# Thomas-Yau conjecture and holomorphic curves

Yang Li

###### Abstract

The main theme of this paper is the Thomas-Yau conjecture, primarily in the
setting of exact, (quantitatively) almost calibrated, unobstructed Lagrangian
branes inside Calabi-Yau Stein manifolds. In our interpretation, the
conjecture is that Thomas-Yau semistability is equivalent to the existence of
special Lagrangian representatives. We clarify how holomorphic curves enter
this conjectural picture, through the construction of bordism currents between
Lagrangians, and in the definition of the Solomon functional. Under some extra
hypothesis, we shall prove Floer theoretic obstructions to the existence of
special Lagrangians, using the technique of integration over moduli spaces. In
the converse direction, we set up a variational framework with the goal of
finding special Lagrangians under the Thomas-Yau semistability asumption, and
we shall make sufficient progress to pinpoint the outstanding technical
difficulties, both in Floer theory and in geometric measure theory.

###### Contents

  1. 1 Foreword
  2. 2 Thomas-Yau conjecture backgrounds
    1. 2.1 The Thomas-Yau-Joyce picture
    2. 2.2 Principal evidence of Thomas-Yau
    3. 2.3 Variant: uniqueness of the Lawlor neck
    4. 2.4 Philosophy of open string mirror symmetry
    5. 2.5 Hermitian-Yang-Mills
    6. 2.6 Deformed Hermitian Yang-Mills
    7. 2.7 Extensions and wall crossing
    8. 2.8 Space of almost calibrated Lagrangians
    9. 2.9 Totally real geometry
    10. 2.10 Analogy with tunneling effect
  3. 3 Moduli space of holomorphic curves
    1. 3.1 Lotay-Pacini picture revisited
      1. 3.1.1 The exact embedded Lagrangian case
      2. 3.1.2 Immersed case
    2. 3.2 Solomon functional revisited
    3. 3.3 Automatic transversality
      1. 3.3.1 Automatically transverse cases
    4. 3.4 Positivity condition
    5. 3.5 Floer theoretic obstructions
    6. 3.6 Towards a Bridgeland stability condition
    7. 3.7 Moduli integral formula for the Solomon functional
      1. 3.7.1 Moduli integral formula for the Solomon functional
      2. 3.7.2 Change of reference Lagrangians formula revisited
      3. 3.7.3 First variation formula revisited
      4. 3.7.4 Speculations on compact Calabi-Yau manifolds
    8. 3.8 More applications of moduli space integrals
      1. 3.8.1 Lotay-Pacini convexity
      2. 3.8.2 Lower bound of the Solomon functional
      3. 3.8.3 Bounded part of the Solomon functional
  4. 4 Continuity, LMCF and variational method
    1. 4.1 Lagrangian mean curvature flow
    2. 4.2 Continuity method
  5. 5 Variational method
    1. 5.1 Compactness and regularity
      1. 5.1.1 Standard geometric measure theory
      2. 5.1.2 Exact Lagrangians under weak regularity
      3. 5.1.3 Almgren’s regularity in the almost Calabi-Yau setting?
    2. 5.2 Quantitative almost calibratedness
      1. 5.2.1 Intrinsic distance bound and potential clustering
      2. 5.2.2 Bounded part of the Solomon functional revisited
    3. 5.3 Variational strategy
    4. 5.4 Floer theory under weak regularity
    5. 5.5 Asymptotes of the Solomon functional
      1. 5.5.1 Thomas-Yau conjecture
    6. 5.6 Minimizers and special Lagrangians
    7. 5.7 Thomas-Yau uniqueness revisited
      1. 5.7.1 Special Lagrangians are minimizers
    8. 5.8 Comparison with Joyce’s LMCF program
  6. 6 Appendix on the Fukaya category
    1. 6.1 Fukaya category for embedded exact Lagrangians
    2. 6.2 Immersed exact Lagrangians

##  1 Foreword

The visionary proposal of Thomas and Yau [65][66] is the philosophy that
_existence and uniqueness questions of certain special Lagrangian branes
inside an almost Calabi-Yau manifold $(X,\omega,\Omega)$ should be governed by
stability conditions in the derived Fukaya category_ (_cf._ section 2.1). This
proposal lies at the intersection of two major mathematical disciplines:

  * •

From the viewpoint of geometric measure theory, the Thomas-Yau proposal
promises a systematic method to prove existence theorems for special
Lagrangians, which are certain absolute volume minimizers within prescribed
homology classes. Currently, the known construction methods are based on
perturbative techniques, symmetry reductions, some special ansatzs, and
integrable system techniques specific to low dimensions [43]. In contrast,
Thomas and Yau suggests the difficult PDE questions may be reducible to
largely topological and algebraic questions in Floer theory.

  * •

From the viewpoint of mirror symmetry, the Thomas-Yau proposal is a window
beyond homological mirror symmetry. Currently, most of the mathematical works
on mirror symmetry are concerned with the duality between pure symplectic
topology and pure complex geometry, and as such have a largely topological and
algebraic flavour. As soon as one simultaneously consider the full Kähler
geometric data on each side of the mirror, then questions of analytic nature
become inevitable, and the Thomas-Yau proposal is a central component of this
larger picture (_cf._ section 2.4).

Two decades have lapsed since their proposal, but we feel only a small part of
the mystery has been unveiled. Part of the problem is that the relevance of
holomorphic curves to the Thomas-Yau proposal is insufficiently understood,
despite their central role in defining the Fukaya category. One of the main
goals of this paper is to propose the following picture, in the more
specialized setting of exact Lagrangian branes inside (almost) Calabi-Yau
Stein manifolds:

  * •

Using $(n+1)$-dimensional universal families of holomorphic curves associated
to certain $(n-1)$-dimensional moduli spaces, one can construct bordism
currents between two $n$-dimensional Lagrangians in the same derived category
class. More generally, one can also sometimes associate bordism currents
between several Lagrangians, the notable example coming from distinguished
triangles. This is a special case of the open-closed map well known to
symplectic geometers, and some lower brow expositions are given in section
3.1.

  * •

Solomon [75] defined a functional on the universal cover of the space of
Lagrangians within the same Hamiltonian deformation class, whose critical
points are precisely special Lagrangians. In the exact setting, we will give a
more homological formula for the Solomon functional, and propose that the
bordism current produced from holomorphic curves allows one to extend the
functional to Lagrangian objects in the same derived category class, which is
well defined without the universal cover problem (_cf._ section 3.2). A
different perspective involving integration over moduli spaces is presented in
section 3.7, and we suggest further how the Solomon functional may generalize
to compact Calabi-Yau manifolds.

  * •

We further specialize to almost calibrated Lagrangians. We propose that given
a distinguished triangle $L_{1}\to L\to L_{2}\to L_{1}[1]$, then there are
Floer theoretic necessary conditions for $L$ to be a special Lagrangian:

$\arg\int_{L_{1}}\Omega\leq\arg\int_{L}\Omega\leq\arg\int_{L_{2}}\Omega.$

Here $L_{1},L_{2}$ are not required to be special Lagrangians, and the
cohomological integral is the only quantitative way they enter into
obstruction criterions. In other words, obstructions are of numerical nature.
We will prove this assuming further that the bordism current from holomorphic
curves satisfies automatic transversality and a _positivity condition_ (_cf._
section 3.3, 3.4, 3.5). The main technique is integration over the
$(n-1)$-dimensional moduli spaces of holomorphic curves. We also give
generalizations to several Lagrangians, to make possible contact with Harder-
Narasimhan decomposition. Compared with the proposal of Joyce [41], the Floer
theoretic obstructions capture some features of the Bridgeland stability
condition, but instead of tackling the entire derived Fukaya category
$D^{b}Fuk(X)$, we restrict to almost calibrated Lagrangian objects, which
morally correspond to the heart of a $t$-structure.

######  Remark 1.1.

While Joyce’s prediction of a Bridgeland stability condition is a very
important heuristic motivation for this paper, in our proposal we will
carefully avoid assuming that the Bridgeland stability condition exists on
$D^{b}Fuk(X)$, or that $D^{b}Fuk(X)$ is idempotent closed, or that the almost
calibrated Lagrangians form an abelian subcategory of $D^{b}Fuk(X)$, except
when we make comparisons with Joyce’s program.

