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Cole Comfort (Mar 03 2025 at 00:10):A 2d-topological quantum field theory (2d TQFT) can be regarded a dagger symmetric monoidal functor . is the category of Hilbert spaces and bounded linear maps, regarded as a dagger symmetric monoidal category with respect to the bilinear tensor product and the Hermitian adjoint. is the dagger symmetric monoidal category whose objects are circles and whose morphisms are cobordisms between circles up to diffeomorphism. The composition is given by glueing; the monoidal structure is given by the product; and the dagger is given by transposing the bordisms.
Since is nothing more than the walking commutative Frobenius algebra, a 2d TQFT is nothing more than the data of a commutative \dag-Frobenius algebra in , ie. the choice of an orthogonal basis for a finite dimensional Hilbert space.
Perhaps more interestingly, following Segal, a 2d conformal field theory (2d CFT) can be regarded as a dagger symmetric monoidal functor from the dagger symmetric monoidal monoidal semicategory of 2 dimensional conformal cobordisms into . One must work with semicategories, because there is no identity conformal cobordism. Intuitively, I understand a conformal cobordism to be a cobordism equipped with a choice of complex structure, regarded as a Kaehler manifold with boundaries. That is to say, one takes your plain old cobordisms, constructs the cotangent bundle and chooses a polarization corresponding to a choice of complex structure. My understanding is that the objects are sent by the CFT to a holomorphic Fock space on a choice of Hilbert space, and the conformal cobordisms are sent to certain "Gaussian contractions" between tensor products of this Fock space.
My question is, are the generators and relations for 2d conformal cobordisms known?
I imagine that conformal cobordisms is the walking semicategory of phased commutative Frobenius algebras. That is to say, if we were to view the connected Frobenius algebras as spiders, here we impose that spiders must be equipped with a positive real parameter, such that when two spiders fuse, the parameters add. This can be made into a category by allowing the phase 0. In other words, I imagine that these parameters would encode a notion of "distance" which is absent in the TQFT picture. I am imagining that the cotangent bundle of the cup cobordism equipped with one choice of complex structure would be the vacuum state; whereas, equipped with the dual complex structure this would be interpreted as the vacuum effect. Is it really this simple, or is my understanding flawed?
John Baez (Mar 03 2025 at 00:46):Various people have worked on this issue, since conformal field theories are extremely popular. See for example the paper by Fiore, and references therein. I'm not sure I'm happy with any existing approach. I'm also not sure I'm unhappy, since I haven't dug into it enough.
John Baez (Mar 03 2025 at 01:06):I have a couple of remarks about what you wrote.
This isn't quite right:
Intuitively, I understand a conformal cobordism to be a cobordism equipped with a choice of complex structure, regarded as a Kaehler manifold with boundaries.
Actually conformal geometry is different from Kaehler geometry.
In conformal field theory we want to think about things like a 2-dimensional smooth manifold equipped with a smoothly varying choice of how to multiply by on each tangent space, obeying an 'integrability condition' that makes the manifold look locally isomorphic to . Such a thing is usually called a 1-dimensional complex manifold (and yes, the dimension just went down by a factor of 2).
A Kaehler manifold is a complex manifold with a complex inner product on each tangent space obeying some niceness conditions: as the Wikipedia article rather nicely explains, this gives the manifold a complex structure, a Riemannian structure (the real part of the complex inner product), and a symplectic structure (the imaginary part of the complex inner product), which all fit together in a compatible way.
In conformal field theory we don't want all this extra structure: we just want the complex structure. In geometrical terms, this means we can measure oriented angles in spacetime, but not distances or areas.
John Baez (Mar 03 2025 at 01:14):My question is, are the generators and relations for 2d conformal cobordisms known?
I've never even seen anyone define the category you're hoping for, which makes me feel there are some essential technical difficulties that keep things from being as simple as you're hoping. The paper by Fiore will immediately thrust you into the realm of things people think about. The section "Motivating example: worldsheets" may be a good place to start.
