On the gauge-invariant path-integral measure for the overlap Weyl fermions in 16 of SO(10)

Abstract

We consider the lattice formulation of SO(10) chiral gauge theory with left-handed Weyl fermions in the 16-dimensional spinor representation ($\underline{16}$) within the framework of the overlap fermion/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 it 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 of admissible link fields. 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 smoothness and locality in the gauge-field dependence of the Weyl fermion measure, but the question is well defined and the necessary and sufficient condition for this property is formulated in terms of the correlation functions of the right-handed auxiliary fields. In the weak gauge-coupling limit at least, all the auxiliary fields have short-range correlations and the question can be addressed further by 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. These include the Eichten–Preskill model, the mirror-fermion model with overlap fermions, the domain-wall fermion model with the boundary Eichten–Preskill term, 4D topological insulator/superconductor with a gapped boundary phase, and the recent studies on the PMS phase/“mass without symmetry breaking”. We clarify the similarities and differences in the technical details and show that our proposal is a unified and well defined testing ground for that basic question.

Gauge Enhanced Quantum Criticality Beyond the Standard Model

“…Even if it condenses, it is not expected to gap out the Weyl fermions if its vaccum expectation value is small (but it will Higgs down the gauge group), so the theory remains gapless in the fermion sector in all phases. However, sufficiently strong Higgs condensation of TrΦ bi (or Φ 1 equivalently) can lead to symmetric mass generation (SMG) [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49] as discussed previously.…”

“…The Symmetric Mass Generation (SMG) mechanism is explored in various references, for some selective examples, by Fidkowski-Kitaev[35] in 0+1d, by Wang-Wen[36,37] for gapping chiral fermions in 1+1d, You-He-Xu-Vishwanath[38,39] in 2+1d, and notable examples in 3+1d by Eichten-Preskill[40], Wen[41], You-BenTov-Xu[42,43], BenTov-Zee[44], Kikukawa[45], Catterall et al[46,47], Razamat-Tong[48,49], etc.12 Here fermions are anti-commuting Grassman variables, so this expression ψψψψ is only schematic. The precise expression of ψψψψ includes additional spacetime-internal representation indices and also includes possible additional spacetime derivatives (for point-splitting the fermions to neighbor sites if writing them on a regularized lattice).…”

Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy

“…Based on the Nielsen-Ninomiya fermion doubling of the free fermion theory, the 16 + ⊕ 16 − can be regarded as the realization of the chiral matter 16 + and the mirror matter 16 − (anti-chiral with complex conjugated representation). Based on a generalization of gapping mirror fermion [50] by suitable nonperturbative interactions (see an overview from [51][52][53][54][55]), References [12,[56][57][58] suggested that the mirror matter 16 − can be fully gapped without breaking the Spin(10) group. It is shown in [12] that the gapping 16 − without breaking Spin( 10) is consistent with the classification of all invertible local and global anomalies from the cobordism classification [12,13].…”

Ultra Unification

“…The issues of mirror fermion doubling [83] on the mirror world, depending on the precise anomaly matching on the mirror sector, may be fully trivially gapped (if anomaly-free), or may be topologically ordered gapped with TQFT. These issues are tackled in many recent (selective) works [13,[84][85][86][87][88][89][90][91][92][93].…”