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Dai-Freed theorem and topological phases of matter

Kazuya Yonekura

TL;DR

This work connects the Dai-Freed theorems for the eta-invariant to a physically transparent framework based on a massive fermion on a manifold with boundary. By examining the ground-state wavefunction, generalized APS boundary conditions, and the determinant line bundle, it derives how boundary data control eta-invariants, gluing formulas, and the associated Berry phases. The formalism encodes anomalies and topological phases of matter, providing concrete tools (determinant-line geometry, Berry connections, and curvature) and illustrating them with the integer quantum Hall effect and Majorana-chain examples. Together, these results bridge mathematical index theory with physical topological phases and anomaly inflow, offering a unified view of boundary conditions, ground-state geometry, and topological responses.

Abstract

We describe a physics derivation of theorems due to Dai and Freed about the Atiyah-Patodi-Singer eta-invariant which is important for anomalies and topological phases of matter. This is done by studying a massive fermion. The key role is played by the wave function of the ground state in the Hilbert space of the fermion in the large mass limit. The ground state takes values in the determinant line bundle and has nontrivial Berry phases which characterize the low energy topological phases.

Dai-Freed theorem and topological phases of matter

TL;DR

This work connects the Dai-Freed theorems for the eta-invariant to a physically transparent framework based on a massive fermion on a manifold with boundary. By examining the ground-state wavefunction, generalized APS boundary conditions, and the determinant line bundle, it derives how boundary data control eta-invariants, gluing formulas, and the associated Berry phases. The formalism encodes anomalies and topological phases of matter, providing concrete tools (determinant-line geometry, Berry connections, and curvature) and illustrating them with the integer quantum Hall effect and Majorana-chain examples. Together, these results bridge mathematical index theory with physical topological phases and anomaly inflow, offering a unified view of boundary conditions, ground-state geometry, and topological responses.

Abstract

We describe a physics derivation of theorems due to Dai and Freed about the Atiyah-Patodi-Singer eta-invariant which is important for anomalies and topological phases of matter. This is done by studying a massive fermion. The key role is played by the wave function of the ground state in the Hilbert space of the fermion in the large mass limit. The ground state takes values in the determinant line bundle and has nontrivial Berry phases which characterize the low energy topological phases.

Paper Structure

This paper contains 25 sections, 116 equations, 2 figures.

Table of Contents

  1. Introduction and summary
  2. Description of the theorems
  3. Theorem 1.
  4. Theorem 2.
  5. Corollary.
  6. Theorem 3.
  7. Summary of the paper
  8. Physical states and boundary conditions of the path integral
  9. The relation of states and boundary conditions
  10. Generalized APS boundary conditions
  11. The standard APS boundary condition.
  12. The wave function of the ground state
  13. The path integral on a manifold with boundary
  14. Derivation of Theorem 1
  15. Derivation of Theorem 2
  16. ...and 10 more sections

Figures (2)

  • Figure 1: Some examples of $X$ and $X^{\rm cut}$.
  • Figure 2: The manifolds $X_w$, $-X_{w+dw}$ and the parallel transport given by the manifold $Y_{dw}$. They glue together to give a manifold without boundary.