Grothendieck group of the category of boundary conditions of topological field theory In this paper also the journal front page, eq. 2.14, it introduces
the Grothendieck group $K^0$ of the category of boundary conditions of topological field theory.


My question is that

*

*what exactly is the Grothendieck group $K^0$ of the category of boundary conditions of topological field theory really mean?

*Does this say that the boundary conditions of topological field theory can be related or classified by a Grothendieck group? Is this an abelian group or nonabelian group? ($K^0(BCondμ(M^{d−1})  )=?$)

*Should the boundary conditions of topological field theory classified by some bimodule of certain modular tensor category? how is this related to a Grothendieck group?


 A: To understand the possible spaces of boundary conditions for a TQFT, it is helpful to start in highest dimension.
Suppose you have a $(d+1)$-dimensional nonanomalous TQFT $\mathcal Q$. (The anomalous story is a bit more complicated, because it means that lots of things you might want to write down are only well-defined up to phase factors.) Given a $d$-manifold $M^d$, it makes sense to ask about boundary conditions placed on $M$. Physically, you imagine a cylinder $M \times \mathbb{R}_{\geq 0}$, and ask that the "bulk" $M \times \mathbb{R}_{>0}$ be flooded with your TQFT $\mathcal Q$, and then you try to write down some valid rules for the "boundary values of fields" (at least if your TQFT does come to you with a description in terms of fluctuating quantum fields).
There are a few things you could do:

*

*You could impose some local conditions on the values of the fields at the boundary. If you think in terms of classical field theory for a moment, then the "bulk" field theory is some $(d+1)$-dimensional PDE that the fields need to solve (this PDE is called the equations of motion EOM), and you could demand that the restrictions of the fields (and their normal derivatives) satisfy some additional $d$-dimensional PDE. You could add some new "boundary fields" that couple to the bulk.

A typical example of this is in the Chern-Simons / WZW
correspondence. Classically, the bulk EOM say that you have a
$G$-bundle with flat connection. This is "topological": the flat connections are flat in any metric. The WZW boundary condition includes a field which is a section of this bundle, and the boundary EOM say that it is a holomorphic section. Holomorphicity is something that you can check locally: it is a PDE on the boundary.


*You could also impose some nonlocal conditions on the boundary. For example, you might ask that some field at some point in $M^d$ have value related in some way to the value of some field at some other far-away point in $M^d$. A more physical example occurs when you ask for the boundary values of a field to satisfy an integral equation, rather than a (partial) differential equation. For example, when $d+1=2$, a typical field theory might ask that the field be, say, a holomorphic function $f$. Given a circle $M^1 = S^1$, an interesting boundary value is to ask that the field extend holomorphically over the disk $D^2 \supset S^1$. Of course, this happens exactly when the negative Fourier modes of $f$ vanish. These Fourier modes are something like $\oint_{S^1} z^{-n} f(z) \mathrm{d}z$.
I bring up this distinction to emphasize that when people talk about the collection of "boundary conditions on $M$", they implicitly mean that nonlocal boundary conditions are allowed.
What type of thing is the collection of boundary conditions on $M^d$? Well, because the boundary conditions are allowed to be nonlocal, I can just as well work with the $(0+1)$-dimensional QFT produced by compactifying along $M$, i.e. $\int_{M}\mathcal{Q} := \mathcal{Q}(M \times -)$. But then we're just discussing boundary conditions for a (topological) quantum mechanics model, since that's what $(0+1)$-dimensional QFT is. A "boundary condition" is then just the same as a state in the model: it is a thing that you can prepare at time $0$ and then allow to propagate forward in time.
So the punchline is that the collection of boundary conditions on $M^d$ is a Hilbert space: specifically, it is the Hilbert space that $\mathcal{Q}$ assigns to $M^d$.
Ok, with that understood, now we can ask about a $(d-1)$-manifold $M^{d-1}$. If I tell you such a manifold, then you could look at the $d$-manifold $M^{d-1} \times \mathbb{R}$, if you want, and ask about boundary conditions that you can place on $M^{d-1} \times \mathbb{R}$. Well, that's not a very good question because of the noncompactness of $\mathbb{R}$ (you end up not defining elliptic operators). Rather, you could ask about boundary conditions placeable on $M^{d-1} \times \mathbb{R}$ which are local in the $\mathbb{R}$-direction. In other words, I will allow you to impose integral equations, but only if the integral is over something that only points in the $M^{d-1}$ direction. I also demand that the boundary conditions be invariant under translation in the $\mathbb{R}$-direction. I might forget to say "translation invariant" below, but I always want it.
Just as before, to understand the structure of this collection, you can compactify along $M$, and consider the now-$(1+1)$-dimensional (T)QFT $\int_{M^{d-1}}\mathcal{Q} := \mathcal{Q}(M^{d-1}\times-)$.
All I want to point out if that this collection is naturally a category. Specifically, suppose you have two local-in-the-$\mathbb{R}$-direction boundary conditions $X$ and $Y$ on $M^{d-1} \times \mathbb{R}$. Place one of them just on $M^{d-1} \times \mathbb{R}_{<0}$ and the other just on $M^{d-1} \times \mathbb{R}_{>0}$. Now you can ask to extend this configuration to something on all of $M^{d-1} \times \mathbb{R}$. Again I will allow you to write down expressions which integrate over $M^{d-1}$, if you want. The extension then looks like an "interface" between the boundary conditions. Anyway, this set of interfaces between $X$ and $Y$ will be $\hom(X,Y)$. If you place an interface at $M \times\{0\}$ and another one at $M \times \{1\}$, and then compactify out that interval $M \times [0,1] \to M$, then you get a (manifestly associative) composition of interfaces.
[Actually, the compactification is subtle, because the interface might secretly depend on a metric in the $\mathbb{R}$-direction. You can do it if the boundary conditions $X,Y$ are "topological in the $\mathbb{R}$-direction", but in general it is harder.]
Ok, fine, so this is why you have a "category of boundary conditions" assigned to $M^{d-1}$. And, just like we had Hilbert spaces in dimension $d$, we have actually linear categories in dimension $d-1$. This linearity is essentially the ability to perform "superpositions" in quantum theory. (In unitary quantum theory, it is in fact a Hilbert category.)
Now I can get to your actual question. First, of course, any linear category has an abelian group as its $K^0$. Let's understand it. Well, if you do have a local-in-the-$\mathbb{R}$-direction boundary condition on $M^{d-1} \times \mathbb{R}$, then, because it is local and translation invariant, it also defines a (partially local) boundary condition on $M^{d-1} \times S^1$. This assignment takes direct sums to sums.
In other words, there is a canonical map of abelian groups $K^0(\mathcal{Q}(M^{d-1})) \to \mathcal{Q}(M^{d-1} \times S^1)$. The codomain is the Grothendieck group of the category of local-and-translation-invariant-in-the-$\mathbb{R}$-direction boundary conditions on $M^{d-1} \times \mathbb{R}$, and the domain is the Hilbert space of boundary conditions on $M^{d-1} \times S^1$.
In many cases, this map induces an isomorphism $K^0(\mathcal{Q}(M^{d-1})) \otimes \mathbb{C} \cong \mathcal{Q}(M^{d-1} \times S^1)$.
This is true, for example, in any (extended) TQFT valued in any member of the Bestiary of 2-vector spaces. You can prove this simply by compactifying down to a $(1+1)$-dimensional TQFT, and then proving a universal statement for $(1+1)$d TQFTs. Cases where it fails have some pathology (nonunitarity or noncompactness or...) that make them less "physical".
