Hermitian Yang--Mills connections on general vector bundles: geometry and physical Yukawa couplings

We compute solutions to the Hermitian Yang-Mills equations on holomorphic vector bundles $V$ via an alternating optimisation procedure founded on geometric machine learning. The proposed method is fully general with respect to the rank and structure group of $V$, requiring only the ability to enumerate a basis of global sections for a given bundle. This enables us to compute the physically normalised Yukawa couplings in a broad class of heterotic string compactifications. Using this method, we carry out this computation in full for a heterotic compactification incorporating a gauge bundle with non-Abelian structure group.

💡 Research Summary

This paper introduces a fully general, machine‑learning‑driven method for numerically approximating Hermitian Yang‑Mills (HYM) connections on holomorphic vector bundles of arbitrary rank and non‑Abelian structure group, a problem of central importance in heterotic string compactifications. The authors formulate the HYM problem as an alternating optimisation task that separately enforces the trace (Abelian) part of the curvature to be harmonic and drives the trace‑free (non‑Abelian) part to satisfy the Λ F = 0 condition.

The starting point is a background Hermitian metric H₀ on the bundle, taken as the generalized Fubini‑Study metric induced by the Kodaira embedding of a sufficiently high twist V(k)=V⊗Lᵏ into a Grassmannian. The full metric is written as H = h · e^{f} · H₀, where f∈C^∞(X,ℝ) is a scalar conformal factor that adjusts the determinant line bundle, and h∈Γ(End V) is a positive‑definite Hermitian endomorphism with pointwise unit determinant that deforms the non‑Abelian sector.

In the “Abelian stage” the authors minimise the L²‑norm of the ∂̄‑adjoint of η = Tr F_{e^{f}H₀} = Tr F_{H₀} + (rank V) ∂ ∂̄ f. This functional is equivalent to forcing η to be a harmonic (1,1) form, i.e. Λ η = constant, thereby producing the unique harmonic representative of the first Chern class of det V.

In the “Non‑Abelian stage” the trace‑free curvature F₀ = F – (1/rank V) Tr F·I is targeted. The authors define the objective E