## January 21, 2007

### Fusion and String Field Star Product

#### Posted by Urs Schreiber

From the point of view of functorial transport #, I describe the structure of the star product of string fields, and, as a special case, the fusion product of loop group representations.

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

(1)$\array{ \bullet & \stackrel{g}{\to} & \bullet \\ g' \downarrow\;\; && \;\; \downarrow g' \\ \bullet & \stackrel{\mathrm{Ad}_{g'}g}{\to} & \bullet } \,.$

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.

Posted at January 21, 2007 8:45 PM UTC

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### Re: Fusion and String Field Star Product

Hi Urs,

This is a nice description of the fusion product… isn’t it basically the same approach though as the classical’ description of fusion by Freed, using the pair of pants, in Quantum Groups from Path Integrals, pages 34-36? It would be instructive to compare the two approaches. A relevant exercise (which I haven’t worked through yet) is Exercise 4.34, which precisely makes reference to the kind of push-forward you are referring to.

By the way, can you explain a bit more about how you got the fundamental groupoid of the trinion by taking the free category on $\{a \rightarrow b \rightarrow c \}$ and then identifying $a, b$ and $c$? Recall that $\pi_1$(trinion)$= \langle a,b,c : a b c = 1 \rangle$ (I think).

It would be nice if you performed the explicit calculation you’re referring to when you do the push-forward to a point. Looking at your notes on this, I’ve convinced myself that indeed you’re right - it will work. It would be cool to actually see the calculation though, perhaps even to compare it to the pushforward Freed is talking about in that exercise.

Posted by: Bruce Bartlett on January 22, 2007 1:00 AM | Permalink | Reply to this

### Re: Fusion and String Field Star Product

Presumably your $a$ is Urs’ generator of the loop $a \to b$, your $b$ is Urs’ generator of $b \to c$, and your $c$ is the reverse path (loop) from Urs’ $c$ to $a$. The figure-of-eight and trinion are homotopy equivalent.

Posted by: David Corfield on January 22, 2007 10:18 AM | Permalink | Reply to this

### Trinion and figure-of-eight

Hi all,

I am finally back online. I’ll answer Bruce’s nice questions now.

I admit that the above entry it too terse, in general. I was writing it on a very slow internet connection and had only that much patience with it. You cannot imagine how long it took me to compile even that terse entry.

Thanks for reading it anyway! :-)

First of all, David is right concerning the interpretation of my notation. I should have mentioned that the category which I cryptically called $\mathrm{par}_2$ represents, in essence, the figure-of-eight.

Or I should have drawn this picture: $\array{ & b & \\ &\nearrow \searrow \\ a &\rightarrow& c } \,.$

This is my “parameter space of two composed strings” for the open string: one string is stretching from (brane) $a$ to (brane) $b$. The next one from $b$ to $c$. And their composite is the one stretching from $a$ to $c$.

For the closed string, I identify all three vertices and think of $\array{ & \bullet & \\ &\nearrow \searrow \\ \bullet &\rightarrow& \bullet } \,.$

Of course, at this point my notation becomes ambiguous, since now the morphisms are no longer identitfied by their source and target. It would be better if I introduced additional labels for the morphisms: $\array{ & \bullet & \\ &C \nearrow \searrow A \\ \bullet &\stackrel{B}{\rightarrow}& \bullet } \,.$

This diagram is still supposed to commute. But now, since these morphisms come back to their sources, we also need to define what it means to compose the strings $A$, $B$ and $C$ with themselves.

In close analogy to what we do for the single closed string, I simply demand that $A$, $B$ and $C$ be invertible and then take the category freely generated from them, subject to the relation expressed by the above triangle.

The result is in fact (isomorphic to) the fundamental group of the trinion, or, as David points out, the fundamental group of the figure-of-eight.

We can think of $C$ as encircling one of the incoming tubes and of $A$ as encircling the other incoming tube. Then their composite $B = A\circ C$ encirles the outgoing tube (all up to homotopy).

Upon request I will try to draw a picture that makes this obvious. But I guess it is clear now.

Part of the fun of the approach that I am kind of promoting here is that we don’t need to think of $\mathrm{par}_2$ as being the fundamental group of the trinion. The formalism I describe works for whatever category $\mathrm{par}_2$ that you feel like addressing as a “parameter space” for something.

Posted by: urs on January 22, 2007 11:05 AM | Permalink | Reply to this

### Re: Trinion and figure-of-eight

Upon request I will try to draw a picture that makes this obvious. But I guess it is clear now.

Another thread you guys had going made me think you should really set up a wiki. Subsequent discussions explicitly floated similar ideas. I’m a passive observer, but understand enough to contribute art work once in a while (time permitting!) :)

Baez was somewhat of a pioneer with SPR and TWF and Urs was somewhat of a pioneer with blogs, the next logical step for you guys is to set up a wiki so that others can contribute.

The questions then is, “Why not start adding stuff to WikiPedia?” for which I do not have a great answer :)

Just something to think about and apologies for the digression.

