If you have a question or two about what terms mean, I’ll be glad to answer them. If you have lots of questions, you may want to read some books on differential geometry. I really enjoyed Choquet-Bruhat *et al*‘s book *Analysis, Manifolds and Physics*.

The term ‘alternation operator’ is a bad choice because I’ve never seen anyone use that term, but it’s still clear (to me) what it must be, because there’s only one interesting operator

An element of is called a **1-form at the point** An element of is called a ** p-form at the point ** We can take the wedge product (also called ‘exterior product’) of a 1-form and a p-form and get (p+1)-form. So, we get a bilinear map

and thus a linear map

Since we can do this at any point , we get a vector bundle map

and that’s what the ‘alternation map’ must be.

]]>but I meant different Laplacians which could be constructed from different connections.

Like if this A operator, whose form I do not understand, because amongst others I do not understand how this alternation map works, and the whole terminology there is alien to me (there are also no further links on that wikipedia page), but which looks on a first glance like the Einstein operator just with a different sign in front of the one-half scalare curvature then this Lichnerowicz formula suggests that one could write the Einstein tensor as some linear combination from the usual A operator and that “A operator on Spin manifolds”. (which seems to be the scalarcurvature according to Wikipedia) or in other

words if this A is what it looks like then subtracting the Einstein tensor from the A operator could give the scalar curvature (i.e. the “A operator on Spin manifolds”).

Anyways I am aware that these are very very vague guesses and I just wanted to drop them here, because eventually this might catalyze some more guesses.

]]>I’m getting tired of this Weitzenböck stuff, so I’ll stop here.

thanks anyways. It was probably the longest discussion about Laplacians I ever had in my life.

This article claims to “demystify” the Weitzenböck tensor, but it would take a little while to read.

This seems to be at most a draft not an article like on page 3 it starts out:

1. Lichnerowicz Laplacians (r⇤T)(X2,…,Xk) = ␣(rEiT)(Ei,X2,…,Xk)

In this section …. Conventions

and given the sparseness of definitions it is for me impossible to follow the paper.

And by very briefly looking at it seems it wouldn’t help too much

anyway, since it seems to mention only different Laplacians constructed from one connection, but I meant different Laplacians which could be constructed from different connections.

So I understood this as that is at least locally linear, i.e. that it is a tensor.

Yes, it’s a tensor, built from the Riemann tensor somehow. The Wikipedia formula for it is indeed poorly explained. Their so-called “alternation map”

is obviously just taking the wedge product of a 1-form and a p-form to get a (p+1)-form—what else could possibly make sense here? Some weird person just decided to call it the “alternation map” instead of its usual name, “wedge product” or “exterior multiplication”.

Similarly, their so-called “universal derivation inverse to on 1-forms”

has got to come from interior multiplication:

by using duality to turn it into a map

Again, there’s nothing else it could reasonably be.

I’m getting tired of this Weitzenböck stuff, so I’ll stop here. By the way, this This article claims to “demystify” the Weitzenböck tensor, but it would take a little while to read. Also by the way, I’ve never heard the term “rough Laplacian” before.

]]>No, that’s not the problem:

so in any situation where we’ve worked out the details carefully enough that d^* d is self-adjoint, we can see that its spectrum is nonnegative.

If you assume positive definiteness and your other definitions yes, but as said one could try to generalize that. Eventually it would be enough to use a hermitian form then you can keep your positive definiteness. But it seems that in order to include magnetism some generalizations may eventually need to be made.

The problem with is that it’s not the Laplacian.

It is not the Hodge Laplacian.

I’ve never thought about this. I needed to look up the Weitzenböck identity to think about this. Maybe some trickery could turn the operator this reference calls A—I guess that’s your ‘Weitzenböck tensor’—into the Einstein tensor.

Yes I meant that operator A.

I called it Weitzenböck tensor because I don’t know how this thing is usually called in various dialects. And I called it tensor because the Wikipedia entry said:

A is a linear operator of order zero involving only the curvature.

So I understood this as that is at least locally linear, i.e. that it is a tensor. But may be this interpretation is wrong. I might have been lured into that interpretation by the fact that the difference of two covariant derivatives is a tensor, if I remember correctly ( : O) ). By the way, since I haven’t thought about this stuff since quite a while it feels quite awkward to talk about this in public. It feels really quite like a public Nadja-Geometry-Amnesia test. : O

Maybe some trickery could turn the operator this reference calls A..