John Baez (Mar 03 2025 at 01:22):However, people understand the space of isomorphism classes of complex structures on a 2-dimensional surface with boundary reasonably well - well enough, anyway, that you should to learn that stuff rather than trying to reinvent it.
The key buzzword is Teichmueller space, named after the odious Nazi who classified complex structures on a compact oriented surface without boundary modulo homeomorphisms that are connected to the identity homeomorphism. The italicized phrase makes the problem much simpler: if you left it out you'd get the "moduli space of Riemann surface", which is a quotient of Teichmueller space by a discrete group - and this is better thought of not as a space, but as a stack!
Teichmueller space is not so bad, and you don't need to get into stacks for that. But what you may really want is the analogue of Teichmueller space for a compact oriented surface with boundary. This too is understood. The best place to start is the 'pair of pants' - a 2-sphere with 3 open disks removed. There is an eminently manageable space of complex structures on this!
In the topological situation, as I'm sure you know, the pair of pants is the 'multiplication' for a Frobenius object. In conformal field theory this is replaced by a parametrized family of 'multiplications'. I can dig up more information if you want - a lot can be found in old issues of This Week's Finds, written when this was a hot topic.
Nathanael Arkor (Mar 03 2025 at 01:22):A slightly tangential question: is there a reason you say that one must work with paracategories rather than [[semicategories]], if the existence of identities is the only issue?
Cole Comfort (Mar 03 2025 at 10:24):Nathanael Arkor said:
A slightly tangential question: is there a reason you say that one must work with paracategories rather than [[semicategories]], if the existence of identities is the only issue?
I just mixed up the terms. Will edit the question.
Cole Comfort (Mar 03 2025 at 10:27):John Baez said:
In conformal field theory this is replaced by a parametrized family of 'multiplications'. I can dig up more information if you want - a lot can be found in old issues of This Week's Finds, written when this was a hot topic.
Thank you for the references and the thrilling historical context. This view of conformal field theory in terms of parametrized families of multiplications is exactly the kind of thing I would really enjoy reading about.
Is this not still a hot topic? Has the buzz died down?
John Baez (Mar 03 2025 at 15:55):It was extremely hot in the early 1990s; there was nowhere to go but down. I haven't heard people talk about it lately. Here's something good to read: week28.
John Baez (Mar 03 2025 at 16:05):This article sketches why it takes 3 numbers to describe a complex structure on the "pair of pants" cobordism (which acts kind of like the multiplication in this funny Frobenius-monoid-like thing).
I forgot to mention something simpler: it takes 1 number to describe a complex structure on the annulus (which acts kind of like the identity). For example, let be the annulus in the complex plane bounded by concentric circles of radius and . This seemingly takes two numbers to describe, but is conformally equivalent to for any , since rescaling doesn't change angles. So if we're doing conformal geometry we only get a 1-parameter family of annuli of this sort.
It's much harder to prove that given any smoothly embedded circle in the plane, and any smaller smoothly embedded circle inside it, the region between those two is conformally equivalent to some . But it's true.
Cole Comfort (Mar 03 2025 at 17:45):@John Baez Very cool! This is lots of food for thought.
It seems reasonable, as a baby example, to take the codomain of the CFT to be the (semi) prop generated by conformal, pants and copants, cups and caps, glued together without allowing for twisting, suspended in a higher dimension to avoid things tangling. Because the glueing just adds more (co)holes, the morphisms then can naturally be represented by graphs whose edges are labelled by positive real numbers with distinguished, disjoint inputs and outputs. To me, this feels reminiscent of circuits of linear resistors: where there is also no identity, because the parameters on the nodes also have to be strictly positive.
I need to try to calculate if these two (semi)props are isomorphic.
John Baez (Mar 03 2025 at 17:51):I believe people know how the associativity law works in this setting. Each pair of pants takes 3 parameters to describe; to glue two together one parameter must match and you need one twisting parameter, but I believe the resulting conformal surface requires fewer than 6 parameters to describe. I.e., we get relations in our presentation. I forget the details. But they're known.