The n-Category Café

His latest, with Dana Fine, gives a rigorous version of the path integral for supersymmetric quantum mechanics.

As they explain in the introduction, this is an extension of a similar construction by Bär and Pfäffle.

Christian Bär gave a talk on their approach a while ago here in Hamburg. One of the key ingredients is an innocent-looking and very obvious construction: they look at piecewise geodesic paths.

I was thinking about such paths a bit recently, while trying how to prove the statement, which seems to be true, that smooth pseudofunctors from the fundamental groupoid of $X$ to some $n$-group $G_{(n)}$ $$\Pi_1(X) \to \mathbf{B}G_{(n)}$$ are equivalent to smooth $n$-functors from $n$-paths in $X$ $$P_n(X) \to \mathbf{B} G_{(n)} \,.$$

One direction is easy: you differentiate the pseudofunctor and obtain the data of a Lie $n$-algebroid morphism $$T X \to g_{(n)}$$ from the tangent algebroid of $X$ to the Lie $n$-algebra underlying $G_{(n)}$.

But integrating that up a again to a pseudofunctor on $\Pi_1(X)$ has global subtleties: for any homotopy class of paths, you’d want to pick one representative and then integrate your form data over it. But the result will depend on which representative you choose.

So apparently one has to choose singled out representatives.

Probably choosing a metric and then looking at geodesics might be good way to do this.

But of course that requires a bit care. The best one can hope for is is to deal with piecewise geoedesic paths.

I was wondering if the apparent need to introduce a metric here is a mere nuisance or maybe actually an inidcation that something interesting is going on.

Because the situations where we can make sense of the QM path integral are precisely the situations where we have a particle which is subject to precisely two effects:

a) a gauge background field, b) background gravity.

The first is encoded in a pseudofunctor. The precise abstract right way to encode the latter is less clear, generally.

And it would be nice if both parts could actually be seen to merge nicely. The point is that the path integral is

$$\int_{P X} \mathrm{hol}(\gamma) d\mu(\gamma)$$

with the hol-part giving the background gauge field and the measure $d\mu$ being the kinetic part, depending on the metric, and coming from a metric. People expect that we should think of this as one measure $$\int_{P X} d\tilde \mu(\gamma) \,.$$

And when I think of the $\mathrm{hol}$ as encoded in a pseudofunctor (pseudo-anafunctor really, but let’s not worry about that right now) it seems I do indeed need think of it always in conjunction with a metric.

Well, this is how far I got with thinking about this right now.

But when I read Bär-Pfäffle and Fine-Sawin, I am feeling there might be something to this.

Cheese whizz, and this link from the guy who would not include my “Quantum Gravity Topological Quantum Gravity Blues,” in his Oxford University Press book “Knots and Quantum Qravity.”

Lyrics below:

I’ve been calculating
I said I’ve been calculating
calculating all night long
Got a quasi-triangular Hopf algebra
and I wrote down the coproduct wrong.

I’ve been integrating
integrating the whole day through
I said I’ve been integrating
integrating the whole day through
Got a Chern-Simons functional integral
and its convergent, too.

I’ve been writing down knot diagrams
converting them to braids
Using the Alexander isotopy
you know I’m not afraid
I’ve been
assigning modules
to each of these six strings
been doin’ it for weeks now
and I still don’t understand a thing.

I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
And without NSF funding I think that you would, too.

********
Now I think I have to vid the calculus rap:
********

I’m a going to tell you all you need to know
To be the master rapper of the calculus show:
Now the first thing you do is to take the limit;
Yes that’s right, that’s how to begin it.
You don’t need to use epsilon delta stuff,
Four simple rules are going to be enough:

The limit of a constant is still a constant
And the limit of a sum is the sum of the limits
The limit of a product is the product of the limits
And the limit of the quotient is the quotient of the limits
provided that the limit of the denominator is not zero.
This last condition is certainly necessary
for in this case you must be clever!

Now the thing upon which we will take the limit
is an object known as the Newton quotient
It’s a simple object so will you repeat it:
Say ef of ex plus delta ex minus ef of ex over delta ex
Say ef of ex plus delta ex minus ef of ex over delta ex
Now this here limit is evavaluated as delta ex approaches zero
Given a function when you take this limit
The resulting quantity is called the derivative
But in general it is quite inefficient
To take the derivative by using the limit
SO a few simple rules will help you in fact
Master the world of the calculus rap.

The power rule is very simple
using it will cure your pimples
See the derivative of ex to the en
is en times ex to the en minus one
en times ex to the en minus one
don’t be frightened don’t be scared
differentiate the function ex squared
Yes the derivative of ex to the two
is two times ex

And any old fat headed thick skulled rube
Can differentiate the function ex cubed
Yes the derivative of ex to the three
is three times ex to the two

The next of the rules that we will employ
Is the product rule so stop calling me “Boy!”
The derivative of how times doo
is do dee how plus how dee doo

Where the conotation of the notation dee
Is to differentiate, naturally.

Now the next of the rules ain’t a rule for fools
this here rule is the quotient rule
Yes the derivative of hi over ho
is ho dee hi minus hi dee ho
over ho ho

And the next few functions will not make you go mental
these are the functions that are called transcendental
Say derivative of the sine of ex is
the cosine of ex.
And the derivative of the cosine of ex is
minus the sine of ex.
And the derivative of ee to the ex is
ee to the ex.
And the derivative of the natural log of ex is
one over ex.

The next of the rules is the golden rule
This here rule is called the chain rule
We apply it to the composite of functions
it is very much like peeling an onion
layer by layer oh yes it’s a fun one:
Now suppose that a function ach of ex
can be written in the form ef of gee of ex
Then the derivative of ach of ex
is the derivative of ef at gee of ex
times the derivative of gee at ex
We often say dee why dee ex
is dee why dee you times dee you dee ex
where why is ef and you is gee
why is ef and you is gee
but it’s alphabet soup if you want to ask me

Now put on your coat and put on your hat
Since you have completed the calculus rap
Yes put on your shoes and put on your pants
Because in the next song we do the calculus dance!

Copywrite 1988 By J. Scott Carter (P. E. Zap)
Published Lobe Current Music and BMI.
All right reserved.

I think I’ll stick with Maxwell:

My soul’s an amphicheiral knot
Upon a liquid vortex wrought
By Intellect in the Unseen residing,
While thou dost like a convict sit
With marlinspike untwisting it
Only to find my knottiness abiding,
Since all the tools for my untying
In four-dimensioned space are lying,
Where playful fancy intersperses
Whole avenues of universes,
Where Klein and Clifford fill the void
With one unbounded, finite homoloid,
Whereby the infinite is hopelessly destroyed.