### An Exercise in Groupoidification: The Path Integral

#### Posted by Urs Schreiber

As we have been reminded of by the last entry a while ago some of us had been very busily thinking here about

What is the quantum path integral

really?

We were trying to understand this by looking at simple finite combinatorial toy models. I can’t tell how far John Baez and Alex Hoffnung have gotten since then, but I know how far I got. Here is where I am coming from:

Extended quantum field theory of $\Sigma$-model type should work like this:

a) the “classical” data is: a target space $X$ together with a nonabelian differential $n$-cocycle $\nabla$ on it, expressed in terms of a parallel transport $n$-functor.

b) the quantization procedure is, roughly: to each piece $\Sigma$ of parameter space assign the result of forming the “space of sections” of the transgression of $\nabla$ to $Maps(\Sigma,X)$.

It’s comparatively clear that and how this works for $dim \sigma \lt n$: transgression of transport is just forming the inner hom and then taking sections.

The more mysterious part is this: with the *really* right way of looking at this, it *should* be true that turning this crank for $dim \Sigma = n$ magically leads to the path integral itself, thus realizing Dan Freed’s old observation that the path integral should be just the top dimension part of a general process which always just transgresses and then takes sections. If this comes out as hoped, one would begin to hope that this provides hints for how we should *really* be thinking of the mystery of the path integral.

Anyway, I had a bunch of ideas about this but didn’t quite get to the point where I was entirely happy. Now here is something which is simple but looks a bit like progress to me. A simple exercise in Groupoidification. I haven’t really had the time to think it true in its entirety. But that’s one reason more for me to share it.

So I want to look at this pathetically simple setup:

**Background/motivation**

target space is a category $P_1(X)$ generated from a *finite graph*.

We fix a finite gauge group $G$ and some representation $\rho : \mathbf{B} G \to \mathrm{Set}$ (where $\mathbf{B}G$ is the one-object groupoid version of $G$). Let’s write $V//G$ for the corresponding action groupoid.

The background field is a functor $\nabla : P_1(X) \to \mathbf{B}G \,.$ A state is a section of this restricted to points, namely a lift of $\nabla|_0 : P_0(X) \to \mathbf{B}G$ through $V \to V//G \to \mathbf{B}G \,.$ So that’s just a choice of element in $V$ over each point.

A bit more interesting, if we transgress to path space by homming the interval category
$(a \to b)$ into everything to get
$tg \nabla : hom((a \to b), P_1(X)) \to hom(a\to b, \mathbf{B}G)
\,.$
Then restricting *that* functor to objects and taking sections in terms of lifts through
$hom(a \to b, V) \to hom(a \to b, V//G) \to hom(a \to b, \mathbf{B}G)$
over objects yields: *over each path a choice of element in $V$ over the endpoints, such that they are related by the parallel transport of $\nabla$ along that path*.

Everybody still following? But to some extent that is just motivation for the following simple situation that I want to look at:

**A span of groupoids**

Let’s build a span of finite groupoids this way:

Let $\Gamma_X := \cup_{x \in P_0(X)} V//G$ be something like the groupoid of the graph of sections over points. Big words – I just mean the disjoint union of one copy of the action groupoid of our rep over each point of target space. To be thought of as a groupoid of sections over target space.

Similarly, let $\Gamma_{P X} := \cup_{\gamma \in hom(a\to b, P_1(x))} \Gamma( tg_\gamma \nabla)$ be the disjoint union over all morphisms in $P_1(X)$ of all sections of $\nabla$ over that path: for each $\gamma$ this groupoid is isomorphic to $V//G$ again, but we think of an object now as a flat section over the path $(x,v_1) \stackrel{(\gamma,g = \nabla(\gamma))}{\to} (y,v_2 = v_1\cdot g) \,.$

Now we build a span from these of the form

$\array{ &&& \Gamma_{P X} \\ & {}^{in}\swarrow &&& \searrow^{out} \\ \Gamma_{X} &&&&& \Gamma_{X} } \,.$