  * •

When there are no destabilizing distinguished triangles, we say the derived
Fukaya category class of $L$ is _Thomas-Yau semistable_ (_cf._ Definition
3.32). This is similar to the original viewpoint of Thomas and Yau [65][66].
Let $X$ be a Calabi-Yau Stein manifold, such that $(\omega,\Omega)$ satisfies
the complex Monge-Ampère equation, and the regularity scale of the Calabi-Yau
manifold tends to infinity asymptotically. Let $\mathcal{L}$ be the geometric
measure theoretic closure of the class of exact, quantitatively almost
calibrated, unobstructed Lagrangians in the $D^{b}Fuk(X)$ class. (Beware that
we include immersed and singular Lagrangians as in Joyce [41], so the
definition of $\mathcal{L}$ is partially conjectural.) According to our
interpretation of the Thomas-Yau conjecture, if $L_{0}\in\mathcal{L}$ is
Thomas-Yau semistable, then there exists a special Lagrangian current in
$\mathcal{L}$ (_cf._ section 5.5.1 for the full version). Morever, it is
desirable that the special Lagrangian current carries unobstructed brane
structure in some formal sense, to represent the $D^{b}Fuk(X)$ class or some
weaker equivalence class.

  * •

We will make a start on the variational approach to tackle the Thomas-Yau
conjecture in the exact and quantitative almost calibrated setting. A summary
of the variational strategy is in section 5.3. We prove a number of uniform
estimates independent of the Lagrangians, notably the potential clustering
property, which implies a uniform bound on the energy of holomorphic curves,
and that the Solomon functional deviates from an elementary functional by a
uniformly bounded amount (_cf._ section 5.1, 5.2). We are not able to complete
the proof of the Thomas-Yau conjecture, but we try to identify the main
technical difficulties to be overcome. To make contact with the a priori
compactness theorems in geometric measure theory, it is natural to set up the
variational program in terms of Lagrangian currents and varifolds. One then
encounters both geometric measure theoretic difficulties, to do with the
existence of enough Lagrangian competitors, and the difficulty to make sense
of Floer theory for Lagrangian currents. We expect that many aspects of Floer
theory will not be robust under passage to such weak limits of Lagrangians,
but as a heuristic principle, we hope that the bordism currents relevant to
our proposal are robust (_cf._ section 5.4).

  * •

Assume there is a suitable compactness theory for Lagrangian objects, then we
expect the existence of special Lagrangians to be essentially equivalent to
the properness of the Solomon functional. The following heuristic principle
would then explain why Thomas-Yau conjecture could be true: for a sequence of
Lagrangians whose Solomon functional diverges to infinity, the underlying
Lagrangian objects should break up into a bounded number of connected
components $L_{i}$, such that the Lagrangian potential on each component has
uniformly bounded oscillation, and the asymptotic behaviour of the Solomon
functional is controlled by Floer theoretic data related to the cohomological
integrals $\int_{L_{i}}\Omega$. We will give evidence for this picture (_cf._
section 5.5), and explain a similar picture for Hermitian Yang-Mills
connections (_cf._ section 2.5).

  * •

Although we restrict to the exact setting in the above, we will leave the room
open for more ambitious speculations in the setting of compact almost Calabi-
Yau manifolds. The almost calibrated condition, on the other hand, is
essential in our proposal, and cannot be dropped within this framework even
with substantial efforts.

Aside from the main goal of making the proposal and presenting evidence, this
paper also contains a large amount of expository content, which aims to
present both a (biased) overview and a critique of the current literature on
the Thomas-Yau conjecture, and to explain how our picture fits into this body
of works. The most relevant works are the original papers of Thomas and Yau
[65][66], and the major update by Joyce [41]. We will devote substantial
attention to their main considerations (_cf._ section 2.1, 2.2, 2.3, 2.4, 2.5,
2.7, 4.1), and the significance of the Thomas-Yau-Joyce proposal to mirror
symmetry (_cf._ section 2.4), and we will constantly draw comparisons between
their pictures and ours. We also touch on a number of other relevant works:

  * •

(_cf._ section 2.3) Joyce-Imagi-Santos [40] gave a characterisation of Lawlor
necks, in which holomorphic curves made an interesting, albeit somewhat
technical appearance.

  * •

(_cf._ section 2.5, 2.6) The Hermitian Yang-Mills equation [27][26][79] was a
major inspiration to Thomas and Yau. The deformed Hermitian Yang-Mills
equation [18][22][23][16] has been advocated as a mirror version of special
Lagrangians.

  * •

(_cf._ section 2.8) Solomon [75][76][77] introduced a functional and a formal
Riemannian metric on the infinite dimensional space of almost calibrated
Lagrangians in a given Hamiltonian isotopy class. This space is too small for
our variational purpose, but his functional is fundamental to our proposal.

  * •

(_cf._ section 2.9) Lotay and Pacini [57][58] introduced a formal GIT picture
for totally real submanifolds. Their notion of ‘geodesic’ was our main
inspiration for the bordism currents between Lagrangians.

  * •

(_cf._ section 4.1, 5.2) Neves [61][62][63] pointed out the inevitability of
finite time singularity in the Lagrangian mean curvature flow, which is
fundamental to Joyce’s proposal [41]. Some of his technical arguments are
adapted to support our variational program in section 5.2.

  * •

(_cf._ section 4.2) Joyce’s earlier work on wall crossing and special
Lagrangian counting problems [42] is in some sense the elliptic cousin to his
Lagrangian mean curvature flow proposal [41].

######  Remark 1.2.

(Prerequisites) While we have endeavoured to survey most of the previous works
directly aimed at the Thomas-Yau conjecture, there is a very extensive
literature on geometric measure theory and symplectic geometry in the
background. We do not assume expertise on these matters, but some previous
exposures such as F. Morgan’s introductory book [60], and the excellent
surveys of Auroux [9] and Smith [73] would be useful. The most important
background facts for our main purpose are also recalled in section 5.1 and the
Appendix on the Fukaya category. While the brief summary therein is not
completely sufficient for all arguments in this paper, we hope the casual
reader could get the main gists, if not some sporadic remarks. The punctilious
reader may wish to refer to the Floer degree and sign convention summarized in
the Appendix, which is different from e.g. Seidel’s book [69]. The various
allusions to Kähler geometry are mainly for motivational purposes, which can
be skipped by readers less interested in these topics.

######  Remark 1.3.

(Rigor) This paper contains a somewhat unconventional mixture of proposals,
expositions and proofs. We have attempted to indicate all speculative elements
as ‘heuristic’ or ‘conjecture’. Other arguments are either complete proofs, or
sketches intended as expositions.

######  Acknowledgement.

The author is currently an MIT CLE Moore Instructor and a Clay Research
Fellow. This paper owes much intellectual debt to the proposal of Thomas-Yau
and Joyce. The author thanks MIT for providing a stimulating research
environment, and especially P. Seidel, S. Rezchikov and T. Collins for useful
discussions.

##  2 Thomas-Yau conjecture backgrounds

###  2.1 The Thomas-Yau-Joyce picture

An almost Calabi-Yau manifold $(X,\omega,J,\Omega)$ is a complete
$n$-dimensional Kähler manifold with a nowhere vanishing holomorphic volume
form. It is called Calabi-Yau if the complex Monge-Ampère equation
$\omega^{n}=\text{const}\Omega\wedge\overline{\Omega}$ holds, whence $X$ is
Ricci flat. A real $n$-dimensional compact submanifold of an almost Calabi-Yau
manifold $\iota:L\to X$ is called special Lagrangian of constant phase angle
$\hat{\theta}\in\mathbb{R}$, 111Different lifts of the phase angle from
$\mathbb{R}/\pi\mathbb{Z}$ to $\mathbb{R}$ shift the grading of the brane
structure, so they are different as objects of the Fukaya category. if

$\omega|_{L}=0,\quad\text{Im}(e^{-i\hat{\theta}}\Omega|_{L})=0.$ (1)

Special Lagrangian submanifolds have unobstructed deformation
theory.222However, obstructions may arise if brane structures are taken into
account. Furthermore, if $X$ is Calabi-Yau, then a special Lagrangian is a
minimal submanifold, and in fact minimizes the area within its homology class,
thanks to the calibration inequality for submanifolds, saturated precisely by
special Lagrangians:

$|\int_{L}\text{Re}(e^{-i\hat{\theta}}\Omega)|\leq|\int_{L}\Omega|\leq\int_{L}dvol(L)=\text{Vol}(L).$
(2)

We will always impose some compactness or convexity at infinity on $X$ to
ensure the Fukaya category of $(X,\omega)$ makes sense.

