3 Examples
This chapter records concrete spectral triples built directly on the Fourier side, bypassing spin-geometry and Sobolev-space machinery from the literature, together with standalone Fredholm-index examples that illustrate Section 2.1 directly. The circle and torus examples share the same diagonal-operator technology (Section 3.1): the circle example (Section 3.2) is a complete, finitely summable, odd spectral triple; the two-torus example (Section 3.3) is a complete, finitely summable, even spectral triple with vanishing index. Sections 3.4 and 3.5 give two further examples with nonzero Fredholm index, \(-1\) and \(k\) respectively, completing the index arc \(0, -1, k\). Sections 3.6 and 3.7 connect the abstract flux-\(k\) model of Section 3.5 back to genuine geometry: together they give a complete, purely algebraic dimension count \(\dim H^0(L_k) = k\), \(\dim H^1(L_k) = 0\) for the degree-\(k\) line bundle on the square torus, with no index theorem and no \(L^2\) analysis. Section 3.8 begins the remaining operator bridge (M3c/M1/M4): the Gaussian-weighted orthogonality of the Hermite polynomials, the foundation of the Hermite-function \(L^2(\mathbb {R})\) basis needed for the Landau/Hermite decomposition route.
3.1 Block-diagonal operators on \(\ell ^2\)
Given a family of finite-dimensional Hilbert spaces \((G_i)_{i \in \alpha }\) and a uniformly bounded family of block operators \(T_i \colon G_i \to G_i\), this section builds the associated block-diagonal operator on \(\ell ^2(\alpha ; G)\) and proves the compactness criterion used to show the Dirac resolvents in Sections 3.2 and 3.3 are compact.
Given a uniformly bounded family of operators \(T_i \colon G_i \to G_i\) with \(\| T_i\| \le C\) for all \(i\), the block-diagonal operator \(\operatorname {diagL}(T) \colon \ell ^2(\alpha ; G) \to \ell ^2(\alpha ; G)\) is defined by \((\operatorname {diagL}(T) a)_i = T_i(a_i)\).
\(\| \operatorname {diagL}(T)\| \le C\) whenever \(\| T_i\| \le C\) for all \(i\).
If only finitely many blocks \(T_i\) are nonzero, then \(\operatorname {diagL}(T)\) is finite-rank, hence compact.
If \(\| T_i\| \to 0\) along the cofinite filter on \(\alpha \), then \(\operatorname {diagL}(T)\) is a compact operator. The finitely-supported truncations of Theorem 28 converge to \(\operatorname {diagL}(T)\) in operator norm.
3.2 The Dirac triple of the circle \(S^1\)
Status: complete. A fully formalized, odd, finitely summable spectral triple on the circle. The algebra representation is generated by a single Fourier shift unitary; since the Dirac eigenvalues are additive in the shift, the commutator of the shift with \(D\) collapses to a scalar multiple of the shift itself, simplifying the boundedness argument relative to the torus case (Section 3.3), which uses the same diagonal-operator technology but with matrix-valued blocks.
The model is \(H = \ell ^2(\mathbb {Z})\), the Fourier side of \(L^2(S^1)\), with Dirac operator \(D = -i\, d/d\theta \) acting diagonally on the orthonormal Fourier basis \((e_n)_{n \in \mathbb {Z}}\) by \(D e_n = n \cdot e_n\), with maximal domain the Sobolev space \(H^1 = \{ a \in \ell ^2(\mathbb {Z}) : \sum n^2 |a_n|^2 {\lt} \infty \} \).
The operator \(D \colon H^1 \to \ell ^2(\mathbb {Z})\), \((D a)_n = n \cdot a_n\).
The circle Dirac operator is self-adjoint.
The resolvent \(R(i, D)\) is a compact operator on \(\ell ^2(\mathbb {Z})\). It is the diagonal operator \(b_n \mapsto b_n / (i - n)\), with block norms \(1/|i-n| \to 0\), via Theorem 29.
The star-subalgebra of \(\mathcal{B}(\ell ^2(\mathbb {Z}))\) generated by the single Fourier shift unitary \(W_1\) (the Fourier image of the coordinate function \(e^{i\theta }\) on \(S^1\)), represented by inclusion into \(\mathcal{B}(\ell ^2(\mathbb {Z}))\).
The circle data \((A, \ell ^2(\mathbb {Z}), \pi , D)\), with \(A\) the algebra of Definition 34, is an odd spectral triple. Since the Dirac eigenvalues are additive in the shift index, the commutator \([D, W_g]\) collapses to the scalar multiple \(g \cdot W_g\) of the shift itself; this bound is propagated through the generated star-subalgebra by StarAlgebra.adjoin_induction.
The circle spectral triple is finitely summable: its Dirac operator has compact resolvent at \(i\).
