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starting an entry on the integer Heisenberg group.
For the moment it remains telegraphic as far as the text is concerned (no Idea-section)
but it contains a slick (I find) computation of the modular transformation of Chern-Simons/WZW states from the manifest modular automorphy of certain integer Heisenberg groups.
Hope to beautify this entry a little more tomorrow (but won’t have much time, being on an intercontinental flight) or else the days after (where I am however at a conference, but we’ll see).
I don’t have time but there might be connections to Ganter’s categorical tori, in how she contracts the crossed module from a lattice that is the kernel of exp, and a central extension.
Not sure what specifically you are thinking of?
Maybe I can amplify that the mere existence of the integer/discrete Heisenberg group is immediate and classical. The point that I meant to record here is that its evident action by the modular group (which is still obvious) induces a modular action on its irrep which is the famous modular functor of abelian Chern-Simons theory that is no longer that obvious and traditionally derived only over pages of analysis of geometric quantization.
This is an observation which I have not seen discussed anywhere. But since its proof is so elementary (as recorded in the entry), I won’t be surprised if this has been noted somewhere before. A reference would be welcome.
in the proof of the modular equivariance of the integer Heisenberg representation (this prop) I have added the previously missing argument that the claimed representation of the modular group really is one in the first place.
In fact, for this to hold I get that the T-operator (here) needs to be rescaled by a funny prefactor ck which seems to be missing in literature like Manoliu 1998a p 67 (?)
finally found that the “level=2” version of the integer Heisenberg group, with its modular action, is discussed in:
Răzvan Gelca, Alejandro Uribe: From classical theta functions to topological quantum field theory [arXiv:1006.3252, slides pdf, GelcaUribe-ThetaFunctionsTQFT.pdf:file]
Răzvan Gelca, Alastair Hamilton: Classical theta functions from a quantum group perspective, New York J. Math. 21 (2015) 93–127 [arXiv:1209.1135, nyjm:j/2015/21-4]
Răzvan Gelca, Alastair Hamilton: The topological quantum field theory of Riemann’s theta functions, Journal of Geometry and Physics 98 (2015) 242-261 [doi:10.1016/j.geomphys.2015.08.008, arXiv:1406.4269]
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