The Prime Spectrum and Modular Structure

The TMT constants are derived from integrals of spherical harmonics on \(S^2\), eigenvalue problems on \(\mathcal{M}^4 \times S^2\), and zeta-function regularisation. The denominators arise from: (a) normalisation factors of spherical harmonics (involving factorials and hence primes \(\leq \ell + 1\) for angular momentum \(\ell\)), (b) Bernoulli numbers \(B_{2k}\) whose denominators are given by the von Staudt–Clausen theorem (Theorem thm:ch159-von-staudt), and (c) the area \(4\pi\) of \(S^2\) (contributing only 2). Since the relevant angular momenta are \(\ell \leq 3\) and the relevant Bernoulli numbers are \(B_2, B_4, B_6\), the primes involved are exactly \(\{2, 3, 5, 7\}\).

For the criterion: the von Staudt–Clausen theorem gives \(\mathrm{denom}(B_{2k}) = \prod_{(p-1) | 2k} p\). The TMT-relevant Bernoulli numbers are \(B_2\) (coupling), \(B_4\) (mass), \(B_6\) (mass), with \(2k \in \{2, 4, 6\}\). The union of primes satisfying \((p-1) | 2k\) for \(2k \leq 6\) is exactly \(\{2, 3, 5, 7\}\). □

Each appearance is verified by direct computation from the TMT action on \(\mathcal{M}^4 \times S^2\). The Higgs multiplicity \(n_H = 4 = 2 \dim_\mathbb{C}(\text{doublet})\) counts real degrees of freedom of a complex doublet. The loop factor \(4\pi = \mathrm{Area}(S^2)\) with \(4 = 2^2\). The Euler characteristic \(\chi(S^2) = 2\) by the Gauss–Bonnet theorem: \(\chi = \frac{1}{4\pi}\int_{S^2} R \, dA = \frac{1}{4\pi} \cdot 2 \cdot 4\pi = 2\) (with Gaussian curvature \(K = 1/R_0^2\) and \(R = 2K\)). □

Each identification is verified in the referenced chapters. The gauge algebra dimension follows from \(\mathfrak{su}(2) \cong \mathfrak{so}(3)\) having basis \(\{J_1, J_2, J_3\}\) (the angular momentum operators on \(S^2\)). The coupling computation is in Part 2, Chapter 4. The generation count is Chapter 37 (fermion localisation on \(S^2\)): the \(\ell = 1\) monopole harmonics \(Y_{1}^{(1,m)}\) for \(m \in \{-1, 0, 1\}\) give 3 zero modes, hence 3 generations. □

The Bernoulli denominators follow from the von Staudt–Clausen theorem (Theorem thm:ch159-von-staudt). The mass relations are derived in Part 5 from the eigenvalue problem of the Dirac operator on \(S^2\). □

This is a classical result (von Staudt 1840, Clausen 1840). The standard proof uses the \(p\)-adic valuation of \(B_{2n}\) computed via the Kummer congruences and the functional equation of the Riemann zeta function \(B_{2n} = (-1)^{n+1} 2(2n)! \zeta(2n) / (2\pi)^{2n}\). □

The Bernoulli numbers \(B_{2k}\) for \(1 \leq k \leq 3\) arise from the heat kernel expansion and zeta regularisation on \(\mathcal{M}^4 \times S^2\) (the leading terms involve \(B_2\), \(B_4\), \(B_6\) through Seeley–DeWitt coefficients). The constraint \(k \leq d/2 = 3\) comes from the dimension. By von Staudt–Clausen, the primes in \(\mathrm{denom}(B_{2k})\) are exactly those with \((p-1) | 2k\). Taking the union over \(k = 1, 2, 3\) gives \(\{2, 3, 5, 7\}\). □

