Timeline for "extended TQFT" versus "TQFT with defects"
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 20, 2014 at 3:09 | comment | added | Kevin Walker | 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. | |
| Oct 20, 2014 at 1:43 | comment | added | 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? | |
| Oct 20, 2014 at 1:35 | comment | added | Alex Turzillo | @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. | |
| Oct 15, 2014 at 14:29 | comment | added | Kevin Walker | 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. | |
| Oct 15, 2014 at 12:28 | comment | added | Urs Schreiber | 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 | |
| Oct 15, 2014 at 6:08 | comment | added | Qiaochu Yuan | I think you want the cobordism hypothesis with singularities (Theorem 4.3.11 in arxiv.org/abs/0905.0465). | |
| Oct 15, 2014 at 2:07 | history | asked | Alex Turzillo | CC BY-SA 3.0 |