The Standard Models contain chiral fermions coupled to gauge theories. It has been a long-standing problem to give such gauged chiral fermion theories a quantum non-perturbative definition. By classification of quantum anomalies and symmetric invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and the existence of symmetric gapped boundary for the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3+1D lattice, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the Standard Models from the 16n-chiral fermion SO(10) Grand Unification can be defined non-perturbatively via a 3+1D local lattice model of bosons or qubits. Furthermore, we propose that Standard Models from the 15n-chiral fermion SU(5) Grand Unification can also be realized by a 3+1D local lattice model of fermions.

Recently Freed and Hopkins [11] proved that there is no parity anomaly in M-theory on pin+ manifolds in the low-energy field theory approximation, and they also developed an algebraic theory of cubic forms. Earlier Witten [33] proved the anomaly cancellation for spin manifolds by introducing the E8-bundle technique. Motivated by the cubic forms and the anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spinc, spinω2 and orientable 12- manifolds respectively. We relate them to η-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spinc or spinω2 . Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine E8.

No abstract uploaded!

In this paper, we propose and analyze a mixed $H^1$-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is non-convex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-$H^1$ very weak function and thus causes many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh-dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of $H^1$-conforming finite elements, namely the $CP_2$-$P_1$ elements, for electric field and multiplier is studied and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the $L$-shaped and cracked domains are presented to illustrate the high performance of the proposed mixed $H^1$-conforming finite element method.

No abstract uploaded!