2.8 Space of almost calibrated Lagrangians [048S]

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2.8 Space of almost calibrated Lagrangians

We now return to the A-side of the mirror, and describe the work of J. Solomon [75][76], generally accepted as the canonical picture on the subject. While dHYM is motivated by the dream of a correspondence (semiflat mirror symmetry) between the two sides of the mirror at the level of classical objects, Solomon is interested in the structural similarities between the infinite dimensional spaces involved in the mirrors, and especially in finding analogues for the HYM equation.³⁶³⁶ 36 Solomon’s work predates the substantial works on the dHYM equation. The reader is thus invited to keep in mind the comparison with section 2.5.

Let $\mathcal{L}^{+}$ be the space of almost calibrated compact immersed Lagrangians in an almost Calabi-Yau manifold $(X,\omega,\Omega)$ ,

\mathcal{L}^{+}=\{\iota:L\to X|-\frac{\pi}{2}<\theta<\frac{\pi}{2}\}.

For the benefit of intuition, we shall loosely identify the Lagrangian immersion with its image. Solomon [76] considers the space $\mathcal{O}$ of Lagrangians which are exact isotopic (aka. local Hamiltonian isotopic)³⁷³⁷ 37 Exact isotopies of immersed Lagrangians differ from global Hamiltonian isotopies, in that the Hamiltonian function on the Lagrangians may depend on the local sheets, so may not always extend smoothly to the ambient space. within $\mathcal{L}^{+}$ to a fixed Lagrangian. It should be borne in mind that unlike the space $\mathcal{H}$ of Hermitian metrics on a bundle, the space $\mathcal{O}$ may have very nontrivial topology; we call its universal cover $\tilde{\mathcal{O}}$ . The formal deformations of the Lagrangians are given by the Hamiltonian functions $h:L\to\mathbb{R}$ , up to the ambiguity of an additive constant. Solomon proposes to fix the constant by the normalisation condition $\int_{L}h\text{Re}\Omega=0$ , and assigns a formal Riemannian metric on

T_{L}\mathcal{O}=\{h:L\to\mathbb{R}|\int_{L}h\text{Re}\Omega=0\},

via the formula

\langle h,k\rangle=\int_{L}hk\text{Re}\Omega.

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This is positive definite because $\text{Re}\Omega$ is a volume form on $L$ by almost calibratedness. The main result of [76] is a computation on the Riemannian curvature of $\mathcal{O}$ , which is found to be non-positively curved, similar to the HYM setting. A further paper [77] computes the geodesic equation with respect to this formal metric, and reinterprets a geodesic between $L_{1},L_{2}\in\mathcal{O}$ in terms of a 1-parameter family of special Lagrangians (of phase $\frac{\pi}{2}$ instead!) with boundary on $L_{0}\cup L_{1}$ .

Another major aspect of Solomon’s work is to look for an analogue of the Donaldson functional. The definition of this Solomon functional does not really require the almost calibrated condition, even though some of the main properties do. Choose some appropriate $\hat{\theta}\in(-\frac{\pi}{2},\frac{\pi}{2})$ so that $\int_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)=0$ . Now take a 1-parameter family of Lagrangians $L_{t}$ in $\mathcal{O}$ with associated Hamiltonian functions $h_{t}:L_{t}\to\mathbb{R}$ . Solomon defines

\mathcal{S}=\int_{0}^{1}dt\int_{L_{t}}h_{t}\text{Im}(e^{-i\hat{\theta}}\Omega).

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His main theorem [75] is

Theorem 2.10.

[75] The functional $\mathcal{S}$ is independent of Hamiltonian deformations of the path of Lagrangians fixing the two ends. In particular, by fixing the starting Lagrangian $L_{0}$ , we obtain a functional $\mathcal{S}$ of the endpoint Lagrangian, which is well defined on the universal cover $\tilde{\mathcal{O}}$ .

Like the Donaldson functional, this well definition is nontrivial. It is however obvious that the critical points in $\mathcal{O}$ are precisely special Lagrangians of phase $\hat{\theta}$ . Furthermore,

Theorem 2.11.

[75] Assume $\hat{\theta}=0$ . Then the second variation of $\mathcal{S}$ at a critical point is positive semidefinite. Furthermore, along a geodesic with respect to Solomon’s formal Riemannian metric, the functional $\mathcal{S}$ is convex.

The analogy with Donaldson’s picture in section 2.5 should be quite clear.

Limitations

Unlike Thomas and Yau who based their bet primarily on the Floer theoretic or categorical aspects, which are closer to the quantum world of topological field theories, Solomon’s picture is predominantly classical, and its chief limitation comes from fixing the topological type of the Lagrangian:

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There is no appearance of the brane structure, or the role of holomorphic curves.
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Solomon works with exact isotopic Lagrangians, but the Thomas-Yau argument suggests it is more natural to work in a derived Fukaya category class.
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The Solomon functional is only well defined by passing to a highly nontrivial universal cover. In the very special case where $\omega$ and $\text{Im}(e^{i\hat{\theta}}\Omega)$ are exact forms on $X$ , Solomon gave a formula [75, Thm 1.3] that shows his functional is well defined on $\mathcal{O}$ . We view the exactness on $\text{Im}(e^{i\hat{\theta}}\Omega)$ as too strong an assumption for applications.
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The infinite dimensional Riemannian structure is incomplete in a much more severe way compared to the B-side analogues. This means that in non-pathological examples, we can reach the boundary of the exact isotopy class within finite distance in the Solomon metric, such that the Solomon functional remains finite.

