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The Z_2-valued spectral flow of a symmetric family of Toeplitz operators

Maxim Braverman, Ahmad Reza Haj Saeedi Sadegh

TL;DR

The paper develops a $oldsymbol{Z}_2$-valued spectral flow theory for $ au$-invariant, odd-symmetric operator families and proves the fundamental equality $sf_{ au}A = ext{ind}_{ au}D_A$ for suspension-type constructions. It then applies this framework to Toeplitz operators on complete manifolds via Callias-type operators, yielding a generalized bulk-edge correspondence and a $oldsymbol{Z}_2$-valued bulk-edge theorem. In even dimensions, the correspondence reduces to a boundary Dirac index on a hypersurface, including pseudo-convex domains where degeneracy is controlled; this recovers the Graf-Porta model for 2D topological insulators with time-reversal symmetry. The results connect half-spectral flow on Toeplitz families to $ au$-indices of Callias-type operators and provide a robust analytical tool for bulk-edge phenomena in topological phases. Overall, the work extends classical index/spectral-flow correspondences to a $oldsymbol{Z}_2$-valued setting with substantial implications for topological insulators and related geometric-analytic frameworks.

Abstract

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary $\mathbb{Z}_2$-valued spectral flow of such families. We prove an analog of the Atiyah-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of $A(t)$ is equal to the secondary $\mathbb{Z}_2$-valued index of the suspension operator $\frac{d}{dt}+A(t)$. Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel's index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII.

The Z_2-valued spectral flow of a symmetric family of Toeplitz operators

TL;DR

The paper develops a -valued spectral flow theory for -invariant, odd-symmetric operator families and proves the fundamental equality for suspension-type constructions. It then applies this framework to Toeplitz operators on complete manifolds via Callias-type operators, yielding a generalized bulk-edge correspondence and a -valued bulk-edge theorem. In even dimensions, the correspondence reduces to a boundary Dirac index on a hypersurface, including pseudo-convex domains where degeneracy is controlled; this recovers the Graf-Porta model for 2D topological insulators with time-reversal symmetry. The results connect half-spectral flow on Toeplitz families to -indices of Callias-type operators and provide a robust analytical tool for bulk-edge phenomena in topological phases. Overall, the work extends classical index/spectral-flow correspondences to a -valued setting with substantial implications for topological insulators and related geometric-analytic frameworks.

Abstract

We consider families of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary -valued spectral flow of such families. We prove an analog of the Atiyah-Singer-Robbin-Salamon theorem, showing that this secondary spectral flow of is equal to the secondary -valued index of the suspension operator . Applying this result, we show that the graded secondary spectral flow of a symmetric family of Toeplitz operators on a complete Riemannian manifold equals the secondary index of a certain Callias-type operator. In the case of a pseudo-convex domain, this leads to an odd version of the secondary Boutet de Monvel's index theorem for Toeplitz operators. When this domain is simply a unit disc in the complex plane, we recover the bulk-edge correspondence for the Graf-Porta module for 2D topological insulators of type AII.
Paper Structure (53 sections, 23 theorems, 129 equations)

This paper contains 53 sections, 23 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. The $\tau$-index
  3. An anti-linear involution
  4. Odd symmetric operators
  5. Graded odd symmetric operators
  6. The $\mathbb{Z}_2$-valued index
  7. Quaternionic vector bundles
  8. Odd symmetric elliptic operators
  9. The spectral flow and the Robbin-Salamon theorem
  10. Some spaces of operators
  11. The spectral flow
  12. The index and the spectral flow
  13. A periodic family of operators
  14. The half-spectral flow of a family of operators on real line
  15. An anti-linear anti-involution
  16. ...and 38 more sections

Key Result

Lemma 2.1

An anti-linear anti-involution $\tau$ does not have non-zero fixed vectors, i.e. $\tau x=x$ iff $x=0$. Further, if $\tau$ acts on a finite-dimensional space $H$, then $\dim H$ is even. Moreover, there exists a subspace $L\in H$ such that $H=L\oplus \tau L$.

Theorems & Definitions (40)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 3.2
  • Remark 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 4.1
  • ...and 30 more