The n-Category Café

I think there are really two issues here:

1) can we handle “nongeometric” spacetimes (i.e. spacetimes which are not manifolds, but something more general, like a noncommutative space)?

2) can we handle the quantum theory of all these nonperturbatively, i.e. without trying to expand the theory around any given point in the space of all these spacetimes?

It seems that when Connes and Marcolli write

[…] there was no geometry at all, and that only after the phase transition was there a spontaneous symmetry breaking […]

they are referring to point 1), while when they say

[…] the people who are trying to develop quantum gravity in a fixed space are on the wrong track.

they are referring to point 2).

There is a proposal in which for every 2-dimensional SCFT of central charge 15 we get a point in a space of possibly noncommutative/generalized geometries, and a prescription for how to make an expansion from there into the rest of the space of geometries.

But it is true that this proposal has been mainly studied at points where the given geometry is commutative.

The Planck temperature, from the eponymous Max Planck, is the unit of temperature, denoted by T_P, in the system of natural units known as Planck units, which any intelligent beings would use, free of arbitrary units (i.e. the kilogram defined by a metal cylinder in Paris).

It is one of the Planck units that represent a fundamental limit of quantum mechanics. The Planck temperature is the fundamental upper limit of temperature; modern science considers it nonsensical to conjecture about anything hotter. It is also, almost inevitably in most theories, the temperature of the Universe during the first unit of Planck time of the Big Bang according to current cosmology.

T_P = {m_P c^2}/k =
sqrt{hbar c^5}/{G k^2}}
= 1.41679(+-.0000011) x 10^32 K

where:

m_P is the Planck mass

c is the speed of light in a vacuum

hbar is the reduced Planck constant (or Dirac’s constant)

k is the Boltzmann constant

G is the gravitational constant

Now, even fairly conservative scientists suggest that the geometry of space-time could have been quite different at this first clock-tick after Big Bang.

LISA will search for the energy released when, hypothetically, additional dimensions curled up into neat Calabi-Yau spaces, breaking whatever symmetry was present beforehand.

Planck temperature is also a kind of threshold regarding quantum foam, hot enough for wormholes to emerge everywhere, and ruining the pseudoreimannian manifold, i.e. by topologically eliminating a well-behaved sense of neighborhood of a point, or even a proper metric for distance at small (plank-length) scales.

The next step, as urs points out, is to generalize to some noncommutative space. I’m also puzzled that he seems (previous thread) to suspect that the Standard Model plus plausible neutrino generations and mass (i.e. oscillation) pops out when one adds a pinch of quaternions and a dash of 3x3 matrices to the good old complex numbers.

First, not having read the 300 pages of cosmology and 300 pages of pure math, I have no idea how this goes beyond well-known use of quaternions to get rotaions of spheres in 3-space, and Pauli matreices for spin in QM.

If they’d needed to go to, say, 5x5 or 11x11 matrices I’d be more surprised.

No question about the leadership of Alain Conne’s noncommutative geometry program. Simply that nothing has trickled down to folks like me, who’ve published on Mathematical Physics in minor venues, and attended the occasional lecture on noncommutative geometry (in my case, at Caltech seminars by visitors in that area).

I’d be interested in Conne’s said that the Baez-Egan work is on the right track.

Congratulations on 300. That means that these threads can now fight off an army of a million Persian theorists, right?

And Happy Shakespeare’s Birthday. And death day. As he put it, in Lear, about the Big Bang: “nothing shall come from nothing.”

Before trying to combine GR and QM, one should first understand the concepts of time in both theories:

1. In GR, time is what clocks show.

2. In QM, time is a parameter rather than an observable. It was apparently shown by Pauli in the 1920s that a self-adjoint time operator leads to an unbounded Hamiltonian. (Thanks to Wolfgang for pointing this out.)

There is obviously a conflict here, since what is observed in a physical clock-reading experiment must be an observable. It seems to me that any approach to QG which does not explain this apparent paradox is still-born.