This is geometrically very significant. In the LMCF approach, this would strongly suggest the formation of finite time singularity, which is a major difference with the HYM case. In the variational viewpoint, this incompleteness would negate all the favourable arguments from the convexity of the functional and the non-positivity of curvature, and suggest instead that the exact isotopy class is not an adequate framework for finding special Lagrangians. We will discuss later that a more promising variational framework needs to incorporate Lagrangians from the same derived Fukaya category class, not just the same exact isotopy class.

Exact isotopy class versus derived category class

The example below is closely related to the most symmetric case of the Lawlor necks (cf. section 2.3). It is also morally related to the Lawlor neck pinching singularity in the Joyce program [41, section 3.5].

Example 2.12.

Consider two almost calibrated Lagrangians $L^{1},L^{2}$ with a unique intersection point $p$ , and there is a Darboux chart around $p$ modelled on $B_{1}\subset\mathbb{C}^{n}$ , such that inside the chart $L^{1},L^{2}$ the local setup is

L^{1}=e^{i\pi/n}\mathbb{R}^{n},\quad L^{2}=\mathbb{R}^{n},\quad\omega=\frac{\sqrt{-1}}{2}\sum dz_{k}\wedge d\bar{z}_{k},\quad\Omega\approx\prod dz_{k}.

Let $(L_{t})_{0<t\ll 1}$ be a 1-parameter family of Lagrangian connected sums with neck length $O(t)$ , which all agree with $L_{1}\cup L_{2}$ except in a compact subset in $B_{1}$ . Inside $B_{1}$ , we take the ansatz

L_{t}=\{(t\gamma(s)x_{1},\ldots,t\gamma(s)x_{n})|x_{1}^{2}+\ldots x_{n}^{2}=1\},

where the curve $\gamma(s):\mathbb{R}\to\mathbb{C}$ can be chosen so that $L_{t}$ is almost calibrated and agrees with $L_{1}\cup L_{2}$ outside $B_{1/2}$ . Clearly, $L_{t}$ are related by scaling inside $B_{1}$ . The Hamiltonian vector field along $L_{t}$ , which is really a section of $(TX/TL_{t})|_{L_{t}}$ , agrees with $(\gamma(s)x_{1},\ldots,\gamma(s)x_{n})$ inside $B_{1}$ and is zero outside.³⁸³⁸ 38 This is consistent because near the boundary of $B_{1}$ , the position vector is a tangent vector of $L$ , so vanishes in the quotient $TX/TL_{t}$ . The corresponding Hamiltonian function $h_{t}$ is $t^{2}$ times a smooth function of one variable $s$ ; a small caveat is that $h_{t}$ converges to two generally different constants along $L_{1}$ and $L_{2}$ . Thus it takes finite distance in the Solomon metric to reach the limit $t\to 0$ , and the Solomon functional remains finite, but the topology changes from $L_{t}$ to $L^{1}\cup L^{2}$ .

Remark 2.13.

The Hamiltonian functions $h_{t}$ along $L_{t}$ can be extended to global functions on $X$ , with

\left\lVert h_{t}\right\rVert_{C^{0}}=O(t^{2}),\quad\left\lVert dh_{t}\right\rVert_{C^{0}}=O(t),\quad\left\lVert\nabla^{2}h_{t}\right\rVert_{L^{\infty}}\leq C.

But in the $t\to 0$ limit, the second derivatives fail to be continuous at the origin, and indeed $L_{t}$ changes topology in the limit.

In this example the essential failure is the breakdown of smoothness. In view of Joyce’s program, this suggests that the remedy is to allow for (Floer theoretically unobstructed) almost calibrated Lagrangians connected to each other not just by exact isotopies, but also surgeries such as Lagrangian connected sums. In these transitions the derived category class of the Lagrangian is unchanged, and the Thomas-Yau argument suggests the $D^{b}Fuk(X)$ class is a natural framework to look for special Lagrangian representatives. The following fundamental question is thus relevant for the compatibility between the geometric and the categorical perspectives:

Question 2.

When are two almost calibrated unobstructed Lagrangian branes isomorphic in $D^{b}Fuk(X)$ connected by exact isotopies with surgeries?

Remark 2.14.

The Joyce program suggests that running the LMCF would result in a sequence of exact isotopies and surgeries, to connect the initial Lagrangian to its infinite time limit, which one hopes to be the unique representative of the Harder-Narasimhan decomposition. In the almost calibrated case, there is no ‘collapsing zero object’ in this process for homological reasons, so the surgeries should be continuous in the geometric measure theory sense. Since any two such Lagrangians within the same $D^{b}Fuk(X)$ class are expected to flow to the same limit, they are supposedly connected to each other through a continuous family of unobstructed Lagrangians.