The String Coffee Table

I’ll very briefly recall some basic ideas from section 6 of the above paper and then make some comments on the general structure appearing here.

Consider some sort of 2-dimensional quantum field theory with boundary, and assume there are some kind of “line like” operator in the theory. For heuristic purposes this shall mean nothing but that there are functionals which depend on the theory’s fields over a given line drawn on the worldsheet, and that we may include these in the path integral when computing correlators.

Clearly, in this heuristic picture - and if everything is assumed to behave well enough - we may imagine inserting such a line operator corresponding to a curve which runs very close to the worldsheet boundary. It is natural to expect that the line operator close to the boundary behaves essentially like a new kind of boundary itself. In other words, we expect the line operator to induce an endomorphism on the space of boundary conditions of the theory.

If we are thinking strings, our boundary conditions are known as branes. So we expect an endomorphism on the space of branes - whatever that will mean in detail.

To get a handle on the nature of a given line operator, it is natural to search for branes which are transformed in a simple manner under the action of the corresponding endomorphism. In the simplest case the brane will be mapped to itself. In the next simplest case it will be mapped to itself only up to some small modification.

If our brane were represented by a vector in some vector space, we’d consider transformations that leave the brane intact up to a linear factor. But branes do not live in vector spaces (which are sets with extra structure), they live in module categories (which are categories with extra structure).

Hence we should probably address some brane as an eigenbrane of the action of our line operator if it gets sent to itself up to some sort of categorified linear factor.

In typical special cases arising in practice, our branes are to be thought of as being represented by certain (locally free) sheaves ($\to$). We may naturally tensor such sheaves with vector spaces. So it is plausible that a good notion of eigenbrane of some line operator is one that is invariant under the action of that operator up to tensor product with some vector space.

This is the definition given in equation (6.11) of Witten’s paper. Identifying branes with sheaves, we may analogously think of eigensheaves. This is apparently the langauge used in

A. Beilinson & V. Drinfeld
Quantization of Hitchin’s Integrable System and Hecke Eigensheaves
available from this site
pdf

(which I haven’t had the chance to look at myself, yet).

That’s already about all from the above paper that I shall consider here. The only additional hint we might want to record is the following.

There are two particular classes of line operators that are considered in the Langlands context of geometric Langlands duality. Wilson loops (traces of holonomies of connections transported along the respective curves) and their duals: “‘t Hooft operators”. The latter are “disorder operators”.

Now, disorder operators (in CFT) create defect lines. Defect lines, on the other hand, are well known, as amplified in particular by FRS formalism ($\to$) to act on the space of branes “much like” linear operators act on vector spaces.

The “much like” in the previous sentence can actually be given a very precise meaning. This shall be the content of the remainder of this post.

As I have remarked before ($\to$) one particuarly entertaining way to look at the step from point physics to string physics, and in particular at the structures appearing in FRS formalism ($\to$) is to consider categorifying quantum mechanics, and in particular categorifying the linear algebra involved. Pursuing this leads naturally to the concept of eigenbranes with respect to disorder operators.

Let me quickly recall the idea.

Let’s be general and forget for a moment that the complex numbers $\mathbb{C}$ are indeed a field, remembering only that they form in particular a semi-ring $R$. Similarly, let’s address complex Hilbert spaces as special $R$-modules and linear operators on Hilbert spaces as $R$-module homomorphism. Alltogether, these live in the catgory

of (left, say) $R$-modules.

Now, let $P_1$ be some category of 1-dimensional (Riemannian) cobordisms, to be thought of as pieces of world-line of some particle. The quantum mechanics of that particle is specified by defining a functor

The functor’s values on points is the system’s Hilbert space at that particular instant, the value on morphisms is the propagator from one instant to another.

This idiosyncratic way of formulating QM is designed to lend itself to the process we are interested in: categorification. We want to take the above structure and internalize it in $\mathrm{Cat}$. When we do so, we obtain the general setup of FRS formalism, hence structures known to apply to topological and conformal 2-dimensional field theory.

The obvious categorification of a semi-ring is an abelian monoidal category. Pick some and call it $C$ - a 2-ring.

There is an obvious notion of a module for a 2-ring - a module category. These naturally live in a 2-category

whose objects are module categories, whose morphisms are functors respecting the (left) $C$-action and whose 2-morphisms are natural transformations between these.

We have a dictionary as follows:

$\bullet$ $C$ is our categorification of the complex numbers $\mathbb{C}$

$\bullet$ a $C$-module category is our categorification of a complex Hilbert space

$\bullet$ a morphism of $C$-module categories is our categorification of a linear operator between complex Hilbert spaces

$\bullet$ a 2-morphism of $C$-module categories is new structure, having no lower-dimensional analog.

In order to get a handle of these structures, it may be useful to consider the following examples.

Let $A$ be any algebra internal to $C$. Then the category $\mathrm{Mod}_A$ of internal right $A$-modules is naturally a left $C$-module:

Moreover, any internal $A$-$B$ bimodule is naturally a $C$-module morphism from $\mathrm{Mod}_A$ to $\mathrm{Mod}_B$.

(This is particularly nice to see when we restrict attention to Kapranov-Voevodsky 2-vector spaces. See this for more details.)

