Real Representatives of Holomorphic Maurer–Cartan Solutions

Determine whether the characteristic fibre of a Maurer–Cartan solution of the holomorphic heterotic theory contains a real representative satisfying conformal balance, gauge primitivity, and the leading-flux condition, and characterize such representatives when they exist.

Background

The characteristic reduction identifies the holomorphic heterotic deformation theory as a quotient of the product variational theory, but it removes directions corresponding to choices of real representative. The physical requirements of conformal balance, gauge primitivity, and leading-flux admissibility are not constant along the characteristic fibres and therefore are not imposed by the quotient itself.

The unresolved issue is whether a given holomorphic deformation can be lifted back to a real representative satisfying all of these additional physical conditions, together with the classification or uniqueness of such lifts. The paper states that resolving this requires geometric and analytic input beyond the characteristic reduction.

References

This leaves a lifting problem that is not addressed by the reduction. Given a Maurer--Cartan solution of the holomorphic theory, one may ask whether its characteristic fibre contains a real representative satisfying balance, primitivity and the leading-flux condition, and how such representatives are parametrized when they exist. Existence and uniqueness of such representatives require additional geometric and analytic input beyond the characteristic reduction.

— Cyclic $L_3$ Models and Holomorphic Reduction in Heterotic $G_2$ Deformation Theory  (2610.03301 - Brongers, 2 Oct 2026) in Section Consequences and open problems