3.3 The Dirac triple of the two-torus \(T^2\)
Status: complete. This is the flagship example: a fully formalized, even, finitely summable spectral triple on the flat torus \(T^2 = \mathbb {R}^2/\mathbb {Z}^2\), whose graded-kernel index is computed and shown to vanish, matching the Atiyah–Singer prediction for the trivial line bundle.
The Hilbert space is \(H = \ell ^2(\mathbb {Z}^2; \mathbb {C}^2)\) (square-summable spinor-valued sequences over the Fourier lattice \(\mathbb {Z}^2\)). On the Fourier mode \((m, n)\) the Dirac operator acts on the spinor fibre \(\mathbb {C}^2\) by the self-adjoint block
with eigenvalues \(\pm 2\pi \sqrt{m^2+n^2}\); the chirality grading is \(\gamma = \sigma _3 = \operatorname {diag}(1,-1)\) fibrewise.
The block-diagonal operator \(D \colon H^1 \to H\), \((D a)_{(m,n)} = D_{(m,n)} (a_{(m,n)})\), on the Sobolev domain \(H^1\).
The torus Dirac operator is self-adjoint.
The resolvent \(R(i, D)\) is a compact operator on \(H\). It is exhibited explicitly as a block-diagonal operator with block norms tending to \(0\) (since \(|D_{(m,n)}| \to \infty \)), via Theorem 29.
The block-diagonal operator \(\gamma \) acting fibrewise by \(\sigma _3 = \operatorname {diag}(1,-1)\).
\(\gamma \) is self-adjoint, \(\gamma ^2 = 1\), and \(D\gamma \xi = -\gamma D\xi \) for \(\xi \in \operatorname {dom}(D)\): each Dirac block is off-diagonal and \(\sigma _3\) is diagonal, so they anticommute fibrewise.
The star-subalgebra of \(\mathcal{B}(H)\) generated by the two coordinate shift unitaries \(W_{(1,0)}, W_{(0,1)}\) (the Fourier image of the trigonometric polynomials \(\mathbb {C}[\mathbb {Z}^2]\) on \(T^2\)), represented by inclusion into \(\mathcal{B}(H)\).
The torus data \((A, H, \pi , D)\), with \(A\) the algebra of Definition 42, is an odd spectral triple; together with the grading \(\gamma \) of Definition 40, it is an even spectral triple. The bounded-commutator condition is checked by propagating a uniform commutator bound for the two generators \(W_{(1,0)}, W_{(0,1)}\) through the generated star-subalgebra.
The torus spectral triple is finitely summable: its Dirac operator has compact resolvent at \(i\).
\(\operatorname {index}(D, \gamma ) = 0\) for the flat torus Dirac triple: the graded kernel of \(D\) is exactly the zero Fourier mode, with \((\ker D)^{+}\) and \((\ker D)^{-}\) each spanned by one of the two spinor basis vectors at that mode, so \(\dim (\ker D)^{+} = \dim (\ker D)^{-} = 1\). This matches the Atiyah–Singer prediction for the trivial (degree-\(0\)) line bundle on \(T^2\).
3.4 The unilateral shift on \(\ell ^2(\mathbb {N})\)
Status: complete. A direct illustration of the Fredholm index machinery (Section 2.1) on a single bounded operator, independent of the spectral-triple framework: the classical forward unilateral shift on \(\ell ^2(\mathbb {N})\), with Fredholm index \(-1\).
The forward unilateral shift \(S\) on \(\ell ^2(\mathbb {N})\): \((Sx)_n = x_{n-1}\) for \(n \geq 1\) and \((Sx)_0 = 0\).
\(\operatorname {ind}(S) = -1\). The shift is injective (\(\ker S = 0\)) and its range is closed with one-dimensional orthogonal complement spanned by the zeroth basis vector \(e_0\), so \(\operatorname {ind}(S) = \dim (\ker S) - \dim (\operatorname {coker} S) = 0 - 1 = -1\).
3.5 The flux-\(k\) magnetic Dirac model
Status: complete (model level). A Landau-level / magnetic-translation model of the flux-\(k\) Dirac operator on \(T^2\), after reduction to \(\ell ^2(\mathbb {N}) \otimes \mathbb {C}^k\), with Fredholm index exactly \(k\). The unitary equivalence between this lowest-Landau-level model and the geometric PDE on \(T^2\) (the degree-\(k\) line bundle) is not formalized here and is deferred analysis; the content proved is that the index is \(k\), and that the \(k\)-dimensional kernel carries the expected magnetic-translation (finite Weyl) structure certifying it as the flux-\(k\) degeneracy.
The backward shift on the Landau-level index \(\mathbb {N}\), tensored with the identity on the \(k\)-dimensional guiding-center factor \(\mathbb {C}^k\): an operator \(D_k\) on \(\ell ^2(\mathbb {N}) \otimes \mathbb {C}^k\).
