The String Coffee Table

In ordinary linear algebra and functional analysis, the condition on an operator $O$ to have something like a complete basis of eigenvectors, or, more precisely, to admit a spectral theorem ($\to$), is that it is normal ($\to$)

For instance $O$ might be unitary $O O^* = \mathrm{Id}$, or hermitean, $O = O^*$.

More generally, there is spectral theory for operators which are semi-normal ($\to$), meaning that the above commutator is not necessarily vanishing, but trace class

There are various indications ($\to$, $\to$), that a suitable categorification of something like this helps to understand 2-dimensional (topological/conformal) field theory.

In that context, the field of complex numbers $\mathbb{C}$ gets replaced by some monoidal category $C$ and the category $\multiscripts{_\mathbb{C}}{\mathrm{Mod}}{}$ of complex vector spaces gets replaced by the 2-category $\multiscripts{_C}{\mathrm{Mod}}{}$ of $C$-module categories.

A $C$-module category plays hence the role of a categorified vector space, while 1-morphisms of $C$-module categories play the role of categorified linear maps.

In nice cases our $C$-module categories are equipped with an internal $\mathrm{Hom}$-functor ($\to$), which can be regarded as a categorified sesquilinear scalar product (items 3 and 4 in Ostrik’s Lemma 5 ($\to$)). In this case we can think of dealing with categorified Hilbert spaces (HDA II).

One easily sees that the category $\mathrm{BiMod}(C)$ of bimodules internal to $C$ sits inside $\mathrm{Mod}_C$, as does $C$ itself:

In nice cases the second arrow is an equivalence ($\to$), which allows us to restrict attention to bimodules.

Moreover, at least in these nice cases we have a very explicit understanding of the internal $\mathrm{Hom}$ on all $C$-module categories, hence of the categorified scalar product.

As shown in FRS II (section 2.4, $\to$), we have, for $N$ and $N'$ being internal left (special symmetric Frobenius algebra-) $A$-modules, that the internal $\mathrm{Hom}$ of $\mathrm{Mod}_A(C)$ looks like

where $N^\vee$ is the dual of $N$, as an object of $C$, with the canonical right $A$ action.

Notice how this realization of the internal $\mathrm{Hom}$ makes the categorified “sesquilinearity” of the scalar product manifest.

For $X \in \mathrm{Obj}(C)$ any object in $C$ (a categorified complex number, if you like), we have

and

You should visualize all these formulas in the context of Kapranov-Voevodsky 2-vector spaces ($\to$). (which is, roughly, the right ambient category for topological 2D field theory).

In that context we have $C = \mathrm{Vect}$ over the ground field $\mathbb{C}$, say. Our internal algebras are $\mathbb{C}^n = \oplus_{i=1}^n \mathbb{C}$ and a left $\mathbb{C}^n$-module is a vector whose $n$ entries are $\mathbb{C}$-vector spaces.

So for instance for $n=2$ we’d have a 2-vector of the form

Using the above definition of scalar product, we find its norm square to be

as it should be.

It has been observed long before (HDA II) that for 2-vector spaces the notion of adjoint functor and adjoint 2-linear map coincide. That’s a direct consequence of interpreting the internal $\mathrm{Hom}$ as the categorified scalar product.

In the present context 2-linear maps are functors of $C$-module categories respecting the (right) $C$-action. If we work in terms of bimodule categories this are nothing but functors that act by (left) tensor-multiplication with a given bimodule.

Again, this is easiest sean in categorified matrix multiplication.

A $\mathbb{C}^2$-bimodule would be a $2\times 2$-matrix whose entries are vector spaces

It acts on 2-vectors as

The adjoint of a 2-linear map $\multiscripts{_A}{\mathrm{Mod}}{}(C) \overset{B}{\to} \multiscripts{_{A'}}{\mathrm{Mod}}{}(C)$ (an $A'$-$A$ bimodule) is $B^\vee$, which is the dual of $B$ as an object of $C$, with the canonical $A$-$A'$ bimodule structure.

For instance

as it should be.

We could agree to write

to emphasize that the dual of a bimodule $B$ (with duality in the sense of duality of objects in $C$, which is supposed to be a monoidal category with all duals) plays the role of an adjoint linear operator.

Notice that vectors of vector spaces are nothing but vector bundles over finite sets. Hence, with little effort, we can generalize the above to a 2-linear algebra where 2-vectors are vector bundles (or locally free sheaves) over some space, and where 2-linear maps are correspondences between two base spaces.

We may imagine, in this categorified setup, to study all the questions familial from ordinary linear algebra or functional analysis. Are the 2-linear maps which have a complete basis of eigenvectors, for instance?

It seems that one nontrivial example for such a situation is that of Hecke operators ($\to$) arising in the context of geometric Langlands duality ($\to$). As Frenkel explains in section 4.4 of his lecture notes ($\to$), we may view Hecke operators as categorified differential operators whose eigen-2-vectors are like categorified exponential functions. The geometric Langlands duality (in the “classical limit” ($\to$) )is like a categorified Fourier transformation which exchanges exponential with delta-distributions.

If we try to abstract away from the particular example of geometric Langlands (which corresponds (only) to a very specific 2D TFT, as Kapustin and Witten explain), we may ask the general question under which conditions the category $\mathrm{Mod}_C$ associated with some 2D QFT admits disorder operators (2-linear maps) that have a spectral decomposition.

Apart from possibly being an interesting question by itself, we may also try to see if the condition to be found here has maybe already arisen in some guise in 2D field theory.

Indeed, maybe it has. In

J. Fröhlich, J. Fuchs, I. Runkel, C. Schweigert
Kramers-Wannier duality from conformal defects
cond-mat/0404051

results are announced which relate algebraic properties of bimodules associated to defect lines ($\to$) with various dualities of rational conformal field theory, among them order/disorder dualities like that found by Kramers-Wannier, but also for instance T-duality.

In brief, the condition for a bimodule $B$ to induce a duality is that

is a direct sum of simple invertible bimodules.

Now, given the above dictionary of 2-linear algebra, we notice that this is an equation relating a linear 2-map with its adjoint.

In fact, from a naïve categorification of the above mentioned condition on semi-normal operators, we would be tempted to ask what this duality condition tells us about possible isomorphisms

In order for that to make sense, we first need a notion of trace for 2-linear maps.

But luckily, one such notion which has several indications of being “right”, already exists. It has been proposed recently by Ganter and Kapranov ($\to$). that the right notion of 2-trace for 2-vector spaces is the functor which sends bimodules $B \otimes B^\vee$ to the 2-morphisms from the identity to $B \otimes B^\vee$:

They find that this trace is nothing but the Hochschild cohomology of $B \otimes_A B^\vee$ (see for instance this pdf for a quick definition of Hochschild cohomology for bimodules. Hm, I realize that I don’t understand precisely how the categorical trace leads to Hochschild cohomology here…).

Therefore, possibly

is the right condition for $B$ to admit a categorified spectral decomposition, and possibly it should be interpreted as saying that the bimodule Hochschild cohomologies of $B \otimes_A B^\vee$ and $B^\vee \otimes_A B$ coincide.

Clearly, this is indeed the case for instance for those duality defects that satisfy the condition that $B \otimes_A B^\vee$ is a direct sum of simple invertible bimodules by simply being equal to the identity bimodule. This are the so-called group-like defects.

If it also holds more generally, I don’t know yet. If it really implies a categorified spectral decomposition as in Hecke operator theory, I don’t know yet.