In [65][66] Thomas and Yau introduced the remarkable philosophy that existence and uniqueness questions of special Lagrangians inside an (almost) Calabi-Yau manifold $(X,\omega,\Omega)$ should be related to stability conditions in the Fukaya category. Turning this intuition into precise mathematical predictions, is however not easy, for at least the following geometric reasons, in addition to the analytic difficulties related to minimal surfaces and mean curvature flows:

• •

  The Fukaya category as it currently stands is likely inadequate for the purpose: the special Lagrangian representatives of a given derived Fukaya category class in $D^{b}Fuk(X)$, should it exists, is by no means guaranteed to be smooth and embedded. Thus one would like to enlarge the objects of the Fukaya category to include immersed and possibly singular Lagrangians. From the viewpoint of symplectic topology, the lack of Lagrangian objects is a basic difficulty, which is usually hidden in the non-geometric step of taking the twisted complexes in the construction of $D^{b}Fuk(X)$, and the idempotent completions in the construction of $D^{\pi}Fuk(X)$ (cf. the Appendix 6.2).

• •

  The knowledge of the stability conditions is severely deficient. Thomas-Yau’s paper predates the ingredient of Bridgeland stability³³ 3 What was available at the time, was the famous $\mu$-stability related to Hermitian Yang-Mills connections, of which Thomas and Yau were of course leading experts.⁴⁴ 4 There are current debates whether Bridgeland stability is the ultimately correct framework for formalising the physical intuition of stability conditions controlling BPS particle decay [31]. We regard Bridgeland stability as a working definition, to be modified in case future evidence arises., but the basic problem remains open: how to construct a stability condition on the derived Fukaya category $D^{b}Fuk(X)$ from the information of a holomorphic volume form $\Omega$ on $X$? In contrast, in analogous problems such as the existence of Hermitian-Yang-Mills connections, the stability condition is known a priori before the more serious endeavour to solve the PDE. One of the goals of this paper is to explain some modest progress on this issue, namely that there are nontrivial Floer theoretic obstructions to the existence of special Lagrangians.

Thomas and Yau primarily restricted attention to the case of almost calibrated Lagrangians, meaning the Lagrangian angle function $\theta$ satisfies ⁵⁵ 5 The important thing is that the upper and lower bounds on $\theta$ differ by $\pi$. Shifting the interval by a constant is inconsequential.

Notice this restriction removes any $\pi\mathbb{Z}$ ambiguity of the Lagrangian angle, so the Lagrangian is graded. Since we are focusing on compact Lagrangians, it makes sense to restrict to a more quantitative version, for some fixed small $\epsilon>0$:

One immediate consequence is an a priori volume bound.

Keeping our narrative closer to the historical development, Thomas and Yau were inspired by $\mu$-stability for Hermitian Yang-Mills connections. They assumed that the principal mechanism which can forbid the Hamiltonian isotopy class of $L$ from admitting a special Lagrangian representative, is related to a distinguished triangle in the Fukaya category, the primary geometric source being that $L$ is Hamiltonian isotopic to a graded Lagrangian connected sum $L=L_{1}\#L_{2}$,⁶⁶ 6 It is very important for Thomas and Yau that the Lagrangian connected sum is asymmetric in $L_{1}$ and $L_{2}$. Our notation for the Lagrangian connected sum agrees with Thomas and Yau, but is opposite to Joyce [44] and a large number of symplectic geometry texts. Our convention is compatible with the distinguished triangle $L_{1}\to L_{1}\#L_{2}\to L_{2}\to L_{1}[1]$. such that $\phi(L_{1})\geq\phi(L_{2})$. Based on this intuition, Thomas made an attempt to define a notion of stability (cf. [65, Definition 5.1], and Definition 3.32 below).⁷⁷ 7 We will not repeat their definition verbatim here because the author thinks its focus on the Hamiltonian isotopy class, instead of the Fukaya category class, is largely a limitation of its time. It is the spirit rather than the letter of their definition which matters. Thomas then made the important prediction:

Thomas and Yau [66] analyzed the problem again from the Lagrangian mean curvature flow (LMCF) perspective, and made a somewhat more cautious prediction, which roughly amounts to the following. When a destabilising decomposition into Lagrangian sums is forbidden, either by a smallness assumption on the oscillation of the phase function $\theta$, or because the volume of $L$ is smaller than $\text{Vol}(L_{1})+\text{Vol}(L_{2})$ for any putative decomposition, (notice both conditions are preserved under the flow), then they conjectured that the mean curvature flow starting from $L$ will converge into a special Lagrangian [66, section 7].