If this is too clueless of a question, feel free to disregard it, but I’m curious how this might be related to what I thought I understood about the *-product way back when I wrote this:

NCG/SUSY/*-Product

Cheers,
Eric

Posted by: Eric on January 22, 2007 4:04 PM | Permalink | Reply to this

### wiki or not wiki

the next logical step for you guys is to set up a wiki

Possibly that’s quite right. From what I have seen of research wikis (mostly here and here) I deduce two things:

a) I might find it quite interesting contributing to one (or participating in one, or whatever the right verb is)

b) it is quite unlikely that I will be the one who sets it up.

For the moment, this blog here is already pretty good a platform.

What we could do is maybe organize the joint discussion more, like David Corfield used to do for the Klein-2-geometry line of discussion. I should maybe post a summary of what has been discussed, what has been achieved and what is still open once in a while.

Maybe I am in the process of preparing something along these lines.

Posted by: urs on January 22, 2007 4:27 PM | Permalink | Reply to this

### star product

Eric pointed to (not for the first time :-)

NCG/SUSY/*-Product

Right. I should think about that!

What I described above is how in certain situations the prescription

take two “string fields” $f$ and $g$ and form a new string field defined on a given string state by applying $f$ and $g$ to all possible ways of decomposing that string into two strings

can be understood as a certain pullback followed by a push-forward.

But in my examples, the “string fields” were really “string 2-fields”: they took values in 1-vector spaces instead of in 0-vector spaces (numbers). This makes their push-forward easier to handle.

For true string fields with values in “numbers” the push-forward will be similar to a path integral and hard to get under control.

Posted by: urs on January 22, 2007 4:36 PM | Permalink | Reply to this

### The n-Category Wiki (Was: Trinion and figure-of-eight)

The questions then is, “Why not start adding stuff to WikiPedia?” for which I do not have a great answer :)

For explaining established facts, that’s a good idea. But Wikipedia isn’t appropriate for new reasearch. More generally, Wikipedia won’t work in any situation where you want to present a perspective that isn’t already well documented, if not widely accepted and understood. To describe either your own new ideas, or your own new interpretation of old ideas (and almost everything discussed here is one or the other), you need your own wiki.

Posted by: Toby Bartels on January 22, 2007 6:37 PM | Permalink | Reply to this

### Re: Fusion and String Field Star Product

Ok thanks I get it now.

Posted by: Bruce Bartlett on January 22, 2007 4:06 PM | Permalink | Reply to this

### Re: Fusion and String Field Star Product

Hi Bruce,

thanks for taking interest. This one was written for you. :-)

It would be nice if you performed the explicit calculation you’re referring to when you do the push-forward to a point.

Right, sure, I should have done that. The only nontrivial thing about my discussion is that push-forward. Here is how it works.

I claimed that the push-forward $\tilde T$ of $T : \mathrm{conf}_2 \to \mathrm{Vect}$ from $\mathrm{conf}_2$ to $\mathrm{conf}$ along $F_3^* : \mathrm{conf}_2 \to \mathrm{conf}$ is the functor $\tilde T$ on $\mathrm{conf}$ that acts as $\tilde T : ( g \stackrel{g'}{\to} \mathrm{Ad}(g')(g) ) \; \mapsto \; \oplus_{g_1 g_2 = g} ( T(g_1,g_2) \stackrel{T(g')}{\to} T(\mathrm{Ad}(g')(g_1),\mathrm{Ad}(g')g_2) ) \,.$

To do so, I need to show that there is an isomorphism of Hom-spaces $\mathrm{Hom}((F_3^*)^* f, T) \simeq \mathrm{Hom}(f, \tilde T)$ for all $f : \mathrm{conf} \to \mathrm{Vect} \,.$

But, luckily, this becomes obvious by simply writing it out:

a morphism on the left is a natural transformation $r$ coming from naturality squares of the form $\array{ f(g_1 g_2) &\stackrel{r(g_1,g_2)}{\to}& T(g_1,g_2) \\ f(g')\downarrow \;\; && \;\, \downarrow T(g') \\ f(\mathrm{Ad}(g')(g_1 g_2)) &\stackrel{r(\mathrm{Ad}(g')(g_1), \mathrm{Ad}(g')(g_2))}{\to}& T(\mathrm{Ad}(g')(g_1), \mathrm{Ad}(g')(g_2)) } \,.$

A morphism on the right is a natural transformation $R$ coming from naturality squares of the form $\array{ f(g) &\stackrel{\oplus_{g_1 g_2 = g}r(g_1,g_2)}{\to}& \oplus_{g_1 g_2 = g}T(g_1,g_2) \\ f(g')\downarrow \;\; && \;\, \downarrow \oplus_{g_1 g_2 = g}T(g') \\ f(\mathrm{Ad}(g')(g_1 g_2)) &\stackrel{\oplus_{g_1g_2 =g}r(\mathrm{Ad}(g')(g_1), \mathrm{Ad}(g')(g_2))}{\to}& \oplus_{g_1 g_2 = g}T(\mathrm{Ad}(g')(g_1), \mathrm{Ad}(g')(g_2)) } \,.$ These are clearly in bijection: I have already indicated the required identification by using the letter $r$ in the second diagram.