Here the functor $in : \Gamma_{P X} \to \Gamma_X$ simply projects out the left end of a path, and the functor $out : \Gamma_{P X} \to \Gamma_X$ the right end. So

$in : ((x,v_1) \stackrel{(\gamma, g = \nabla(\gamma))}{\to} (y,v_2)) \mapsto (x,v_1)$

and

$out : ((x,v_1) \stackrel{(\gamma, g = \nabla(\gamma))}{\to} (y,v_2)) \mapsto (y,v_2) \,.$

Okay, now lets do groupoidily linear algebra and see how bundles of sets over $\Gamma_X$ pull-push through this span.

Let me pick one single point $x$ in target space and one section over it, $v \in V$. Define

$\array{ \{\bullet\} \\ & \searrow \\ && \Gamma_X }$

to be the functor of groupoids which sends the single object of the terminal groupoid to that object $(x,v)$ in $\Gamma_X$.

The pull-push

$\array{ &&&&& in^* \{\bullet\} \\ &&&&& \downarrow \\ \{\bullet\}&& &&& \Gamma_{P X} &&&&& \int in^*\{\bullet\} \\ & \searrow && {}^{in}\swarrow &&& \searrow^{out} && \swarrow \\ &&\Gamma_{X} &&&&& \Gamma_{X} }$

produces first $in^* \{\bullet\} \to \Gamma_{P_X}$: that has precisely one point sitting over each labeled path which starts at $x$ and is labeled there by $v$.

Then it produces $\int in^* \{\bullet\} \to \Gamma_X$: this has over the point $y$ with label $v'$ one point per path $x \to y$ which is labeled by $v$ over $x$ and by $v'$ over $y$.

To see what this means, fix one point $y$ in $X$. Then we get one point over each label $v'$ for each path from $x$ to $y$ labeled by $v$ on the left and by $v'$ on the right.

But since the labels of the paths are sections of the transgressed transport functor over these paths, which just means that these are flat sections of the original transport over these paths, it means that the $v'$ appearing here are of the form $\nabla(g) v$ for $g$ the parallel transport over the given path.

So the “total” space of $\int in^* \{\bullet\}$ over $y \in X$ is the $V$-colored set $\cup_{\gamma : x \to y} \{\bullet_{\nabla(\gamma) v} \} \,,$

where I write $\bullet_v$ for an element colored by $v$.

If $V$ has an additive structure, for instance if it is a vector space, we have the cardinality operation on $V$-colored sets $|\cdot| : Set_V \to V$ and get that the cardinality of the above is

$\sum_{\gamma : x \to y} {\nabla(\gamma) v} \,,$

But that’s the right path integral kernel for propagation from $x$ to $y$ acting on the state $v$ over $x$.

**So what?**

About all ingredients of the above we have talked before, in one way or another. Lots of ingredients from John’s discussion of groupoidification and John an Jeffrey’s “categorified” quantum mechanics appear. But somehow I feel that I have not before put things together in the picture as above. To me, I had the feeling this clarified some things that had been a bit mysterious to me before:

a) the fact that the path integral should be “taking sections at codimension 0”;

b) the natural connection of a) to groupoidification;

c) the natural and automatic appearance of $V$-colored sets.

But I have to stop here. If Konrad or Hisham read this, or some of the other people waiting for me getting back to them with tasks finished, they’ll be unhappy to see me instead talk about foundational abstract nonsense here. But I needed to relax a bit. :-)

## Re: An Exercise in Groupoidification: The Path Integral

You no doubt saw my comment here and no doubt could guess what I was thinking :)

I still stand by my old comment here, but extend its (admittedly highly speculative) scope to what you are talking about here as well.

I’m slowly catching up, which means you must be getting slow in your old age ;) Just kidding! :) I am finding that I’m understanding more and more about things discussed here lately, which somehow seems miraculous. Thanks!