Dependencies

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 operator \(D \colon H^1 \to \ell ^2(\mathbb {Z})\), \((D a)_n = n \cdot a_n\).

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 \((\operatorname {diagL}(T) a)_i = T_i(a_i)\).

For a (partially defined) linear operator \(D \colon \operatorname {dom}(D) \to \mathcal{H}\), the kernel \(\ker D \subseteq \mathcal{H}\) is the subspace of vectors \(\xi \in \operatorname {dom}(D)\) with \(D\xi = 0\).

An even spectral triple \((A, \mathcal{H}, \pi , D, \gamma )\) consists of an odd spectral triple \((A, \mathcal{H}, \pi , D)\) together with a bounded operator \(\gamma \in \mathcal{B}(\mathcal{H})\) (the grading operator) such that:

• \(\gamma \) is self-adjoint: \(\gamma ^* = \gamma \);

• \(\gamma \) is an involution: \(\gamma ^2 = 1\);

• \(\gamma \) commutes with \(\pi (a)\) for every \(a \in A\);

• \(\gamma \) maps \(\operatorname {dom}(D)\) into \(\operatorname {dom}(D)\);

• \(D \gamma \xi = -\gamma D \xi \) for every \(\xi \in \operatorname {dom}(D)\).

An odd spectral triple \((A, \mathcal{H}, \pi , D)\) is finitely summable if the resolvent \(R(i, D)\) at the imaginary unit \(i \in \Bbbk \) is a compact operator on \(\mathcal{H}\). By Theorem 11, the hypothesis \(i \in \rho (D)\) holds automatically whenever \(\operatorname {Im}(i) \neq 0\) (e.g. when \(\Bbbk = \mathbb {C}\)), and is not required as part of the definition.

The Fredholm index of a linear map \(f \colon E \to F\) is the integer

The (super)index of an even Dirac datum \((D, \gamma )\) is the integer

This is the Fredholm index of the chiral operator \(D^{+}\) from Definition 24 in the special case \(p = 1\). The corresponding invariant of an even spectral triple is .

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.

\(H^0(L_k)\): 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.

\(H^0(L_{-k})\): 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.

Let \((A, \mathcal{H}, \pi , D, \gamma )\) be an even spectral triple and let \(p \in M_n(A)\) be a self-adjoint idempotent (a projection representing a class in \(K_0(A)\)). The index pairing of \(p\) with \((A, \mathcal{H}, \pi , D, \gamma )\) is the Fredholm index

where \(D^{+}\) denotes the restriction of \(D\) to the \((+1)\)-eigenspace of \(\gamma \).

A linear map \(f \colon E \to F\) is Fredholm if \(\ker f\) is finite-dimensional, the range of \(f\) is closed, and the range of \(f\) has finite codimension (the cokernel \(F / \operatorname {ran} f\) is finite-dimensional).

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 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 \(\ell ^2(\mathbb {N}) \otimes \mathbb {C}^k\).

Let \(\mathcal{H}\) be a Hilbert space over \(\Bbbk \in \{ \mathbb {R}, \mathbb {C}\} \), let \(A\) be a \(*\)-algebra over \(\Bbbk \), and let \(\pi \colon A \to \mathcal{B}(\mathcal{H})\) be a \(*\)-algebra homomorphism. A possibly unbounded linear operator \(D \colon \operatorname {dom}(D) \to \mathcal{H}\), with \(\operatorname {dom}(D) \subseteq \mathcal{H}\) a linear subspace, together with the data \((A, \mathcal{H}, \pi , D)\), is an odd spectral triple if:

• \(D\) is self-adjoint (in particular, \(\operatorname {dom}(D)\) is dense and \(D\) is closed);

• for every \(a \in A\), \(\pi (a)\) maps \(\operatorname {dom}(D)\) into \(\operatorname {dom}(D)\);

• for every \(a \in A\), the commutator \([D, \pi (a)]\), viewed as an operator on \(\operatorname {dom}(D)\), is bounded on the closed unit ball of \(\operatorname {dom}(D)\).