#### Thomas-Yau’s proposal

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 stability333 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. 444There
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 555The
important thing is that the upper and lower bounds on $\theta$ differ by
$\pi$. Shifting the interval by a constant is inconsequential.

$-\frac{\pi}{2}<\theta<\frac{\pi}{2}.$

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 $\epsilon>0$:

$-\frac{\pi}{2}+\epsilon\leq\theta\leq\frac{\pi}{2}-\epsilon.$ (3)

One immediate consequence is an a priori volume bound.

######  Lemma 2.1.

If $L$ satisfies (3), then
$\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re }\Omega.$

###### Proof.

From
$e^{i\theta}dvol_{L}=\Omega|_{L}=\text{Re}\Omega|_{L}+\sqrt{-1}\text{Im}\Omega|_{L},$
we see $\text{Re }\Omega$ is an orientation form on $L$, and we arrive at

$\text{Vol}(L)=\int_{L}\frac{1}{\cos\theta}\text{Re
}\Omega\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re }\Omega$

as required. ∎

######  Remark 2.1.

Another immediate consequence of the almost calibrated condition, is that the
complex number $Z(L)=\int_{L}\Omega$ is nonzero, with
$\phi(L)=\frac{1}{\pi}\arg Z(L)\in(-\frac{1}{2},\frac{1}{2})$.

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
$L=L_{1}\\#L_{2}$,666It 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).777We 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:

######  Conjecture 2.2.

[65, Conj 5.2] A graded Lagrangian has a special Lagrangian representative in
its Hamiltonian class if and only if it is stable, and this special Lagrangian
representative is unique.

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].

#### Bridgeland stability, Joyce’s proposal

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].

######  Definition 2.3.

A Bridgeland stability condition $(Z,\mathcal{P})$ on a triangulated category
$\mathcal{D}$ consists of a group homomorphism (‘central charge’) $Z$ from the
(numerical) Grothendieck group $K(\mathcal{D})$ to $\mathbb{C}$, and full
additive subcategories $\mathcal{P}(\phi)\subset\mathcal{D}$ for each
$\phi\in\mathbb{R}$, whose objects are called ‘semistable objects of phase
angle $\pi\phi$’,888In the convention of Bridgeland, the phase is $\phi$. We
have instead opted to call $\pi\phi$ the phase, which is more naturally
identified with the phase angle of special Lagrangians. satisfying the
following axioms:

  * •

(Phase) If $L\in\mathcal{P}(\phi)$ then $Z(L)=m(L)\exp(i\pi\phi)$ for some
$m(L)>0$,

  * •

(Shift) For all $\phi\in\mathbb{R}$,
$\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]$,

  * •

(Monotonicity) If $\phi_{1}>\phi_{2}$ and $L_{j}\in\mathcal{P}(\phi_{j})$,
$j=1,2$ then $Hom_{\mathcal{D}}(L_{1},L_{2})=0$,

  * •

For each nonzero object $L\in\mathcal{D}$ there are a finite sequence of real
numbers $\phi_{1}>\phi_{2}>\ldots>\phi_{N}$ and a Harder-Narasimhan
decomposition

$0=\mathcal{E}_{0}\to\mathcal{E}_{1}\to\ldots\to\mathcal{E}_{N}=L,$ (4)

with distinguished triangles

$\mathcal{E}_{i-1}\to\mathcal{E}_{i}\to L_{i}\to\mathcal{E}_{i-1}[1]$

such that $L_{j}\in\mathcal{P}(\phi_{j})$.

  * •

(Calibration) For any fixed norm on the finite dimensional vector space
$K(\mathcal{D})\otimes_{\mathbb{Z}}\mathbb{R}$, we have a uniform constant
$C$, such that any semistable object $L$ satisfies $\left\lVert
L\right\rVert\leq C|Z(L)|.$

######  Remark 2.2.

For the heuristic but naïve geometric meaning, one can imagine that
$\mathcal{D}$ is the derived Fukaya category, $K(\mathcal{D})$ is the
sublattice of $H_{n}(X)$ generated by the Lagrangians, the central charge is
$Z(L)=\int_{L}\Omega$, the subcategory $\mathcal{P}(\phi)$ is generated by the
special Lagrangians of constant phase angle $\theta=\pi\phi$, the Harder-
Narasimhan decomposition means a multiple Lagrangian connected sum with
decreasing phase angles

$L\simeq L_{1}\\#L_{2}\\#\ldots\\#L_{N},$

and the calibration property comes from the fact that the total mass of a
special Lagrangian is equal to $|Z(L)|$, and the mass bounds any norm of
$[L]\in H_{n}(X)$ using Poincaré duality, assuming $H_{n}(X)$ is finite
dimensional.

######  Remark 2.3.

We write the subcategory generated by all the $\mathcal{P}(\phi)$ within the
interval $\phi\in(\phi_{0},\phi_{1}]$ as
$\mathcal{P}(\phi_{0}<\phi\leq\phi_{1})$. The important property is that
$\mathcal{P}(\phi_{0}<\phi\leq\phi_{0}+1)$ is an abelian category, known as
the heart of a bounded $t$-structure. The advantage is that while in general
triangulated categories only distinguished triangles make sense, for abelian
categories we can talk about exact sequences, subobjects and quotients.

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 999The 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_
$\mathcal{P}(-\pi/2<\theta\leq\pi/2)\subset D^{b}Fuk(X)$,101010Here 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.

#### Why Thomas-Yau has predictive power

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:

######  Question 1.

Fix the holomorphic volume form $\Omega$. Let $\omega_{1},\omega_{2}$ be two
generic Kähler forms, differing only by the differential of a compactly
supported 1-form. Can we define a count of special Lagrangian rational
homology spheres, such that the numbers agree for $\omega_{1}$ and
$\omega_{2}$?

######  Remark 2.4.

As we shall see, the main evidence of Thomas-Yau picture (_cf._ section 2.2)
does not really use the complex Monge-Ampère equation.111111The almost Calabi-
Yau setting is desirable not only for the sake of generality, but may be
essential to achieve suitable genericity. As an additional motivation on the
side of physics, the SCFT condition translates into a Kähler condition on the
target space metric, which satisfies the Ricci-flatness only approximately
[38, section 14.2.4]. On the mirror side, as a consequence of the
$\mu$-stability characterisation, the existence of Hermitian Yang Mills
connections on a compact Kähler manifold does not depend on the choice of the
Kähler form within a fixed Kähler class.

###  2.2 Principal evidence of Thomas-Yau

Thomas and Yau offered a number of arguments in support of their picture. We
now describe the evidence from _first principles_ , and will later return to
the evidence from _analogies_.

The following elementary observation is a first indication about how special
Lagrangian objects resemble stable objects of a Bridgeland stability
condition:

######  Proposition 2.4.

[66, section 5.2] Let $L$, $L^{\prime}$ be unobstructed Lagrangian branes
whose supports are compact special Lagrangian manifolds, with Lagrangian
angles $\theta_{L}>\theta_{L^{\prime}}$. Then $HF^{k}(L,L^{\prime})=0$ for
$k\leq 0$.

