January 21, 2009

New beginnings

Today was my first lecture of the new semester, there was some sort of hubbub on The Mall in Washington, and Jacob Lurie gave the first of a series of lectures on “Extended” Topological Quantum Field Theory and a proof of (some might say a precise statement of) the Baez-Dolan Cobordism Hypothesis.

According to Atiyah, a $d$-dimensional TQFT is a tensor functor, $Z$, from the category, $Cob(d)$, whose objects are closed $(d-1)$-manifolds1, and whose morphisms are bordisms $Hom (M,N) = \{B: \partial B \simeq M\amalg \overline{N}\}/\text{diffeomorphisms}$ to $\Vect$, the category of complex vector spaces. $Cob(d)$ is a symmetric monoidal category, with $\otimes$ given by disjoint union of manifolds, and $Z$ preserves tensor products $\begin{gathered} Z(M\amalg N) \simeq Z(M)\otimes Z(N)\\ Z(\emptyset) \simeq \mathbb{C} \end{gathered}$

The vague idea of extended TQFT is to replace $Cob(d)$ with some sort of $n$-category (where $n=d$), consisting of manifolds of all dimension $m\leq d$, and replace $Vect$ with a similarly fancied-up $n$-category, $\mathcal{C}$.

Over the course of several lectures, Jacob proposes to tell us exactly what these all are, but the vague version is as follows:

For $m\leq d$, a $d$-framing of a manifold, $M$, of dimension $m$, is a trivialization of $T_M\oplus \mathbb{R}^{d-m}$.

$Cob(d)^{\text{framed}}_{\text{ext}}$ is a symmetric monoidal $d$-category (with tensor product given by disjoint union of manifolds).

• Objects are $0$-manifolds with a $d$-framing.
• Morphisms are $d$-framed bordisms between $d$-framed $0$-manifolds.
• 2-Morphisms are $d$-framed bordisms between $d$-framed $1$-manifolds.
• $d$-morphisms are $d$-framed $d$-manifolds (with corners) $/\text{diff}$

The statements which he proposes to prove are that

Given a $d$-category, $\mathcal{C}$, with tensor product, an ETQFT “with values in $\mathcal{C}$” is a tensor functor

1. $Z:\, Cob{d}^{\text{framed}}_{\text{ext}}(d) \to \mathcal{C}$
2. The “fully dualizable” objects, $X\in \mathcal{C}$ are given by $X = Z(\bullet)$.

In some sense, the whole ETQFT is determined by knowing what $Z$ of a point is. Here, “fully dualizable objects” is some condition analogous to demanding that the vectors spaces in an ordinary TQFT, associated to a $(d-1)$ manifold, are finite-dimensional.

There is, of course, a 110 page paper providing “an outline” of the idea. Probably, it will be more intelligible than my summary.

1 Here, and below, a “manifold” is smooth, compact and oriented. Usually, it will be a manifold with boundary. When we want to denote a manifold without boundary, we’ll call it “closed.”

Posted by distler at January 21, 2009 2:07 AM

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formalizing it

(some might say a precise statement of) the Baez-Dolan Cobordism Hypothesis

My impression is that nobody would claim that the original statement was already technically precise, nor that it was suggested to be. Part of the aspect of proving it is to find the natural formalization that makes it naturally true. In this respect the tangle hypothesis is not unlike the homotopy hypothesis, I’d think.

Posted by: Urs Schreiber on January 21, 2009 7:47 AM | Permalink | Reply to this

Re: formalizing it

In our paper on this subject, James Dolan and I explained that we used the term ‘hypothesis’ instead of ‘conjecture’ because the theory of $n$-categories wasn’t sufficiently developed to state a precise conjecture. The goal was to get people to develop enough $n$-category theory to turn these hypotheses into conjectures and then into theorems.

It seems to be working. But unfortunately, a Google search under “Baez Dolan” still corrects this string to “Baez Dylan”.

Posted by: John Baez on January 23, 2009 1:02 PM | Permalink | Reply to this

Re: New beginnings

I’d like to understand the import of the framing in this story. Most manifolds aren’t framable after all (referring to the top dimension where we don’t get any extra stuff to play around with). Later on in the notes, Lurie gives a version where the framing is removed and reasonably arbitrary other structures are substituted. So, I guess I’m wondering if the framing is just a math trick or is something deeper. Of course, I’m only on pg 50 of the notes, so maybe this is addressed later.

Posted by: Aaron Bergman on January 21, 2009 12:17 PM | Permalink | Reply to this

Re: New beginnings

There are many flavors of cobordism theory, and framed cobordism theory is the ‘simplest’ in a certain conceptual sense, though one of the most complicated to compute. So, the $n$-categorical description of cobordisms is simplest for framed cobordisms. So we should study this one first, and then others.

A bit more precisely, the spectrum for framed cobordism theory is the ‘sphere spectrum’. This is a cleverly defined limit of the $n$-fold loop space of the $n$-sphere as $n \to \infty$. The simple nature of the $n$-sphere gives rise to the simplicity of the sphere spectrum.

Again, this simplicity is conceptual rather than calculational: the homotopy groups of the sphere spectrum are famously tough to compute! But here’s the reason for this conceptual simplicity:

The fundamental $d$-groupoid of the $n$-sphere should have a nice universal property: it should be the free $d$-groupoid on an $n$-loop. Here an $n$-morphism is called an ‘$n$-loop’ if it’s an automorphism of the identity of the identity of the identity… of the identity morphism of some object.

Why should the $n$-sphere have this property? It boils down to this: if you draw a diagram of an $n$-loop, it looks just like an $n$-sphere.

Calculations show this idea is not so goofy as it may sound.

Starting from this hypothesized universal property of the fundamental groupoid of the $n$-sphere, one can heuristically derive a universal property of the sphere spectrum, and also of the framed cobordism $n$-categories. That’s what Jim and I did on page 28-29 here.

Here’s another way to say what’s so great about the sphere spectrum: it’s the initial object in the category of ring spectra, just as the integers is the initial object in the category of rings.

Posted by: John Baez on January 23, 2009 2:31 PM | Permalink | Reply to this

Re: New beginnings

hi,
i got on a search quest today and found your site.
I suppose you heard of the “1982 aspect experiment” ?
Can you tell me what it actually was… the experiment itself?

more specifically, what does math and the experiment have to do with each other?

People say its proves this and that…
other say no.
or it brings up more questions than it answers.

anyway, thanks for a reply :)

Posted by: Timeshares on January 30, 2009 4:22 AM | Permalink | Reply to this

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