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...espondence between Broadhurst and Deligne [84] resulting e.g. in ref. [85], work of Belkale–Brosnan [86] or more recently by Brown [87] and others. One should also mention work of Connes and Marcolli =-=[88]-=- in this direction. – 5 – thereof [26, 27]. As harmonic polylogarithms are just a special case of the multiple polylogarithms introduced at the beginning of this section, all HPL’s through weight thre...

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...mately related to twisted spectral triples and we refer to [8, 10, 9, 27, 26, 21] for detailed discussions. Also it is closely related to the spectral action computations in the presence of a dilaton =-=[5, 6]-=-. For noncommutative four tori T4Θ, the scalar curvature is computed in [15] and it is shown that flat metrics are the critical points of the analog of the Einstein-Hilbert action. Also noncommutative...

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...ility). The operator Q is called the supercharge operator, its square the Hamiltonian. Remark 3.2. Restricting to the even subalgebra A+ of A, the above definition is essentially Connes [6] (see also =-=[7]-=-) definition of a (even) spectral triple (A+, H,Q). This is the fundamental object for index theorems and evaluating on K-theory elements. In this case the supercharge Q is traditionally called Dirac ...

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.../ is the Dirac operator on the spinor bundle. For even dimensional spin-manifolds there exists a grading γ on L2(M,S). A spectral triple can have additional structure such as reality. Definition 2.3 (=-=[11]-=-, Definition 1.124). A real structure on a spectral triple (A,H, D) is an anti-unitary operator J : H → H, with the property that J2 = ε, JD = ε′DJ, and Jγ = ε′′γJ, (even case), where the numbers ε,ε′...

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... construction is intimately related to idempotent analysis 5 (cf. [13], [16]). In fact our analogue in characteristic one of the Witt construction provides the inverse process of the “dequantization” of idempotent analysis, a fact which justifies the word “quantization” appearing in the title of this paper. In conclusion we conjecture that the extension Run of R is the natural home for the “values” of the many ~-dependent physical quantities arising in quantum field theory. This fits with the previous understanding of renormalization from the Riemann-Hilbert correspondence (cf. [4], [5], [8], [9]). 3. Sum of Teichmuller representatives Our goal in this section is to recall a formula which goes back to Teichmuller [20], for the sum of Teichmuller representatives in the context of strict p-rings. We begin by recalling briefly the simplest instance of the polynomials with integral coefficients which express the addition in the Witt theory (We refer to [19] and [17] for a quick introduction to that theory). One defines the polynomials with integral coefficients Sn(x, y) ∈ Z[x, y] by the equality (1− tx)(1− ty) = ∞∏ n=1 (1− Sn(x, y)tn) (28) The first few are of the form S1 = x+ y S2 = −...

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...[70]. There is a statistical mechanics interpretation of the Riemann zeta function as a partition function given in Bost and Connes [40], with subsequent developments described in Connes and Marcolli =-=[64]-=-. For some other views on arithmetical dynamical systems related to zeta functions, see Lagarias [190], [192] and Lapidus [197, Chapter 5]. Here we note the coincidence that in formulations of the Rie...

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.... . . . . . . group scheme ofn-th roots of unity (n integer) Part A. QSM-ISOMORPHISM OF NUMBER FIELDS 1. Isomorphism of QSM systems We recall some definitions and refer to [7], [11], and Chapter 3 of =-=[12]-=- for more information and for some physics background. After that, we introduce isomorphism of QSM-systems, and prove they preserve KMS-states (cf. infra). 1.1. Definition. A quantum statistical mecha...