###### Proof.

After $C^{\infty}$-small Hamiltonian perturbation we may assume any
intersection point $p$ between $L$ and $L^{\prime}$ is transverse. We can
write the tangent planes of $L,L^{\prime}$ inside $T_{p}X\simeq\mathbb{C}^{n}$
in the standard form

$T_{p}L=\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad
T_{p}L^{\prime}=(e^{i\phi_{1}},\ldots
e^{i\phi_{n}})\mathbb{R}^{n}\subset\mathbb{C}^{n},\quad 0<\phi_{i}<\pi.$

The Floer degree of the intersection point $p$ is

$\mu_{L,L^{\prime}}(p)=\frac{1}{\pi}(\theta_{L}-\theta_{L^{\prime}}+\sum_{1}^{n}\phi_{i})>0,$

whence $HF^{k}(L,L^{\prime})=0$ for $k\leq 0$. ∎

######  Remark 2.5.

As is clear from the proof, the special Lagrangian condition can be relaxed to
$\inf_{L}\theta_{L}>\sup_{L^{\prime}}\theta_{L^{\prime}}$.

######  Remark 2.6.

Our Floer degree convention follows Joyce [41], but is opposite to that of
Seidel [69] and many other symplectic geometry texts. In our convention,
shifting the Lagrangian phase by $k\pi$ for $k\in\mathbb{Z}$ is equivalent to
considering the shifted object $L[k]$. Thus a slight extension of the above is
that if $\theta_{L}>\theta_{L^{\prime}}+l$, then

$HF^{k}(L,L^{\prime})=HF^{k-l}(L,L^{\prime}[l])=0,\quad k\leq l.$

The most compelling evidence due to Thomas and Yau is

######  Theorem 2.5.

(Thomas-Yau uniqueness) [66] Let $L,L^{\prime}$ be unobstructed Lagrangian
branes supported on embedded special Lagrangians of the same phase, which
define isomorphic objects in $D^{b}Fuk(X)$, then their supports coincide.

######  Remark 2.7.

The most general Thomas-Yau uniqueness, which includes immersed Lagrangians,
seems to be due to Imagi [39]. The original Thomas-Yau theorem is phrased in
terms of uniqueness in the Hamiltonian isotopy class, even though it can be
cast in more general categorical terms. The categorical perspective is
preferred, because it is closer to the spirit of homological mirror symmetry,
and because one derived Fukaya category class may contain several Hamiltonian
isotopy classes. If immersed Lagrangians are allowed, then isomorphism in
$D^{b}Fuk(X)$ would also identify certain embedded Lagrangian objects with
immersed objects of different topologies. When this happens, an interesting
corollary of Thomas-Yau uniqueness is that at most one of these Hamiltonian
classes contains special Lagrangian branes.

###### Proof.

(Sketch) Assume the suppports do not coincide. After small Hamiltonian
perturbations, we can ensure the perturbed Lagrangians
$\tilde{L},\tilde{L}^{\prime}$ have transverse intersections, and still define
the same isomorphic objects in $D^{b}Fuk(X)$. Under the special Lagrangian
assumption, through judicious choice of the Hamiltonian via Morse theory as in
Thomas-Yau [66, Thm 4.3], or by using genericity arguments based on real
analyticity as in Joyce-Imagi-Santos [40, section 4.3], one can ensure there
is no intersection point $\tilde{L}\cap\tilde{L}^{\prime}$ of degree $0,n$, so
in particular $HF^{0}(L,L^{\prime})\simeq
HF^{0}(\tilde{L},\tilde{L}^{\prime})=0$. However, this implies the
cohomological unit of $HF^{*}(L,L)$ is zero, so the Floer cohomology ring of
$L$ is zero, namely $L$ is a zero object in $D^{b}Fuk(X)$, contradiction. ∎

The Thomas-Yau argument reveals the relevance of the Fukaya category to
special Lagrangian geometry. The role of holomorphic curves, which are a
central ingredient in the Fukaya category, is however rather opaque in this
argument; their only appearance is to make the Floer cohomology defined.

###  2.3 Variant: uniqueness of the Lawlor neck

A variant of the Thomas-Yau argument121212Abouzaid and Imagi have mentioned in
their talks some other interesting applications on the topology of special
Lagrangians inside the cotangent bundle of a special Lagrangian with some
fundamental group conditions, using another variant of the Thomas-Yau
argument. We look forward to the appearance of their paper.  appears in the
subsequent work of Joyce-Imagi-Santos [40] on the uniqueness of Lawlor necks,
where holomorphic curves and the algebraic structures of the Fukaya category
appear in a more prominent, albeit somewhat technical way.

#### Lawlor necks

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

$\Pi_{0}=\mathbb{R}^{n},\quad\Pi_{\phi}=(e^{i\phi_{1}},\ldots
e^{i\phi_{n}})\mathbb{R}^{n},\quad 0<\phi_{i}<\pi,\quad\sum\phi_{i}=\pi.$

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

$L_{\phi,A}=\\{(z_{1}(y)x_{1},...,z_{n}(y)x_{n}):y\in\mathbb{R},x_{k}\in\mathbb{R},x^{2}_{1}+\ldots+x_{n}^{2}=1\\}.$
(7)

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

$p(x)=(1+a_{1}x^{2})\ldots(1+a_{n}x^{2})-1,\quad P(x)=\frac{p(x)}{x^{2}}.$

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

$\phi_{k}=a_{k}\int_{-\infty}^{\infty}\frac{dx}{(1+a_{k}x^{2})\sqrt{P(x)}},\quad
A=\int_{-\infty}^{\infty}\frac{dx}{2\sqrt{P(x)}}$

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

$z_{k}(y)=e^{i\psi_{k}(y)}\sqrt{a_{k}^{-1}+y^{2}},\quad\text{where
}\psi_{k}(y)=a_{k}\int_{-\infty}^{y}\frac{dx}{(1+a_{k}x^{2})\sqrt{P(x)}},$

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

$|f|=O(|\vec{x}|^{2-n}),\quad|\partial^{k}f|=O(|\vec{x}|^{2-n-k}).$

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

#### Joyce-Imagi-Santos uniqueness theorem

######  Theorem 2.6.

[40] Let $n\geq 3$. 131313The theorem is also true in complex dimension 2,
proved earlier by Lotay and Neves [56] by other means not involving Floer
theory. Assume $L$ is a smooth embedded exact special Lagrangian of phase
zero, asymptotic at rate $<0$ to the union of the two planes
$\Pi_{0}\cup\Pi_{\phi}$ with $\sum\phi_{i}=\pi$, then $L$ is $L_{\phi,A}$ for
some $A>0$.

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.

#### Ideal triangles

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.141414A 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
$x_{k}\to 0$.

######  Example 2.8.

In the most symmetric case $\phi_{1}=\phi_{2}=\ldots\phi_{n}=\frac{\pi}{n}$,
the Lawlor neck is invariant under $SO(n,\mathbb{R})$, and these holomorphic
triangles are up to $SO(n,\mathbb{R})$ rotation, simply the triangle inside
the first coordinate line $\mathbb{C}\subset\mathbb{C}^{n}$ enclosed by the
three Lagrangians.

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

$\int_{\Sigma}\omega=\int_{\partial\Sigma}\lambda=\int_{\partial\Sigma\cap
L_{\phi,A}}df_{L_{\phi,A}}=A,$

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.

###  2.4 Philosophy of open string mirror symmetry

The Thomas-Yau conjecture is largely motivated by mirror symmetry, based on
the analogy between special Lagrangian geometry on the $A$-side
(‘symplectic’), and stability conditions in the derived category of coherent
sheaves on the $B$-side (‘holomorphic’). It is worth emphasizing that mirror
symmetry is quantum by nature, and only sometimes admits classical geometric
interpretations. The mathematical predictions of open string mirror symmetry
chiefly fall into the following three classes:

  * •

(Homological mirror symmetry, _i.e._ the categorical approach) Given a
(compact) Calabi-Yau manifold $X$ viewed as a symplectic manifold, one expects
to find a (compact) Calabi-Yau manifold $X^{\vee}$ viewed as a complex
manifold (sometimes over the Novikov field), such that the (derived,
idempotent closed) Fukaya category $D^{\pi}Fuk(X)$ is equivalent as a
triangulated category to the derived category of coherent sheaves
$Coh(X^{\vee})$. Frequently, there is a preferred $A_{\infty}$ enhancement on
both sides, such that $D^{\pi}Fuk(X)\simeq D^{b}Coh(X^{\vee})$ holds as an
$A_{\infty}$-equivalence. The concrete implication is that Lagrangian branes
$L$ correspond to objects $E$ of $D^{b}Coh(X^{\vee})$, such that the Floer
groups are identified with the $Ext$ groups, which explains the name
‘homological’:

$HF^{*}(L,L^{\prime})\simeq Ext^{*}(E,E^{\prime}).$

Homological mirror symmetry has many ramifications beyond Calabi-Yau
manifolds, involving for instance Fano manifolds and Landau-Ginzburg models.
As a general feature, homological mirror symmetry only uses pure symplectic
topology (vis. a vis. complex geometry) on either side of the mirror. It is
the best understood aspect of mirror symmetry, with many special cases proven.

