Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

TL;DR

<3-5 sentence high-level summary> Quantum anomalies are used as a diagnostic of interaction effects on fermionic symmetry-protected topological (SPT) phases in (3+1)D. The authors analyze two exemplar bulk SPTs by enforcing symmetries on their 2D surface theories via orientifold projections: (i) CP-symmetric TIs with U(1) flux exhibit a global U(1) gauge anomaly signaling a Z2 bulk classification, and (ii) reflection-symmetric crystalline TSCs reveal a gravitational (modular) anomaly whose structure across flux and parity twists encodes the Z16 collapse under interactions. The approach relies on precise modular and flux-twisted partition functions for surface Dirac/Majorana fermions on toroidal geometries, and on analyzing their behavior under large gauge transformations and diffeomorphisms. The results provide an anomaly-based, complementary view of how interactions modify non-interacting classifications and establish connections to bulk topological distinctions that persist when symmetries are preserved. This work deepens the understanding of surface-bulk correspondence in interacting SPT phases and motivates further exploration of anomalies in other symmetry settings and higher-dimensional analogues.

Abstract

Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the non-interacting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)-dimensional fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\times$ reflection) and electromagnetic $\mathrm{U}(1)$ symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large $\mathrm{U}(1)$ gauge transformations, but picks up an anomalous sign, signaling a $\mathbb{Z}_2$ topological classification. Similarly, for the second example, which is related to, by CPT-theorem, time-reversal symmetric topological superconductors, we discuss the invariance/non-invariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (3d modular transformations). The connection to the collapse of the non-interacting classification by an integer ($\mathbb{Z}$) to $\mathbb{Z}_{16}$, in the presence of interactions, is discussed.

Figures (1)

• Figure 1: The three-torus and its (unorientable) descendants generated by the orientifold projection. While the $y$-boundary condition is twisted by $G_y=1$ or $\mathscr{G}_f$, the ${\tau}$- and $x$- boundary conditions are twisted by $({G}_{\tau}, G_{x})=$$(\mathscr{G}_f^{2a_{\tau}}, \mathscr{G}_f^{2a_{x}})$, $(\mathscr{P}\mathscr{G}_f^{2a_{\tau}}, \mathscr{G}_f^{2a_{x}})$, $(\mathscr{G}_f^{2a_{\tau}}, \mathscr{P}\mathscr{G}_f^{2a_{x}})$, $(\mathscr{P}\mathscr{G}_f^{2a_{\tau}}, \mathscr{P}\mathscr{G}_f^{2a_{x}})$, as shown in figures ($i$)--($iv$), respectively. (Un)twisted boundary conditions are represented by arrows with the same color.