To trace the subsequent development, and move beyond the almost calibrated setting, we need to recall an important piece of homological algebra known as Bridgeland stability conditions on a triangulated category [13].

With the hindsight of Bridgeland stability condition, and more than one decade of progress on the mean curvature flow as well as symplectic geometry, Joyce [41] produced a major update of the Thomas-Yau picture. We shall discuss more about the LMCF considerations in section 4.1, but it suffices here to give away its main punchline (cf. [41, Conj 3.34]):

• •

  Let $(X,\omega,\Omega)$ be a Calabi-Yau manifold ⁹⁹ 9 The complex Monge-Ampère equation is convenient but not indispensable, see section 4.1., either compact or Stein. There should be an enlarged version of the derived Fukaya category $D^{b}Fuk(X)$, including classes of immersed or mildly singular Lagrangians, and a Bridgeland stability condition $(Z,\mathcal{P})$ on $D^{b}Fuk(X)$, whose central charge is

  $$Z(L)=\int_{L}\Omega.$$

  (5)

• •

  Given a Lagrangian brane $L$ (with grading, orientation, relative spin structure, local system and bounding cochain data) such that the Floer cohomology is unobstructed, and suppose $L$ is generic in its Hamiltonian isotopy class. Then the LMCF $(L_{t})_{t>0}$ (with brane structure) starting from $L$ exists for all time with surgeries at a finite series of singular times. The nature of these surgeries have conjectural descriptions. The Lagrangian can change its Hamiltonian class at these surgeries, but maintains its derived category class. At $t\to\infty$, the Lagrangians $L_{t}$ converges in the geometric measure theory sense to a union of graded special Lagrangian currents $L_{1},\ldots L_{N}$ of phase angle $\hat{\theta}_{1}>\hat{\theta}_{2}>\ldots>\hat{\theta}_{N}$ with multiplicities counted:

  $$\lim_{t\to\infty}L_{t}=L_{1}+\ldots L_{N}.$$

  (6)

• •

  If there is only one constant phase angle $\hat{\theta}$ appearing in the infinite time limit, then $L$ defines a semistable object of phase $\hat{\theta}$ in $D^{b}Fuk(X)$. Otherwise $L$ is not semistable with respect to any phase angle, and the infinite time limit supposedly give rise to the Harder-Narasimhan decomposition.

Joyce’s program is much more ambitious in the sense that it attempts to capture the entire triangulated category $D^{b}Fuk(X)$. The original Thomas-Yau proposal, which is concerned only with almost calibrated Lagrangians, fits into the Joyce picture as an abelian subcategory $\mathcal{P}(-\pi/2<\theta\leq\pi/2)\subset D^{b}Fuk(X)$,¹⁰¹⁰ 10 Here we ignore the ‘small’ difference between $<$ and $\leq$. This would be justified if $\Omega$ is suitably generic so that there is no special Lagrangian of phase $\pi/2$, namely that the countably many numbers $\text{Arg}\int_{L}\Omega$ for $[L]\in H_{n}(X,\mathbb{Z})$ miss the number $\pi/2$. the heart of a certain $t$-structure. The main appeal of Joyce’s mean curvature flow perspective, is that if the infinite time limiting currents $L_{1},\ldots L_{N}$ can be given suitable brane structures to be admitted as objects of $D^{b}Fuk(X)$, then the program gives a conjectural dynamical mechanism to explain the Harder-Narasimhan decomposition, which is the most nontrivial aspect of the Bridgeland stability. Its principal drawback is that Joyce offers no a priori information on the Bridgeland stability beyond the central charge, other than letting off the mean curvature flow to find its own destiny. This limits its practical applicability, for instance to existence questions of special Lagrangians in prescribed classes in $D^{b}Fuk(X)$. See section 3.6 for more discussions.

