The String Coffee Table

Good morning! :)

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

two mottos:

a) “geometry is supersymmetric quantum theory”

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

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

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

1) general formalism:

Dimakis & Müeller Hoissen formalism:

N=2 generalized spectral ‘tuple’

exterior calculus $\Omega(\mathcal{A},d)$ over some algebra $\mathcal{A}$

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

now turn $\Omega(\mathcal{A})$ into inner product space in order to get generalized spectral triple $(\mathcal{A},d,\langle\cdot|\cdot\rangle)$

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

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

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

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

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

0-forms self-adjoint $\leftrightarrow$ space is flat

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

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

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

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

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

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

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

2) application to mathematics ——————-

get NCG version of Hodge star operator:

$\star A|0\rangle = A^\dagger|vol\rangle$

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

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

easy to check that this Hodge has correct continuum limit

Nice :)

self-adjoint, no fermion doubling slightly non-local

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

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

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

answer: further background fields, like Kalb-Ramond fields

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

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

string field theory: ———————-

BRST operator is exterior derivative on gauge group:

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

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

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

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

5) conclusion ———————

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

Neil Armstrong? :)

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

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

Best of luck,
Eric