The String Coffee Table

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 $$B (a_{0}, \cdots , a_{k}) = \sum (-1)^{ki} (1.a_{i} , \cdots , a_{k}, a_{0}, \cdots , a_{i-1})$$ 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 $$(a_{0}, \cdots , a_{k}) \mapsto u^{-k} \int STr (a_{0}e^{\sigma_{0} D^{2} / u} [D,a_{1}] \cdots [D,a_{k}] e^{\sigma_{k} D^{2} / u}) d \sigma$$ 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 $(\partial + u B)$ 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 $$d a + a \cup a$$ It can do things like recover the data of a spectral triple!