On the cobordism classification of symmetry protected topological phases

TL;DR

The work proves a precise classification of d-dimensional, unitary invertible TQFTs with symmetry H_d by cobordism invariants, showing a 1:1 correspondence with ${\rm Hom}(\Omega_d^H, {\rm U}(1))$ after tuning the Euler term to fix sphere partition functions. It develops a rigorous framework based on Atiyah's TQFT Axioms, locality, unitarity, and spin-statistics, and demonstrates both the invariance of partition functions under bordism and a constructive reconstruction of a TQFT from any given cobordism invariant. The results provide a solid, explicit bridge between physical notions of SPT phases, anomalies, and theta-terms and the mathematical cobordism classification, under a finite-generation assumption on the relevant bordism group. This furnishes a robust, deformation-class-level classification important for understanding topological phases of matter in relativistic and gravitational contexts. The techniques hinge on Morse-theoretic decompositions, reflection positivity, and a careful treatment of boundary and fermionic structure, culminating in an explicit functorial construction of the TQFT from cobordism data.

Abstract

In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological (SPT) phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant. Two theories having the same cobordism invariant partition functions are isomorphic.

Figures (14)

• Figure 1: A manifold $X$ which consists of $X_1$ and $X_2$ glued at their common boundary $Y$.
• Figure 2: The bordism $\tau_{Y,Y'}$ from $Y \sqcup Y'$ to $Y' \sqcup Y$. As an $H_d$-manifold, it is just $([0,1] \times Y) \sqcup ([0,1] \times Y')$.
• Figure 3: The composition \ref{['eq:Scomposition']}.
• Figure 4: The trace \ref{['eq:traceformula']}. Notice that we have to use ${c}_{\overline{Y}}$ instead of $c_Y$.
• Figure 5: The bordism $C$ from $X_0$ to $X_1$. The $C$ is sliced by the values of the function $f$. Here it is sliced at $f^{-1}(t)$ which is assumed not to contain any critical points $p_a$.

Theorems & Definitions (56)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Definition 2.9