\(\dim \ker (D_k) = k\): the kernel is exactly the lowest Landau level, identified with \(\mathbb {C}^k\).
\(\operatorname {ind}(D_k) = k\): the operator is surjective (trivial cokernel), so the index equals the kernel dimension.
The magnetic clock \(\hat C\) and cyclic shift \(\hat S\) operators on the guiding-center factor \(\mathbb {C}^k\), acting by the phase \(\omega _k = e^{2\pi i / k}\) and by cyclic permutation respectively.
The magnetic clock and shift satisfy the finite Weyl relation \(\hat C \hat S = \omega _k \hat S \hat C\), and both commute with \(D_k\) — certifying that the \(k\)-dimensional kernel of Theorem 49 carries the genuine flux-\(k\) Heisenberg degeneracy.
3.6 Theta sections on the square torus
Status: complete (lower bound). This section proves the lower-bound half of the square-torus flux-\(k\) Landau-level computation directly on the geometric side: \(k\) explicit holomorphic theta sections for the degree-\(k\) line bundle on the square torus \(\mathbb {C} / (\mathbb {Z} + i \mathbb {Z})\) are exhibited and shown linearly independent, giving \(\dim \ker D^{+} \geq k\) for the geometric Dirac operator twisted by the degree-\(k\) bundle (as opposed to the magnetic-translation model of Section 3.5). The matching upper bound \(\dim \ker D^{+} \leq k\) and the vanishing of the cokernel are completed in Section 3.7. The resulting identification of the geometric index with the model index \(k\) (Theorem 50) — which additionally needs the \(L^2\)-to-holomorphic bridge (elliptic regularity for \(\bar\partial \), not in Mathlib) and the unitary equivalence to the magnetic model — is deferred; see SpectralTriples/docs/INDEX_PAIRING.md for the roadmap.
For \(k \geq 1\) and \(a \in \{ 0, \dots , k-1\} \), the theta section
built from Mathlib’s two-variable Jacobi theta function \(\vartheta \) at modulus \(\tau = ki\).
Each \(\theta _{k,a}\) is differentiable (holomorphic) on \(\mathbb {C}\).
\(\theta _{k,a}(z+1) = \theta _{k,a}(z)\) (periodicity under the lattice generator \(1\)) and \(\theta _{k,a}(z+i) = e^{-\pi i k (2z+i)} \, \theta _{k,a}(z)\) (the degree-\(k\) automorphy factor under the generator \(i\)): together, these are the defining quasi-periodicity conditions for a holomorphic section of the degree-\(k\) line bundle on \(\mathbb {C}/(\mathbb {Z}+i\mathbb {Z})\).
\(\theta _{k,a}(z + 1/k) = \omega _k^{a} \, \theta _{k,a}(z)\), where \(\omega _k = e^{2\pi i / k}\): translation by \(1/k\) diagonalizes the theta sections, with the same \(k\)-th roots of unity that appear in the magnetic clock operator of Definition 51.
The \(k\) theta sections \(\theta _{k,0}, \dots , \theta _{k,k-1}\) are linearly independent over \(\mathbb {C}\): they are eigenvectors of translation by \(1/k\) with pairwise distinct eigenvalues \(\omega _k^{0}, \dots , \omega _k^{k-1}\) (the \(k\) distinct \(k\)-th roots of unity), hence independent.
3.7 The Fourier-coefficient recursion: an exact dimension count
Status: complete (M3a, M3b). This section completes the dimension count begun in Section 3.6 (M2) by an entirely algebraic route, with no index theorem and no \(L^2\) analysis: the upper bound \(\dim H^0(L_k) \leq k\) (M3a) and the cokernel vanishing \(\dim H^1(L_k) = 0\) (M3b). Both rest on a single new analytic lemma not in Mathlib — the contour shift — from which a Fourier-coefficient recursion is derived algebraically. See SpectralTriples/docs/INDEX_PAIRING.md for the full roadmap; the remaining gap (M3c/M4: the \(L^2\)-to-holomorphic bridge and the unitary equivalence to the magnetic model of Section 3.5) is the genuinely analytic frontier and is not addressed here.
For an entire \(1\)-periodic function \(f\), the period integral \(\int _0^1 f(x+iy)\, dx\) does not depend on the height \(y\). (Cauchy–Goursat on the rectangle \([0,1] \times [y_1,y_2]\): the two vertical sides cancel by periodicity, leaving the two horizontal integrals equal.)
\(H^0(L_k)\): the space of entire functions \(f\) with \(f(z+1) = f(z)\) and the degree-\(k\) automorphy factor \(f(z+i) = e^{-\pi i k(2z+i)} f(z)\) — the same quasi-periodicity conditions satisfied by the theta sections of Definition 53.