In fact, under some nice conditions it can be shown that all of ${}_C \mathrm{Mod}$ comes from internal modules this way. More precisely, under some nice conditions ${}_C\mathrm{Mod}$ is equivalent to the 2-category $\mathrm{BiMod}(C)$ of bimodules internal to $C$

This goes back to a theorem by Ostrik ($\to$,$\to$).

In any case, the message is that we should think of categories of (right) internal $A$-modules as the analog of vector spaces, and of internal bimodules as the analog of linear operators on such vector spaces.

Equipped with this intuition for the categorification we are dealing with, we perform the obvious next step and define a 2-dimensional quantum theory to be given by a functor

from surface elements to ${}_C\mathrm{Mod}$.

So this gadget assigns categorified Hilbert spaces to points, categorified linear operators to paths between points, and certain 2-morphisms to pieces of worldsheet.

And it turns out, that, indeed, for suitable choice of $C$ such a 2-functor does describe 2-dimensional (topological, conformal) field theory ($\to$). More precisely, it can be shown that locally trivializing such a 2-functor gives rise to (at least parts of) the structure used in the FRS description of TQFT/CFT.

This relation provides us with an extension of our above dictionary. Not only do we know that objects of ${}_C\mathrm{Mod}$ are to be interpreted as categorified vector spaces, but it is moreover known that they encode boundary conditions for 2-dimensional field theories.

The argument behind this can be summarized using some quiver logic.

Indicate every D-brane of the theory by a point. Draw an edge connecting two points, whenever there is supposed to be a type of open string stretching between the two corresponding branes. Finally, “represent” the resulting structure (i.e. the graph) in the category $C$, which, recall, serves as our categorification of the complex numbers.

The representation will associate an algebra $A_{bb}$ internal to $C$ to every point $b$. Moreover, it will associate an $A_bb$-$A_{b'b'}$-bimodule to every edge stretching between $b$ and $b'$. ($\to$).

The algebras $A_{bb}$ are the OPE algebra of open $b-b$ strings. The bimodules are the spaces of states of string stretching between $b$ and $b'$.

So in particular, once we fix any brane $b$ and the corresponding algebra $A_{bb}$, all other branes $b'$ are characterized by the (right) $A_{bb}$ modules associated to the edge $b'-b$.

So branes are modules. Computing open string correlators for open strings attached to a brane $b'$ amounts to assigning the corresponding module to the boundary of the worldsheets.

But we also learned that these $A$-modules are to be thought of as objects in the categorified vector space $\mathrm{Mod}_A$ which lives inside ${}_C \mathrm{Mod}$. Hence they are to be thought of as categorified vectors.

We also said that $A$-$B$ bimdodules are to be thought of as categorified linear operators between these higher vector spaces. Moreover, it turns out that, in this sort of language, CFT defect lines are labeled by precisely such bimodules.

(Using the general relationship between surface holonomy of gerbes and the way CFT can be described by locally trivialized 2-functors with values in ${}_C\mathrm{Mod}$, these defect lines can be seen to play the role of transition data between different local trivializations. ($\to$)).

This, finally, provides us with all the ingredients that we are looking for. There is a precise way in which branes are like categorified vectors (“states”, if you think in terms of QM), and in which defect lines (and hence discorder operators) act on these like linear operators. This precise way is categorified linear algebra. And FRS formalism uses this to describe (rational) conformal field theory.

Update, May 8, more details:

Once this is set up, there is nothing to stop us from doing all our familar linear algebra in this categorified context.

In particular, we may study categorified eigenvalue problems. By using the dictionary described above, a categorified eigenvector of a categorified linear operator represented by some $A$-$A$ bimodule $N$ would be a right $A$ module $B$ such that

for some categorified number $V \in \mathrm{Obj}(C)$.

As an example, consider the monoidal category $C = \mathrm{Vect}$, whose objects are vector spaces and whose morphisms are linear maps between these. The 2-category ${}_C\mathrm{Mod}$ contains all categories of algebra modules internal to $C$ as objects. But an algebra internal to $C = \mathrm{Vect}$ is just an ordinary algebra in the familiar sense.

If we restrict attention to those algebras that arise as function algebras over topological spaces we find inside ${}_C\mathrm{Mod}$ a version of the 2-category of Kapranov-Voevodsky 2-vector spaces.

As one can easily see ($\to$) our modules are now simply vector bundles over these topological spaces, while bimodules are vector bundles with two different projections on two (possibly different) spaces. The bimodule tensor product is simply the pullback along common projections, i.e. the fiberwise tensor product of vector bundles.

When one works this out, one finds that hence a categorified vector in this context is a vector whose entries are vector spaces. A categorified linear map is a matrix whose entries are vector spaces. Composition is the usual matrix product, with addition and multiplicaiton of numbers replaced everywhere by direct sum and tensor product of vector spaces.

So in this case our categorified eigenvector equation demands that we find a vector bundle, which, when acted on by a categorified linear map, transforms into itself up to tensor product with a vector space.

This is precisely the situation considered in the above paper.

Of course, there are other operations familiar from linear algebra that one might want to adopt in the categorified setup. Rather recently Kapranov has studied trace operations in this context, for instance ($\to$). This was motivated by an attempt to realize equivariant string theory in the sense of equivariant elliptic cohomology. In fact, the possibly most prominent application of Kapranov-Voevodsky 2-vector spaces has been in the attempt by Baas, Dundas & Rognes to realize elliptic cohomology as a categorified K-theory ($\to$).