The n-Category Café

Let $$\mathrm{par} := \{ a \to b\}$$ be the category that models (the parameter space of) a string. For the open string we think of this as the category with two objects and one nontrivial morphism going between them.

For the closed strings we set $a = b$ and think of this as the category $\Sigma(\mathbb{Z})$, with a single object and freely generated from a single nontrivial automorphism of that object. In this case, it is useful to think of this category as the fundamental groupoid of of the circle: $$\mathrm{par} = \Pi(S^1) \,.$$

In a similar manner, we may model the composite of two strings by a category of the form $$\mathrm{par}_2 := \{a \to b \to c\} \,.$$ This describes a string stretching from $a$ to $b$, concatenated with a string stretching from $b$ to $c$.

Again, in the case where all of these strings are closed, we set $a= b$ and $b = c$. Then we should think of $\mathrm{par}_2$ as the fundamental groupoid of the trinion, the sphere with three disks cut out: $$\mathrm{par}_2 = \Pi(\mathrm{trinion}) \,.$$

I’ll concentrate on closed strings in the following.

There are three basic non-trivial maps of the closed string to the trinion, i.e functors $$F : \mathrm{par} \to \mathrm{par}_2 \,,$$ namely those that send the closed string to one of the three boundary components of the trinion. I’ll call these three functors $F_1$, $F_2$ and $F_3$.

Now, assume all these strings propagate on some target space $$\mathrm{tar} \,.$$ Then we say that the space of maps from parameter space to target space $$\mathrm{conf} = [\mathrm{par},\mathrm{tar}]$$ is the configuration space of the closed string.

I shall motivate and illustrate all constructions in this post here by the example where target space is a group. Or rather, where we set $$\mathrm{tar} = \Sigma(G) \,,$$ the category with a single object and one morphism for each element of the group $G$.

In this case configuration space is $$\mathrm{conf} = [\Sigma(\mathbb{Z}),\Sigma(G)] = \Lambda G \,,$$ which is the loop groupoid of $G$. As Simon Willerton explains, this is, in many important respects, the categorical incarnation of the loop group of $G$.

An object in configuration space is a string, stretching along an element $g \in G$: $$\bullet \stackrel{g}{\to} \bullet \,.$$ A morphism in configuration space $$\array{ g \\ \;\;\downarrow g' \\ \mathrm{Ad}_{g'} g }$$ is a square

We can play the same game with maps from the parameter space $\mathrm{par}_2$ of two concatenated strings: $$\mathrm{conf}_2 = [\mathrm{par}_2,\mathrm{tar}] \,.$$

A configuration now is a pair of group elements $$\bullet \stackrel{g_1}{\to} \bullet \stackrel{g_2}{\to} \bullet \,,$$ and a morphism in configuration space $$\array{ (g_1,g_2) \\ \;\;\downarrow g' \\ (\mathrm{Ad}_{g'} g_1, \mathrm{Ad}_{g'} g_2) }$$ is a diagram $$\array{ \bullet &\stackrel{g_1}{\to}& \bullet &\stackrel{g_2}{\to}& \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' && \;\; \downarrow g' \\ \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet &\stackrel{\mathrm{Ad}_{g'}g_2}{\to}& \bullet } \,.$$

Using the three maps from the closed string to the trinion, we may pull back any configuration of two composable strings to one of the single string: $$F_i^* : \mathrm{conf}_2 \to \mathrm{conf}$$ simply by precomposing with that map: $$F_i^* \gamma : \mathrm{par} \stackrel{F_i}{\to} \mathrm{par}_2 \stackrel{\gamma}{\to} \mathrm{tar} \,.$$

This simply means that we may read out the configuration of any of the three circles of the trinion.

The crucial point now is the following: there will be an $n$-vector bundle with connection on the configuration space of the string, encoding its quantum dynamics.

For illustration purposes, I shall assume that we have an ordinary vector bundle on configuration space. This is given by a functor $$\mathrm{tra} : \mathrm{conf} \to \mathrm{Vect} \,.$$

For our example of configuration space above, this is nothing but a representation of the loop groupoid of the group $G$. Simon Willerton teaches us that this is to be thought of as a representation of the loop group.

There is an obvious and immediate monoidal structure on the category of all such representations: the tensor product inherited from $\mathrm{Vect}$.