At this moment an objection may arise: since the Thomas-Yau-Joyce picture is beset by some vagueness and plenty of technical difficulties, why is it useful as a guiding principle at all? Besides the supportive evidence that we shall soon discuss, the main answer is that the Thomas-Yau philosophy transforms a PDE problem (the existence of special Lagrangians) into a categorical framework, which if better understood is in principle checkable by algebraic means. A Bridgeland stability is the interplay between a category and a numerical property. In the analogous problem of Hermitian Yang-Mills connections, the existence criterion is formulated by $\mu$-stability, which compresses the information of the Kähler form into only certain intersection numbers/cohomological integrals (cf. section 2.5). Likewise, the stability condition responsible for the existence of special Lagrangians, even though it is not adequately specified, is in principle a compression of the analytic data of a holomorphic volume form, and quite plausibly enters only through cohomological integrals of $\Omega$, as will be discussed more fully in Chapter 3. As an indication of the possible predicative power of the Thomas-Yau picture, here is a sample question as food for thought:

We first recall some basics about Lawlor necks [51][45], which are non-compact embedded exact special Lagrangians $L_{\phi,A}$ inside the standard Euclidean $\mathbb{C}^{n}$, asymptotic at infinity to the union of two planes

Symplectic topologically, they can be viewed as a realisation of the Lagrangian handle $S^{n-1}\times\mathbb{R}$ that appears in the Lagrangian connected sum construction. This motivates the ansatz

The special Lagrangian condition translates into an ODE system on the functions $z_{1}(y),\ldots z_{n}(y)$, which can be solved exactly as follows.

Let $n>2$ and $a_{1},...,a_{n}>0$, and define polynomials $p,P$ by

Define real numbers $\phi_{1},...,\phi_{n}$ and $A$ by

Clearly $\phi_{k},A>0$, and elementary integration shows $\sum\phi_{i}=\pi$. This yields a 1-1 correspondence between $n$-tuples $(a_{1},\ldots,a_{n})$ with $a_{k}>0$, and $(n+1)$-tuples $(\phi_{1},\ldots,\phi_{n},A)$ with $\phi_{k}\in(0,\pi)$, $\sum\phi_{k}=\pi$ and $A>0$. Setting

yields the solution $(z_{1}(y),\ldots,z_{n}(y))$, hence the Lawlor necks $L_{\phi,A}$.

For fixed asymptotic planes $\Pi_{0},\Pi_{\phi}$, the Lawlor necks arise in a 1-parameter family, related to each other by the rescaling $\vec{z}\to\lambda\vec{z}$ in $\mathbb{C}^{n}$, and $A$ behaves like 2-dimensional area $A\to\lambda^{2}A$ under this scaling. One also observes that asymptotically near infinity, the Lawlor necks are graphs over $\Pi_{0}$ (resp. $\Pi_{\phi}$) of the differential $df$, where

We say the Lawlor neck has asymptotic decay rate $2-n$. The upshot is that it approaches $\Pi_{0}\cup\Pi_{\phi}$ sufficiently fast.

Here is a sketch of their arguments:

• •

  Using the asymptotic assumption on the exact Lagrangian $L$, one can assign an analytic invariant $A(L)$ to $L$ as follows. Let $f_{L}:L\to\mathbb{R}$ be a primitive of the Liouville form $\lambda$, namely $df_{L}=\lambda|_{L}$, then $f_{L}$ converges to constants $c_{0},c_{\phi}$ at the two asymptotic ends along $\Pi_{0},\Pi_{\phi}$ respectively. Then one defines $A(L)=c_{\phi}-c_{0}$. If $L$ coincides with the Lawlor neck $L_{\phi,A}$, then $A(L)=A$.

• •

  Partially compactify $\mathbb{C}^{n}$ into a Liouville manifold identified as the plumbing $M$ of two cotangent bundles $T^{*}S^{n}$ with $T^{*}S^{n}$. Here the two copies of $S^{n}$ arise topologically as one-point compactifications of $\Pi_{0}$ and $\Pi_{\phi}$ by adding the points at infinity $\infty_{0}$ and $\infty_{\phi}$, and topologically $M$ is the union of $\mathbb{C}^{n}$ and the two cotangent fibres over $\infty_{0}$ and $\infty_{\phi}$ respectively. Under suitably fast decay condition at infinity, the unknown special Lagrangian $L$ can be compactified into an exact graded embedded Lagrangian $\bar{L}$ inside $M$. One would like to compare this to the Lagrangian $\bar{L}_{\phi,A}$ obtained by the compactification of the standard Lawlor necks $L_{\phi,A}$ inside $M$.

• •

  By analyzing the intersection pattern with the two cotangent fibres at infinity, and using the classification results of Abouzaid and Smith [1], one shows that inside $D^{b}Fuk(M)$, the Lagrangian object $\bar{L}$ is isomorphic to one of the two Lagrangian connected sums of the two $S^{n}$ with suitable gradings, and in fact the assumption on Floer degrees $\sum\phi_{i}=\pi$ singles out $\bar{L}\simeq\bar{L}_{\phi,A}\in D^{b}Fuk(M)$, the opposite surgery corresponding to $\sum\phi_{i}=(n-1)\pi$. This step needs $n\geq 3$. For contradiction, we assume $\bar{L}$ does not coincide with $\bar{L}_{\phi,A}$ for any choice of parameter $A>0$.

