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

#### Posted by urs

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

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

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

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

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

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

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

two mottos:

a) “geometry is supersymmetric quantum theory”

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

- contents -

0) introduction

1) general formalism

2) application to mathematics

3) applications to field theory

4) application to string theory

5) conclusion

——

1) general formalism —————————————–

Dimakis & Mü Hoissen formalism:

N=2 generalized spectral ‘tuple’:

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

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

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

direct interpretation:

$\Omega (\mathcal{A},d)$ is (one) NCG generalization of exterior algebra

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

start with simplest inner product and then deform

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

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

${d}^{\u2020}$ is exterior coderivative

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

Clifford algebra and spinors constructible in Dirac-Kaehler formalism

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

get NCG version of Hodge star operator:

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

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

easy to check that this Hodge has correct continuum limit

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

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

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

but rho is nothing but the trivial *holonomy 1-form*

so introduce non-trivial holonomies H

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

so YM action on curved space simply

spinors and fermion doubling:

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

self-adjoint, no fermion doubling slightly non-local

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

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

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

answer: further background fields, like Kalb-Ramond fields

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

string field theory: ————————–

BRST operator is exterior derivative on gauge group:

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

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

5) conclusion —————————–

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

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

long standing problem of NCG Hodge star finds elegant solution

based on Connes central insight: geometry = supersymmetric quantum theory

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

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

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

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

Good morning! :)

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

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.

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.

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)

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.

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

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

Nice :)

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

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.

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.

Neil Armstrong? :)

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

Best of luck,

Eric