### Project: 2-NCG

#### Posted by Urs Schreiber

In this entry here I want to develop some notes on *categorification* of **N**ot necessarily **C**ommutative **G**eometry by means of spectral triples in the sense of Connes, and, eventually and if possible, relations of this to superstrings, loop space geometry and maybe generalized cohomology.

This entry will be work in progress for a while since I plan to develop these notes as I go along.

The original reason for developing these notes in the public is actually the following: I usually type all my notes on my notebook computer. Currently I am visiting John Baez and his group at UC Riverside. It seems that the screen of my notebook computer did not survive the security checks at the airport, unfortunately, so I am left without my crucial thinking tool. This weblog must now serve as a substitute.

But maybe it is an interesting experiment in its own right. I’d be grateful for anyone interested in accompanying me on my quest for seeing more of the big picture of which ordinary NCG, superstrings, loop spaces, Dirac operators, 2-bundles, etc. are jigsaw pieces. Collaboration on papers not excluded.

I had been thinking and talking about related ideas for a long time already. But it was not before last Sunday when I was sipping hot chocololate at Starbucks late at night that something occurred to me which suddenly made all the pieces fall in place.

The crucial catalyst was John Baez’s remark that what I was talking about seemed to call for the 2-Hilbert spaces, $2-{\u2102}^{*}$-algebras and the categorification of the Gelfand-Naimark theorem he discussed in [1]. Meditating over that paper I realized that it should be possible to start with a bosonic 2-${\u2102}^{*}$-algebra and form its abstract differential 2-calculus by throwing in a formal $d$-functor. By an analogous procudure one can obtain nice ordinary spectral triples, as I have discussed together with Eric Forgy in [2].

Therefore I propose that a **2-spectral triple** should be a triple

consisting of an 2-algebra $A$, a 2-Hilbert space $H$ on which this algebra acts functorially and on which a 2-Dirac operator $D$ is represented.

My first step shall be to develop in detail a simple but interesting example for such a 2-spectral triple, namely one describing discrete superstrings. Depending on how that works out there are many directions to follow and things to work out.

` Last modified: March 20`

Here is an outline of some ideas:

Consider the ordinary commutative case first. The algebra of sunctions from the manifold to the complex numbers obviously must become a category of functors from some 2-space to some target.

For simplicity, assume the 2-space to be a graph category $Q$ for the moment. In HDA2 it has been suggested that the correct target is Vect. However, $(\mathrm{Vect}{)}^{Q}=\mathrm{mod}A$ (where A is the path algebra of the graph) is more like a categorification of functions taking values in the natural numbers. In order to fix that throw in multiplicative inverses to obtain the category $\mathrm{bimod}A$ of $A$ bimodules with the product being the tensor product. This is still lacking addtive inverses. In order to get these we need to know how to decompose objects. Given any object B in an abelian category we can consider any exact sequence $A\to B\to C$ as a decomposition of $B$ in $A$ and $C$. This suggests to let the objects be chain complexes in bimod A. Hence we are led to take the derived category $D(\mathrm{bimod}A)$ as the correct categorification of the algebra of complex functions.

In the uncategorified case we use the algebra of functions to construct vector bundles as suitable modules of this algebra. The wave function of the system itself is a section of such a bundle.

Hence we should be looking for 2-modules of the 2-algebra $D(\mathrm{bimod}A)$. The simplest one is just $D(\mathrm{mod}A)$. This suggests to interpret elements of $D(\mathrm{mod}A)$ as states of the categorified mechanics.

Indeed, these objects are known to describe D-brane states with strings stretching between them. More here.

## Re: Project: 2-NCG

Urs,

How was the talk? :)

I have a small problem with Section 1.2. Basically, the

orderin which things happens is a little off.What came first? The chicken or the egg?

I mean, the inner product or the adjoint? :)

I spent a good fraction of my graduate research trying to find a nice inner product from which I could determine the adjoint. It seems like a natural enough thing to do, but then you came along and said, “Hey, why don’t we just define an adjoint first?” That one simple observation is what made everything fall in place.

Once you have an adjoint, you can construct the inner product in an obvious way (which is also outlined in one of Baez’ papers).

Once you have any adjoint, you construct more general inner products by deforming the original adjoint, not the inner product directly. This places more emphasis on the algebra.

I know it is not a big deal and both chronologies are probably mathematically equivalent, but in terms of “naturalness”, I think cooking up adjoints first is a better way to approach new avenues.

Just a thought…

Best regards,

Eric