### Adjunctions and String Composition

#### Posted by Urs Schreiber

At one of our Café tables # Jim Stasheff listed a couple of different notions of paths with composition. In particular, motivated by string field theory, people have considered categories whose

- morphisms are parameterized paths $\gamma : [0,1] \to M$
- composition $\stackrel{\gamma_1}{\to}\stackrel{\gamma_2}{\to}$ of which is defined whenever a portion $\gamma_c$ of the end of $\stackrel{\gamma_1}{\to} = \stackrel{\gamma_{1l}}{\to}\stackrel{\gamma_{c}}{\to}$ is the reverse $\bar \gamma_c$ of a portion of the beginning of $\gamma_2$ $\stackrel{\gamma_2}{\to} = \stackrel{\bar \gamma_{c}}{\to}\stackrel{\gamma_{2r}}{\to} \,,$ in which case the composition is the result of cancelling both copies of this portion, like this: $\array{ (\stackrel{\gamma_1}{\to})(\stackrel{\gamma_2}{\to}) \\ \;\;\Downarrow = \\ (\stackrel{\gamma_{1l}}{\to}\stackrel{\gamma_{c}}{\to}) (\stackrel{\bar \gamma_{c}}{\to}\stackrel{\gamma_{2r}}{\to}) \\ \Downarrow \\ \stackrel{\gamma_{1l}}{\to}\stackrel{\gamma_{2r}}{\to} \\ \;\;\Downarrow = \\ \stackrel{\gamma_2 \circ \gamma_1}{\to} } \,.$

If we agree that the cancelled portion is always exactly one half of the string, we are left with the associative composition going back to

Edward Witten
*Noncommutative Geometry And String Field Theory.*

Nucl.Phys.B268:253,1986 .

If, however, we allow arbitrary portions to be cancelled we get a nontrivial associator.

Here I just want to notice that all these compositions can be nicely understood as coming from special ambidextrous adjunctions in the 2-category $P_2(X)$ whose objects are points of $X$, whose morphisms are parameterized paths in $X$ and whose 2-morphisms are, for instance, thin homotopy classes of 2-paths.

What I am saying here is not supposed to be very deep. But I believe it provides a useful way to look at this situation. We need not restrict ourselves to a particular choice of 2-morphisms in $P_2(X)$. All we need to require is that the 2-morphisms are such that a path $\gamma$ is adjoint to its reverse $\bar \gamma$ $\begin{aligned} \gamma &: \sigma \mapsto \gamma(\sigma) \\ \bar \gamma &: \sigma \mapsto \gamma(1-\sigma) \,, \end{aligned}$ such that the adjunction is ambidextrous and special.

(We may for instance choose 2-morphisms to be thin homotopy classes of 2-paths.)

If we arrange for that, then we may “cancel” a path with it’s reverse by inserting the counit 2-morphism $\epsilon_\gamma$ $\array{ x \stackrel{\gamma}{\to} y \stackrel{\bar \gamma}{\to} x \\ \;\;\;\Downarrow \epsilon_\gamma \\ x \stackrel{\Id}{\to} x } \,.$ Similarly, we may create a path and its reverse “from nothing” by inserting the unit $i_\gamma$ $\array{ x \stackrel{\Id}{\to} x \\ \;\;\;\Downarrow i_\gamma \\ x \stackrel{\gamma}{\to} y \stackrel{\bar \gamma}{\to} x } \,.$ Composition and co-composition of paths using these cancellations and co-cancellations follows the rules of products and coproducts in special Frobenius algebras. From this point of view, the fact that the associator satisfies a pentagon is more or less automatic.

The co-composition operation is particularly interesting. It should induce the product operation on string fields.

That needs to be worked out in details. But here is a sketch.

For simplicity, consider the case where we decide to only cancel and create portions of paths that are precisely half of a former path. Then the above setup yields a category $P_1^{LW}(X)$ which is at the same time a co-category.

There are many different choices of co-category structure we can use, each determined by a choice of path for each point of $X$. Later we will want to “integrate over all these choices”, but for the moment just fix one.

To define a string field, let $\mathrm{tra} : P_1(X) \to \mathrm{Bim}(\mathrm{Vect}_\mathbb{C})$ be a gerbe with connection on $X$ #. Let $[0,1]$ be the category obtained from regarding the interval as a poset. Then the functor category $[[0,1],P_1(X)]$ is the configuration space of our string #, and a string field is a section of the gerbe with connection pulled back to this configuration space.

It sends each path $\stackrel{\gamma}{\to}$ to a morphism $e_\gamma$ $\array{ \mathbb{C} &\stackrel{\mathrm{Id}}{\to}& \mathbb{C} \\ \downarrow &\Downarrow e_\gamma& \downarrow \\ A_x &\stackrel{\mathrm{tra}(\gamma)}{\to}& A_y } \,.$ which is essentially a section of some kind of bundle over path space.

Now, given two such sections, $e_1$ and $e_2$, say, we may use the co-composition on paths to produce a new section - their product: $P_1^{LM}(X) \stackrel{\text{co-compos.}}{\to} P_1^{LM}(X) {}_t \times_s P_1^{LM}(X) \stackrel{e_1 \times e_2}{\to} \tilde T {}_t \times_s \tilde T \stackrel{\text{compos.}}{\to} \tilde T \,.$

If we had a measure on our paths (like is assumed in string field theory), we could try to integrate this operation over all choices of co-category structures. The result should be the star-product of string field theory #.