Finite volume calculation of \(K\)-theory invariants.

*(English)*Zbl 1384.46048The pairing between a \(K\)-theory and a \(K\)-homology class is defined as the index of a certain Fredholm operator built from the data. This usually involves operators on infinite-dimensional vector spaces. This article identifies this index, in an important and typical case, with one half times the signature of a certain invertible matrix. This signature may be efficiently computed numerically. Such numerical computations are useful to study the topological phases of topological insulators. Here one also needs analogous index pairings in the even case and in various real cases. The latter may take values in \(\mathbb Z\) or \(\mathbb Z/2\). The authors briefly discuss how to carry over their method to these situations. In the special case \(d=1\), they also link their result to \(\eta\)-invariants and the spectral flow of a certain path of self-adjoint operators.

The \(K\)-homology class treated in the article is the fundamental class of the \(d\)-torus \(\mathbb T^d\) for odd \(d\). Let \(\Gamma_1,\dotsc,\Gamma_d\in \mathbb M_k(\mathbb C)\) be anti-commuting, self-adjoint unitaries, so that they generate a representation of the Clifford algebra. Let \(X_1,\dotsc,X_d\) be the position operators on \(\ell^2(\mathbb Z^d,\mathbb C^k)\), which multiply pointwise with the coordinate functions. The Dirac operator on the \(d\)-torus is unitarily equivalent to the self-adjoint operator \(\sum_{j=1}^d X_j \Gamma_j\) on \(\ell^2(\mathbb Z^d,\mathbb C^k)\). This operator pairs with elements in the odd \(K\)-theory of the torus \(\mathbb T^d\). These are given by continuous functions \(A: \mathbb T^d \to \mathrm{Gl}_N(\mathbb C)\). Let \(P\) be the spectral projection for \(D\) and the interval \([0,\infty)\). Let \(P_N = P \otimes 1_{\mathbb C^N}\) and \(A_k = A \otimes 1_{\mathbb C^k}\) be the operators on \(\ell^2(\mathbb Z^d,\mathbb C^{N\cdot k})\) obtained by taking a direct sum of several copies of \(P\) and \(A\), respectively. The pairing of the \(K\)-homology class of \(D\) with the \(K\)-theory class of \(A\) is the index of the Fredholm operator \(P_N A_k P_N + 1-P_N\). We may also start with \(A\) and \(D\) both acting on the same Hilbert space \(\ell^2(\mathbb Z^d,\mathbb C^k)\). But then we must assume that the values of \(A\) commute with the Clifford matrices \(\Gamma_j\) to ensure that \([P,A]\) is compact. This assumption is forgotten in the article.

Fix \(\kappa,\rho>0\). Let \(\mathbb D_\rho\subseteq\mathbb Z^d\) be the subset of all points with \(\| x\|\leq\rho\). Let \(L_{\kappa,\rho}\) be the compression of the operator \[ \begin{pmatrix} \kappa D_N&A_k\\A_k^*&-\kappa D_N \end{pmatrix} \] to the subspace \(\ell^2(\mathbb D_\rho)\otimes \mathbb C^{2 N k}\). This is a finite matrix. If \([A,D]\) is bounded and \(\kappa^{-1}\) and \(\kappa\cdot\rho\) are sufficiently large, then the signature of the matrix \(L_{\kappa,\rho}\) is twice the index of \(P_N A_k P_N + 1-P_N\). The lower bounds on \(\kappa^{-1}\) and \(\kappa\cdot\rho\) are explicit and depend only on the norms of \([D,A]\), \(A\) and \(A^{-1}\).

The \(K\)-homology class treated in the article is the fundamental class of the \(d\)-torus \(\mathbb T^d\) for odd \(d\). Let \(\Gamma_1,\dotsc,\Gamma_d\in \mathbb M_k(\mathbb C)\) be anti-commuting, self-adjoint unitaries, so that they generate a representation of the Clifford algebra. Let \(X_1,\dotsc,X_d\) be the position operators on \(\ell^2(\mathbb Z^d,\mathbb C^k)\), which multiply pointwise with the coordinate functions. The Dirac operator on the \(d\)-torus is unitarily equivalent to the self-adjoint operator \(\sum_{j=1}^d X_j \Gamma_j\) on \(\ell^2(\mathbb Z^d,\mathbb C^k)\). This operator pairs with elements in the odd \(K\)-theory of the torus \(\mathbb T^d\). These are given by continuous functions \(A: \mathbb T^d \to \mathrm{Gl}_N(\mathbb C)\). Let \(P\) be the spectral projection for \(D\) and the interval \([0,\infty)\). Let \(P_N = P \otimes 1_{\mathbb C^N}\) and \(A_k = A \otimes 1_{\mathbb C^k}\) be the operators on \(\ell^2(\mathbb Z^d,\mathbb C^{N\cdot k})\) obtained by taking a direct sum of several copies of \(P\) and \(A\), respectively. The pairing of the \(K\)-homology class of \(D\) with the \(K\)-theory class of \(A\) is the index of the Fredholm operator \(P_N A_k P_N + 1-P_N\). We may also start with \(A\) and \(D\) both acting on the same Hilbert space \(\ell^2(\mathbb Z^d,\mathbb C^k)\). But then we must assume that the values of \(A\) commute with the Clifford matrices \(\Gamma_j\) to ensure that \([P,A]\) is compact. This assumption is forgotten in the article.

Fix \(\kappa,\rho>0\). Let \(\mathbb D_\rho\subseteq\mathbb Z^d\) be the subset of all points with \(\| x\|\leq\rho\). Let \(L_{\kappa,\rho}\) be the compression of the operator \[ \begin{pmatrix} \kappa D_N&A_k\\A_k^*&-\kappa D_N \end{pmatrix} \] to the subspace \(\ell^2(\mathbb D_\rho)\otimes \mathbb C^{2 N k}\). This is a finite matrix. If \([A,D]\) is bounded and \(\kappa^{-1}\) and \(\kappa\cdot\rho\) are sufficiently large, then the signature of the matrix \(L_{\kappa,\rho}\) is twice the index of \(P_N A_k P_N + 1-P_N\). The lower bounds on \(\kappa^{-1}\) and \(\kappa\cdot\rho\) are explicit and depend only on the norms of \([D,A]\), \(A\) and \(A^{-1}\).

Reviewer: Ralf Meyer (GĂ¶ttingen)