I wrote:

Maybe last weekend I had an idea for how to proceed here, maybe even a good idea. I am waiting for that idea to crystallize a little more,

Oops, looks like I am disregarding my own rule of conduct here. After all, the best way to help an idea crystallize is to make a fool of yourself and start talking about it to others. ;-)

So here is the question: “What is a Dirac operator, really? What is supergeometry, really?”

(Here “really” is defined like this: you know you really know what it is if and only if it can be blindly categorified.)

I had vaguely thought about this question every now and then in the past. But recently I was alerted again to the urgency of this question by the following observation:

The superpoint is essentially nothing but the infinitesimal interval $T$ of synthetic differential geometry.

This is in the sense that, if $\nearrow$ denotes either the superpoint or the infinitesimal interval, then, for any space $X$, the morphism space $$T X := \mathrm{Hom}(\nearrow,X)$$ behaves like the tangent bundle to $X$.

For $\nearrow = T$ the synthetic infinitesimal interval, this is true essentially by construction. For $\nearrow = \mathbb{R}^{0|1}$ the $N=1$ superpoint, this is a well known fact, appearing for instance as Lemma 3.2 on p. 29 of the thesis

Florin Dumitrescu, Superconnections and parallel transport.

The only slight lie about this statement is that $\mathrm{Hom}(\mathbb{R}^{0|1},X)$ is really the odd tangent bundle. That is an important issue, which however I will no further get into here.

What I find exciting about this is that it suggests a way to naturally approach supergeometry in categorical terms. That’s because I think I know a nice way to approach synthetic differential geometry in categorical terms.

Here is how I conceive the “infinitesimal interval” this way: I think of it as the terminal 1-category, that with a single morphism. Accordingly, I shall argue that this is the $N=1$-superpoint for me.

More generally, I would then regard the $(N=n)$-superpoint as the terminal $n$-category: that with a single $n$-morphism.

I’ll try to convince you in a moment that this actually might make good sense. But before continuing, just notice that for $(N=0)$ this just says that the non-super point is just the 1-element set.

Which it is.

(If Toby is reading this, we could indulge in using this to favour the world with a discussion of $(N=-1)$ and $(N=-2)$ supersymmetry .)

The idea was this:

think of any space $X$ in terms of its strict $\infty$-groupoid of paths $$P(X) \,.$$ $n$-morphism are thin-homotopy classes of maps $[0,1]^n \to X$ cobounding two $(n-1)$-morphisms.

Then $$P_n(X) := \mathrm{Hom}_{n\mathrm{Cat}} ( \nearrow_n , P(X) ) \,,$$ with slight abuse of notation, where I now write $\nearrow_n$ for my $(N=n)$-superpoint, as before.

What looks like notational overkill here is turned into a cool fact by using this cool fact: strict smooth $p$-functors $$f : P_p(X) \to \Sigma^p(U(1))$$ are in bijection with differential $p$-forms.

But this tells us that in the world of categories, functors, transformations, modifications, etc (i.e. in the best of all worlds ;-) we are entitled to identitfy $$T^{\vee n} X := P_n(X)$$ the $n$th exterior power of the tangent bundle with the $n$-path $n$-groupoid of $X$.

In fact, we can just as well use the $n$-groupoid $(U(1))^n$ coming from the $n$-crossed module $U(1) \to U(1) \to \cdots \to U(1)$ and find the entire exterior bundle up to degree $n$ $$\oplus_{i=0}^n \Omega^n(X) \simeq \mathrm{Hom}_{n\mathrm{Cat}}( P_n(X),(U(1))^n) \,.$$

(Only thing I am still puzzled about is how to naturally see the wedge product operation on these functors. Must be something obvious, but I don’t see it yet).

Time is running out. I need to be more brief.

