If you haven't found something strange: July 2013

Sunday, July 14, 2013

2 dimensional (T)(C)FT choose at least 1

Theo, Zach and Dan B-E at tea: February 2013

What does a TFT do to every topological surface?

It gives an invariant. Let us say that invariant lives in V.

What does a CFT do to every topological surface?

It gives a function on the moduli space of Riemann surfaces with that genus. The function will take value in some V.

What about a TCFT?

Well now we want a locally constant function on the moduli. In this way a lot of the dependence on conformal structure is lost. But large transformations, namely the mapping class group, can still do stuff. Another way to say that is it is a closed degree 0 form. But V might have a Q on it making it differential graded vector space.
So what we are really looking at is $(\Omega^\bullet , d ) \otimes (V,Q)$

There are maps to TFT and CFT.

If you have a choice of volume form on the moduli stack you can integrate the CFT and TCFT results. This is what you do in perturbative string theory where you need to integrate out all the choices you made about conformal structure on the worldsheet.

Koszul Duality

I have a f.d. semisimple Lie algebra L. What is it's structure? I could give you a basis and structure constants and then you could check that that works.

$[x_i , x_j ] = c_{ij}^k x_k$

Or I could give you this chain complex where the differential is as follows

$0 \to k \to \mathfrak{l^*} \to \wedge^2 \mathfrak{l^*} \to \cdots$

$d (x_i^*) = c_i^{jk} x_j^* \wedge x_k^*$

This chain complex is generated in degree 1 so that is all I had to give you. If you compute

, you get Jacobi Identity. This is Chevalley-Eilenberg.

So the data of a Lie algebra is the same as a semifree differential graded-commutative algebra generated in degree 1.

Aside:
You can change 1 to 1-n to get a Lie n-algebra, but then you would need to specify the higher brackets because of the other generators. If you replace it with just n, you get a n-Lie algebra. Removing all restrictions on generators whatsover lands you in $L_\infty$ .

Now let's flip the arrows and go the other way.

I have a free graded Lie algebra structure on something called SC^*. If you see the condition for it to be a complex, you see it forces C to be a commutative algebra.

Again we could play with where it is generated to get the rest of the E_n type structures instead of just Comm.