\stackrel{˜}{e}}_{m}\mid \right)}^{†}$

so that

(3)$〈{e}_{p}\mid {H}_{\mathrm{int}}\mid {e}_{q}〉=\frac{1}{N}\mathrm{cos}\left(\frac{2\pi \left(p-q\right)m}{N}\right)$

I think :)

Eric

Posted by: Eric on May 21, 2004 4:44 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

My friend Jesus Clemente has recently collaborated on some initiative of Talasila and v. der Shaft to study geometrically Hamiltonian mechanics on discretised spaces.

We have been speaking about it and he pointed out that the crux is to have a (q?-)bracket, thus a (q?-)Lie derivative, for discrete vector fields.

Posted by: alejandro rivero on October 8, 2004 12:44 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

We have been speaking about it and he pointed out that the crux is to have a (q?-)bracket, thus a (q?-)Lie derivative, for discrete vector fields.

We do! In the approach that Eric and I have been persuing everything about geometry is obtained from the two Dirac operators. We construct an exterior derivative $d$ on the discrete space as well as operators of exterior and interior multiplication with discrete differential forms. Let ${e}_{\mu }$ be the operator of interior multiplication with the discrete differential form ${\mathrm{dx}}^{\mu }$ and let ${v}^{\mu }$ be any ‘vector’ field on the discrete space. In the continuum the Lie derivative ${ℒ}_{v}$ with respect to $v$ is given by

(1)${ℒ}_{v}=\left\{d,{v}^{\mu }{e}_{\mu }\right\}\phantom{\rule{thinmathspace}{0ex}}.$

Since Eric and mine formulation is completly ‘mimetic’ we can simply copy this formula to the discrete setting. Thus a discrete Lie derivative on the discrete space would simply be

(2)${ℒ}_{v}=\left\{d,{v}^{\mu }{e}_{\mu }\right\}$

as above, but now with $d$ and $e$ being the operators on the space of discrete differential forms as defined in our paper.

So this is one way to obtain a discrete Lie derivative. You then get the discrete deformation of the Lie bracket $\left[v,w\right]$ from

(3)${ℒ}_{\left[v,w\right]}:=\left[{ℒ}_{v},{ℒ}_{w}\right]\phantom{\rule{thinmathspace}{0ex}}.$

This is one possible discrete version of the Lie bracket. Of course, there are many deformations of the continuous Lie bracket that can be expressed as discrete Lie brackets. Which one of these one chooses will depend on the applications and intentions. Our discrete Lie bracket has the nice property that, by construction, it mimetically satisfies the same algebraic relations in terms of operators on differential forms as in the continuum, namely those given above.

Of course it is well possible that these are not the algebraic relations crucial for somebody else’s application. Our framework is modeled after the principles of quantum theory on configuration space. That’s where it is most naturally applied. I could imagine that applying it to a discretized phase space might not necessarily result in the desired operation.

Posted by: Urs Schreiber on October 14, 2004 2:23 PM | Permalink | PGP Sig | Reply to this

### Re: The search for discrete differential geometry

But, are we sure that the discrete Lie derivative of a discrete vector field is still a genuine discrete vector field?

In the more naive implementations of discretisation, the result is more of a composition of derivation plus displacement.

Posted by: Alejandro Rivero on November 3, 2004 1:16 PM | Permalink | Reply to this

### Re: The search for discrete differential geometry

But, are we sure that the discrete Lie derivative of a discrete vector field is still a genuine discrete vector field?

That depends on how many properties of a continuum vector field you want to preserve. Most generally, a discrete vector field in the context that I am talking about is nothing but a form annihilation operator of grade -1. This property is indeed preserved by the discrete Lie derivative that I talked about, since that is grade 0.

On the other hand, I realized that in my previous reply I maybe was not careful enough. I haven’t thought about if the discrete vector field $u$ defined by ${ℒ}_{u}\equiv \left[{ℒ}_{v},{ℒ}_{w}\right]$ always exists.

Posted by: Urs Schreiber on November 3, 2004 2:59 PM | Permalink | PGP Sig | Reply to this

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