Embedded TQFT?

So, a subject rather near and dear to the hearts of many of my fellow co-bloggers is that of 1+1-dimensional TQFT: that is, of monoidal functors from the category of 1-manifolds with morphisms given by smooth cobordisms to the category of vector spaces over your favorite field .

There’s a rather remarkable theorem about such functors, which really deserves a post of its own for proper explanation, but I’ll spoil the surprise here.

Any such functor associates a vector space to a single circle, and to the “pair of pants” cobordism, it assigns a map , which one can check is a commutative multiplication.

Furthermore, the cap, thought of as a cobordism from the empty set to a circle gives a map , which gives a unit of this algebra. Thought of as a cobordism from the circle to the empty set, it gives us a map which we call the counit or Frobenius trace.

Theorem. A commutative algebra with counit arises from a TQFT if and only if kills no left ideal of .

Well, I bet you didn’t think 2-d TQFTs would prove to be so easy, huh? Unfortunately, that is something of dumb luck. It happens to be in that sweet spot where things are interesting but not particularly hard (mostly because 2-manifolds are in the same sweet spot. 3-manifolds are much harder).

But one can try to dress up this picture a bit. For example, Lauda and Pfeiffer considered a generalization where line segments and open cobordisms are allowed and classified these sorts of theories in terms of objects they call “knowledgeable Frobenius algebras” which is a trickier, but a similar sort of idea.

I’d like to take a moment to suggest a different generalization which showed up in some research I’ve been doing with Catharina Stroppel (hopefully more on that later). Rather than allowing more cobordisms, I’d like to be more careful in distinguishing them. Consider the category where

• objects are closed 1-d submanifolds of the plane.
• morphisms are cobordisms embedded in the plane times the interval, considered up to isotopy (I’m going to wimp out and not use higher morphisms).

Whereas before all abstractly isomorphic surfaces were the same, now they might give different maps if embedded differently.  This is a perfectly good monoidal category (as usual, tensor product is just putting things next to each other), and one could ask what monoidal functors from it to vector spaces look like.

To keep my head from exploding, let’s assume for now that all collections of circles still go to the tensor product of the corresponding number copies of some vector space .  There may well be some wacky way of weakening this condition.

I’m not so sure myself, but here’s as far as I got.

1. Now there are 3 basic sorts of cobordisms: the cap and two pairs of pants, one in the normal embedding for pants, and with one leg stuck down the middle of the other so that it results in two nested circles, rather than unnested ones as the standard pants do. (Does anyone have a good suggestion what to call this? It’s rather hard to picture, as I can’t think of a place it occurs in the real world).
2. The first two types can still be used to construct a Frobenius algebra, since the proof only uses embedded isotopies.
3. The third move gives a “strange” automorphism of the algebra , given by putting the identity on the outer circle of the “cuffs” of the pants. This is not embedded isotopic to the identity, and so could be non-trivial.

I think I can prove that the example of with the trace given by and equipped with the “strange” automorphism sending is a real example.

So, my question to you guys is as follows: what are the conditions you need for consistency? Are there any other worthwhile examples?