Chiral fermions, Gravity and GUTs

TL;DR

The paper identifies a global gravitational anomaly affecting chiral fermions in the Standard Model and GUTs, proposing conditions for anomaly cancellation that favor GUTs with 16 Weyl fermions per generation, like SO(10).

Contribution

It reveals a global anomaly in chiral fermion coupling to gravity and establishes anomaly cancellation conditions that select specific GUT structures.

Findings

Standard Model with 15 fermions per generation is inconsistent on certain manifolds.

Adding an extra Weyl fermion can restore consistency.

GUTs with multiples of 16 Weyl fermions per generation are favored.

Abstract

We discuss a global anomaly associated with the coupling of chiral Weyl fermions to gravity. The Standard Model based upon $SU(3){\times}SU(2){\times}{U(1)}$ which has 15 fermions per generation is shown to be inconsistent if all background spin manifolds with signature invariant $\tau=8k$ are allowed. Similarly, GUTs based on odd number of Weyl fermions are inconsistent. Consistency can be achieved by adding an extra Weyl fermion which needs to couple only to gravity. For arbitrary $\tau$'s, generalized spin structures are needed, and the global anomaly cancellation requires that the net index of the total Dirac operator with spin and internal gauge connections be even. As a result GUTs with fundamental multiplets which contain multiples of 16 Weyls per generation are selected. The simplest consistent GUT is the SO(10) model with a multiplet of 16 Weyls per generation. The combined gravity and internal symmetry gauge group of the theory is then $[Spin(3,1){\times}Spin(10)]/Z_2$. Physical implications of these results are commented on.