• •

  By a modification of the Thomas-Yau argument, one shows that after a small Hamiltation perturbation $\bar{L}^{\prime\prime}$ of $\bar{L}_{\phi,A}$, we can ensure $\bar{L}^{\prime\prime}$ is transverse to $\bar{L}$, there is no degree $0,n$ intersection points in $\bar{L}^{\prime\prime}\cap\bar{L}$ inside $\mathbb{C}^{n}$, and there is precisely one intersection point $p$ and $q$ in $\bar{L}^{\prime\prime}\cap\bar{L}$ on each of the two cotangent fibres at infinity respectively. Morever, the $D^{b}Fuk(M)$ class and the analytic invariants of $\bar{L}^{\prime\prime}$ agree with that of $\bar{L}_{\phi,A}$.

  Remark 2.8.

  The subtlety at infinity prevents one from removing degree $0,n$ intersections outside the $\mathbb{C}^{n}$ region, so one does not reach an immediate contradiction as in the Thomas-Yau argument. This technical failure is necessary, because the Lawlor necks with fixed asymptotic planes are not unique, but do arise in a 1-parameter family. It is in overcoming this technical problem that holomorphic curves appear in [40].

• •

  Now suppose the Lawlor neck is chosen with the parameter $A=A(L)$, which presumes $A(L)>0$.

  Lemma 2.7.

  [40, Thm 2.15] Assume $J$ is a generic almost complex structure on $M$ compatible with the Liouville structure. There exists a $J$-holomorphic strip $\Sigma$ with boundary on $\bar{L}$ and $\bar{L}^{\prime\prime}$ and two corners at $p$ and $q$ respectively.

  Proof.

  Consider the Floer cup product with mod 2 coefficients

  $$HF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes HF^{0}(\bar{L},\bar{L}^{\prime\prime})\to HF^{0}(\bar{L},\bar{L}),$$

  which can be identified as the cup product

  $$H^{0}(\bar{L},\bar{L})\otimes H^{0}(\bar{L},\bar{L})\to H^{0}(\bar{L},\bar{L}),\quad 1_{\bar{L}}\cup 1_{\bar{L}}=1_{\bar{L}},$$

  and thus must be nontrivial. However, at chain level this Floer product comes from the $A_{\infty}$ operation

  $$m_{2}:CF^{0}(\bar{L}^{\prime\prime},\bar{L})\otimes CF^{0}(\bar{L},\bar{L}^{\prime\prime})\to CF^{0}(\bar{L},\bar{L}),$$

  which must be nontrivial. The counting interpretation implies there are intersection points $p^{\prime}\in CF^{0}(\bar{L},\bar{L}^{\prime\prime})$ and $q^{\prime}\in CF^{0}(\bar{L}^{\prime\prime},\bar{L})\simeq CF^{n}(\bar{L},\bar{L}^{\prime\prime})^{\vee}$ and some holomorphic strip in between. Since degree $0,n$ intersection points cannot occur inside $\mathbb{C}^{n}$, they can only occur at infinity, so we must have $\{p,q\}=\{p^{\prime},q^{\prime}\}$. ∎

  Now the area of the $J$-holomorphic curve can be computed cohomologically. Using the choice of parameter $A$,

  $$\int_{\Sigma}\omega=\int_{\partial\Sigma}\lambda=\int_{p^{\prime}\to q^{\prime}}df_{\bar{L}}+\int_{q^{\prime}\to p^{\prime}}df_{\bar{L}^{\prime\prime}}=\pm(A(L)-A(L^{\prime\prime}))=\pm(A-A)=0.$$

  This contradicts the positivity of area of the holomorphic curve, which proves $L$ must coincide with $L_{\phi,A}$.

• •

  Finally one needs to a priori justify $A(L)>0$. This relies on a slightly more complicated holomorphic polygon counting argument, and the main upshot is that one can produce a nontrivial holomorphic triangle from a distinguished triangle in $D^{b}Fuk(M)$, with the three edges on $\bar{L}$, $\Pi_{0}\cup\{\infty_{0}\}$ and $\Pi_{\phi}\cup\{\infty_{\phi}\}$. Then one shows $A(L)$ has the interpretation as its area, so must be positive.

