Cyclic Cohomology, the Resolvent Trace, and Form (12)
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2026
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| _version_ | 1866901648485384192 |
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| author | Buchanan, Paul |
| author_facet | Buchanan, Paul |
| contents | <p>Form (12) of the MNZI programme states: kappa = 0 if and only if lim_{lambda->0+} lambda * Tr((Delta_J + lambda)^{-1}) = 0, where Delta_J = A*A is the J-Laplacian and the trace is the ordinary Hilbert-space trace. We show that this resolvent limit is precisely the pairing of the cyclic cocycle Phi_J (associated to the J-trace) with the unit of the algebra generated by the Eisenstein scattering operator. Form (12) is therefore the statement that the cyclic cocycle Phi_J annihilates the identity — a condition in Connes' cyclic cohomology that is equivalent to the J-index vanishing. We prove the cocycle property of Phi_J, compute the pairing explicitly for the Eisenstein family, and show that the pairing <Phi_J, 1> = 0 unconditionally from the prime number theorem.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19160793 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Cyclic Cohomology, the Resolvent Trace, and Form (12) Buchanan, Paul cyclic cohomology, resolvent trace, J-Laplacian, Eisenstein scattering, prime number theorem, noncommutative geometry <p>Form (12) of the MNZI programme states: kappa = 0 if and only if lim_{lambda->0+} lambda * Tr((Delta_J + lambda)^{-1}) = 0, where Delta_J = A*A is the J-Laplacian and the trace is the ordinary Hilbert-space trace. We show that this resolvent limit is precisely the pairing of the cyclic cocycle Phi_J (associated to the J-trace) with the unit of the algebra generated by the Eisenstein scattering operator. Form (12) is therefore the statement that the cyclic cocycle Phi_J annihilates the identity — a condition in Connes' cyclic cohomology that is equivalent to the J-index vanishing. We prove the cocycle property of Phi_J, compute the pairing explicitly for the Eisenstein family, and show that the pairing <Phi_J, 1> = 0 unconditionally from the prime number theorem.</p> |
| title | Cyclic Cohomology, the Resolvent Trace, and Form (12) |
| topic | cyclic cohomology, resolvent trace, J-Laplacian, Eisenstein scattering, prime number theorem, noncommutative geometry |
| url | https://doi.org/10.5281/zenodo.19160793 |