### Non-Commutative Geometry School I

#### Posted by Guest

I never meant to say much about this week, but in a completely unexpected turn of events yesterday, Getzler pulled a rabbit out of the hat. Remember that cryptic Kapranov-Getzler exchange from the Streetfest? Well, after days of lectures on cyclic cohomology and K-theory and operator algebra Getzler starts doing iterated integrals. He says that if you care about iterated integrals and equivariant cohomology (like all good physicists should) you would be naturally led to do cyclic cohomology. The rest of NCG makes much more sense to me now (that’s relatively speaking, folks).

Anyway, here’s how Getzler’s lectures went:

First, in the 1940’s, Eilenberg and MacLane (yes, the category theorists) discovered the bar construction. Let $A$ be an associative algebra with unit 1. $\overline{A}$ as a vector space is $A$ modulo constants. Let $M$ be a right $A$-module and $N$ a left $A$-module. The bar complex is built from the vector spaces $${B}_{k}(M,A,N)=M\otimes {\overline{A}}^{\otimes k}\otimes N$$ and a homological differential $\partial $.

Now look at bimodules: in particular $A$ as an $A$-bimodule. There is a complex built from the spaces $${C}_{k}(A)=A\otimes {\overline{A}}^{\otimes k}$$ with a sort of Hochschild differential $\partial $ defined with a cyclic term $$\partial (a,{a}_{1},{a}_{2},\cdots ,{a}_{k})=\cdots +(-1{)}^{k}({a}_{k}a,{a}_{1},\cdots )$$ giving cyclic Hochschild homology.

If $A$ is the smooth functions on $M$ (which is compact Riemannian or whatever as necessary) then one obtains the de Rham complex this way, but the differential doesn’t map nicely. So, we introduce a new differential $$B({a}_{0},\cdots ,{a}_{k})=\sum (-1{)}^{\mathrm{ki}}(1.{a}_{i},\cdots ,{a}_{k},{a}_{0},\cdots ,{a}_{i-1})$$ and $\partial +B$ acts on the even/odd pieces of the full $C(A)$ complex. This is the NC replacement of the de Rham complex.

However, we would like to do integration. How? Well, there’s a lot to know about Dirac operators and so on, which I’m sure Urs would be most happy to go into, but to cut a long story short, for the smooth functions case one can define an integral (depending on an operator $D$) and it looks like $$({a}_{0},\cdots ,{a}_{k})\mapsto {u}^{-k}\int \mathrm{STr}({a}_{0}{e}^{{\sigma}_{0}{D}^{2}/u}[D,{a}_{1}]\cdots [D,{a}_{k}]{e}^{{\sigma}_{k}{D}^{2}/u})d\sigma $$ for some parameter $u$ which could be one if you like. The integral is over the standard $k$-simplex using coordinates ${\sigma}_{i}$. It uses a supertrace. The differential $(\partial +uB)$ satisfies a Stoke’s theorem.

Now back to the bar construction. There is a map from $B(\Omega (M),\Omega (M),\Omega (M))$ to forms on the path space of $M$, where we double grade $B$ by picking out pieces of fixed total degree. Then there are two differentials, $\partial $ and $d$. It is $d$ that gives the differential on the path space side.

The beauty of all this is that it’s tailor made for reparameterisation invariance of paths, hence it’s appearance in Kapranov’s work. In the last five minutes Getzler mentioned some applications…you guessed it: holonomies for connections! There is a really general way of defining a curvature in terms of a cup product on the cocycles that looks like $$da+a\cup a$$ It can do things like recover the data of a spectral triple!

## Re: Non-Commutative Geometry School I

Hi, greetings from the Cardigan Bay and now from New Quay!

I wish I had more details on what you are describing here. I am familiar with parts of this stuff, but from your account I am not sure exactly how and where what Getzler talked about goes beyond that.

So for instance I know

Getzler, Jones & Petrack:

Differential Forms on Loop Spaces and the Cyclic Bar Complex: (online available here)and related things.

In particular, I know how given a couple of $p$-forms ${\omega}_{i}$ on $M$ we get a form called $({\omega}_{1},{\omega}_{2},\dots )$ of grade ${\sum}_{i}({p}_{i}-1)$ on $\mathrm{LM}$ by pulling all these forms back to loops and iteratively integrating them over. The exterior derivative $d$ on loop space acts on these ‘

Chen forms’ as I used to call them aswhere

and where $m$ is the Hochschild differential on the string of $\omega $-symbols, i.e. what you seem to call $\delta $ above.

This stuff is the prerequisite for our proof that a 2-connection in a strict principal 2-bundle is locally given by a 1-form ${A}_{i}\in {\Omega}^{1}({U}_{i},\U0001d524)$ and a 2-form ${B}_{i}\in {\Omega}^{2}({U}_{i}\U0001d525)$, as detailed in sections 83 and 11.5 of my thesis, for instance.

So I would love to know in what respect what you were talking about is just a review of things like in the above mentioned paper and in which respect they are new results.