The n-Category Café

Almost everything I mention here is also described in The Globular Extended QFT of the String propagating on the Classifying Space of a strict 2-Group.

Before talking about the string, which is the first interesting simple example of a quantum charged $n$-particle (at least when we restrict attention to kinematics, as I shall do), we should first get a feeling for what an ordinary particle looks like, when regarded as a special case of a charged $n$-particle.

The charged (1-)particle.

On a (“target”) space $$P_1(X)$$ we have a vector bundle with connection, $(E,\nabla)$, given by its parallel transport(-ana-)functor: $$\mathrm{tra} : P_1(X) \to \mathrm{Vect} \,.$$

Coupling a particle $\{\bullet\}$ to this bundle amounts to specifying the configurations of the particle in (target) space, i.e. to choosing a sub-category $$\mathrm{conf} \subset [\{\bullet\},P_1(X)] \simeq P_1(X) \,.$$

The importance of this choice of sub-category concerns the morphisms. Every isomorphism in $P_1(X)$ that we retain in $\mathrm{conf}$ will make the source and target configuration isomorphic.

So we only want to retain those morphisms in $\mathrm{conf}$ that connect “gauge equivalent” configurations.

For instance, if $P_1(X)$ were the groupoid of paths in an orbifold, regarded itself as a groupoid, we would want to retain all those morphisms in $\mathrm{conf}$ that relate points which are identitfied under the orbifold action.

So in particular, if $P_1(X)$ is just the ordinary path groupoid on an ordinary space, we keep only the identity morphisms in $\mathrm{conf}$, since all points in $X$ we then want to regard as different configurations of our particle $\{\bullet\}$.

This is why we would set configuration space equal to the discrete category over $X$: $$\mathrm{conf} = \mathrm{Disc}(X) \,.$$

Now, the general procedure for finding the quantum kinematics of the particle

$$\{\bullet\}$$

on target space

$$P_1(X)$$

with configurations

$$\mathrm{Disc}(X)$$

and coupled to the background field

$$\mathrm{tra} : P_1(X) \to \mathrm{Vect}$$

is to pull-push $\mathrm{tra}$ through the correspondence

$$\array{ & & \mathrm{Disc}(X)\times \{\bullet\} \\ & {}^{\mathrm{ev}}\swarrow\;\; && \searrow \\ P_1(X) &&&& \{\bullet\} } \,.$$

In our case here this just means that we

- first restrict $\mathrm{tra}$ to the constant paths

- and then push it forward to a point.

The result of this is the functor $$q(\mathrm{tra}) : \bullet \mapsto \Gamma(E) \,,$$ which sends the point particle to the vector space of sections of the original vector bundle $E$. This push-forward is described in The First Part of the Story of Quantizing by Pushing to a Point….

More details on the above treatment of the charged 1-particle, including a discussion of how this looks like when we locally trivialize and make everything smooth, is given in section 1.2 of Quantum 2-States: Sections of 2-vector bundles.

Interlude: how to compute the $n$-space of sections.

The main point in determining the quantum theory of a charged $n$-particle is to figure out the result of pull-pushing an $n$-functor on target space $\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}$ through a correspondence $$\array{ & & \mathrm{conf}\times \mathrm{par} \\ & {}^{\mathrm{ev}}\swarrow\;\; && \searrow \\ \mathrm{tar} &&&& \mathrm{par} } \,.$$

For computing this quantization, it is convenient to proceed as follows.

We take the product $\mathrm{conf} \times \mathrm{par}$ to be adjoint to the internal hom of our notion of $n\mathrm{Cat}$ #. Then $$[\mathrm{conf} \times \mathrm{par},\mathrm{phas}] \simeq [\mathrm{conf},[\mathrm{par},\mathrm{phas}]] \,.$$ The image of $\mathrm{tra} : \mathrm{tar} \to \mathrm{phas}$ under $$\array{ [\mathrm{tar},\mathrm{phas}] & \stackrel{\mathrm{ev}^*}{\to} & [\mathrm{conf}\times \mathrm{par},\mathrm{phas}] & \stackrel{\sim}{\to} & [\mathrm{conf},[\mathrm{par},\mathrm{phas}]] \\ \mathrm{tar} && \mapsto && \mathrm{tar}_* }$$ is simply postcomposition with $\mathrm{tra}$: $$\mathrm{tra}_* \;\; : \;\; ( \mathrm{par} \stackrel{\gamma}{\to} \mathrm{tar} ) \mapsto \;\; ( \mathrm{par} \stackrel{\gamma}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} ) \,.$$ The push-forward $$[\mathrm{conf}\times \mathrm{par},\mathrm{phas}] \stackrel{\bar{p_1^*}}{\to} [\mathrm{par},\mathrm{phas}]$$ then corresponds to the push-forward $$\array{ \mathrm{conf} & \stackrel{p}{\to} & \{\bullet\} \\ [\mathrm{conf},[\mathrm{par},\mathrm{phas}]] & \stackrel{\bar{p^*}}{\to} & [\{\bullet\},[\mathrm{par},\mathrm{phas}]] & \stackrel{\sim}{\to} & [\mathrm{par},\mathrm{phas}] \\ \mathrm{tar}_* && \mapsto && q(\mathrm{tra}) } \,.$$ This $q(\mathrm{tra})$ is then defined by $$\mathrm{Hom}(p^* f_\bullet, \mathrm{tra}_*) \simeq \mathrm{Hom}(f_\bullet, q(\mathrm{tra})) \,,$$ for any $f_\bullet : {\bullet} \to [\mathrm{par},\mathrm{phas}]$.

