Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action

Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 2 April 2026 V1 Latest version Share on Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action Author : David Alfyorov 0009-0003-6027-7837 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.177516440.00576784/v1 268 views 94 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract We prove that the traced Seeley-DeWitt coefficients tr(a 2n) for the Dirac operator on a Ricci-flat four-manifold decompose as tr(a 2n) = f n (p) + f n (q), where p = |C + | 2 and q = |C − | 2 are the squared norms of the self-dual and anti-self-dual Weyl tensors. The proof rests on a chirality theorem: the spin connection generators σ rs = 1 4 [γ r, γ s ] commute with the chirality operator γ 5 in four dimensions, rendering the curvature endomorphism Ω µν block-diagonal in the chiral basis. We establish a crossed chirality assignment-the left-handed spinor block couples exclusively to the anti-self-dual Weyl tensor, and the right-handed block to the self-dual partusing the 't Hooft symbol decomposition. At the quartic level (mass dimension 8), the theorem implies that tr(a 8) contains only the combination c(p 2 + q 2) with zero pq cross-term, fixing the ratio (C 2) 2 : (* CC) 2 = 1:1 exactly. The result extends to the full Standard Model spectral triple on the almost-commutative geometry M × F. By a Molien series analysis, the three parity-even quartic Weyl invariants reduce first to two via the Cayley-Hamilton identity, and then to one effective structure via chirality. This reduces the three-loop finiteness problem for the spectral action from an overdetermined system to a single ratio condition. We show that the strict spectral renormalizability conjecture for gravity is disproven: the graviton propagator is chirality-blind (a scalar on the traceless-transverse sector), and multi-eigenvalue sums at two loops and above generate pq cross-terms outside the spectral function space. The precise obstruction is the multi-trace structure of divided differences in the van Nuland-van Suijlekom framework applied to gravity, where the metric modifies the Dirac operator itself rather than entering as an inner fluctuation. However, the failure is quantitatively negligible: the cross-terms are suppressed by α 3 C /(16π 2) 3 ≈ 3.2 × 10 −10, and the spectral action provides effective UV finiteness within the perturbative regime to this precision. All algebraic results are verified by independent numerical computation on ensembles of random Weyl tensors with precision ≤ 10 −12 . Supplementary Material File (sct_chirality_theorem.pdf) Download 286.75 KB Information & Authors Information Version history V1 Version 1 02 April 2026 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords cayley-hamilton identity chirality dirac operator extended dirac operator heat kernel molien series noncommutative geometry seeley-dewitt coefficients self-dual and anti-self-dual decomposition spectral action standard model three-loop finiteness weyl tensor Authors Affiliations David Alfyorov 0009-0003-6027-7837 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 268 views 94 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation David Alfyorov. Chirality of the Seeley-DeWitt coefficients and quartic Weyl structure in the spectral action. Authorea . 02 April 2026. 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