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Ricci-flat Kähler tangent connection is Hermitian–Yang–Mills

Physical scope

Purely mathematical verification for a supplied Ricci-flat Kähler metric; it proves no metric existence theorem.

Statement

Let $(X, J)$ be a complex threefold with a supplied smooth Kähler metric $g$ , meaning a positive $J$ -invariant Riemannian metric whose fundamental form $ω (u, v) = g (J u, v)$ is closed. In a holomorphic chart set $h_{i \overset{ˉ}{j}} = g_{C} (\partial_{z^{i}}, \partial_{{\overset{ˉ}{z}}^{j}})$ , where $g_{C}$ is the complex-bilinear extension. Define its Ricci form locally by $ρ (ω) = - i \partial \overset{ˉ}{\partial} \log \det h$ , with the local forms glued as verified below, and assume $ρ (ω) = 0$ (Ricci-flatness). Equip $T^{1, 0} X$ with the Hermitian metric induced by $g$ , and let $\nabla$ be its Chern connection, the unique Hermitian connection whose $(0, 1)$ part is the holomorphic structure; write $F_{\nabla}$ for its curvature. In complex dimension three, define the Hermitian–Yang–Mills equations for this connection by $F_{\nabla}^{0, 2} = 0$ and $ω^{2} \land F_{\nabla} = 0$ , with the endomorphism-valued wedge product interpreted entrywise. Then $\nabla$ satisfies both equations. This is conditional on the metric being supplied.

Statement references

Kähler class, canonical bundle, and Ricci form

Facts & Assumptions

Given: A complex threefold $(X, J)$ , its Kähler form $ω$ , a supplied metric $g$ with $ρ (ω) = 0$ , its holomorphic tangent bundle with the Hermitian metric induced by $g$ , and the Chern connection described in the Statement.

[F1]

Kähler class, canonical bundle, and Ricci form fixes these conventions on compact complex manifolds. Here they are defined locally on arbitrary complex threefolds, with no characteristic-class identification: $ω = i h_{i \overset{ˉ}{j}} d z^{i} \land d {\overset{ˉ}{z}}^{j}$ and $ρ = - i \partial \overset{ˉ}{\partial} \log \det h$ . Indeed a coordinate change with holomorphic invertible Jacobian $C$ gives $h^{'} = C^{T} h \overline{C}$ and $\log \det h^{'} = \log \det h + \log ∣ \det C ∣^{2}$ . Locally $\det C$ has a holomorphic logarithm, so the last term has zero $\partial \overset{ˉ}{\partial}$ . Thus the Ricci forms agree on overlaps without compactness or an application of the supplier outside its stated domain.

[A1]

The Chern curvature convention is $F_{i \overset{ˉ}{j}} = [\nabla_{i}, \nabla_{\overset{ˉ}{j}}]$ in a holomorphic coordinate frame. Reversing this curvature order changes an overall sign but not either vanishing equation.

Proof

Given: The Kähler metric and connection data in the Statement, with [F1] and [A1].

1.1givenA1algebra

Write $h_{i \overset{ˉ}{j}} = g_{C} (\partial_{i}, \partial_{\overset{ˉ}{j}})$ and use repeated-index summation. Metric compatibility and the specified $(0, 1)$ part give the Chern coefficients $Γ^{k}_{i ℓ} = h^{k \overset{ˉ}{m}} \partial_{i} h_{ℓ \overset{ˉ}{m}}$ and $\nabla_{\overset{ˉ}{j}} = \partial_{\overset{ˉ}{j}}$ in a holomorphic frame. Thus $F^{0, 2} = 0$ . The $(2, 0)$ curvature is zero as well: substituting these coefficients into $\partial_{i} Γ_{j} - \partial_{j} Γ_{i} + [Γ_{i}, Γ_{j}]$ and differentiating the inverse metric cancels its terms. Hence $F$ has type $(1, 1)$ and [A1] gives $F_{i \overset{ˉ}{j}}^{k}_{ℓ} = - \partial_{\overset{ˉ}{j}} (h^{k \overset{ˉ}{m}} \partial_{i} h_{ℓ \overset{ˉ}{m}})$ .

2.1givenF1A1step 1.1algebraconstruct

Fix a point. A complex-linear coordinate change makes $h_{i \overset{ˉ}{j}} = δ_{i j}$ there. Closedness of $ω$ gives $\partial_{i} h_{ℓ \overset{ˉ}{m}} = \partial_{ℓ} h_{i \overset{ˉ}{m}}$ , so $Γ^{k}_{i ℓ}$ is symmetric in $i, ℓ$ . The holomorphic substitution $z^{k} = w^{k} - \frac{1}{2} Γ^{k}_{i ℓ} (0) w^{i} w^{ℓ}$ has identity derivative and makes the transformed connection coefficients zero at the point. Metric compatibility then makes all first derivatives of $h$ zero there. At this point the curvature formula and the holomorphic and antiholomorphic Kähler derivative symmetries give $\sum_{i} F_{i \overset{ˉ}{i}}^{k}_{ℓ} = - \sum_{i} \partial_{\overset{ˉ}{i}} \partial_{i} h_{ℓ \overset{ˉ}{k}} = - \partial_{ℓ} \partial_{\overset{ˉ}{k}} \sum_{i} h_{i \overset{ˉ}{i}} = - \partial_{ℓ} \partial_{\overset{ˉ}{k}} \log \det h$ , where the last equality uses $h = I$ and $d h = 0$ at the point. Since $ρ = - i \partial \overset{ˉ}{\partial} \log \det h = 0$ , this contraction vanishes. Tensoriality and the arbitrary point give $h^{i \overset{ˉ}{j}} F_{i \overset{ˉ}{j}} = 0$ everywhere.

3.1givenstep 1.1step 2.1algebra∎

In a unitary frame write $ω = i \sum_{a = 1}^{3} d z^{a} \land d {\overset{ˉ}{z}}^{a}$ . For any $(1, 1)$ -form $F = F_{i \overset{ˉ}{j}} d z^{i} \land d {\overset{ˉ}{z}}^{j}$ , off-diagonal terms wedge to zero with $ω^{2}$ , and expanding the diagonal terms gives $ω^{2} \land F = \frac{1}{3 i} (\sum_{i} F_{i \overset{ˉ}{i}}) ω^{3}$ . This identity holds entrywise for endomorphism-valued forms. Step 2.1 therefore gives $ω^{2} \land F_{\nabla} = 0$ , while step 1.1 gives $F_{\nabla}^{0, 2} = 0$ , proving both equations for the supplied metric.

Depends on

Kähler class, canonical bundle, and Ricci form

Dependency tree · 2 levels

Sources

Angel M. Uranga, String Theory: A Graduate Course, Lecture 9: Calabi–Yau compactification and Heterotic String phenomenology