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A gluing formula for the $Z_2$-valued index of odd symmetric operators

Maxim Braverman, Ahmad Reza Haj Saeedi Sadegh, Junrong Yan

TL;DR

This work develops a $\mathbb{Z}_2$-valued index theory for odd symmetric (tau) Dirac-type operators on involutive manifolds with boundary. It proves a Z2-splitting formula that relates the $\tau$-index on a closed manifold to a boundary-value problem on the cut manifold and shows that, when the cut hypersurface separates the manifold, the $\mathbb{Z}_2$-index equals the mod $2$ reduction of the usual Atiyah-Patodi-Singer index on one side, yielding a cohomological expression. The authors construct a robust framework of boundary operators, adapted boundary data, and hybrid Sobolev spaces, and they develop a deformation/extension-operator toolkit to prove the splitting theorem in the tau-setting. In the product-case, they further derive a cohomological formula for the $\tau$-index, connecting it to the integral of $\hat{A}(TM)\,\operatorname{ch}(E/S)$ corrected by a half-dimension term of the kernel, with implications for bulk-edge correspondences of $\mathbb{Z}_2$ invariants.

Abstract

We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index.

A gluing formula for the $Z_2$-valued index of odd symmetric operators

TL;DR

This work develops a -valued index theory for odd symmetric (tau) Dirac-type operators on involutive manifolds with boundary. It proves a Z2-splitting formula that relates the -index on a closed manifold to a boundary-value problem on the cut manifold and shows that, when the cut hypersurface separates the manifold, the -index equals the mod reduction of the usual Atiyah-Patodi-Singer index on one side, yielding a cohomological expression. The authors construct a robust framework of boundary operators, adapted boundary data, and hybrid Sobolev spaces, and they develop a deformation/extension-operator toolkit to prove the splitting theorem in the tau-setting. In the product-case, they further derive a cohomological formula for the -index, connecting it to the integral of corrected by a half-dimension term of the kernel, with implications for bulk-edge correspondences of invariants.

Abstract

We investigate Dirac-type operator on involutive manifolds with boundary with symmetry, which forces the index of to vanish. We study the secondary -valued index of elliptic boundary value problems for such operators. We prove a -valued analog of the splitting theorem: the -valued index of an operator on a closed manifold equals the -valued index of a boundary value problem on a manifold obtained by cutting along a hypersurface . When divides into two disjoint submanifolds and , the -valued index on is equal to the mod 2 reduction of the usual -valued index of the Atiyah-Patodi-Singer boundary value problem on . This leads to a cohomological formula for the -valued index.
Paper Structure (35 sections, 10 theorems, 112 equations)

This paper contains 35 sections, 10 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Boundary value problems for Dirac-type operators
  3. A Dirac-type operator
  4. A boundary-defining function
  5. An adapted operator on the boundary
  6. The product case
  7. Sobolev spaces on the boundary
  8. The hybrid Soblev spaces
  9. Boundary value problems
  10. Atiyah-Patodi-Singer boundary conditions
  11. The Atiyah-Patodi-Singer index theorem
  12. The splitting theorem
  13. The $\tau$-index
  14. An anti-linear involution
  15. Odd symmetric operators
  16. ...and 20 more sections

Key Result

Lemma 2.9

Over ${\partial M}$, the isomorphism $c(dt):E_{{\partial M}}\to E_{{\partial M}}$ induces an isomorphism $\hat{H}(A^+)\to\check{H}(A^-)$. In particular, the sesquilinear form is a perfect pairing of topological vector spaces.

Theorems & Definitions (24)

  • Remark 2.4
  • Definition 2.8
  • Lemma 2.9
  • Definition 2.11
  • Remark 2.12
  • Definition 2.13
  • Remark 2.16
  • Theorem 2.18
  • Definition 3.2
  • Definition 3.4
  • ...and 14 more