  * •

(SYZ mirror symmetry, _i.e._ the moduli space approach) Taken in a somewhat
generalised and very optimistic sense 151515The SYZ paper [78] focuses
primarily on special Lagrangian torus fibrations, but hints at more general
special Lagrangian branes. Some more discussions can be found in Fukaya’s work
[34, section 8.5]., the Strominger-Yau-Zaslow philosophy says that after
taking into account the quantum instanton corrections, then the moduli space
of (semi)stable Lagrangian branes within fixed homology classes acquires a
modified complex structure, and can be identified with certain moduli spaces
of (semi)stable objects from the mirror side. The stable Lagrangian branes are
believed to be related to special Lagrangians via the Thomas-Yau conjecture.
Most of the literature concentrates on the restricted case of (special)
Lagrangian torus fibrations arising from the large complex structure limit, in
which case the moduli space of Lagrangian branes supported on the fibres
inside $X$ should recover the moduli space of points on $X^{\vee}$, namely the
mirror manifold $X^{\vee}$ itself 161616This SYZ mirror construction has seen
the intense research effort on the symplectic side by Auroux [8], Abouzaid,
Fukaya, among many others. The very recent progress [83] achieves a non-
archimedean mirror from Fukaya categorical considerations, but the nature of
singular fibres is still not adequately understood. The skeptics may
reasonably doubt if special Lagrangians exist at all in the region with large
curvature inside the Calabi-Yau manifold. Even if they do exist, there is
insufficient evidence that the fibration structure persists. This is maybe the
weakest point of the SYZ conjecture.. Beyond the torus fibration case, the
moduli space of Lagrangian branes defined by the Mauer-Cartan equation
typically has singularities, which is compatible with the fact that the moduli
space of semistable objects in $D^{b}Coh(X^{\vee})$ typically is very
singular.

  * •

(DT theory, _i.e._ the counting approach) In the special case of Calabi-Yau
3-folds, the moduli spaces/stacks of semistable objects have virtual dimension
zero, so one can hope to extract enumerative invariants, which can be thought
of as integrals over the moduli space, the most basic version being the Euler
number. Donaldson-Thomas theory is a highly elaborate framework for extracting
such numbers using virtual techniques, on the side of $D^{b}Coh(X^{\vee})$. It
is suggested (e.g in [34, section 8.5]) that one can assign DT invariants to
the moduli space of semistable objects in the Fukaya category171717This would
presuppose suitable properness and algebraicity on the moduli space of special
Lagrangian objects, which is a highly nontrivial claim related to the Thomas-
Yau conjecture. For instance, in the SYZ special Lagrangian torus fibration
case, compactifying the moduli space requires good understanding in the
singular region, which is far beyond current knowledge., and these should be
equivalent to the DT invariants on the mirror $X^{\vee}$. The role of Thomas-
Yau conjecture is that in principle it transforms a problem about counting
special Lagrangian objects, into a categorical framework involving Calabi-Yau
$A_{\infty}$-categories and stability conditions, which then supposedly
181818modulo heroic efforts feeds into the grand machinery of Kontsevich and
Soibelman [46][47][48][49]. This DT perspective may be regarded as the
ultimate goal of Thomas-Yau conjecture 191919As an analogy, the practical
calculation of Donaldson’s ASD instanton invariants depends largely on the
solution of Hermitian-Yang-Mills equation on algebraic surfaces., and is close
to the physical applications involving BPS states counting, although on the
$A$-side it is the most removed from mathematical attempts.

It is worth emphasizing that taking stability questions into account requires
one to go beyond homological mirror symmetry. While the Fukaya category (vis.
a vis. $D^{b}Coh(X^{\vee})$) depends only on the symplectic (holomorphic)
data, the stability condition remembers information about the holomorphic
volume form (polarisation class), which now sees the $B$-side ($A$-side). As
such, neither the above strong version of the SYZ mirror symmetry, nor the DT
mirror symmetry are consequences of homological mirror symmetry alone, as
indicated already in the SYZ paper [78].202020In the words of SYZ, they
required the full equivalence of the two type II string theories, including
the full BPS spectrum. A very general framework that encompasses all the three
aspects of mirror symmetry is in Kontsevich and Soibelman [46], but how to fit
the $A$-side into this picture is wildly conjectural.

#### Caveats for differential geometers

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.

###  2.5 Hermitian-Yang-Mills

Hermitian Yang-Mills (HYM) connections have long been established as the
epitome of how stability conditions control the existence questions of
geometric PDEs, but we shall attempt to say a few new words besides the
customary hommage. We consider Hermitian metrics $H$ on a holomorphic vector
bundle $E\to X^{\vee}$ over a compact Kähler manifold $X^{\vee}$, 212121If we
are interested in noncompact $A$-model target spaces, the mirror is in fact
not a compact Kähler manifold. We hope the reader will excuse us on this
issue, since mirror symmetry is only used as motivations in this paper.
inducing the Chern connection $\nabla_{H}$ and the curvature
$F_{H}\in\Omega^{1,1}(End(E))$. The HYM equation can be written as

$\omega_{X^{\vee}}^{n-1}\wedge\frac{\sqrt{-1}}{2\pi}F_{H}=\frac{\mu(E)}{\int_{X^{\vee}}\omega_{X^{\vee}}^{n}}Id_{E}\otimes\omega_{X^{\vee}}^{n},\quad\mu(E)=\frac{\int_{X^{\vee}}c_{1}(E)\wedge\omega_{X^{\vee}}^{n-1}}{rk(E)}=\frac{deg(E)}{rk(E)}.$

This implies the Yang-Mills inequation $\nabla_{H}^{*}F_{H}=0$, and in fact
HYM connections are absolute minimizers of the Yang-Mills energy among all
unitary connections on the Hermitian vector bundle $E$. The $\mu$-stability
(resp. semistability) means that for all proper coherent subsheaf
$E^{\prime}\subset E$, the slope $\mu(E^{\prime})<\mu(E)$ (resp.
$\mu(E^{\prime})\leq\mu(E)$). The bundle $E$ is called $\mu$-polystable if it
is a direct sum of $\mu$-stable bundles of the same slope.

The famous Donaldson-Uhlenbeck-Yau theorem says

######  Theorem 2.9.

On a compact Kähler manifold, the holomorphic bundle $E$ admits a HYM metric
if and only if $E$ is $\mu$-polystable.

######  Remark 2.9.

Certain parallels between HYM and special Lagrangians are already known to
Thomas and Yau. The Chern connection $\nabla_{H}$ is analogous to the
Lagrangian submanifold $L$ (with $U(1)$ local systems), the HYM equation as a
first order equation on $\nabla_{H}$ is analogous to the special Lagrangian
condition on $L$, the second order Yang-Mills equation is analogous to the
minimal surface equation on $L$, and the Yang-Mills energy minimization is
analogous to the volume minimization. If the Lagrangian $L$ is represented as
the graph of an exact 1-form, then the potential function defining $L$ could
be viewed as analogous to the Hermitian metric $H$. Finding a mirror analogue
resembling the $\mu$-stability was one of Thomas and Yau’s principal
motivations.

####  $\mu$-stability and its wider context

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.222222It 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 $X^{\vee}$. 232323Path
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.