For a \(1\)-periodic function \(f\), the Fourier coefficient \(a_m = \int _0^1 f(x)\, e^{-2\pi i m x}\, dx\), computed on the real period.
For \(f \in H^0(L_k)\), the Fourier coefficients satisfy
Comparing Fourier coefficients of the degree-\(k\) automorphy relation, via the contour shift (Theorem 58) relating \(f\)’s Fourier coefficients on different horizontal lines, turns the quasi-periodicity condition into this purely algebraic recursion: the whole coefficient sequence is determined by \((a_0, \dots , a_{k-1})\).
If \(f\) is entire and \(1\)-periodic with all Fourier coefficients \(a_m = 0\), then \(f = 0\): lifting \(f\) to the circle, Mathlib’s Fourier completeness gives \(f\) vanishes on \(\mathbb {R}\), and the identity theorem for the entire function \(f\) then gives \(f \equiv 0\) on \(\mathbb {C}\).
\(H^0(L_{-k})\): the same definition as \(H^0(L_k)\) (Definition 59) but with the opposite-sign automorphy factor \(f(z+i) = e^{\pi i k(2z+i)} f(z)\). By Serre duality this represents \(H^1(L_k)\), the cokernel of the degree-\(k\) Dirac operator.
For any entire \(1\)-periodic \(f\), the Fourier coefficients \(a_m \to 0\) as \(m \to \infty \) (and as \(m \to -\infty \)): a consequence of \(\ell ^2\)-summability of \(|a_m|^2\) (Parseval, via the \(L^2\) lift to the circle).
\(H^0(L_{-k}) = 0\) for \(k {\gt} 0\) (equivalently \(\dim H^1(L_k) = 0\), the cokernel-vanishing half of \(\operatorname {index} = k\), via Serre duality \(h^1(L_k) = h^0(L_k^{-1})\)). The opposite-sign automorphy gives the opposite-sign recursion \(a_{m+k} = e^{\pi (2m+k)} a_m\), whose growth factor has modulus \({\gt} 1\); this clashes with the Parseval decay of Lemma 66 unless every coefficient is already \(0\), and then Theorem 62 gives \(f = 0\).
3.8 The Hermite functions: Gaussian-weighted orthogonality
Status: partial (orthogonality only). This section is the foundation of Route B for the M3c/M1/M4 operator bridge of Section 3.7 (see SpectralTriples/docs/INDEX_PAIRING.md): the recommended next step there is to build the Hermite functions \(h_n(x) = c_n \cdot H_n(x) \cdot e^{-x^2/2}\) as an orthonormal basis of \(L^2(\mathbb {R})\), the gating lemma for the Landau/Hermite decomposition that identifies the geometric magnetic Dirac operator with the already-formalized model (Section 3.5). Mathlib has the probabilists’ Hermite polynomials and the Rodrigues identity, but neither the Gaussian-weighted orthogonality integral nor the \(L^2\) basis. This section proves the orthogonality integral; the normalization constants \(c_n\) and the completeness of \(\{ h_n\} \) in \(L^2(\mathbb {R})\) (needed for the HilbertBasis) remain to be built.
The probabilists’ Hermite polynomials satisfy \(H_{n+1}' = (n+1) \cdot H_n\). (Mathlib has the three-term recursion \(H_{n+1} = X H_n - n H_{n-1}\) but not this derivative form.)
Any polynomial \(p\) times the Gaussian weight \(e^{-x^2/2}\) is integrable on \(\mathbb {R}\).
For \(m \neq n\), \(\int _{\mathbb {R}} H_m(x) H_n(x) e^{-x^2/2}\, dx = 0\). From the Rodrigues identity \((H_n \cdot w)' = -H_{n+1} \cdot w\) (with \(w = e^{-x^2/2}\)) and a single integration by parts, the weighted pairing of a polynomial \(P\) against \(H_{n+1}\) satisfies \(\langle P, H_{n+1}\rangle = \langle P', H_n\rangle \); iterating shows \(\deg P \leq n\) forces \(\langle P, H_{n+1}\rangle = 0\), which gives off-diagonal vanishing without \(n\)-fold integration by parts.
\(\int _{\mathbb {R}} H_n(x)^2 e^{-x^2/2}\, dx = n! \sqrt{2\pi }\). The same integration-by-parts recursion gives \(\langle H_{n+1}, H_{n+1}\rangle = (n+1) \langle H_n, H_n \rangle \) (Lemma 68), reducing to the base case \(\int _{\mathbb {R}} e^{-x^2/2}\, dx = \sqrt{2\pi }\) (the Gaussian integral).
\(\int _{\mathbb {R}} H_m(x) H_n(x) e^{-x^2/2}\, dx = n! \sqrt{2\pi } \cdot \delta _{mn}\), combining the off-diagonal and diagonal cases.