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started some minimum at vacuum amplitude. Briefly mentioned relation to a) generating functionals for correlators and b) to zeta functions and c) to expected evanishing in supersymmetric theories
Remarked that in view of b) and c) one is tempted to expect some relation between 1-loop vacuum amplitudes of supersymmetric field/string theories with the Riemann hypothesis. Added a pointer to the article ACER 11 which seems to find just that.
If anyone has further pointers to literature relating vanishing of susy 1-loop vacuum amplitudes and (generalized) Riemann hypotheses, please drop me a note.
related PO question
Did you look at who’s citing the ACER paper, e.g., these?
Not yet, am out of time and battery for the moment. I am hoping somebody might point me to a specifically clean insight. But otherwise I’ll try to dig around, of course.
Eluding SUSY at every genus on stable closed string vacua, Sergio L. Cacciatori, Matteo A. Cardella (arxiv) suggests there have been extensions of ACER asymptotics to higher genus.
the error term turns out to be remarkably connected to the Riemann hypothesis [CC1],[CC2]
CC1] S. Cacciatori M. Cardella, “Equidistribution rates, closed string amplitudes, and the Riemann hypothesis,” JHEP12 (2010) 025. [CC2] S. L. Cacciatori and M. A. Cardella, “Uniformization, Unipotent Flows and the Riemann Hypothesis,” arXiv:1102.1201 [math.NT].
Thanks, David, for all these pointers.
I did now look at ACER11 in a bit more detail, the statement there (the other references you list revolve around the same statement) is that by the Rankin-Selberg-Zagier method the partition function of a (super-)string is usefully expanded for small proper time as a constant (the vacuum energy) plus a sum of decaying oscillatory terms, one for each non-trivial zero of the Riemann zeta function. By estimates on the decay strength this implies certain asymptotic vanishing results, too.
This is nice (I made a note at Riemann hypothesis and physics) but it is not yet quite what I was hoping for. I was thinking that since the vacuum amplitude itself is like a zeta function, there should be a “generalized Riemann hypothesis” applying to it directly, and describe its vanishing directly.
Maybe I am off here, or maybe it just hasn’t materialized. But also this is not going to be my high priority for the moment, it was just an thought that occured to me while fine-tuning the table of analogies.
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