• 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
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Conformal Quantum Field Theories as Chain-complex valued TQFTs?

+ 4 like - 0 dislike

We will use the reference http://arxiv.org/abs/math/0412149v7.

Let \(\mathcal{M}\) be Segal's category of Riemann surfaces. Let \(C_{*}\) be a symmetric monoidal functor from \(\mathbf{Top}\) to the category of complexes of \(\mathbb{K} \)-vector spaces, with \(\mathbb{K}\) a base field of characteristic \(0\). Let \(\mathbf{Ch}_{n}\) denote the category of \(n \)-dimensional chain complexes. A CQFT is defined as a symmetric monoidal functor \(C_{*}(\mathcal{M})\to \mathbf{Ch}\). Note that \(C_{*}(\mathcal{M})=\mathbf{Ch}_{3}\), so the CQFT is a functor \(\mathbf{Ch}_{3}\to\mathbf{Ch}\). The CQFT maps \(\mathbf{Ch}_{3}\) to some subset \(\mathrm{Ch}_3\subset\mathbf{Ch}_{3}\). Using Lurie's classification allows us to define a CTQFT (a TQFT with the structure of a CQFT) as a functor \(\mathcal{C}:3\mathrm{Cob}\to\mathbf{Ch}_3\). This is a chain-complex valued TQFT, hence allowing us to conclude that

Idea: A CTQFT is a chain-complex valued TQFT. I.e., let \(\widetilde{3\mathrm{Cob}}\) denote the \((\infty,1)\) category with objects as two-dimensional manifolds, morphisms as bordisms, \(2\)-morphisms as diffeomorphisms, and \(3\)-morphisms as isotopies. Then, the functor \(\mathcal{C}:\widetilde{3\mathrm{Cob}}\to\mathbf{Ch}_3\bigcap n\mathrm{Hilb}\) is a CTQFT.

My question is, can this idea be extended to the more general class of CQFTs? I.e., can we get Idea 2?

Proposed Idea 2: A CQFT is a chain-complex valued (T)QFT, in the sense mentioned above.

asked Apr 2, 2014 in Mathematics by SDevalapurkar (285 points) [ no revision ]

Hi Sanath Devalapurkar; nice to see you participating here! 

Interesting idea... hope you get an answer soon!  

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