If it is a tensor then locally, lets assume that one has invertibility then it seems (again if I remember correctly) one could just compose one tensor (as a linear map) with the inverse of the other and get a linear map between them. So if this “tensor” interpretation is correct then locally this seems to work always (apart from divergencies due to noninvertibility etc.), but what about globally? I mean apart from that plus/minus sign and modulo the fact that I don’t know what is exactly meant by these alternation map and this universal derivation inverse to θ on 1-forms and what they could spoil up this Weitzenböck “tensor” looks really like the Einstein tensor.

If I remember correctly ( : O ) the Einstein tensor was constructed as a generalization to the laplacian in Poisson equation for gravity, so I was hesitating to assume that it could be related to the *difference* of two Laplacians, thats what was meant by my “replystammer”. So I wonder wether there exists some way, with which one could express the Einstein tensor as a linear expression (meaning here a difference or a mean or something in that sense) of two different laplacians, just as the Weitzenböck “tensor” is the *difference* between the Hodge laplacian and the rough laplacian.

]]>[Robert Kotiuga]: Beno, there is something I really don’t understand about Hermann Weyl.

[Beno Eckmann]: What is it?

RK: Well, in his collected works, there are two papers about electrical circuit theory and topology dating from 1922/3. They are written in Spanish and published in an obscure Mexican mathematics journal. They are also the only papers he ever wrote in Spanish, the only papers published in a relatively obscure place, and just about the only expository papers he ever wrote on algebraic topology. It would seem that he didn’t want his colleagues to read these papers.

BE: Exactly!

RK: What do you mean?

BE: Because topology was not respectable!

RK: Why was topology not respectable?

BE: Hilbert!

So you say that in a theory of electricity which behaves with integrity and discretion must be zero.

That’s your interpretation… I prefer to repeat what I actually said, which sounds much less interesting, but has the advantage of being surely true:

If maps 0-forms to (-1)-forms, and all (-1)-forms are zero, and is a 0-form, then .

(Some people say “there are no (-1)-forms”, but a mathematically more sophisticated approach is to say all (-1)-forms are zero, and that’s the approach I usually take.)

But if you postulate this anyways then less then ever I understand why you would need the positive definiteness of the inner product on or ?!

I will let you worry about this; I don’t feel like it. What I know is that:

1) I have a specific formula for the inner product on which works nicely for electrical circuits; this inner product is positive definite.

2) There’s a standard formula for the inner product on (0-forms on a compact manifold), which for Riemannian manifolds is positive definite.

3) We need the inner product on zero-forms to be positive definite for this standard calculation in Hodge theory to work:

in the case where is a *1-form*.

What is the problem with just taking ? The possibly negative spectrum on a compact manifold?

No, that’s not the problem:

so in any situation where we’ve worked out the details carefully enough that is self-adjoint, we can see that its spectrum is nonnegative.

The problem with is that it’s not the Laplacian.

For example, we have a pre-existing definition of the Laplacian of a vector field on Euclidean , and thus, using the metric to turn a vector field into a 1-form, we get a definition of the Laplacian of a 1-form on . It’s just the obvious thing: take the Laplacian of each component using the standard basis of 1-forms .

But this Laplacian does not agree with For example take . In this case of any 1-form is zero!

On the other hand, this pre-existing definition of the Laplacian of a 1-form on *does* agree with

There are lots of other reasons why everyone defines to be the Laplacian. Mostly, it makes Hodge theory work (just read the section on ‘Riemannian Hodge theory’, which is nice and short and lists the key results).

If you look at the Weitzenböck tensor (difference between your usual Laplacian and the rough Laplacian) then this looks “almost” like the Einstein tensor. Can one formulate the “Einstein operator” (here I mean Einstein tensor + cosmological term – Energy momentum tensor, or may be just the Einstein tensor for simplicity) as the difference of the laplacians of two different (co)derivatives ?

I’ve never thought about this. I needed to look up the Weitzenböck identity to think about this. Maybe some trickery could turn the operator this reference calls —I guess that’s your ‘Weitzenböck tensor’—into the Einstein tensor. But I don’t instantly see how to do it. Real experts on general relativity should know if it’s possible. It’s a nice idea.

