Complete classification of non-perturbative global gauge anomalies in four-dimensional chiral gauge theories for arbitrary compact gauge groups and fermion representations

In 1982 Witten showed that a four-dimensional SU(2) gauge theory with an odd number of left-handed Weyl fermion multiplets in representations of isospin $2r+1/2$ is mathematically inconsistent: the first non-perturbative (global) gauge anomaly, detected by $\pi_4(SU(2))=\mathbb{Z}_2$ and the mod 2 index of a five-dimensional Dirac operator. Witten's 1982 paper establishes the SU(2) criterion and checks specific other theories but leaves open the general question it naturally raises: for an arbitrary compact gauge group $G$ (in each of its global forms, e.g. quotients $G/\Gamma$), for each admissible background structure on spacetime (spin, spin-c, or generalized spin-$G$ structures), and for arbitrary chiral fermion representations, exactly when does such a global obstruction arise? Later literature advanced this substantially but did not complete it: the Freed-Hopkins invertible-phase/cobordism framework supplies the general principle (global anomalies of four-dimensional fermionic theories are classified by torsion bordism groups $\Omega^5_\xi(BG)$ paired with mod 2 indices and eta-invariants); Wang-Wen-Witten closed the SU(2) case in 2018, giving analogs for Sp(2N), by discovering a second, previously missed SU(2) anomaly that arises only when the theory is placed on spin-SU(2) structures; and Davighi-Gripaios-Lohitsiri and Wan-Wang completed the Standard Model, grand-unified, and many beyond-Standard-Model cases. Nevertheless, no uniform complete criterion covering all compact gauge groups, all global forms, and all structure types exists in the literature, computations continue case-by-case into 2025-2026, and the completeness of the bordism classification itself is established only under effective-field-theory assumptions. The problem is to complete this classification and to establish its foundations.

Background

A four-dimensional chiral gauge theory couples left-handed Weyl fermions (two-component spinor fields of a single handedness) to a gauge field taking values in the Lie algebra of a compact gauge group $G$, with the fermions transforming in unitary representations $R_i$ of $G$. The quantum theory is consistent only if gauge transformations act trivially on its observables; failures of this consistency are called gauge anomalies. The perturbative or local anomaly is a one-loop effect detected by an anomaly polynomial computable from the $R_i$, and a theory with a nonzero local anomaly is inconsistent outright. A subtler failure, first exhibited by Witten's 1982 paper, is the global (non-perturbative) anomaly: the fermion path integral (the determinant or Pfaffian of the Dirac operator coupled to the gauge field) can change sign under a gauge transformation that is connected to the identity only through a loop in the space of gauge fields, with no signal at any order in perturbation theory. For $G=SU(2)$ this is controlled by the fourth homotopy group $\pi_4(SU(2))=\mathbb{Z}_2$ (homotopy groups $\pi_k$ record equivalence classes of maps from the $k$-sphere into a space): the nontrivial class of gauge transformations on a compact spatial three-manifold flips the sign of the determinant precisely when the number of Dirac zero modes in an instanton background (a finite-action topologically nontrivial solution of the Euclidean gauge-field equations) is odd. Witten's 1982 criterion states that an SU(2) theory with an odd number of left-handed Weyl multiplets in representations of isospin $2r+1/2$ (the fundamental doublet being $r=0$) is mathematically inconsistent; the obstruction is computed by the mod 2 index, a mod-2 analogue of the classical index theorem for elliptic operators, counting zero modes of a five-dimensional Dirac operator on a mapping torus (the closed five-manifold obtained from $M\times[0,1]$ by gluing the two ends via a symmetry $\varphi$). Witten's paper verified the consistency of the Standard Model fermion content (each family contains four SU(2) doublets: three colored quarks plus one lepton) but did not give a criterion for arbitrary gauge groups and arbitrary fermion representations, and it is this gap that the present problem inherits. Two later developments define what a complete answer must contain. First, the modern framework of anomalies as invertible field theories (theories whose partition-function values are invertible complex numbers, so that they contribute only a phase or weight; anomalies of a $d$-dimensional theory are encoded by such a theory in $d+1$ dimensions, a mechanism known as anomaly inflow) led Freed and Hopkins to the classification of deformation classes of invertible topological phases with symmetry type $(H,\xi)$ by the stable homotopy theory of Thom bordism spectra (spectra whose homotopy groups compute bordism groups; $\xi$ denotes the structure required on spacetime, such as a spin structure or a spin-$G$ structure). Applied to four-dimensional fermionic gauge theories, the torsion anomaly classification is the bordism group $\Omega^5_\xi(BG)$, where $BG$ is the classifying space that parametrizes principal $G$-bundles: classes of closed five-manifolds carrying the structure $\xi$ and a $G$-bundle, paired with the fermion content via mod 2 indices and eta-invariants (the spectral asymmetry of the Dirac operator, which computes anomaly phases). This is a general criterion in principle, but for any given $G$ it requires computing $\Omega^5_\xi(BG)$, typically via the Atiyah-Hirzebruch spectral sequence (a machine computing bordism from the cohomology of $BG$ together with its Stiefel-Whitney classes $w_k$, the characteristic classes that obstruct orientations ($w_1$), spin structures ($w_2$) and related structures on vector bundles), and the published computations cover specific families of groups rather than all compact $G$. Second, Wang, Wen and Witten showed in 2018 that even the SU(2) story of 1982 was incomplete: if the theory is defined on all manifolds admitting a spin-SU(2) structure, $(Spin(4)\times SU(2))/\mathbb{Z}_2$ (possible when all fermions carry half-integer and all bosons integer isospin, so that no gauge-invariant local field is a fermion), then an odd number of multiplets in representations of isospin $4r+3/2$ suffers a second, new global anomaly, detected by the invariant $(-1)^{\int_Y w_2(TY)\,w_3(TY)}$; and the two $\mathbb{Z}_2$ anomalies are exhaustive for SU(2) because the corresponding cobordism invariant group is $\mathbb{Z}_2\times\mathbb{Z}_2$. A general criterion must therefore specify not only $G$ and the representations but also the global form of the group (for a fixed Lie algebra, its quotients $G/\Gamma$ by central or discrete subgroups) and the structure $\xi$ on which the theory is defined (spin; spin-c, a weakening of a spin structure using a U(1) line bundle, which every orientable four-manifold admits; or spin-$G$ structures), and must track the interplay between local and global anomalies (in quotients of the Standard Model gauge group, local anomaly cancellation data can eliminate would-be global anomalies and vice versa). The problem is the completion of this classification.