######  Remark 2.10.

The $\mu$-semistability condition can be recast in terms of the central charge
$Z(E)=-deg(E)+\sqrt{-1}rk(E)$, as saying $\arg Z(E^{\prime})\leq\arg Z(E)$ for
all nonzero proper subsheaves $E^{\prime}\subset E$. It is worth emphasizing
that except for the case of Riemann surfaces, $\mu$-stability does _not_ give
rise to a Bridgeland stability condition on $D^{b}Coh(X^{\vee})$, since
skyscrapper sheaves will generally have zero rank and zero degree, hence zero
central charge. A similar but more subtle failure of Bridgeland stability
happens in the context of the deformed Hermitian-Yang-Mills connections (_cf._
[18, section 4]). Such a failure does not spell doom for the PDE applications,
nor for DT theoretic applications.242424R. Thomas defined DT invariants for
$\mu$-stability long before the insight of Bridgeland. Even though Bridgeland
stability seems to be a plausible framework for special Lagrangians in the
light of Joyce’s proposal, it is probably advisable to maintain a more
flexible attitude to stability conditions.

#### Donaldson functional

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 $H\in\mathcal{H}$:

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

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

#### Proof idea of Donaldson-Uhlenbeck-Yau theorem

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.

######  Remark 2.11.

An alternative approach by Donaldson [27] in the projective manifold case, is
also based on the flow method, but uses a dimensional induction in which the
stability condition appears indirectly through alternative algebro-geometric
characterisations. The approach of Uhlenbeck and Yau [79] uses the continuity
method instead, where the stability condition appears in a way similar to the
above.

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),252525What 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,262626The 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 $\mathcal{H}$.

######  Remark 2.12.

The nonlinear analysis concerning special Lagrangians is more difficult than
HYM. For instance, the finite time singularities of the LMCF are inevitable.
The HYM equation can be viewed as a toy model which shares some, but by no
means all, of the high level features with the special Lagrangian equation.

#### Infinite dimensional GIT picture, and possible lack of mirror analgoue

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,272727Thomas 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.282828Thomas’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.

###  2.6 Deformed Hermitian Yang-Mills

Much recent research concentrates on a more nonlinear cousin of HYM, known as
the deformed Hermitian Yang-Mills equation (dHYM), expertly surveyed in [18].
Although the techniques involved in dHYM are more akin to other areas of
Kähler geometry, such as Kähler-Einstein metrics and the J-equation
292929Indeed, the recent breakthrough of Gao Chen [16] on dHYM is largely
based on the methods he developed for the J-equation. The work of Collins et
al [23] has strong analogy with the homogeneous complex Monge-Ampère equation
important for CSCK metrics., its motivation is in part to find an improved
mirror analogue of special Lagrangians, with the distant goal of constructing
the Bridgeland stability on the B-side of the mirror.

The equation can be motivated differential geometrically from semiflat mirror
symmetry [22]. To start, one imposes _toric symmetry_ , so that over a local
$n$-dimensional base $D$, the $A$ and $B$ sides are respectively the
$T^{n}$-bundles $T^{*}D/\Lambda$ and $TD/\Lambda^{\vee}$ equipped with the
canonical symplectic/complex structure, where $\Lambda$ and $\Lambda^{\vee}$
are dual lattices, so that the torus fibres are naturally dual. A _section_
$L$ of $T^{*}D/\Lambda$ over $D$ fibrewise determines a flat $U(1)$-connection
on the corresponding fibre of $TD/\Lambda$, and the Lagrangian condition on
$L$ is equivalent to these fibrewise connections fitting into a connection $A$
on a line bundle $E$ over $TD/\Lambda$, with the integrability condition
$F_{A}^{0,2}=0$. Now a Hessian metric on $D$ induces a Kähler structure on
both sides. The condition for $L$ to be a special Lagrangian of phase
$\hat{\theta}$, translates into the mirror condition on the holomorphic line
bundle:

$\text{Im}(e^{-i\hat{\theta}}(\omega_{X^{\vee}}+F_{A})^{n})=0.$

Here the curvature $F_{A}$ of the $U(1)$-connection is an imaginary valued
$(1,1)$-form. Up to rescaling $\omega_{X^{\vee}}$, the global version is the
dHYM equation for a Hermitian metric on a holomorphic line bundle $E$ over
$X^{\vee}$, thought of as a PDE on the potential function $\phi$:

$\text{Im}(e^{-i\hat{\theta}}(\omega_{X^{\vee}}+\sqrt{-1}\alpha_{\phi})^{n})=0,\quad[\alpha_{\phi}=\alpha+\sqrt{-1}\partial\bar{\partial}\phi]\in-
c_{1}(E).$ (11)

For an alternative view, we can pointwise simultaneously diagonalize
$\omega_{X^{\vee}}$ and $\alpha_{\phi}$, to extract the eigenvalues
$\lambda_{i}$,

$\omega_{X^{\vee}}=\frac{\sqrt{-1}}{2}\sum dz_{i}\wedge
d\bar{z}_{i},\quad\alpha_{\phi}=\frac{\sqrt{-1}}{2}\sum\lambda_{i}dz_{i}\wedge
d\bar{z}_{i}.$

The dHYM equation is then

$\Theta_{\omega}(\alpha)=\sum_{i}\arctan\lambda_{i}=\hat{\theta}\mod\pi\mathbb{Z}.$
(12)

Thus the equation actually separates into several discrete branches depending
on the choice of $\hat{\theta}$, and changing $\hat{\theta}$ by $k\pi$ can
drastically alter the behaviour of the equations. Replacing $\hat{\theta}$ by
$-\hat{\theta}$ (so $E$ is replaced by its dual) does not change the problem,
so without loss of generality $0\leq\hat{\theta}<\frac{n\pi}{2}$. There are
two significant limiting cases:

  * •

When $\lambda_{i}\gg 1$ for any $i$, so
$\hat{\theta}\approx\frac{n\pi}{2}-\sum\frac{1}{\lambda_{i}}$, the equation
becomes approximately the J-equation

$\alpha_{\phi}^{n-1}\wedge\omega_{X^{\vee}}=\frac{\int\alpha^{n-1}\wedge\omega_{X^{\vee}}}{\int\alpha^{n}}\alpha_{\phi}^{n}.$

Most mathematical works on dHYM assume some lower bound such as
$\hat{\theta}>\frac{(n-2)\pi}{2}$, which may be morally interpreted as being
near this large phase limit.

  * •

When $|\lambda_{i}|\ll 1$ for any $i$, so
$\hat{\theta}\approx\sum\lambda_{i}$, the equation is approximately the line
bundle case of the HYM equation

$\alpha_{\phi}\wedge\omega_{X^{\vee}}^{n-1}=\frac{\int\alpha\wedge\omega_{X^{\vee}}^{n-1}}{\int\omega_{X^{\vee}}^{n}}\omega_{X^{\vee}}^{n}.$

This is also known as the large volume limit, in the sense that the Kähler
metric length scale is much larger than the curvature scale of the bundle.

#### Towards a Bridgeland stability condition

The existence of dHYM solutions has algebraic obstructions. Let $V$ be a
$p$-dimensional subvariety of $X^{\vee}$, with $1\leq p<n$. Under the large
phase assumption $\hat{\theta}>\frac{(n-2)\pi}{2}$, a pointwise consideration
of eigenvalues shows [18, Prop. 3.4]

$\text{Im}\big{(}\int_{V}e^{-\sqrt{-1}(\hat{\theta}-(n-p)\frac{\pi}{2})}(\omega_{X^{\vee}}+\sqrt{-1}\alpha_{\phi})^{n}\big{)}>0.$

More suggestively, denote

$Z_{V}(E)=-\int_{V}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)\in\mathbb{R}_{>0}e^{\sqrt{-1}\phi_{V}(E)},\quad\phi_{V}(E)\in(0,\pi),$
(13)

so the algebraic obstruction can be rewritten as

$\text{Im}(\frac{Z_{V}(E)}{Z_{X^{\vee}}(E)})>0,\quad\emph{i.e.}\quad\phi_{V}(E)>\phi_{X^{\vee}}(E)=\hat{\theta}-\frac{(n-2)\pi}{2}.$

This bears some resemblance to a Bridgeland stability condition with central
charge $Z_{X^{\vee}}(E)$ , even though on the technical level there are some
discrepancies [18, section 3.2]. In the simplest understood examples, such as
the blow-up of $\mathbb{CP}^{2}$ in a point, the obstruction criterion for
dHYM seems to refine Bridgeland stability, in the sense that every dHYM stable
object is Bridgeland stable, but not conversely. It is not entirely clear how
to interpret this (_cf._ Remark 2.10).

