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I have created an entry modular equivariant elliptic cohomology.
The subject barely exists, for the moment the entry is to serve two purposes:
first, to highlight that by results of Mahowald-Rezk, Lawson-Naumann, Hill-Lawson this exists as a rather compelling generalization of KR-theory;
second, that the close the relation of KR-theory to type II string theory on orientifolds has previously been conjectured to correspond in the lift of the latter to F-theory to a modular equivariant universal elliptic cohomology.
So while the subject hasn’t been studied yet (it seems), both its construction and plenty of motivation for it already exists. And now also an $n$Lab entry for it does. :-)
added a section Definition extracting the main statement, theorem 9.1, from (Hill-Lawson 13)
I have expanded a bit more:
added more details to the statement of the Hill-Lawson construction of the modular-equivariant $tmf$-spectrum;
added a paragraph on plausibility checks from physics that indeed in F-theory one might expect a “genuine equivariant” cohomology version of S-duality, acting both on spacetime and on the worldsheet. Here I am still looking for more, currently I have
which discusses in detail how target space S-duality is accompanied by a conformal transformation on the worldsheet. So this is qualitatively exactly what the equivariant-spectrum story suggests, but I haven’t tried to check that it is also quantitatively (i.e.in actual detail) the same.
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