Timeline for 1d TQFT minus connection =?
Current License: CC BY-SA 4.0
6 events
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| May 29, 2020 at 17:26 | comment | added | Adrien | As for your last question, rephrasing what Mike said, giving an HQFT, i.e. a functor $\prod_1(X)\rightarrow Vect$, is the same as giving a vector space $V_x$ for each point $x \in X$, and an iso $V_x \cong V_y$ for every path between $x$ and $y$ which depends only on the homotopy class of that path in a way compatible with composition, so this induces a locally constant sheaf (aka locall system) on $X$ in a fairly tautological way (every choice of a contractible $U$ containing $x,y$ identifies $V_x$ and $V_y$ canonically). | |
| May 29, 2020 at 11:30 | comment | added | Adrien | In general you can always talk about representations of the fundamental groupoid, which are the same as locally constant sheaves on $X$. | |
| May 29, 2020 at 11:15 | comment | added | mme | A flat connection on a bundle is a lift of the structure group from $G$ to $G^\delta$, the group made discrete. (This is a topological notion, no smoothness needed.) This is equivalent to a representation $\pi_1(X) \to G$ which gives rise to a $G$-bundle isomorphic to your given bundle. | |
| May 29, 2020 at 9:40 | comment | added | მამუკა ჯიბლაძე | Many thanks for the correction, and the last part is especially interesting! So, what is a flat vector bundle over a non-smooth manifold? And, could you please indicate (say, give a reference for) how to reconstruct it from an $X$-HQFT? | |
| May 29, 2020 at 9:17 | history | edited | Adrien | CC BY-SA 4.0 |
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| May 29, 2020 at 9:12 | history | answered | Adrien | CC BY-SA 4.0 |