Timeline for Relation between Gerstenhaber bracket and Connes differential
Current License: CC BY-SA 3.0
7 events
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| Jul 9, 2011 at 11:57 | vote | accept | Kevin Walker | ||
| Jul 18, 2011 at 12:24 | |||||
| Jul 8, 2011 at 16:25 | comment | added | David Ben-Zvi | That sounds very reasonable.. I'm thinking within the full cobordism hypothesis framework, which requires the algebra smooth proper to get a framed 2d TFT (the smoothness and properness follows from allowing pictures corresponding to the disc and the saddle). I don't have a clear picture what TFT structures are allowed without full dualizability (eg what hypotheses you need on an algebra to define a noncompact framed theory). | |
| Jul 8, 2011 at 16:04 | comment | added | Kevin Walker | ...higher genus. The surfaces also have a foliation by oriented intervals, corresponding to the direction of "time". Homology classes in the topological space of such surfaces give operations (e.g. the four operations mentioned in the question), and one can deduce all the relations I mention above from easy homology calculations for these spaces of surfaces. I don't think I need to put any restrictions on the algebra C for all this to work, but perhaps I've overlooked something? | |
| Jul 8, 2011 at 15:58 | comment | added | Kevin Walker |
Yes, something like that. In my set-up, I consider circles divided into incoming and outgoing regions (instead of framings). HH^* corresponds to a circle composed of one incoming interval and one outgoing interval (i.e. a bigon). HH_* corresponds to the entire circle being outgoing (a 0-gon). In general one could have a circle with n incoming regions alternating with n outgoing regions (a 2n-gon). The 2-dimensional part of the TFT-ish structure is a colored operad, sort of like the little disks operad except that the circles are "colored" by 2n-gons and the surfaces could be...
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| Jul 8, 2011 at 15:32 | comment | added | David Ben-Zvi | Great! I haven't tried but this must follow formally from TFT, at least for smooth proper algebras (which is presumably how you arrived at it?) - i.e., we can act with one 2-framed circle (HH^*) on another (HH_*), and ask how the product interacts with rotation of the circle.. | |
| Jul 8, 2011 at 15:20 | comment | added | Kevin Walker |
Thanks, that does seem relevant. In particular, the middle part of equation (0.1) of that paper (which they state for polyvector fields and differential forms) is the same as the relation I'm asking about. Presumably the generalization to arbitrary Hochschild [co]homology follows from one of their results about G-infinity structures.
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| Jul 8, 2011 at 14:41 | history | answered | David Ben-Zvi | CC BY-SA 3.0 |