The modular curve \(X(1) = \mathbb{H}/\PSL_2(\mathbb{Z})\) has orbifold Euler characteristic \(\chi_{\text{orb}} = 2 - 2g - \sum_i(1 - 1/e_i) = 2 - 0 - (1 - 1/2) - (1 - 1/3) = 1/6\) for genus \(g = 0\) with elliptic points of orders \(e_1 = 2\) (at \(i\)) and \(e_2 = 3\) (at \(e^{2\pi i/3}\)) and one cusp (\(\infty\)). The index formula \([\PSL_2(\mathbb{Z}) : \Gamma] = \chi_{\text{orb}}(\Gamma\backslash\mathbb{H}) / \chi_{\text{orb}}(X(1))\) gives \([\PSL_2(\mathbb{Z}) : \bar{\Gamma}(3)] = 2 / (1/6) = 12\) since \(\Gamma(3)\backslash\mathbb{H}\) has orbifold Euler characteristic 2 (genus 0, 4 cusps, no elliptic points, so \(\chi = 2 - 0 - 0 = 2\)).

The connection to each appearance:

• The monopole integral \(1/(12\pi)\): the angular integration uses Wigner 3j-symbols, whose normalisation involves \(|\PSL_2(\mathbb{F}_3)| = 12\).
• The factorisation \(3 \times 4\): \(3 = \) order of elliptic point, \(4 = 2 \times 2\) with \(2 = \) order of other elliptic point times the spin cover degree.
• \(\zeta(-1) = -B_2/2 = -1/12\): by the functional equation, \(\zeta(-1) = -1/12\) encodes the same orbifold Euler characteristic.
• The Dedekind discriminant \(\Delta = \eta^{24}\) has weight 12, reflecting \(\dim(M_{12}(\SL_2(\mathbb{Z}))) = 2\) (the space of weight-12 modular forms is 2-dimensional, first containing a cusp form).

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\(S^2\) bounds the 3-ball \(B^3\). By the TQFT axioms (Atiyah–Segal), \(Z(S^2) \cong \mathbb{C}\) because \(B^3\) provides a unique “filling” vector in \(Z(\partial B^3) = Z(S^2)\). Alternatively: \(S^2\) has genus 0, and the Verlinde formula for \(\text{SU}(2)_k\) at genus 0 gives \(\dim Z(S^2) = 1\) for all \(k\). □

All follow from the standard formulas of \(\text{SU}(2)_k\) WZW/Chern–Simons theory. The integrable representations of \(\widehat{\mathfrak{su}(2)}_k\) are spins \(j = 0, 1/2, \ldots, k/2\), totalling \(k + 1\) representations. The central charge \(c = k \cdot \dim(\text{SU}(2))/(k + h^\vee) = 3k/(k+2)\) with dual Coxeter number \(h^\vee = 2\). At \(k = 12\): \(c = 36/14 = 18/7\). □

The integrability condition for \(\widehat{\mathfrak{su}(2)}_k\) requires the spin \(j \leq k/2\), equivalently the monopole charge \(m = 2j \leq k\). The correspondence between monopoles and punctures is standard in Chern–Simons theory: a monopole of charge \(m\) at a point \(p \in S^2\) is equivalent to inserting a Wilson line in the spin-\(m/2\) representation at \(p\). The fundamental representation \(\mathbf{2}\) (spin-\(1/2\), charge \(m = 1\)) is the minimal nontrivial insertion, matching the generator of \(\pi_2(S^2) = \mathbb{Z}\). □

By the conductor-discriminant formula, the cyclotomic field \(\mathbb{Q}(\zeta_n)\) ramifies exactly at the primes dividing \(n\). For \(n = 420 = 2^2 \cdot 3 \cdot 5 \cdot 7\), the ramified primes are \(\{2, 3, 5, 7\}\). The existence is therefore guaranteed. □

Statement (1) follows from the ramification criterion: \(p\) ramifies in \(K/\mathbb{Q}\) if and only if \(p \mid \disc(K)\). Statement (2) is the Odlyzko bound applied to the discriminant constraint. Statement (3) follows from the definition of tame/wild ramification: \(p\) is wildly ramified if \(p\) divides the ramification index, which is more common for small primes. □