For \(z \in \rho (D)\), the resolvent \(R(z, D) \colon \mathcal{H}\to \mathcal{H}\) is the (everywhere-defined, linear) inverse of \(z - D\).

Let \(D \colon \operatorname {dom}(D) \to \mathcal{H}\) be a (partially defined) linear operator, where \(\operatorname {dom}(D) \subseteq \mathcal{H}\) is a linear subspace. For \(z \in \Bbbk \), write \(z - D\) for the partially defined linear map \(\xi \mapsto z\xi - D\xi \) with domain \(\operatorname {dom}(D)\). The resolvent set \(\rho (D) \subseteq \Bbbk \) of \(D\) is the set of \(z \in \Bbbk \) for which \(z - D \colon \operatorname {dom}(D) \to \mathcal{H}\) is bijective.

The forward unilateral shift \(S\) on \(\ell ^2(\mathbb {N})\): \((Sx)_n = x_{n-1}\) for \(n \geq 1\) and \((Sx)_0 = 0\).

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 \(\tau = ki\).

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 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 block-diagonal operator \(\gamma \) acting fibrewise by \(\sigma _3 = \operatorname {diag}(1,-1)\).

If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then \(\operatorname {dom}(D)\) is dense in \(\mathcal{H}\).

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.)

If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then for every \(a \in A\) there is a real constant \(C \geq 0\) such that

for every \(\xi \in \operatorname {dom}(D)\) with \(\| \xi \| \leq 1\).

If a continuous linear map on an inner product space has finite-dimensional range, so does its adjoint.

A bijective linear map with closed range is Fredholm, of index \(0\).

If \((A, \mathcal{H}, \pi , D, \gamma )\) is an even spectral triple and \(\xi \in \ker D\), then \(\gamma \xi \in \ker D\). Consequently \(\ker D\) splits as \(\ker D = (\ker D)^{+} \oplus (\ker D)^{-}\) into the \((\pm 1)\)-eigenspaces of \(\gamma \) restricted to \(\ker D\).

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 \((a_0, \dots , a_{k-1})\).

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).

If \(D\) is self-adjoint and \(\operatorname {Im}(z) \neq 0\), then \(z - D \colon \operatorname {dom}(D) \to \mathcal{H}\) is injective.

Any polynomial \(p\) times the Gaussian weight \(e^{-x^2/2}\) is integrable on \(\mathbb {R}\).

If \((A, \mathcal{H}, \pi , D)\) is an odd spectral triple, then \(D\) is a closed operator.

\(\| \operatorname {diagL}(T)\| \le C\) whenever \(\| T_i\| \le C\) for all \(i\).

If \(D\) is self-adjoint and \(z \in \Bbbk \), then for every \(\xi \in \operatorname {dom}(D)\),

For \(z \in \rho (D)\), the range of \(R(z, D)\) equals \(\operatorname {dom}(D)\).

\(\gamma \) 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.

If \(D\) is self-adjoint and \(\operatorname {Im}(z) \neq 0\), then \(z - D\) is in fact bijective, i.e. \(z \in \rho (D)\).

This strengthens Lemma 10 from injectivity to bijectivity: the range of \(z - D\) is closed (the bounded-below estimate makes preimage sequences Cauchy, and the graph of \(D\) is closed) and dense (its orthogonal complement is trivial, via the adjoint), hence all of \(\mathcal{H}\). In particular \(i \in \rho (D)\), so the resolvent-set hypothesis of a finitely summable spectral triple holds automatically at \(z = i\).

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 circle spectral triple is finitely summable: its Dirac operator has compact resolvent at \(i\).

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.

\(i \in \rho (D)\), by the basic criterion of self-adjointness (Theorem 11) applied to Theorem 31.

Each \(\theta _{k,a}\) is differentiable (holomorphic) on \(\mathbb {C}\).

