“extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?

According to the Atiyah-Segal axioms, a d-dimensional TFT is a symmetric monoidal functor

\begin{equation*}\text{Bord}_d\rightarrow\mathcal{C}\end{equation*}

where the target category $\mathcal{C}$ is a symmetric monoidal category, typically $\text{Vect}$. The objects of $\text{Bord}_d$ are closed $(d-1)$-manifolds and map to the vector spaces of $\text{Vect}$ under the TFT. The morphisms are d-dimensional bordisms of the closed $(d-1)$-manifolds and map to the linear maps of $\text{Vect}$.

This description can be "extended down" to define a d-dimensional n-extended TFT as a symmetric monoidal $n$-functor

\begin{equation*}\text{Bord}_d^n\rightarrow\mathcal{C}\end{equation*}

where $\mathcal{C}$ is now an symmetric monoidal $n$-category. The objects of $\text{Bord}_d^n$ are $(d-n)$-manifolds (for a "fully extended" TFT, i.e. $n=d$, the bordism objects are points) and map to the objects of $\mathcal{C}$, which can be thought of as $(n-1)$-categories. One should really specify $\mathcal{C}$ as some extension of $\text{Vect}$, but I am ignoring this technicality by thinking of the n-category of all (small) $(n-1)$-categories. More generally, $k$-morphisms are $(d-n+k)$-dimensional bordisms between $(d-n+k-1)$-manifolds and map to $(n+k-1)$-categories. There are also technicalities surrounding how one manages "manifolds with corners" in $\text{Bord}_d^n$, but allow me to gloss over them here.

Meanwhile, one uses higher categories to decorate manifolds with defects. In this picture, $k$-dimensional manifolds are decorated with $k$-dimensional (extended?) TFTs called "$k$-defects." Lower dimensional submanifolds can be decorated with "defects within defects." The $k$-defect assigned to a $k$-dimensional "boundary" between two $(k+1)$-dimensional regions, each with attached $(k+1)$-defects, amounts to a morphism of $(k+1)$-defects. Fusion of defects and sub-defects endows the set of $k$-defects with the structure of a $k$-category. (For details, see http://arxiv.org/abs/1002.0385.) If we again imagine the $k$-category of (small) $(k-1)$-categories, we can understand decoration as an assignment of $(k-1)$-categories to $k$-manifolds.

This construction of TFT with defects feels "upside down" compared to extended TFT: $k$-manifolds are decorated by $(k-1)$-categories and map to $(d-k-1)$-categories under the TFT functor. I realize that decoration is not a functor from a bordism category, but is it a functor in some other sense (from some category where higher degree morphisms are lower dimensional submanifolds)? Defects may have an interpretation as inserted operators (Wilson loops, surface operators, etc); can these two formalisms be combined to compute path integrals with operator insertions in an extended theory? In general, I am curious about the relation between how higher categories are used to define an extended TFT and how they are used to characterize TFTs with defects.

Here is a particular problem from http://arxiv.org/abs/1309.1489. Consider a fully extended 4D TFT. The TFT functor assigns $1$-categories to Riemann surfaces $\Sigma$. Meanwhile, in the defect description, 2D TFTs form a $2$-category. Restricting ourselves to a single object leaves us with a symmetric monoidal category of $1$-dimensional domain walls (essentially boundary conditions) of the 2D theory. Is this category of boundary conditions identified with the category assigned by the TFT functor (as is claimed)? It seems that this identification is only possible since $\dim\Sigma=2=\text{codim }\Sigma$ in dimension four and does not reflect a general relation between extended TFT and TFT with defects.

• I think you want the cobordism hypothesis with singularities (Theorem 4.3.11 in arxiv.org/abs/0905.0465). – Qiaochu Yuan Oct 15 '14 at 6:08
• Yes, representations of cobordisms with singularities encode TFTs with boundaries and defects. For "pre-quantum" field theory this is discussed in sections 3.9.14.4 to 3.9.14.6 of arxiv.org/abs/1310.7930 (improved version in preparation at ncatlab.org/schreiber/show/Local+prequantum+field+theory). For quantization of this: arxiv.org/abs/1402.7041 . This is based on discussion with Domenico Fiorenza and Alessandro Valentino that recently appeared as arxiv.org/abs/1409.5723 – Urs Schreiber Oct 15 '14 at 12:28
• Forget about TQFTs for a moment and consider an $n$-category $C$. To a $k$-morphism $x$ of $C$ we can, of course, associate the $k$-morphism $x$ itself. We can also associate to $x$ the $(n-k)$-category of endomorphisms of $x$ in $C$. The "upside-down-ness" you are noticing is closely related to this familiar example. – Kevin Walker Oct 15 '14 at 14:29
• @KevinWalker Following your hint, the upside-down can be turned rightside-up, and the degrees and dimensions work out; however, I am still unclear about how to identify the data of the TFT with the defect data. For a d-dim theory Bord->C, the defects on a k-fold M form a k-category, the objects (defect theories) of which might be interpreted as k-morphisms of C. Endomorphisms in C of a defect theory form a (n-k-1)-category of boundary conditions (or an (n-k)-category with one object) of the defect theory. Somehow this might be identified with the (n-k-1)-category assigned to M by the TFT. – Alex Turzillo Oct 20 '14 at 1:35
• Perhaps the following question needs to be answered first: a defect theory (an object of the k-category of defects that decorates a k-fold) is itself a TFT. How is this object realized as a TFT? In particular, given a d-dim theory Bord->C, how does one characterize the (d-1)-dim TFTs on boundaries (or domain walls)? And how are these boundary theories related to the defect decoration? – Alex Turzillo Oct 20 '14 at 1:43