We now recall why the HYM equation implies $\mu$-semistability. Let $E^{\prime}\subset E$ be a holomorphic subbundle (or more generally, a proper coherent subsheaf). A basic feature of holomorphic geometry is that pointwise curvature decreases in subbundles: the Chern curvature $F_{E^{\prime},H}$ for the restricted Hermitian metric $H|_{E^{\prime}}$ satisfies

$$\sqrt{-1}F_{E^{\prime},H}\leq\sqrt{-1}F_{E,H}|_{E^{\prime}}.$$

Wedging both sides with $\omega_{X^{\vee}}^{n-1}$, taking the trace, and integrating over ${X^{\vee}}$, we get

$$\int_{X^{\vee}}c_{1}(E^{\prime})\wedge\omega_{X^{\vee}}^{n-1}\leq\mu(E)rk(E^{\prime}),$$

namely $\mu(E^{\prime})\leq\mu(E)$, which is $\mu$-semistability. Although this argument is very transparent, we wish to summarize its key features:

• •

  Even though connections and curvatures make sense in a more general setting, we need the integrability of Kähler geometry to obtain pointwise positivity.²²²² 22 It would be interesting if the physicists can explain this positivity from supersymmetry, which is closely related to the Kähler condition.

• •

  To derive $\mu$-stability, one integrates over $X^{\vee}$, which can be interpreted as the moduli space of constant maps into $X^{\vee}$. ²³²³ 23 Path integrals on the topological B-model typically localizes to the moduli space of constant maps. This suggests a worldsheet interpretation, which will be more apparent on the mirror side.

• •

  The input from complex geometry can be interpreted as a short exact sequence

  $$0\to E^{\prime}\to E\to E/E^{\prime}\to 0,$$

  which has a categorical meaning in $D^{b}Coh(X^{\vee})$ as a distinguished triangle.

• •

  The role of the Kähler form enters via cohomological integrals.

• •

  There is no need for the complex Monge-Ampère equation.

Most of these features are not specific to HYM connections, but similar arguments give rise to obstructions for a large class of PDEs involving holomorphic bundles, such as the deformed Hermitian Yang-Mills equation.

The reverse direction, that $\mu$-stability implies the existence of HYM connections, is the hard part of the subject, and a key ingredient is the Donaldson functional. We make the not very essential simplification that $c_{1}(E)=0$. Donaldson [26][27] defined a functional $\mathcal{M}$ on the infinite dimensional space $\mathcal{H}$ of Hermitian metrics on the fixed bundle $E$, by prescribing its first variation at any point $H\in\mathcal{H}$:

$$\delta\mathcal{M}=\int_{X^{\vee}}\Tr(\sqrt{-1}F_{H}\delta HH^{-1})\wedge\omega_{X^{\vee}}^{n-1}.$$

(8)

Obviously from the definition, the critical points are precisely the HYM metrics. Less obviously, this functional is well defined up to an additive constant fixed by the choice of a reference Hermitian metric $H_{0}$. Different choices are related by

$$\mathcal{M}_{H_{0}}(H)=\mathcal{M}_{H_{0}^{\prime}}(H)+\mathcal{M}_{H_{0}}(H_{0}^{\prime}).$$

(9)

The space $\mathcal{H}$ can be formally assigned a Riemannian structure with non-positive curvature:

$$|\delta H|^{2}=\int_{X^{\vee}}\Tr(\delta HH^{-1})^{2}\wedge\omega_{X^{\vee}}^{n}.$$

(10)

The geodesics in $\mathcal{H}$ are given by $e^{tA}H$ where $A$ is self adjoint with respect to $H$, and the Donaldson functional is convex along geodesics.

In very sketchy terms, one standard proof of the Donaldson-Uhlenbeck-Yau theorem (closest to Simpson’s approach [72]) proceeds via the heat flow. It has two principal steps:

• •

  Consider the HYM heat flow

  $$\partial_{t}HH^{-1}=-\sqrt{-1}\Lambda F_{H}.$$

  Using certain parabolic maximum principles, one proves long time existence by showing that all derivatives of $H$ remain bounded for any given finite time. Furthermore, $\sup_{X^{\vee}}|\Lambda F|$ and $\left\lVert F\right\rVert_{L^{2}}$ are non-increasing in time, so remain uniformly bounded for all time. This step does not use stability.

• •

  The Donaldson functional is non-increasing in time almost by definition, so has a uniform upper bound for all time. Together with the pointwise bound on $\Lambda F$, which is like a Laplacian bound, one eventually shows that if $H(t)$ fails to be $L^{\infty}$ bounded for all time, then there exists an $L^{2}$-subbundle of $E$ with destabilizing properties, which is then interpreted algebraically as a subsheaf. (Roughly, the destabilizing sheaf comes from the eigensubspaces of $E$ corresponding to the small eigenvalues of $H(t)$ with respect to a fixed reference metric; compare the variational viewpoint below.) The stability condition rules out this case; one then shows that $H(t)$ actually converges smoothly at infinite time to a solution of the HYM equation.

We now discuss the variational perspective to the HYM equation, even though no proofs have been constructed along such lines. One would try to compactify $\mathcal{H}$ in some weaker topology (which is not known),²⁵²⁵ 25 What is known is how to compactify the space of Kähler potentials via psh functions, see Boucksom [11] for its fantastic application to Kähler-Einstein metrics. In that context, the boundary at infinity is related to non-archimedean geometry. extend the Donaldson functional to this compactification, attempt to find a minimizer of the functional, and then prove its regularity. Since the Donaldson functional is convex, it is natural to expect the existence of minimizer is equivalent to the properness of $\mathcal{M}$, or roughly equivalently $\mathcal{M}$ should grow at the infinity of $\mathcal{H}$.

