On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)

TL;DR

This work develops a non-perturbative lattice formulation of SO(10) chiral gauge theory with Weyl fermions in the $\{16}$ representation by constructing a manifestly gauge-invariant path-integral measure in the overlap/Ginsparg-Wilson framework. The right-handed sector is completely saturated by inserting $SO(10)$-invariant 't Hooft vertices, while the left-handed sector yields the correct chiral determinant; the induced action is CP invariant and the left U(1) is anomalous due to the measure. locality of the measure's gauge-field dependence remains an important, non-perturbative issue, though the authors argue it is tractable in the weak-coupling regime via Monte Carlo methods without a sign problem. The paper also situates this construction relative to established and emerging approaches to decouple doublers/mirror fermions—such as Eichten-Preskill, GW Mirror-Fermion, Domain Wall with EP terms, and TI/TSC gapped-boundary frameworks—and positions the model as a concrete testing ground for such fundamental questions about non-perturbative chiral gauge theories on the lattice.

Abstract

We consider the lattice formulation of SO(10) chiral gauge theory with left-handed Weyl fermions in the sixteen dimensional spinor representation ($\underline{16}$) within the framework of the Overlap fermion/the Ginsparg-Wilson relation. We define a manifestly gauge-invariant path-integral measure for the left-handed Weyl field using all the components of the Dirac field, but the right-handed part of which is just saturated completely by inserting a suitable product of the SO(10)-invariant 't Hooft vertices in terms of the right-handed field. The definition of the measure applies to all possible topological sectors. The measure possesses all required transformation properties under lattice symmetries and the induced effective action is CP invariant. The global U(1) symmetry of the left-handed field is anomalous due to the non-trivial transformation of the measure, while that of the right-handed field is explicitly broken by the 't Hooft vertices. There remains the issue of locality in the gauge-field dependence of the Weyl fermion measure, but the question can be addressed in the weak gauge-coupling expansion at least using Monte Carlo methods without encountering the sign problem. We also discuss the relations of our formulation to other approaches/proposals to decouple the species-doubling/mirror degrees of freedom. Those include Eichten-Preskill model, Ginsparg-Wilson Mirror-fermion model, Domain wall fermion model with the boundary Eichten-Preskill term, 4D Topological Insulator/Superconductor with gapped boundary phase, and the recent studies on the PMS phase/"Mass without symmetry breaking". We clarify the similarity and the difference in technical detail and show that our proposal is a well-defined testing ground for that basic question.

Figures (12)

• Figure 1: The schematic picture of the SO(10) model in terms of the overlap formalism
• Figure 2: The eigenvalue spectra of the matrices $(u^{\rm T} \, i \gamma_5 C_D {\rm T}^a E^a u )$ and $( u^\dagger \Gamma^{10} \Gamma^{a} E^{a} u )$ with a randomly generated spin-field configuration for the case of the trivial link field. The lattice size is $L=4$ and the boundary condition for the fermion field is periodic. For reference, the eigenvalue spectrum of the matrix $(\bar{v}_k D v_i)$ is also shown with green x symbol for the same boundary condition.
• Figure 3: The eigenvalue spectra of the matrices $(u^{\rm T} \, i \gamma_5 C_D {\rm T}^a E^a u )$ and $( u^\dagger \Gamma^{10} \Gamma^{a} E^{a} u )$ with a randomly generated spin-field configuration for the case of the trivial link field. The lattice size is $L=4$ and the boundary condition for the fermion field is anti-periodic. For reference, the eigenvalue spectrum of the matrix $(\bar{v}_k D v_i)$ is also shown with green x symbol for the same boundary condition.
• Figure 4: The eigenvalue spectrum of the matrix $( u^\dagger \Gamma^{10} \Gamma^{a} E^{a} u )$ with a spin-field configuration in the class $E_\ast^a(x)$ for the case of the trivial link field. The lattice size is $L=4$ and the boundary condition for the fermion field is periodic.
• Figure 5: The eigenvalue spectra of the matrices $(u^{\rm T} \, i \gamma_5 C_D {\rm T}^a E^a u )$ and $( u^\dagger \Gamma^{10} \Gamma^{a} E^{a} u )$ with a randomly generated spin-field configuration for the case of the representative SU(2) link field of the topological sector with $Q=-2$. The lattice size is $L=4$ and the boundary condition for the fermion field is periodic.