Fermion Path Integrals And Topological Phases

TL;DR

Witten develops a unifying framework linking fermionic SPT phases to global and perturbative anomalies, using anomaly inflow to relate boundary modes on defect manifolds to bulk invertible spin topological quantum field theories (sTQFTs). He classifies fermions into pseudoreal, real, and complex representations and derives how bulk η-invariants, mod 2 indices, and cobordism invariants govern the boundary-bulk consistency, including the prominent θ=π electromagnetic response in 3D topological insulators. The Dai-Freed theorem is invoked to handle complex fermion anomalies on unorientable spacetimes, ensuring well-defined combined boundary-bulk partition functions and explaining the mod 16 reduction in interacting 3D topological superconductors. Across TI/SC settings, the work clarifies how cobordism invariants control the topological classification, the role of APS boundary conditions, and the precise bulk-boundary matching required for unitarity and symmetry preservation. The results illuminate how free-fermion band topology is encoded in anomaly data and provide a rigorous, cobordism-based language for SPT phases in fermionic systems, with concrete implications for theta-angle physics and Majorana boundary modes.

Abstract

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic $θ$-angle is equal to $π$.

Figures (9)

• Figure 1: Spectral flow for a Dirac operator. The vertical axis parametrizes an eigenvalue $\lambda$ and the horizontal axis parametrizes a parameter $s$ on which the eigenvalues depend. In the case shown, the spectrum is the same at $s=1$ as at $s=0$, but there is a net upward flow of one eigenvalue through $\lambda=0$ between $s=0$ and $s=1$. This leads to a sign change of the fermion path integral $Z_\psi$.
• Figure 2: A mirror-symmetric construction of a 3-manifold $Y$, by gluing together identical pieces $Y_1$ and $Y_2$ along their common boundary $M$, with reversed orientation for $Y_2$.
• Figure 3: A topological insulator supported on the right half of ${\mathbb{R}}\times S^2$ (solid color).
• Figure 4: A topological invariant ${\mathfrak I}$ defined for a manifold $X$ is said to be a cobordism invariant if it vanishes whenever $X$ is the boundary of a manifold $Z$ of one dimension more. If $X$ has some structure (such as an orientation, a spin structure, or a ${\mathrm U}(1)$ gauge field) that is required in defining ${\mathfrak I}$, then this structure is required to extend over $Z$.
• Figure 5: Here we give the simplest illustration of the statement that cobordism invariance leads to relations that are natural in topological field theory. The connected sum$M$ of two $D$-manifolds $M_1$ and $M_2$ is made by cutting small holes out of $M_1$ and $M_2$ and gluing them together along their boundaries. If the space of physical states on a sphere $S^{D-1}$ is 1-dimensional (as expected in a unitary topological field theory), one can deduce a universal relation between partition functions: $Z_M = g Z_{M_1}Z_{M_2}$, where $g$ is a constant characteristic of the theory. Cobordism invariance implies such a relation with $g=1$, because $M$ is cobordant to the disjoint union of $M_1$ and $M_2$. This cobordism is sketched in this figure (for the case $D=1$ with all manifolds being circles). Topological field theories derived from cobordism invariants always satisfy the condition that the space of physical states on any $D-1$-manifold has dimension 1, which is why they lead to a relation of the given kind.