Suppose now that $E$ fits into an extension sequence

$$0\to E_{1}\to E\to E_{2}\to 0.$$

Take arbitrary Hermitian metrics $H_{E}$ and $H_{E_{2}}$ on $E$ and $E_{2}$ respectively,²⁶²⁶ 26 The fact that the choice of Hermitian metrics will not ultimately matter is an expected feature, analogous to the relation between Kähler potentials and non-archimedean potential theory. and regard $H_{E_{2}}$ as a semi-Hermitian metric on $E$. We can then produce a 1-parameter family of Hermitian metrics on $E$, via $H(s)=e^{-s}H_{E}+H_{E_{2}}$. For $s\gg 1$, the metric $H(s)$ can be understood as equal to $e^{-s}H_{E}$ when restricted to $E_{1}$, and almost equal to $H_{E_{2}}$ on the orthogonal complement of $E_{1}$. In terms of bundles with connections, in the limit $s\to\infty$ we get $E_{1}\oplus E_{2}$ with the Chern connection for $H_{E}|_{E_{1}}\oplus H_{E_{2}}$. As such, it is an easy exercise to show that to leading order

$$\mathcal{M}(H(s))\sim-2\pi s\int_{X^{\vee}}c_{1}(E_{1})\wedge\omega_{X^{\vee}}^{n-1}.$$

Recall we have assumed $c_{1}(E)=0$ to simplify the definition of the Donaldson functional. Thus the subbundle $E_{1}$ destabilizes $E$, precisely when the degree of $E_{1}$ is positive, so $\mathcal{M}(H(s))$ goes to $-\infty$ along this 1-parameter family.

If one wants to turn the variational approach into an actual proof, one needs to further show that all possible ways to approach the infinity of (the metric completion of) $\mathcal{H}$ can be approximated by such 1-parameter families of algebraic origin. For our motivational purpose, it suffices to emphasize the following conceptual points:

• •

  The interesting limiting behaviour of the Donaldson functional occurs at an infinite distance boundary of (the metric completion of) $\mathcal{H}$. This sits well with the non-positive Riemannian curvature of $\mathcal{H}$.

• •

  The stability condition controls the asymptotic behaviour of the Donaldson functional near the boundary of $\mathcal{H}$.

The HYM equation famously fits into a formal geometric invariant theory (GIT) framework. For this, we slightly change viewpoint, and consider the bundle $E$ equipped with a fixed Hermitian structure, while the $(0,1)$-connection $\bar{\partial}_{E}$ encoding the holomorphic structure is allowed to vary. Each $\bar{\partial}_{E}$ uniquely determines the Chern connection $\nabla_{E}$. The group of complex gauge transformations $GL(E,\mathbb{C})$ acts on the space of integrable $(0,1)$-connections, via $\bar{\partial}_{E}\mapsto g\circ\bar{\partial}_{E}\circ g^{-1}$. This action is analogous to a complex reductive group action on a finite dimensional Kähler manifold. The subgroup $U(E)$ of unitary gauge transformations is analogous to the maximal compact subgroup. The space $GL(E,\mathbb{C})/U(E)$ can be formally identified with the space of Hermitian metrics on $E$. The HYM equation arises naturally from considerations of the moment map, and the Donaldson-Uhlenbeck-Yau theorem can be formally motivated from this picture [27].

This kind of infinite dimensional GIT framework has successfully suggested the answer in many problems within Kähler geometry, so it is only natural that many people have attempted to find an analogue suitable for special Lagrangian geometry. One such attempt is as follows. Thomas [65, section 3] considered the space

$$\mathcal{Z}=\{(L,A):L\subset X\text{ is Lagrangian},A\text{ is a flat $U(1)$-connection on }L\}$$

(not up to gauge equivalence!). The tangent space is $Z^{1}(L)\oplus Z^{1}(L)$, where $Z^{1}(L)$ is the space of closed 1-forms on $L$. This suggests an almost complex structure on $\mathcal{Z}$

$$J=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}.$$

With some hesitation,²⁷²⁷ 27 Thomas was aware of the possible objections, and did not use the GIT analogy as the principal basis of his proposal. Thomas attempted to complexify the Hamiltonian group action into a complex infinite dimensional group action. Unfortunately, there seems to be no natural way to do this in general, and $J$ is not quite an integrable complex structure. On the other hand, the moduli space of special Lagrangians with $U(1)$-local systems does have a natural complex structure induced from $J$, which is however naïve in the sense that the complex structure of the moduli space of Lagrangian branes is subject to further quantum corrections due to holomorphic curves, known also as ‘worldsheet instantons’ [8].

There seems to be no agreed interpretation, but in the author’s view, this suggests the infinite dimensional GIT framework is itself inadequate for the purpose of special Lagrangian geometry.²⁸²⁸ 28 Thomas’s suggestion is not the only possible way to achieve a GIT analogy. However, the other proposals [28][57] do not exhibit the mirror analogy in the same intuitively plausible way. As we explained in section 2.4, the main predictions of mirror symmetry are quantum in nature, and one should be cautious about taking an overly classical perspective. From our perspective, one underlying reason why the infinite dimensional GIT framework is successful in Kähler geometry, is that on the B-side one does not see the worldsheet instanton effect directly. On the A-side we have no such luxury.

As a word of console, although GIT is very good at suggesting the correct stability conditions for a PDE problem in Kähler geometry, it is almost never involved in the actual proofs of existence and uniqueness results for PDEs.