The String Coffee Table

Back again. More on Kapranov…

For the full dg-algebra $B$ we have generators $x_{I}$ of degree $(- p + 1)$ where $I$ is an index set of $p$ elements. The differential $$d(x_{I}) = \sum_{I=J \coprod J} \varepsilon (J,K) [ x_{J} , x_{K} ]$$ (where $\varepsilon$ is the sign of the shuffle) satisfies $d^{2} = 0$.

Theorem: $B$ is a free NC resolution of $\mathbf{C} [x_{1} , \cdots , x_{n} ]$ and with respect to $d$ the cohomology is $$H^{j} (B) = \mathbf{C} [x_{1} , \cdots , x_{n} ]$$ for $j = 0$ and zero otherwise.

Apparently the proof uses that the Lie algebra is a Harrison complex of a free graded commutative algebra, but don’t ask me what that is.

We want to realise $C_{2}$ inside $B$, but the horizontal pasting of two diamonds gives two possible results, leading to a definition of $D_{2}$ as the quotient of $B$ by $d$ of the commutators $[ x_{ij} , x_{kl} ]$. Then $C_{2}$ quotiented by translations fits into $D_{2}$.

Sigh. Now the continuous version. Extend the NC power series in $z_{1} \cdots z_{n}$ by $z_{ij}$, $z_{ijk}$ and so on, as above. Let $$\Omega = \sum z_{i} dt + \sum_{i \lt j} z_{ij} dt_{i} dt_{j} + \cdots$$ of total degree 1. The claim is that $d \Omega + \frac{1}{2} [ \Omega , \Omega ] = 0$.

What has this got to do with connections on gerbes? Let $G^{0}$ acting on $G^{-1}$ be a 2-group (Urs has mentioned these often enough). For $\mathbf{g}^{i}$ the Lie algebras we have a dg-Lie algebra in degrees -1 and 0. Now let $M$ be a manifold and $$F = d \Omega + \frac{1}{2} [ \Omega , \Omega ] = F_{2} + F_{3}$$ for $F_{2} \in \Omega^{2}_{M} \otimes \mathbf{g}^{0}$ and similarly for $F_{3}$.

If $F = 0$ then it so happens that for all membranes $\sigma$ there exists an $H(\sigma)$ in the universal enveloping algebra of $\mathbf{g}$ with $$d H(\sigma) = H(\gamma_{1}) - H(\gamma_{2})$$ which apparently only depends on $\sigma$ up to reparameterization.

Then onto Chen’s holonomy, but I’m going to skip that bit. To cut a long story short: smooth membranes in $\mathbf{R}^{n}$ correspond to the universal enveloping algebra of the appropriate Lie algebra quotiented by the commutator piece like above.

What else happened yesterday? Well - they opened the university bar for us - but seriously: Berger talked about Iterated Wreath Products with theorems such as a Quillen equivalence between $\mathbf{Top}_{\ast}^{\Delta^{op}}$ and just $\mathbf{Top}_{\ast}$. He then described a recursive family for n-fold loop spaces: start with the one point set dense in $\mathbf{Set}$. Apply the magic wreath business and get something dense in $\mathbf{Cat}$, and so on. Then simplicial objects in the nth iterate happen to form a classifying topos for n-discs with the unique arrow being ‘convex subcells of trees’. For those who know more about this than me, these Quillen equivalences have the left adjoint a n-fold Segal functor.

We also had some relatively physical talks. Bondal spoke about derived categories of toric varieties.