### Tuesday at the Streetfest I

#### Posted by Guest

Hello again! Wednesday already here. I already have 50 pages of notes. It’s quite overwhelming. Now let’s see. Kapranov spoke about a Non-commutative Fourier Transform and Chen’s iterated integrals. For those into membranes, keep reading…

Consider two variables $x$ and $y$ that do not commute. Monomials ${x}^{i}{y}^{j}{x}^{k}{y}^{l}$ can be represented by paths on a lattice in ${R}^{2}$ starting at the origin. Similarly for $n$ variables. Therefore, a polynomial is represented by a summation over paths $\gamma $.

We want to consider the continuous version of this. Imagine subdividing the lattice; allow non-integer powers ${x}_{i}^{\frac{1}{m}}$ and let $m$ go to infinity. Now let $${x}_{i}^{\frac{1}{m}}={e}^{\frac{1}{m}{z}_{i}}$$ in the complex algebra of power series $A$.

$A$ is a ‘connection’. For $\Omega =\sum {z}_{i}{\mathrm{dt}}_{i}$ let ${E}_{\gamma}(z)$ be the holonomy of $\Omega $. Now we can define the NC Fourier Transform using this NC exponential.

Kapranov went on to consider the problem for higher dimensional membranes instead of paths. For a lattice box in the variables ${x}_{1}$ and ${x}_{2}$ there is now a 2-cell filling the box $${x}_{1}{x}_{2}\Rightarrow {x}_{2}{x}_{1}$$ Introduce these ${x}_{\mathrm{ij}}$ of degree -1 in a dg-algebra ${B}_{2}$ with $$d({x}_{\mathrm{ij}})={x}_{i}{x}_{j}-{x}_{j}{x}_{i}$$

Let ${C}_{2}$ be the 2-category generated by the 2-skeleton of the cubical lattice in ${R}^{n}$. The pasting of the half cube for ${C}_{2}$ gives a rule in ${B}_{2}$. But there are 2 choices of half cube. The difference is used to extend to ${B}_{3}$ by adding ${x}_{\mathrm{ijk}}$ now of degree -2 with $$d({x}_{\mathrm{ijk}})=[{x}_{\mathrm{ij}},{x}_{k}]+[{x}_{j},{x}_{\mathrm{ik}}]+[{x}_{\mathrm{jk}},{x}_{i}]$$

so ${B}_{3}$ is the universal enveloping algebra of a dg-Lie algebra. And so on … onto the continuous version of this.

Phew. Must be off.

Marni

P.S. Browser options here aren’t great. We apologise for errors that we are unable to correct at this stage.

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 $$d({x}_{I})=\sum _{I=J\coprod J}\epsilon (J,K)[{x}_{J},{x}_{K}]$$ (where $\epsilon $ is the sign of the shuffle) satisfies ${d}^{2}=0$.

Theorem: $B$ is a free NC resolution of $C[{x}_{1},\cdots ,{x}_{n}]$ and with respect to $d$ the cohomology is $${H}^{j}(B)=C[{x}_{1},\cdots ,{x}_{n}]$$ 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 $[{x}_{\mathrm{ij}},{x}_{\mathrm{kl}}]$. 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 ${z}_{\mathrm{ij}}$, ${z}_{\mathrm{ijk}}$ and so on, as above. Let $$\Omega =\sum {z}_{i}\mathrm{dt}+\sum _{i<j}{z}_{\mathrm{ij}}{\mathrm{dt}}_{i}{\mathrm{dt}}_{j}+\cdots $$ 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 ${g}^{i}$ 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 ${F}_{2}\in {\Omega}_{M}^{2}\otimes {g}^{0}$ 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 $g$ with $$dH(\sigma )=H({\gamma}_{1})-H({\gamma}_{2})$$ 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 ${R}^{n}$ 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 ${\mathrm{Top}}_{ast}^{{\Delta}^{\mathrm{op}}}$ and just ${\mathrm{Top}}_{ast}$. He then described a recursive family for n-fold loop spaces: start with the one point set dense in $\mathrm{Set}$. Apply the magic wreath business and get something dense in $\mathrm{Cat}$, 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.