October 16, 2006

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 #.

Posted at October 16, 2006 11:51 AM UTC

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