(Unless I am mixed up, that is. Please don’t trust my use of the words “obvious”, “clearly”, etc. but check everything yourself. If you think I making a mistake somewhere, please let me know.)

Probably my notation here is not really optimized. Let me try to look at the same situation from a more general point of view with slightly more transparent notation:

Let $p : C \to D$ be a morphism of categories which is surjective on morphisms and such that

a) every morphism in $D$ has precisely one lift for every lift of its source object

and

b) every morphism in $D$ has precisely one lift for every lift of its target object.

Notice that this is the situation for our configuration spaces of strings on $\Sigma(G)$ above. The morphism $\array{ g \\ g'\downarrow \;\; \\ \mathrm{Ad}(g')(g) }$ has precisely one lift, once we specify either the lift of the source or the target.

So assume, generally, that we have a morphism $p : C \to D$ as above and want to compute the push-forward along that morphism of a functor

(1)$\mathrm{tra} : C \to \mathrm{Vect} \,.$

This means we need to find $\tilde \mathrm{tra} : D \to \mathrm{Vect}$ such that $\mathrm{Hom}(p^* f, \mathrm{tra}) \simeq \mathrm{Hom}(f, \tilde \mathrm{tra})$ for all $f : D \to \mathrm{Vect}$.

I claim that $\tilde \mathrm{tra} : (d_1 \stackrel{\kappa}{\to} d_2) \mapsto \oplus_{(c_1 \stackrel{\gamma}{\to} c_2) | p(\gamma)=\kappa} \left( \mathrm{tra}(c_1) \stackrel{\gamma}{\to} \mathrm{tra}(c_2) \right) \,.$

Let’s check that.

A morphism on the left of $\mathrm{Hom}(p^* f, \mathrm{tra}) \simeq \mathrm{Hom}(f, \tilde \mathrm{tra})$ is a natural transformation $r$ coming with naturality squares of the form $\array{ f(p(c_1)) &\stackrel{r(c_1)}{\to}& \mathrm{tra}(c_1) \\ f(p(\gamma))\downarrow\;\; && \;\; \downarrow \mathrm{tra}(\gamma) \\ f(p(c_2)) &\stackrel{r(c_2)}{\to}& \mathrm{tra}(c_2) }$ for every morphism $c_1 \stackrel{\gamma}{\to} c_2$ in $C$.

A morphism on the right is a natural transformation coming from naturality squares of the form $\array{ f(d_1) &\stackrel{\oplus_{p(c_1)=d_1}r(c_1)}{\to}& \oplus_{p(c_1)=d_1}\mathrm{tra}(c_1) \\ f(\kappa)\downarrow\;\; && \;\; \downarrow \oplus_{p(\gamma) = \kappa}\mathrm{tra}(\gamma) \\ f(d_2) &\stackrel{\oplus_{p(c_2)=d_2}r(c_2)}{\to}& \oplus_{p(c_2)=d_2}\mathrm{tra}(c_2) }$ for every morphism $d_1 \stackrel{\kappa}{\to} d_2$ in $D$.

Again, I have used the symbol $r$ for my components in the second diagram in order to manifestly indicate how these two spaces of natural transformations are isomorphic.

Posted by: urs on January 22, 2007 11:53 AM | Permalink | Reply to this

### Understanding

This is a nice description of the fusion product… isn’t it basically the same approach though as the classical’ description of fusion by Freed, using the pair of pants, in Quantum Groups from Path Integrals, pages 34-36? It would be instructive to compare the two approaches.

Maybe it is. To some extent, much of what I am thinking about lately, not the least through your influence, is what Freed is actually doing there.

I am fond of the fact that my discussion of fusion above is essentially elementary and based on rather general and clear functorial concepts (as described in section 1.2 here). (But I might be biased ;-).

I feel to be in line here with the way Simon Willerton reduces lots of apparently sophisticated technology to a clear, elementary idea by identifying the right structure (those reading this who don’t know what I am talking about here should follow this link).

To amplify this, I quote this verse from Simon’s paper, beginning of section 3:

We can now see how the general theory of the previous section easily recovers many facts about [X]. Indeed it is only through this point of view that I understand [X].

That’s my stance here. I tried to show how the general theory of functorial $n$-transport recovers facts about fusion. Indeed, it is only through this point of view that I understand fusion.

I think Simon Willerton explains how the loop group of $G$ is best thought of as the action groupoid of $G$ on itself, which I am fond of thinking as a special case of a functorial way to realize the configuration space of a closed string.

What I tried to add above is how the general idea of two strings merging into a single one then also very nicely explains the fusion product of loop group representations.

A relevant exercise (which I haven’t worked through yet) is Exercise 4.34, which precisely makes reference to the kind of push-forward you are referring to.

Right. Thanks for reminding me of that exercise!

Posted by: urs on January 22, 2007 12:31 PM | Permalink | Reply to this
Read the post The Globular Extended QFT of the Charged n-Particle: Definition
Weblog: The n-Category Café
Excerpt: Turning a classical parallel transport functor on target space into a quantum propagation functor on parameter space.
Tracked: January 24, 2007 9:19 PM