#### Main achievements on dHYM

Some of the most significant results on the dHYM equation are:

  * •

There is a well developed formal GIT picture [23, section 2][19][21],
including a complexified group action on an infinite dimensional Kähler
manifold, a moment map, a negatively curved symmetric space, and a Donaldson
type functional which is convex along geodesics.

  * •

The geodesic equation with prescribed boundary data have $C^{1,1}$-solutions
and viscosity solutions under a large phase assumption [18, Thm 2.7]. This
leads to nontrivial algebraic obstructions [18, section 3].

  * •

Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$ and the existence of a
subsolution in the suitable sense, then the dHYM equation can be solved [18,
Thm 5.2].

  * •

Assume large phase $\hat{\theta}>\frac{(n-2)\pi}{2}$, then the existence of
dHYM solution can be formulated in terms of algebraic obstruction conditions
[16].

#### Will dHYM lead to the mirror Bridgeland condition?

Despite substantial progress, many essential difficulties still need to be
overcome before the dHYM equation can give rise to a Bridgeland stability
condition on $D^{b}Coh(X^{\vee})$:

  * •

Can one relax the _large phase assumption_?

  * •

Is there a generalisation of dHYM equation to _higher rank vector bundles_?
(Currently, there is no well established PDE, let alone how to solve it.)

  * •

Can this story be extended to _complexes_ of vector bundles?

  * •

How can one compare the answer with the A-side of the mirror?

Time will tell how far one can push in this program, but we would like to
momentarily play the skeptic’s advocate:

  * •

The differential geometric motivation 303030There is an independent physics
motivation for dHYM from the Dirac-Born-Infeld action. The DBI action is
however not an exact result, but depends on the assumption that certain
derivative terms of the curvature can be ignored. Unlike the A-model side
where the central charge $Z(L)=\int_{L}\Omega$ is believed to hold exactly, on
the B-side the central charge formula
$Z_{X^{\vee}}(E)=-\int_{X^{\vee}}e^{-\sqrt{-1}\omega_{X^{\vee}}}ch(E)$ is
believed to be subject to worldsheet instanton corrections.  of dHYM assumes
semiflat ambient Kähler metrics. Such metrics only arise naturally if the
manifold has toric symmetry, or as a good approximate description for
degenerating Calabi-Yau metrics near the large complex structure limit/large
volume limit. Near this limit, the dHYM equation may be an improvement on the
HYM equation as the mirror version of special Lagrangians. But far away from
such limits, the instanton corrections cannot be ignored, and indeed a general
Kähler metric has no toric symmetry.

  * •

The reliance on the large phase assumption is a possible indication of a
breakdown once we move too far from a large phase limit. In the closely
related special Lagrangian graph equation, relaxing the phase condition is
known to result in rather severe singularities for the viscosity solutions.

  * •

Being the section of a torus fibration is a serious assumption on the topology
of a Lagrangian.

  * •

The difficulty with higher rank vector bundles is a possible indication that
$Coh(X^{\vee})$ may be not the natural abelian subcategory of
$D^{b}Coh(X^{\vee})$ suited for dHYM. Furthermore, there is no a priori
guarantee for arbitrary stability conditions to correspond to PDEs,313131If a
Bridgeland stability condition arises from a PDE, there is no a priori
guarantee that its deformations also come from PDEs. or to have any classical
geometrical interpretation at all. In particular, the $B$-side mirror to the
hypothetical special Lagrangian stability condition may well be an abstract
stability condition with no particular PDE interpretation.

  * •

Due to the lack of imagination, it is hard to see how PDEs can know about the
degree shifts of a complex of vector bundles, and how it can detect homotopy
equivalence of complexes.

If we take this skeptic view, then we do not expect an exact comparison
between the hypothetical Bridgeland stabilities on both sides of the mirror,
without adding very substantial assumptions such as toric symmetry. But when
HYM shares the same qualitative features as dHYM, which admit a mirror
interpretation, it lends more plausibility for these features to appear also
on the special Lagrangians.

###  2.7 Extensions and wall crossing

Part of Thomas and Yau’s insights is that one should focus on the
_categorical_ aspect of mirror symmetry when we look for an analogy between
the A-side and the B-side. Their starting observation is that _Lagrangian
connected sums are analogous to bundle extensions_ [65, section 3,4].

#### Extension bundle vs. Lagrangian connection sum

An essential aspect of $Coh(X^{\vee})$ is that new bundles can be constructed
from extensions of known bundles $E_{1},E_{2}$, namely

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

Such extension sequences are classified by the complex vector space
$\text{Ext}^{1}(E_{2},E_{1})$. Due to the $\mathbb{C}^{*}$-scaling, the choice
of $E$ is parametrised by the projective space
$\mathbb{P}(\text{Ext}^{1}(E_{2},E_{1}))$. In general, extensions are _not
symmetric_ in $E_{1}$ and $E_{2}$. The extensions

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

are classified by $\text{Ext}^{1}(E_{1},E_{2})$, which is a quite different
space. Extensions can also be viewed as distinguished triangles in
$D^{b}Coh(X^{\vee})$.

In the mirror picture, exact sequences do not make a priori sense, but one can
talk about distinguished triangles, whose geometric sources are the graded
Lagrangian connected sums $L_{1}\\#L_{2}$, fitting into a distinguished
triangle

$L_{1}\to L_{1}\\#L_{2}\to L_{2}\to L_{1}[1].$

Such distinguished triangles are classified by $HF^{1}(L_{2},L_{1})$.323232The
caveat is that unlike bundles, the neck length of the Lagrangian connected sum
cannot be arbitrarily large. Again there is a scaling symmetry related to the
neck size of the Lagrangian connected sum, and there is an asymmetry between
$L_{1}\\#L_{2}$ and $L_{2}\\#L_{1}$.

This analogy is a prime example of _homological mirror symmetry_. On either
side, only pure complex geometry/pure symplectic geometry appears.

#### Wall crossing

Wall crossing in the categorical context refers to the following phenomenon
when the stability condition varies in a 1-parameter family, with central
charges $Z_{t}$. For $t>0$, the objects $E_{1},E_{2},E$ in the extension
sequence are all stable, so necessarily

$\arg Z_{t}(E_{1})<\arg Z_{t}(E)<\arg Z_{t}(E_{2}),\quad
Z_{t}(E)=Z_{t}(E_{1})+Z_{t}(E_{2}).$

At $t=0$, the phase angles become equal, and for $t<0$, the phase angle
inequality is reversed, and $E$ becomes unstable. This phase alignment occurs
on a codimension one locus in the space of stability condition, and thus they
are called walls. Every extension sequence potentially gives rise to a wall,
and the walls can be dense in general.

A notable special case is $\mu$-stability of bundles for a 1-parameter family
of Kähler classes $[\omega_{t}]$, and the slopes
$\mu(E_{1}),\mu(E_{2}),\mu(E)$ become equal precisely for $t=0$. On one side
of the wall, the HYM connections exist on $E_{1},E_{2},E$, and on the other
side $E$ becomes unstable and no longer admits any HYM connection.

