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Stream: learning: questions

Topic: string diagrams with rolled up dimensions

Jules Hedges (Jan 04 2021 at 11:12):

One way to think about the free braided monoidal category on a signature is that its morphisms are (isotopy classes of) string diagrams in $\mathbb R^3$. The first dimension is the "composition direction", the second dimension is the "tensor direction" and the third dimension is the "braiding direction" and is only used only to "implement" the braiding, ie. so strings can cross each other without touching. I informally think of the braiding dimension as "small", so the string diagram lives inside $[0,1] \times [0,1] \times (-\varepsilon, \varepsilon)$ (but everything is only defined up to isotopy so I think that's strictly informal). Now, if you roll up the tensor dimension into a cylinder, so now it lives in $[0,1] \times S^1 \times (-\varepsilon, \varepsilon)$, that describes the free braided spherical category (if I remember my Selinger correctly). My question is, what happens if you instead roll up the braiding dimension, so it lives in $[0,1] \times [0,1] \times S^1$? Can the resulting thing be axiomatised a la spherical categories?

Jules Hedges (Jan 04 2021 at 11:14):

(This is originally motivated by the "small compactified dimensions" of string theory, although I very highly doubt there's any connection left to string theory so I'm asking out of pure curiosity instead)

Nathaniel Virgo (Jan 04 2021 at 11:59):

Here's a random attempt, just going off intuition. I don't know if these are all the axioms you need but I think an approach like this would work.

Equip the category with a natural family of isomorphisms, which I'll represent by $\bullet$, with the inverses represented by $\circ$. The interpretation is that we represent $S^1$ by an interval with the two ends stitched together, and $\bullet$ and $\circ$ represent a string passing across the cut, in the "downwards" or "upwards" direction, like this:

image.png

Being an isomorphism means that a string that goes up and then down across the cut can be transformed into one that doesn't cross the cut, and similarly for one that goes down and then up.

image.png

Being natural means that morphisms can be transported across the cut. We also demand

image.png

and $\circ_I = \bullet_I = \mathsf{id}_I$.

Then aside from the axioms of a braided monoidal category I think possibly all you need are these and their mirror images:

image.png

The first diagram is justified by noting that when a wire goes "down" across the cut then it ends up on top, so when you slide it past the braid it changes it from passing underneath to passing over. The second diagram is the same thing but upside-down.

I'm not 100% sure that you don't need any other axioms, but this seems at least roughly right.

Jules Hedges (Jan 04 2021 at 16:43):

Very nice!! Thanks!

Matteo Capucci (he/him) (Jan 04 2021 at 20:20):

Jules Hedges said:

(This is originally motivated by the "small compactified dimensions" of string theory, although I very highly doubt there's any connection left to string theory so I'm asking out of pure curiosity instead)

Well, there's a reason string theory is called like that! The topological observations one makes about strings in string diagrams are the same one can make about 'physical' strings, so I guess the connection is there.
Indeed, this suggests a nice line of research: what happens to string calculi when drawn on arbitrary 2-manifolds? I guess cobordism has something to say.

Matteo Capucci (he/him) (Jan 04 2021 at 20:22):

Also, how does one treat 'arbitrary small dimensions' rigorously?

Simon Burton (Jan 05 2021 at 01:39):

yeah, i think these are braid groups on other manifolds.. it looks like they are called "surface braid groups".

Joe Moeller (Jan 07 2021 at 18:03):

This reminds me, I saw a talk where someone was talking about affinization of monoidal categories, which seemed to me to be where you take the string diagrams and wrap them around a cylinder. They gave some nice description of this, and some examples where it shows up naturally in rep theory.