So what’s the superline then? Do functors on superlines reproduce superconnections? In particular, do we get supersymmetric quantum mechanics from functors on super-1-cobordisms?

Here is a brief sketch:

Like our superpoint is really a point with lots of paths emanating from it, with the magic of smooth functors on these paths ensuring that only the tangent vector of these paths at that point matters, so a supercurve is a superpoint in curve space, i.e. a supercurve with a tangent vector at every point.

As you have noticed, superification here is actually categorification. The super 0-point is actually a 1-category.

Same for the supercurve now.

It’s a curve as usual $$\array{ \downarrow \\ \downarrow }$$ but now we remember tangency classes of surfaces emanating from that (or rather: this is what our smooth 2-functors will see).

So, schematically, our supercuve looks like an infinitesimally thin strip of surface $$\array{ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow \\ &\to& \\ \downarrow &\Downarrow& \downarrow }$$ (rather, like a finite such surface, but our smooth 2-functors will only care about the tangents).

Suppose we think of this as the worldline of a superparticle. If it were an ordinary particle we’d want to look at smooth functors from the curve to $U(H)$, for $H$ some Hilbert space of states.

Now we need a 2-functor mimicking that. The obvious choice is to use instead the 2-group coming from the crossed module $$\mathrm{id} : U(H) \to U(H) \,,$$ i.e. the 2-group $\mathrm{INN}(U(H))$. Let’s assume that’s right.

Then what’s a smooth 2-functor applied to the above supercuve?

Now some magic happens, which is making me think that this is on the right track.

We know that such a smooth 2-transport comes from

- a 1-form $\tilde D$ on the strip (which represents our supercurve) with values in $u(H) = \mathrm{Lie}U(H)$

- a 2-form $\tilde H$ on the strip, with values in $u(H)$.

First thing to notice: it looks like these two differential forms take values in the same Lie algebra. But if we are sufficiently pedantic, then in fact they don’t! Rather, $\tilde D$ takes value in $u(H)$ regarded as being odd graded, while $\tilde H$ takes values in $u(H)$ regarded as being even graded.

The categorification of Lie algebras, and its equivalence to $L_\infty$-algebras naturally brings a grading into the game here. Without us having put this in by hand, the formalism knows that

$\tilde D$ is odd.

$\tilde H$ is even.

It’s god-given.

Second thing to notice:

the formalism tells us that these two differential forms are not independent.

Since we are thinking quantum mechanics here, and time-independent quantum mechanics for simplity, we assume this 1-form $\tilde D$ and 2-form $\tilde H$ are in fact constant along our curve. Then the compatibility condition (known sometimes as “fake flatness”) demands that $$\tilde H = \tilde D^2 \,.$$

And that’s god-given. It’s not put in by hand.

Third thing to notice is this:

our 2-functor which we apply to the strip can be regarded as follows: it assigns a 1-form to each curve on the strip, and then assigns to the entire strip the path-ordered (in curve space!) exponential of that 1-form over the strip.

That 1-form turns out to be, in turn, the path-ordered exponential, along the curve, of $$H := \tilde H(t,\cdot) \,,$$ where $t$ is the horizontal tangent on our strip.

But since we are just looking at that one curve, and care about the strip emanating from it only in as far as this tangent vector is concerned, we finally find that our 2-functor with values in $\mathrm{INN}(U(H))$ (here $H$ denoted a Hilbert space, recall) sends the supercurve $\gamma$ to $$e^{ |\gamma| H}$$ where the Hamiltonian $H$ is an even operator that is required (that’s precisely what distinguishes our 2-functor on the strip with values in $\mathrm{INN}(U(H))$ from a mere 1-functor with values in $U(H)$!) to be that square of the odd operator $D$ $$H = D^2 \,.$$

And there we go: $U(H)$-transport along supercurves is in bijection with supersymmetric quantum mechanics.

Okay, so much for now.

On the occasion of his 150th birthday, here Max Planck on film.