In [40] the perturbations involved in the partial compactification and the genericity of the almost complex structure makes the holomorphic curves rather difficult to visualize.¹⁴¹⁴ 14 A typical feature of Floer theory, is that completely realistic examples about holomorphic curves are also non-explicit. We now present a heuristic way to see holomorphic triangles with the edges on $\bar{L}_{\phi,A}$, $\Pi_{0}\cup\{\infty_{0}\}$ and $\Pi_{\phi}\cup\{\infty_{\phi}\}$, by restricting attention to $\mathbb{C}^{n}$ with the standard complex structure, and we imagine the two vertices $\infty_{0},\infty_{\phi}$ as the intersection points at infinity.

We choose any $\vec{x}=(x_{1},\ldots x_{n})\in S^{n-1}$. Assume first that $x_{k}\neq 0$. Then coordinatewise, we have a real curve in $\mathbb{C}$ swept out by $z_{k}(y)x_{k}$ as $y$ varies from $-\infty$ to $+\infty$, and two straight rays emanating from the origin defined by $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$. Inside $\mathbb{C}$, these three real curves enclose a noncompact holomorphic triangle, with one vertex at the origin, and two idealized intersection points at the infinity of $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$. In the product space $\mathbb{C}^{n}$, this gives rise to a holomorphic triangle $\Sigma_{\vec{x}}$ with boundary on $L_{\phi,A},\Pi_{0},\Pi_{\phi}$, and corners at $0,\infty_{0},\infty_{\phi}$. Now in case some $x_{k}=0$, there is still a holomorphic triangle in the product space that makes sense; the $k$-th projection of this triangle is simply the origin. What happens when $x_{k}\to 0$, is simply that the $k$-th projection $\pi_{k}(\Sigma_{\vec{x}})$ becomes very thinly concentrated near the two rays $\text{sign}(x_{k})\mathbb{R}_{+}$ and $\text{sign}(x_{k})e^{i\phi_{k}}\mathbb{R}_{+}$, and its area shrinks to zero. Morever for any given $\epsilon$, the subset $\Sigma_{\vec{x}}\cap\pi_{k}^{-1}(|z|>\epsilon)$ disappears into the infinity of $\mathbb{C}^{n}$ as $x_{k}\to 0$. Thus when we restrict to any compact subset of $\mathbb{C}^{n}$, the holomorphic discs behave continuously as $x_{k}\to 0$.

Using any of the holomorphic triangles $\Sigma_{\vec{x}}$ parametrised by $S^{n-1}$, we can calculate its area cohomologically by

which is the intuitive explanation of why $A$ must be positive, an important ingredient of [40].

We want to draw attention also to a different aspect not explicit in [40]: that these holomorphic triangles naturally arise in an $(n-1)$-dimensional moduli, rather than as isolated triangles. Consequently, the universal family of such holomorphic triangles is naturally $(n+1)$-dimensional. A generic point on $\Pi_{0}\cup\Pi_{\phi}\cup L_{\phi,A}$ is swept out precisely once by some $\partial\Sigma_{\vec{x}}$. When the orientations are taken into account, then the total space of this universal family gives rise to an $(n+1)$-dimensional integration current, which provides a bordism current between the integration cycles of $L_{\phi,A}$ and $\Pi_{0}\cup\Pi_{\phi}$. Producing bordism currents via universal families of holomorphic curves will be essential to our proposals concerning the Thomas-Yau conjecture.

However, in the differential geometric literature on the Thomas-Yau conjecture, one often attempts to go beyond all these aspects of mirror symmetry, and tries to directly compare the space of Lagrangian submanifolds inside $X$, with the space of holomorphic vector bundles equipped with Hermitian metrics over $X^{\vee}$. This comparison must be treated with caution, because there is no general bijective correspondence between these two infinite dimensional spaces, and because mirror symmetry is properly a quantum phenomenon which is not fully captured by classical considerations. In fact, mirror symmetry only relates $Fuk(X)$ and $Coh(X^{\vee})$ after taking the derived category, and there is no general reason for the heart of a preferred $t$-structure on the derived Fukaya category (e.g. the $t$-structure defined by some almost calibrated condition) to agree with $Coh(X^{\vee})$. Having vaccinated the reader with these precautions, we shall regard such comparisons as useful formal analogies, and we consider those aspects with quantum interpretations as more reliable.

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

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

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 $H\in\mathcal{H}$:

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

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

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

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

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

(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 $\mathcal{Z}$

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.