In practice, it is often sufficient to consider this equivalence for $f_\bullet = 1_\bullet$, the tensor unit in $[\{\bullet\},[\mathrm{par},\mathrm{phas}]]$.

This is what I shall do in the following example.

The string on $B G_2$.

I now define a “charged $n$-particle” $$\left( \mathrm{par} \stackrel{\gamma \in \mathrm{conf}}{\to} \mathrm{tar} \stackrel{\mathrm{tra}}{\to} \mathrm{phas} \right)$$ as follows:

I model the string by the parameter space $$\mathrm{par} = \Sigma(\mathbb{Z}) \,,$$ which is the category with a single object, $\bullet$, one nontrivial morphism $$\bullet \to \bullet$$ and all its composites (and all inverses of that, but this will be irrelevant in the present context).

For the “target space” that I want this string to propagate on I choose any strict 2-group $G_2$ and set $$\mathrm{tar} = \Sigma(G_2) \,,$$ which is nothing but that 2-group, now regarded as a 2-groupoid with a single object.

The reason why I keep calling this setup a “string on the classifying space of $G_2$” is that by sending any configuration $$\gamma : \Sigma(\mathbb{Z}) \to \Sigma(G_2)$$ to the world of topological spaces, by applying the nerve functor and geometrically realizing the resulting simplicial map $$|\gamma| : |\Sigma(\mathbb{Z})| \to |\Sigma(G_2)|$$ it becomes a map from the circle (with basepoint) to $B G_2$: $$|\gamma| : S^1_\bullet \to B G_2 \,.$$ But beware that this geometrically realized description is somewhat misleading. As we shall see, the categorical formalism (“globular extended QFT”) remembers the difference between a circular open string and a closed string.

All that remains to be specified, now, is the 2-bundle with connection on target space that we wish to couple our 2-particle to. For the moment, we I be content with understanding the simple case where this 2-bundle is trivial, of rank one and has trivial connection.

This means that I take $$\mathrm{tra} : \Sigma(G_2) \to 2\mathrm{Vect} = {}_{\mathrm{Vect}}\mathrm{Mod}$$ to take everything to the identity on $\mathrm{Vect}$ itself $$\mathrm{tra}(\cdots) = \mathrm{Id}_\mathrm{Vect} \,.$$

As always, I consider this as taking values not in all possible $\mathrm{Vect}$-linear categories, but only in those well-behaved ones that live inside the image of the canonical inclusion $$\mathrm{Bim} \hookrightarrow {}_{\mathrm{Vect}}\mathrm{Mod} \,.$$ In this picture, a 2-vector space $V$, i.e. an object in ${}_{\mathrm{Vect}}\mathrm{Mod}$ is represented by an ordinary algebra $A$, namely as the category $\mathrm{Mod}_A$ of modules of that algebra.

In other words, I take the space of phases to be $$\mathrm{phas} = \mathrm{Bim} \,.$$

But notice that the results mentioned at the very beginning of this entry are formulated by implciitly applying the embedding $\mathrm{Bim} \hookrightarrow {}_{\mathrm{Vect}}\mathrm{Mod}$.

Finally, in order to completely specify our charged $n$-particle, we need to choose a subcategory of configurations inside the category of all maps of the string into target space $$\mathrm{conf} \subset [\Sigma(\mathbb{Z}),\Sigma(G_2)] \,.$$ Remember, from the example of the charged particle above, that the point of this choice of subcategory is mainly to to identify those configurations, that we want to regard as “gauge equivalent”.

