• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,079 questions , 2,229 unanswered
5,348 answers , 22,758 comments
1,470 users with positive rep
819 active unimported users
More ...

  "extended TQFT" versus "TQFT with defects"

+ 6 like - 0 dislike

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


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


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.

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Alex Turzillo
asked Oct 15, 2014 in Theoretical Physics by Alex Turzillo (30 points) [ no revision ]
retagged Nov 15, 2014
I think you want the cobordism hypothesis with singularities (Theorem 4.3.11 in arxiv.org/abs/0905.0465).

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Qiaochu Yuan
Yes, representations of cobordisms with singularities encode TFTs with boundaries and defects. For "pre-quantum" field theory this is discussed in sections to 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

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Urs Schreiber
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.

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Kevin Walker
@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.

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Alex Turzillo
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?

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Alex Turzillo
One approach to this is discussed in detail at arxiv.org/abs/1009.5025 . See sections 2 and 6, and the remark about defects near the start of section 6.7.

This post imported from StackExchange MathOverflow at 2014-11-15 11:29 (UTC), posted by SE-user Kevin Walker

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights