Wang and Wen proved that the SO(10) Grand Unified Theory Standard Model, with 16 Weyl fermions per family in the 16-dimensional spinor representation of Spin(10), admits a nonperturbative definition as a 3+1D local lattice model of bosons (qubits): the proof establishes that a Spin(10)-symmetric gapped boundary exists which removes the unwanted mirror fermions, but it does not provide any explicit interaction that produces this gap, and the proposed Higgs-orientation-fluctuation mechanism remains unverified numerically. The open problem is to make this definition constructive: explicitly design local, finite-range, Spin(10)-symmetry-preserving interactions that fully gap the mirror sector without symmetry breaking or ground-state degeneracy while leaving the chiral sector intact, and verify the gapping analytically or numerically. This is the central instance of the broader 'symmetric mass generation' program, which by 2026 has produced continuum existence results, lower-dimensional examples, anomaly-based necessary conditions, and sign-problem-free simulation strategies in 1+1D, but no explicit verified construction in 3+1D.

Background

A chiral gauge theory is a quantum field theory whose fermions couple with unequal left- and right-handed content; the Standard Model of particle physics is one, and it has never had a fully rigorous nonperturbative (lattice) definition. The obstacle is fermion doubling: any free lattice fermion discretization that preserves the basic symmetries produces unwanted extra copies of the fermion spectrum, and removing the doublers by an ordinary mass term either breaks the chiral symmetry or leaves a vector-like (non-chiral) theory. The classic mirror-fermion strategy of Eichten and Preskill (1986) tried to pair each physical fermion with an unwanted 'mirror' partner and gap the mirrors by strong interactions; numerical and analytic studies culminating in Golterman, Petcher and Rivas (1993) found that the mirrors do not decouple in the models tried, and the approach stalled. Domain-wall and overlap fermion constructions define chiral determinants in vector-like lattice gauge theories but do not directly define the Standard Model's chiral gauge dynamics in its own dimension.

Wang and Wen (2020) revisited the mirror strategy with modern tools from condensed matter and topology. The key notions: a symmetry-protected topological (SPT) phase is a gapped quantum many-body state that is trivial in the bulk but only becomes trivial at its boundary if a symmetry is broken; an invertible topological order is a gapped phase with a unique ground state on every closed spatial manifold and no anyonic excitations; its partition function defines an invertible topological quantum field theory (iTQFT), a TQFT whose partition function on any closed spacetime manifold is a pure phase. Freed and Hopkins (2016/2021) classified the invertible topological orders (equivalently, the possible 't Hooft anomalies — obstructions to gauging a global symmetry on a lattice in the same dimension) via cobordism theory, the study of manifolds glued along boundaries. Wang and Wen combined this classification with a proof that a symmetric gapped boundary exists whenever all anomalies vanish, and concluded that a Spin(10) chiral fermion theory — Spin(10) being the double cover of the SO(10) Grand Unified Theory rotation group, a symmetry that contains the Standard Model gauge group, whose 16-dimensional spinor representation holds exactly one family of Standard Model fermions plus a right-handed neutrino — can be put on a 3+1D bosonic lattice, with one boundary carrying the chiral fermions and the other carrying mirror partners that must be gapped without breaking Spin(10). Their proof of existence of the gapping interactions is non-constructive; the interactions themselves — multi-fermion terms strong enough to gap the mirrors while preserving every symmetry — were not written down. Gapping fermions by interactions with no symmetry breaking and no mass bilinear is now called symmetric mass generation (SMG), where 'no symmetry breaking' includes fermion parity — the (−1)F(-1)^{F} symmetry under which every state with an odd number of fermions changes sign, which cannot be spontaneously broken; it requires anomaly cancellation (e.g. the famous mod-16 anomaly: sixteen 16-fermion copies make certain global anomalies vanish, mirroring why SO(10) GUT fermion content is anomaly-free). By 2026 SMG has become an active subfield connecting condensed matter (topological insulator surfaces), lattice gauge theory, and beyond-Standard-Model physics, but the founding question — an explicit, verified gapping of the mirror sector in the 3+1D Spin(10) lattice model — remains unanswered.

Problem Statement

Within Wang and Wen's 3+1D local bosonic lattice model, whose boundary realizes 16n16n Weyl fermions in the 16-dimensional spinor representation of Spin(10) — one family of fermions is 16 Weyl fermions, n≥1n\geq 1 is the number of families, the chiral sector sits on one boundary and its mirror copy on the other — construct explicitly the interactions that their existence proof leaves unspecified, and verify that they work. Concretely: write down local, finite-range interaction terms (a lattice Hamiltonian, or a Euclidean lattice action) added to the free mirror-sector fermions such that (i) every term preserves the full onsite Spin(10) symmetry containing fermion parity, (ii) all mirror-sector Weyl fermions acquire a mass gap with a finite excitation gap above the ground state in the thermodynamic limit, (iii) no symmetry is spontaneously broken and the ground state carries no symmetry-breaking or topological degeneracy, and (iv) the physical chiral boundary sector is left intact at low energy with the mirror sector decoupled. The verification may be analytic (e.g. a rigorous confinement or strong-coupling argument) or numerical (e.g. a sign-problem-free simulation), but it must confirm the gapping in the physically required regime identified in the source paper — the induced mirror mass comparable to the fermion bandwidth, not the infinite-coupling limit, which fails because it generates a mirror 'dead layer' (a neighboring layer that decouples from the gapped region and becomes a new mirror sector with fermion doublers). A valid answer may alternatively supply the missing lattice-side of the source's own proposed mechanism (gapping via smooth orientation fluctuations of a Higgs field) with evidence of its validity. This problem does not include, and should not be conflated with, the source paper's separate open issues: the post-gapping dynamics of the gauged chiral gauge theory, the mathematical lattice realization of cobordism-classified invertible TQFTs, a bosonic realization of the SU(5) model, or the phenomenology of the predicted extra Z2 gauge sector.

References

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