But here I want something very easy to handle, so I simply take $$\mathrm{conf} = [\Sigma(\mathbb{Z}),\Sigma(G_2)] \,.$$

Okay, that completes the definition of that charged 2-particle to which I gave the name “string on $B G_2$”. In summary, it looks like this: $$\left( \Sigma(\mathbb{Z}) \stackrel{\gamma \in [\Sigma(\mathbb{Z}),\Sigma(G_2)]}{\to} \Sigma(G_2) \stackrel{1}{\to} \mathrm{Bim} \right)$$

Clearly, there are plenty of ways to make this setup more interesting and more involved, while still remaining inside the idea of a “string on $B G_2$”. But for the moment, let’s restrict attention to this very simple setup.

Now quantize.

To do so, I follow the prescription described in the above interlude. While not really difficult, this is something that does not really want to be explained inside a blog entry. My original derivation is in Categorical Trace and Sections of 2-Transport. A more focused description appears in 2-Monoid of Observables on String-G, which also contains the interpretation of this quantization for the case that $G_2 = \mathrm{String}_k(G)$.

So here I just state the result:

One finds that the space of sections of $\mathrm{tra}_* = 1_*$ (which is our 2-bundle on $\Sigma(G_2)$ from the point of view of configuration space) is the category of loops inside the representations of the loop groupoid of $G_2$.

$$\mathrm{Hom}(1_* , 1_*) \simeq \Lambda \mathrm{Rep}(\Lambda G_2) \,.$$

This means the following:

$\Lambda G_2$ is a slight generalization of Simon Willerton’s loop groupoid. It is the categegory $$\Lambda G_2 := [\Sigma(\mathbb{Z}),\Sigma(G_2)]/\sim$$ obtained by identifying isomorphic 1-morphisms in our configuration space. Using Simon Willerton’s explanation of the ordinary loop groupoid, we see that we can think of $\Lambda \mathrm{String}_k(G)$ as an incarnation of the central extension of the loop group of $G$. For finite groups, this is made precise by Simon’s parmesam theorem.

As for any groupoid, we can consider the category of its representations $\rho$ $$\mathrm{Rep}(\Lambda G_2) = [\Lambda G_2, \mathrm{Vect}] \,.$$

The point of 2-Monoid of Observables on String-G was to argue that these representations are nothing but the $G$-equivariant twisted vector bundles on $G$.

Now, a single section of our 2-bundle on the configuration space of our 2-particle is a loop in this category, in the sense that it is an automorphism of one of these representations: $$\mathrm{Obj}(\mathrm{Hom}(1_*,1_*)) = \left\{ \rho \stackrel{L}{\to} \rho \right\} \,.$$

Pushing forward this space of sections to a point, amounts, in our context, to finding an algebra $A$ (“living over the point $\bullet$”) and an $A$-bimodule $N$ (“living over the string $\bullet \to \bullet$”), such that the category of bimodule homomorphism $$\array{ \C\mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \C\mathbb{C} \\ \psi(\bullet)\downarrow\;\;\; & \;\;\;\;\Downarrow \psi(\bullet\to\bullet) & \;\;\;\downarrow \psi(\bullet) \\ A &\stackrel{N}{\to}& A }$$ is equivalent to the space of sections $\mathrm{Hom}(1_*,1_*)$ just discussed.

It is easy to see that this is solved by letting $$A = \mathbb{C}[\Lambda G_2]$$ be the groupoid algebra of $\Lambda G_2$…

(This is the algebra generated, over $\mathbb{C}$, from the morphisms in $\Lambda G_2$, subject to the relation that the product of two generators is their composite, as morphisms, if that exists, or is zero otherwise.)

… and by letting $$N = A$$ be the identity morphism in the category of $A$-bimodules, namely $A$ regarded as a bimodule over itself, in the obvious way.

As a result, then, we find that the globular extended quantum field theory of our 2-particle on $B G_2$ is the functor $$q(1) : \Sigma(\mathbb{Z}) \to \mathrm{Bim}$$ that acts as $$q(1) : (\bullet \to \bullet) \mapsto \mathbb{C}[\Lambda G_2] \stackrel{\mathrm{Id}}{\to} \mathbb{C}[\Lambda G_2]$$ on the string. If we interpret this under the canonical injection $\mathrm{Bim} \hookrightarrow 2\mathrm{Vect}$, it looks like $$q(1) : (\bullet \to \bullet) \mapsto \mathrm{Mod}_{\mathbb{C}[\Lambda G_2]} \stackrel{\mathrm{Id}}{\to} \mathrm{Mod}_{\mathbb{C}[\Lambda G_2]} \,.$$

I think I should pause here.