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 \(\mathbb {C}\).

A compact operator \(K\) on a Hilbert space is approximable, in operator norm, by finite-rank operators: for every \(\varepsilon {\gt} 0\) there is a finite-rank \(F\) with \(\| K - F\| \leq \varepsilon \). The approximant is built by composing \(K\) with the orthogonal projection onto the span of a finite \(\varepsilon \)-net of the (relatively compact) image of the closed unit ball.

If \(D\) is self-adjoint with compact resolvent at \(i\), then \(\ker D\) is finite-dimensional. Consequently \(\operatorname {index}(D, \gamma )\) (Definition 22) is a difference of genuine finite dimensions: for \(\xi \in \ker D\), the resolvent \(R(i, D)\) satisfies \(R(i,D)(i\xi ) = \xi \), so \(\ker D\) embeds into the \(i^{-1}\)-eigenspace of the compact operator \(R(i, D)\), which is finite-dimensional by Riesz theory.

\(\operatorname {ind}(D_k) = k\): the operator is surjective (trivial cokernel), so the index equals the kernel dimension.

\(\operatorname {ind}(S) = -1\). 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 \(\operatorname {ind}(S) = \dim (\ker S) - \dim (\operatorname {coker} S) = 0 - 1 = -1\).

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.

Combining the upper bound of Theorem 63 (M3a) with the lower bound of Theorem 57 (M2, the theta sections span a \(k\)-dimensional subspace) gives \(\dim H^0(L_k) = k\) exactly — a complete dimension theorem with no index theorem and no \(L^2\) analysis.

The restriction map \(f \mapsto (a_0, \dots , a_{k-1})\) is an injective linear map \(H^0(L_k) \to \mathbb {C}^k\): by the recursion (Lemma 61), \(a_0 = \cdots = a_{k-1} = 0\) forces every Fourier coefficient to vanish, hence \(f = 0\) by Theorem 62. Injectivity gives \(\dim H^0(L_k) \leq k\).

\(H^0(L_{-k}) = 0\) 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 \(f = 0\).

In the situation of Definition 24, the operator \(\pi (p) D^{+} \pi (p)\), restricted to \(\pi (p)\mathcal{H}\cap \operatorname {dom}(D)\), is Fredholm, so that \(\langle [p], (\mathcal{H}, \pi , D, \gamma ) \rangle \in \mathbb {Z}\) is well defined.

If \(K\) is a compact operator on a Hilbert space, then its adjoint \(K^{\dagger }\) is also compact: the adjoints of finite-rank approximants of \(K\) (Theorem 16) are themselves finite-rank (Lemma 17), hence compact, and converge to \(K^{\dagger }\) in norm since the adjoint operation is norm-preserving.

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.

If only finitely many blocks \(T_i\) are nonzero, then \(\operatorname {diagL}(T)\) is finite-rank, hence compact.

The structural part of the classical Riesz–Schauder theorem: if \(K\) is compact on a Hilbert space, then \(1 - K\) is Fredholm. The kernel is finite-dimensional (it is the eigenspace of \(K\) at the eigenvalue \(1\)); the range is closed, via a bounded-below estimate on the orthogonal complement of the kernel established by a contradiction argument using compactness of \(K\); and the cokernel is finite-dimensional, via the adjoint identity \((1-K)^{\dagger } = 1 - K^{\dagger }\) (Theorem 18) together with \((\operatorname {ran}(1-K))^{\perp } = \ker ((1-K)^{\dagger })\). (The further classical fact that the index is \(0\) is not proved here.)

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.

\(\dim \ker (D_k) = k\): the kernel is exactly the lowest Landau level, identified with \(\mathbb {C}^k\).

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.)

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.

\(\theta _{k,a}(z+1) = \theta _{k,a}(z)\) (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 torus Dirac operator is self-adjoint.

\(\operatorname {index}(D, \gamma ) = 0\) 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 \(T^2\).

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 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 \(i\).