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 k.

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 A to a single circle, and to the “pair of pants” cobordism, it assigns a map m:A\otimes A\to A, 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 i:k\to A, which gives a unit of this algebra. Thought of as a cobordism from the circle to the empty set, it gives us a map \mathrm{tr}:A\to k which we call the counit or Frobenius trace.

Theorem. A commutative algebra with counit (A,\mathrm{tr}) arises from a TQFT if and only if \mathrm{tr} kills no left ideal of A.

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 A.  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 A, 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 k[x]/(x^2) with the trace given by \mathrm{tr}(1)=0 and \mathrm{tr}(x)=1 equipped with the “strange” automorphism sending x\mapsto -x 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?

10 thoughts on “Embedded TQFT?

  1. Hhave you considered orienting your circles and cobordisms? You can’t turn a circle inside out in the plane, so it seem to me that a clockwise oriented circle and a counter-clockwise oriented circle should be two nonisomorphic objects, with different vector spaces assigned to them.

  2. Sigh, that was dumb. Piecing together two of your nested pairs of pants lets me turn a clockwise circle into a counterclockwise circle.

  3. There’s a whole theory of unoriented 1-manifolds and unoriented cobordisms, which is the TQFT of “orientifolds”. It has a second map, C: k → A, and a corresponding trace, given by the mobius band. These satisfy some further consistency relations.

    I don’t think this is related to your theory, but I thought I would mention it.

  4. David, I don’t see the how to turn a clockwise circle into a counterclockwise one. You can two-color the complement of your cobordism based on parity of surrounding layers, and the circles are oriented according to whether the odd component is on the outside or inside.

    I’m also having trouble seeing why funny pants gives an algebra automorphism. I only see two distinct algebra structures, related by some bizarro thing.

  5. I’m also not seeing the automorphism from the “cuffs”. It seems like you can’t put a cap on the “outer cuff” because then the inner cuff would have to pass through the cap, clearly not an embedding.

    Also if the monoidal structure is just placing things next to eachother, can it be shown that the value of nested circles is just the tensor product of ordinary circles? I don’t see why this should be true.

  6. Well, here I show how to turn one clockwise circle into a clockwise circle and a counter clockwise circle, I think. If this is valid, you can then have the counter clockwise circle absorb the clockwise one in the same manner.

    The reason I am not sure is that I don’t know whether or not to take Ben literally when he says that we are only supposed to do embedded cobordisms, or whether disconnect components of the cobordism are allowed to pass through each other. If the former, then my construction above is buggy. If the latter, than I don’t know why Ben says that we associate A^{\otimes n} to any union of n circles — he should only say that this happens when the circles are disjoint.

  7. Also Noah’s comment is important too. If you want to do this precisely you should specify the braiding. So for example if you have embedded 1-manifold A next to embedded 1-manifold B you need an embedded surface connecting them to “B next to A”. The obvious choice is to just take the identity bordism and twist it clockwise or counter clockwise. This is a choice, and neither gives a symmetric monoidal structure.

    A priori this gives a lot of structure. For example you can take any knot and thicken it up to a small embedded tube. This is a nontrivial morphism in the bordism category.

    However you are mapping into a symmetric monoidal category, and presumably you are only considering strong monoidal functors. This means the braiding in the cobordism is sent exactly to the braiding for vector spaces.

    It also means the endomorphisms corresponding to knots, which are interesting in the bordism category, are sent to rather uninteresting endomorphisms of the underlying vector spaces. It seems like you can’t see the knots.

    It’s like taking a non-commutative algebra and mapping it into a commutative algebra. You’re not really going to be able to detect much of the non-commutivity, especially if you only look at one particular map.

  8. I once asked Aaron Lauda a similar question
    about the chaps version of Ben’s category, where the objects are intervals in the plane and the morphisms are composed from embedded chaps or pants.

    My guess was that this should be the ribbon category with Frobenius object freely generated by a single object.

    Aaron replied that was indeed true and gave the reference

    http://www.worldscinet.com/jktr/05/0503/S0218216596000229.html

    which unfortunately I have never gotten around to reading. It looks interesting though and seems to address many of the things being discussed here.

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