So given one representation $$\mathrm{tra}_1 : \mathrm{conf} \to \mathrm{Vect} \,.$$ and another one $$\mathrm{tra}_2 : \mathrm{conf} \to \mathrm{Vect} \,,$$ we can form their tensor product $$\mathrm{tra}_1 \otimes \mathrm{tra}_2 : \mathrm{conf} \to \mathrm{Vect}$$ simply by tensoring the images of $\mathrm{tra}_1$ and $\mathrm{tra}_2$ “pointwise”.

But there is more. Notice that we may also think of the functors $\mathrm{tra}_i$ as “string fields”:

they associate with every configuration of the string a certain “amplitude” (or $n$-amplitude, if you like).

The parameter space $\mathrm{par}_2$ describes how two strings merge into a single one. This induces the “star product” on the corresponding string fields. As follows:

first pull back $\mathrm{tra}_1$ along $F_1^*$ to a field on the configuration space of the trinion. Then pull back $\mathrm{tra}_2$ along $F_2$ to the configuration space of the trinion.

Then take the ordinary tensor product of these string fields.

Then push the result along $F_3^*$ forward, back to the configuration space of the single string.

In formulas: $$\mathrm{tra}_1 \star \mathrm{tra}_2 := \mathrm{pushforward}\;\mathrm{along}\; F_3^* \;of ( (F_1^*)^* \mathrm{tra}_1 \otimes (F_2^*)^* \mathrm{tra}_2 ) \,.$$

In our example, this “string field star product” reproduces the fusion product of representations of the loop groupoid.

Fusion = composition of strings

Let’s unwrap the above definition of the star product to see how this works.

First of all, the pullback field $$(F_1^*)^* \mathrm{tra}_1 : \mathrm{conf}_2 \stackrel{F_1^*}{\to} \mathrm{conf} \stackrel{\mathrm{tra}_1}{\to} \mathrm{Vect}$$ evaluates on a configuration of the trinion $$\array{ \bullet &\stackrel{g_1}{\to}& \bullet &\stackrel{g_2}{\to}& \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' && \;\; \downarrow g' \\ \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet }$$ by first forgetting the configuration of the second string and only remembering that of the first one $$\array{ \bullet &\stackrel{g_1}{\to}& \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' \\ \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet }$$ and then evaluating the original string field on that $$\array{ \mathrm{tra}_1(g_1) \\ \mathrm{tra}_1(\mathrm{Ad}_{g'}) \downarrow \;\;\; \\ \mathrm{tra}_1(\mathrm{Ad}_{g'}g_1) } \,.$$

Analogously for $(F_1^*)^* \mathrm{tra}_1$. As a result, the tensor product of these two pullbacks is the string field on the configuration space of the trinion which sends the configuration $$\array{ \bullet &\stackrel{g_1}{\to}& \bullet &\stackrel{g_2}{\to}& \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' && \;\; \downarrow g' \\ \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet &\stackrel{\mathrm{Ad}_{g'}g_1}{\to}& \bullet }$$ to $$\array{ \mathrm{tra}_1(g_1)\otimes \mathrm{tra}_2(g_2) \\ \downarrow \\ \mathrm{tra}_1(\mathrm{Ad}_{g'}g_1) \otimes \mathrm{tra}_2(\mathrm{Ad}_{g'}g_2) } \,.$$

That is the obvious part. Slightly more subtle is the pushforward. The pushforward along $F_3^*$ achives, in words, a sum over all field values on configurations of two incoming strings that map to the same configuration of the single outgoing string.

The reasoning behind this is essentially the same that determines the pushforward to a point in general: all contributions from all points that get mapped to the same target point are “added up”.

In our example, this means that contributions from all configurations $$\bullet \stackrel{g_1}{\to} \bullet \stackrel{g_2}{\to} \bullet$$ with the same value of $g_1 \cdot g_2$ add up. As a result, the value of $$\mathrm{tra}_1 \star \mathrm{tra}_2$$ on a configuration $$\array{ \bullet &\stackrel{g}{\to}& \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' \\ \bullet &\stackrel{\mathrm{Ad}_{g'}g}{\to}& \bullet } \,.$$ is $$\oplus_{g_1 g_2 = g} \array{ \mathrm{tra}_1(g_1)\otimes \mathrm{tra}_2(g_2) \\ \downarrow \\ \mathrm{tra}_1(\mathrm{Ad}_{g'}g_1) \otimes \mathrm{tra}_2(\mathrm{Ad}_{g'}g_2) } \,.$$

This is the fusion product of representations of the loop groupoid.

Alternatively, if we decategorify this once, we get the star product of two string fields.