Problem Statement

Determine a complete general criterion for non-perturbative global gauge anomalies in four-dimensional chiral gauge theories, of the kind discovered by Witten in 1982 for SU(2). Concretely, for an arbitrary compact gauge group $G$ — including every global form of a given Lie algebra, i.e. the quotients $G/\Gamma$ by central and discrete subgroups — and for each admissible background structure $\xi$ on spacetime on which the fermionic theory can be defined (spin, spin-c, and generalized spin-$G$ structures such as $(Spin(4)\times G)/\mathbb{Z}_2$): (i) compute the group of possible global anomalies, i.e. the torsion bordism group $\Omega^5_\xi(BG)$, with explicit invariants and generating closed five-manifolds; (ii) derive from it the sharp criterion on any set of left-handed Weyl fermion representations $\{R_i\}$ with vanishing perturbative anomalies, deciding whether the theory is globally anomalous — the analogue, for arbitrary $G$, of Witten's 'odd number of isospin $2r+1/2$ multiplets' criterion for SU(2), including how the criterion changes between global forms and structure types and how it interacts with local anomaly cancellation; and (iii) determine whether, and under precisely which hypotheses, this bordism/invertible-phase classification is complete, i.e. whether (and which) global obstructions exist beyond those it detects, or else show that no such completeness statement holds — the classification being currently guaranteed only under effective-field-theory and deformation-class assumptions. The consistency classification of four-dimensional chiral gauge theories is a single question whose criteria hold uniformly across gauge groups rather than only case-by-case: a uniform computation of (\Omega^5_\xi(BG)) across the cases in scope yields (ii) as a consequence of (i), while (iii) is independent of both, asking whether the bordism/invertible-phase classification is complete under explicitly stated hypotheses or whether some definite theory's global anomaly is not detected by (\Omega^5_\xi(BG)).

References

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