]]>By the way, Nad: in this particular conversation, please try to post your comments so they appear below the comment you’re replying to. You keep posting them so they show up far above it, and I keep fixing them.

I am sorry. I thought I always clicked on the reply button right next to your answers, but may be not, let’s see where this here will end up.

First of all thanks a lot for giving me such detailled answers. It is very rare that you find someone who are allowed to ask “holes into the stomach” (Löcher in den Bauch) (means probably you ask so much that the person you ask gets really hungry). Just as a warning: during my studies I was told by a person that one of the professors already complained to this person about my “stupid questions”, in this conversation this professor also said that I am “no big light” (“Sie ist keine grosse Leuchte”). So your answers are very likely pearls before breakfast, unless someone else reads this here too and learns by your answers or might be inspired by my stupid questions (Like an artist or so…).

… is simply that you don’t “need” it, you have it: 0-cochains are automatically coclosed: is zero for any 0-cochain , unless we’re in a weird situation where there exist nonzero (-1)-cochains!

So you say that in a theory of electricity which behaves with integrity and discretion must be zero.

But if you postulate this anyways then less then ever I understand why you would need the positive definiteness of the inner product on or ?!

is, up to a sign, the usual definition of the Laplacian on p-forms.

What is the problem with just taking ? The possibly negative spectrum on a compact manifold?

By the way in this context I may ask what I always wanted to know. If you look at the Weitzenböck tensor (difference between your usual Laplacian and the rough Laplacian) then this looks “almost” like the Einstein tensor. Can one formulate the “Einstein operator” (here I mean Einstein tensor + cosmological term – Energy momentum tensor, or may be just the Einstein tensor for simplicity) as some linear combination of the Laplacians of two different (co)derivatives ? If this would be the case then this could eventually be a useful ingredient for the discretization of the field equations (on which I believe probably tons of people have been working at, at least already when I was still a student there were a couple of people interested in this.

]]>John wrote:

The power consumed by each wire equals the square of the current along that wire times the resistance of that wire. If the resistance were negative this power could be negative!

Nad wrote:

But I was talking about and and not and . There could be different inner products on these spaces.

True. I was just pointing out the physical reason why I assume the inner product on —and thus dually on the isomorphic space —is positive definite. If you can discover or invent a physical interpretation of the inner product on and , you will be able to decide whether you want those to be positive definite too.

John wrote:

Think of as a -form and as the Laplacian.

Nad replied:

I don’t know why you suddenly define the Laplacian in this way, because in your circuit paper it is just …

is, up to a sign, the usual definition of the Laplacian on p-forms. But when dealing with circuits, all the spaces in our cochain complex

are zero-dimensional except and : there are just vertices and edges in our graph, no higher-dimensional stuff. So in this particular case vanishes on 0-cochains, so

for any 0-cochain I took advantage of this simplification in my paper. But now we’re talking about more general things, such as electromagnetism and p-form electromagnetism, where we need 2-cochains, 3-cochains, etc.

Indeed, the basic lemma of Hodge theory

is completely trivial in the case of electrical circuits! In that case, either is a 0-cochain and is always zero, or is a 1-chain and is always zero. So the answer to this question of yours:

But why do I need coclosedness?

is simply that you don’t “need” it, you *have* it: 0-cochains are *automatically* coclosed: is zero for any 0-cochain , unless we’re in a weird situation where there exist nonzero (-1)-cochains!

I am still trying to get an image in my head what people may mean with higher gauge theory and string theory.

The basic idea is just this: you can integrate a 1-form A over the worldline of a particle, and that gives you the action for a particle coupled to the electromagnetic field. Similarly, you can integrate a 2-form B over the worldsheet of a string, and that gives you the action for a string coupled to the ‘higher electromagnetic field’, which is usually called the Kalb-Ramond field. It’s all just the usual story with the dimensions increased by one.

My paper with John Huerta, An invitation to higher gauge theory, was supposed to be a friendly way to start learning about this stuff. But I’ve switched over to working on quite different subjects, so I’m not really interested in talking about it anymore. I mainly keep Derek Wise’s work in mind because it uses a lot of the same math that I’m now applying to electrical circuits.

]]>