Thread by @getjonwithit

How many holes does a straw have?

Topologically, of course, it has 1: it's homeomorphic to a punctured disk. But intuitively it has 2: one at the top and one at the bottom. And this answer lies at the heart of the most rigorous axiomatization of quantum field theory. (1/20)

In this intuitive picture, the two "holes" of the straw are 1-dimensional circles, and they're connected by a 2-dimensional cylinder (the straw itself). Mathematically, this relationship is called a "cobordism". Two n-dimensional manifolds are "cobordant" if they form... (2/20)

...the boundary of some n+1-dimensional manifold (like the two circles forming the boundary of the cylinder). And cobordisms give one a natural framework for thinking about time evolution in physics. Suppose we have two moments in time: t1 and t2, with t1 < t2. (3/20)

We can think of these moments as being 0-dimensional manifolds, and the interval [t1, t2] as being a 1-dimensional cobordism between them. We can call this collection of moments and intervals a "1-dimensional cobordism category" since it satisfies the axioms of a category. (4/20)

Namely, the composition/union of intervals is associative (since it doesn't matter in what order you take unions) and has an identity operation (the trivial interval [t, t] can act as a "do nothing" operation under composition). And in quantum mechanics, each moment... (5/20)

...of time t1 can be mapped to a vector space V1 (the "space of quantum states" at that time), and each time interval [t1, t2] can be mapped to a unitary operator U: V1 -> V2 (obtained from the Schrödinger equation, by exponentiating the time integral of the Hamiltonian). (6/20)

The union of intervals maps to a composition of unitary operators. The identity interval [t, t] maps to the identity operator. We call this map from the category of moments and intervals to the category of vector spaces and unitary operators a "cobordism functor"... (7/20)

...since it is a structure-preserving map (i.e. a functor) between cobordism categories. So time evolution in QM can be recast as a cobordism functor (there is a lot of other algebraic structure here that I'm neglecting, e.g. tensor products, duals, Hermitian adjoints). (8/20)

But this all seems a bit heavy-handed: why bother with cobordism categories when we're just dealing with 0-dimensional manifolds and 1-dimensional cobordisms? Well, in QM, we don't need to. But what happens when we try to make this description relativistically invariant? (9/20)

In relativity, we can't think in terms of single moments of time: we have to think in terms of *simultaneity surfaces* (i.e. spacelike hypersurfaces). So our cobordism category now consists of 3-dimensional spacelike hypersurfaces, joined by 4-dimensional spacetime... (10/20)

...volumes (in place of our previous 1-dimensional intervals). Our cobordism functor now maps each spacelike hypersurface to a space of quantum states, and each spacetime volume (or "foliation") to a unitary transformation. So this functor describes a covariant version... (11/20)

...of QM, i.e. a quantum field theory. Except... not quite. Because, unlike in the non-covariant case with 0-dimensional moments of time, in the covariant case there isn't just *one* foliation of spacetime connecting two spacelike hypersurfaces (i.e. a preferred frame). (12/20)

There exists an infinite family of spacetime foliations connecting any two spacelike hypersurfaces (all related by smooth gauge transformations), so our cobordism category must somehow contain an infinite family of cobordisms connecting any two 3-dimensional manifolds. (13/20)

So how do we include these gauge transformations? Well, a (linear) gauge transformation relating two 4-dimensional spacetime foliations can *itself* be formalized as a 5-dimensional cobordism between spacetimes. In other words, a cobordism between cobordisms. (14/20)

We've now entered the recursive world of higher category theory: cobordism 2-categories containing both cobordisms between manifolds *and* cobordisms between cobordisms, plus cobordism 2-functors relating those 2-categories together. But it doesn't stop there. (15/20)

We've so far only encoded the *linear* gauge transformations: for full covariance, we have to keep going, adding 6-dimensional cobordisms between the 5-dimensional ones, 7-dimensional ones between the 6-dimensional ones, etc. All the way up to infinity. (16/20)

So we finally reach a cobordism ∞-functor between cobordism ∞-categories: one encoding hypersurfaces and their spacetime transformations, the other encoding spaces of quantum states and their unitary transformations. In QFT, this functor is called the "path integral". (17/20)

This idea that QFT could be formalized in terms of cobordisms and ∞-categories goes back to Segal (for conformal field theory) and Atiyah (for topological field theory), through Baez and Dolan (the cobordism hypothesis), to Lurie, Grady and Pavlov in the present day. (18/20)

It has been the most successful program for formalizing QFT to date, generalizing earlier results such as Wightman's axioms and the Haag-Kastler axioms. But it still works only for very restricted cases: countable degrees of freedom (TQFT) or conformal symmetry (CFT). (19/20)

Could such a formalization work for the "real" QFTs that appear to be of relevance to our actual universe (i.e. Yang-Mills theories)? Nobody knows. But straws, cobordisms, and ∞-categories seem like a promising place to look. (20/20)