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  

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.



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






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   .

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.

Doplicher Roberts

The statement in Chari and Pressley Guide to Quantum Groups

Associated to every local quantum field theory in four or more spacetime dimensions is a compact group G whose representation ring is isomorphic to the fusion ring of the theory.

Also the appendix of
http://arxiv.org/pdf/math-ph/0602036v1.pdf

Need to unpack this into a post.

Relative QFT

http://arxiv.org/pdf/1212.1692.pdf

Expectation 5.3: Given a real Lie algebra, invariant inner product such that all coroots have length square 2, and a full lattice in the Cartan such that on the lattice the inner product is integral and even, there should exist a 7 dimensional topological quantum field theory and a 6 dimensional theory relative to it.

This is supposed to be the big goal of the (2,0) siX theory.

A relative QFT is an extended field theory in one higher dimension. To be continued...

Enriching some categories

Start with a plain old category C. You have some objects A and B and Hom(A,B) to describe all the arrows from A to B.

What structure does Hom(A,B) have?

Not much. Ok, now I'm stuck. I can't say any more.

If the Hom(A,B) were just sets, it would be locally small. Not good enough.

If I somehow knew they were abelian groups, then I would be in pre-additive categories, and I would start to have some hope about doing all the abelian category chases.

What are the structures I could possibly ask Hom(A,B) to have? I need to have a well defined map Hom(B,C) x Hom(A,B) to Hom(A,C) so that tells me that it has to be monoidal in order for the left hand side to make sense.

And there should be the identity in Hom(A,A).  That means there should be a morphism from the monoidal unit to Hom(A,A) in this monoidal category.

AND coherence relations time:

Now another example, what if I enrich over chain complexes of complex vector spaces. Each hom is now differential graded. So this is a dg-category.

I can relax conditions a little bit more and give you an example of an A_\infty category. The specific one I have in mind is the Fukaya category of some symplectic manifold.

Picture time

Draw more holomorphic polygons to give the higher maps that mess up the dg-category structure.

Youtube videos better than NYAN cat

Cool people saying cool things

Dimensions and Categorification

So Witten's Khovanov paper reflects this idea that categorifying in math corresponds to adding dimensions in physics. He goes from a three dimensional theory to a five dimensional theory in order to go from Jones polynomial to Khovanov homology. More brane-y, more dimensions, more categorical. See http://math.ucr.edu/home/baez/diary/fqxi_narrative.pdf

There is a lot going on here (quantum groups, 2-groups, infuriating! weird combinatorics) and I am putting this post up here mainly so I am forced to learn this soon enough to fulfill my promise to tell you about it.

Can anybody explain to me the classical-quantum correspondence in terms of categorification to me?? Is that even possible? It just seemed like very similar arguments. Danke.

Representable Functors

Take your favorite category C. For each object A there is a functor,



from op-C to Set. (This is a contravariant functor and a lower index. NOT the same convention used for contravariant vectors in physics). We could do the other one by switching A and -. That tells you what happens to objects. What happens to morphisms, the only things that you can do, pre and post composition respectively.

In fact the assignment from A to the functor is also functorial. It is a functor from C to the functor category Fun(C-op,Set) where morphisms are natural transformations. I hear you like morphisms so we put some morphisms on morphisms. That is just pretentious abstract nonsense talk for associativity of reading functions.

This functor is the Yoneda embedding, It embeds the category C into the category Fun[C-op,Set]. Why bother translating these trivialities into such foul language?

The reason is that say we have a functor we want to understand like the functor from Rings to Set that takes a ring to the zero set of some polynomial where the variables are taken from that ring.



This functor is isomorphic to one of those above. Namely take A to be



So instead of understanding the equation, understand this functor. Things that seem artificial to do to the equation are more readily explained in the category language. So essentially this language shows you are not pulling all your constructions out of your ass.

We will use this to explain why manifolds are shiny by describing the category of smooth or analytic manifolds.