Dai-Freed anomalies in particle physics

TL;DR

This work extends anomaly analysis in high-energy theories by applying the Dai-Freed framework, computing bordism groups for classifying spaces of gauge groups, and evaluating η-invariants to detect both traditional and refined (Dai-Freed) anomalies. It shows that the Standard Model, its SU(5) GUT, and Spin(10) GUT are anomaly-free under this extended criterion, while certain discrete symmetries (notably baryon triality) exhibit nontrivial mod-9 Dai-Freed constraints unless the number of generations is appropriately chosen. The paper also links these constraints to Ibañez–Ross linear conditions, explores UV-sensitivity and nonlinear Dai-Freed constraints, and discusses potential cures via Green–Schwarz terms or coupling to topological quantum field theories, including the appearance of Spin$^{\mathbb{Z}_4}$ structures and domain-wall physics. Finally, it addresses K-theoretic θ angles and summarizes the implications for model building and UV completions, outlining future directions for a fuller classification of discrete and higher-form symmetry anomalies.

Abstract

Anomalies can be elegantly analyzed by means of the Dai-Freed theorem. In this framework it is natural to consider a refinement of traditional anomaly cancellation conditions, which sometimes leads to nontrivial extra constraints in the fermion spectrum. We analyze these more refined anomaly cancellation conditions in a variety of theories of physical interest, including the Standard Model and the $SU(5)$ and $Spin(10)$ GUTs, which we find to be anomaly free. Turning to discrete symmetries, we find that baryon triality has a $\mathbb{Z}_9$ anomaly that only cancels if the number of generations is a multiple of 3. Assuming the existence of certain anomaly-free $\mathbb{Z}_4$ symmetry we relate the fact that there are 16 fermions per generation of the Standard Model - including right-handed neutrinos - to anomalies under time-reversal of boundary states in four-dimensional topological superconductors. A similar relation exists for the MSSM, only this time involving the number of gauginos and Higgsinos, and it is non-trivially, and remarkably, satisfied for the $SU(3)\times SU(2) \times U(1)$ gauge group with two Higgs doublets. We relate the constraints we find to the well-known Ibañez-Ross ones, and discuss the dependence on UV data of the construction. Finally, we comment on the (non-)existence of K-theoretic $θ$ angles in four dimensions.

Figures (22)

• Figure 1: The Dai-Freed construction computes the phase of the fermion path integral on a manifold $X$ via an auxiliary manifold $Y$ such that $\partial Y=X$.
• Figure 2: The $\eta$ invariant behaves nicely under gluing as illustrated in the picture.
• Figure 3: To obtain the phase for a configuration $A_0^g$ starting from $A_0$, we may just attach $X\times [0,1]$ as shown in the picture. The additional contribution to the phase is identical to $\eta$ evaluated on the mapping tours obtained by gluing the two sides of $X\times [0,1]$.
• Figure 4: Traditional global anomalies are studied via the $\eta$ invariant on mapping tori (left figure). The general Dai-Freed anomaly can be regarded as a generalization in which we allow the "mapping torus" to have holes or other nontrivial topologies. In the same way that the traditional mapping torus follows a nontrivial loop in configuration space, the new anomaly can be regarded as coming from new nontrivial loops that arise once topology change is allowed, as one might expect to happen in quantum gravity.
• Figure 5: The two manifolds $Y_1$ and $Y_2$ are bordant if $Y_1\sqcup \bar{Y_2}$ is boundary of another manifold $Z$.