Thomas [65] interpreted a gluing construction of Joyce as the mirror analogue
of the wall crossing phenomenon for bundles.333333It is quite remarkable that
Thomas and Yau knew before Bridgeland, that stability conditions make sense
categorically beyond $\mu$-stability, and the mirror of the hypothetical
special Lagrangian stability condition does not need to be $\mu$-stability. In
the simplest case, let $n\geq 3$, fix a symplectic structure $\omega$, and
vary the holomorphic volume form $\Omega_{t}$ in a 1-parameter family while
keeping the almost Calabi-Yau condition. Let $L^{1}_{t},L^{2}_{t}$ be smooth
special Lagrangians with respect to $\Omega_{t}$ with phase $\theta_{t}^{1}$
and $\theta_{t}^{2}$, intersecting transversely at precisely one point
$p_{t}$, with Floer degree $\mu_{L_{t}^{2},L_{t}^{1}}(p_{t})=1$, such that
$\theta_{t}^{2}-\theta_{t}^{1}$ increases past zero at $t=0$, and in
particular $\theta_{0}^{1}=\theta_{0}^{2}$. Thus at $t=0$, the tangent planes
can be put into the standard form inside $\mathbb{C}^{n}$:

$T_{p}L_{0}^{1}=(e^{i\phi_{1}},\ldots e^{i\phi_{n}})\mathbb{R}^{n},\quad
T_{p}L_{0}^{2}=\mathbb{R}^{n},\quad 0<\phi_{k}<\pi,\quad\sum\phi_{k}=\pi,$
$\omega=\frac{\sqrt{-1}}{2}\sum dz_{k}\wedge d\bar{z}_{k},\quad
e^{-i\theta_{0}^{1}}\Omega=a^{n}dz_{1}\wedge\ldots dz_{n},\quad a>0.$
343434The constant $a$ comes from the almost Calabi-Yau structure, and $a=1$
in the Calabi-Yau case.

This provides the appropriate framing data to topologically glue in a Lawlor
neck $L_{\phi,A}$ (_cf._ section 2.3) to desingularize $L_{t}^{1}\cup
L_{t}^{2}$ for small $t$, so that the glued Lagrangian has the topology
$L_{t}^{1}\\#L_{t}^{2}$. Joyce [44, Thm 9.10] shows that for each small $t>0$,
the glued Lagrangian can be perturbed into a special Lagrangian of phase
$\theta_{t}\approx\theta_{0}^{1}$. As $t\to 0$, these special Lagrangians
converge as currents to $L_{t}^{1}\cup L_{t}^{2}$, while for $t<0$ this gluing
strategy does not produce any new special Lagrangian.

The central charges are

$Z(L_{t}^{i})=\int_{L_{t}^{i}}\Omega_{t}^{i}=R_{t}^{i}e^{\sqrt{-1}\theta_{t}^{i}},\quad
Z(L_{t}^{1}\\#L_{t}^{2})=Z(L_{t}^{1})+Z(L_{t}^{2})=R_{t}e^{\sqrt{-1}\theta_{t}}.$

In particular
$R_{t}^{1}\sin(\theta_{t}^{1}-\theta_{t})=-R_{t}^{2}\sin(\theta_{t}^{2}-\theta_{t})$.
The asymmetry between $t>0$ and $t<0$ in Joyce’s gluing construction, comes
from an approximate formula for the Lawlor neck parameter valid for small $t$
353535For a heuristic short derivation see [42, section 6]. Beware that
Joyce’s Lagrangian connected sum has the opposite convention.

$R_{t}^{1}\sin(-\theta_{t}^{1}+\theta_{t})=a^{m}A>0.$

Thus $\theta_{t}^{1}<\theta_{t}$ is needed in the gluing construction. This is
strongly reminiscent of a Bridgeland stability condition. What happens when
$t$ crosses zero can be interpreted as wall crossing.

To summarize, the mirror analogy of wall crossing phenomenon is of categorical
nature. However, it goes beyond homological mirror symmetry as soon as it
involves the stability conditions, and the new problems involve analytical
aspects, beyond purely topological issues. The similarity between the Joyce
gluing and the bundle case is a strong motivation for Thomas and Yau. However,
the Joyce analysis is only valid in a perturbative regime, and what is missing
here is a non-perturbative understanding of when the $D^{b}Fuk(X)$ class of
the Lagrangian connected sum can admit special Lagrangians.

###  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.363636Solomon’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)373737Exact 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

$\langle h,k\rangle=\int_{L}hk\text{Re}\Omega.$ (14)

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

$\mathcal{S}=\int_{0}^{1}dt\int_{L_{t}}h_{t}\text{Im}(e^{-i\hat{\theta}}\Omega).$
(15)

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:

  * •

There is no appearance of the brane structure, or the role of holomorphic
curves.

  * •

Solomon works with exact isotopic Lagrangians, but the Thomas-Yau argument
suggests it is more natural to work in a derived Fukaya category class.

  * •

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.

  * •

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.383838This 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.

###  2.9 Totally real geometry

There is another interesting framework due to Lotay and Pacini [57] [58],
which makes the holomorphic curves appear on the forefront, and exhibits good
analogy with the classical GIT picture, by enlarging the space of Lagrangians
into the space of totally real submanifolds.

Let $(M,\omega,J)$ be a Kähler manifold. A real $n$-dimensional submanifold
$L$ is called totally real, if at every point $T_{p}L$ is transverse to
$JT_{p}L$. Lotay and Pacini introduce a formal principal bundle
$\mathcal{P}\to\mathcal{T}$, where $\mathcal{P}$ is the space of totally real
immersions $\iota:L\to X$ isotopic to a given immersion, and $\mathcal{T}$ is
formally its quotient by the orientation preserving diffeomorphism group
$Diff^{+}(L)$ of $L$.393939Lotay and Pacini did not worry about analytical
issues involving the quotient; their picture is entirely formal. There is a
horizontal distribution, which at each point $\iota\in\mathcal{P}$ assigns the
transverse bundle $JT\iota(L)$, so gives a way to lift tangent vectors from
$\mathcal{T}$ to $\mathcal{P}$. They then define a geodesic to be a curve
$\iota_{t}$ in $\mathcal{P}$, such that the tangent vector field
$\partial_{t}\iota_{t}$ is parallel with respect to the horizontal
distribution. As a caveat, here the word ‘geodesic’ does not suggest a
Riemannian metric. An alternative characterisation [57, Lem 2.2] of a
geodesic, is a 1-parameter family of totally real submanifolds $\iota_{t}:L\to
X$, such that there is a fixed vector field $Y\in\Gamma(L,TL)$, with

$\frac{d}{dt}\iota_{t}=J\iota_{t*}Y,\quad[\iota_{t*}Y,J\iota_{t*}Y]=0.$

Geometrically, one can imagine $(\iota_{t})_{0\leq t\leq 1}$ sweeps out an
$(n+1)$-dimensional submanifold with boundary, foliated into complexified
integral curves of $Y$, which are holomorphic curves inside $X$.

Lotay and Pacini also define the J-volume functional on $\mathcal{T}$.
Pointwise on a totally real submanifold $L$, a real cotangent vector of $L$
corresponds to a $(1,0)$-form in $\Lambda^{(1,0)}X|_{L}$, so taking the $n$-th
wedge power, we have a canonical isomorphism between
$\Lambda^{n}L\otimes_{\mathbb{R}}\mathbb{C}$ and $K_{X}|_{L}$. Now the Kähler
structure on $X$ induces a Hermitian metric on $K_{X}$, and pointwise on $L$
an element of $K_{X}$ with unit Hermitian norm uniquely specifies a volume
form $dvol_{J}$ on $L$. Their $J$-volume functional is

$\text{Vol}_{J}(L)=\int_{L}dvol_{J}.$

It is easy to show $\text{Vol}_{J}$ provides a lower bound
$\text{Vol}_{J}(L)\leq\text{Vol}(L)$ to the Riemannian volume of $L$, with
equality precisely when $L$ is Lagrangian.

In case $X$ is Calabi-Yau, this construction is particularly transparent. The
totally real condition is equivalent to the pointwise non-vanishing of
$\Omega$ when restricted to $L$, and

$|\int_{L}\Omega|\leq\text{Vol}_{J}(L)=\int_{L}e^{-i\theta}\Omega\leq\text{Vol}(L),$

where the phase factor is chosen to make $e^{-i\theta}\Omega$ an orientation
form on $L$. Consequently, any totally real submanifold in a Calabi-Yau
manifold with $\theta=\hat{\theta}$ constant, with no need for the Lagrangian
condition, is an absolute minimizer of the $\text{Vol}_{J}$ functional. This
enormous space of critial points is closely related to the non-ellipticity of
the critical point equation. The situation is somewhat better in a negative
Kähler-Einstein ambient space, where the critical points coincide with minimal
Lagrangians [57, section 5.5].

One of their main results is

######  Proposition 2.13.

[57, Thm 5.10] In a Kähler-Einstein ambient manifold with non-positive Ricci
curvature, the $J$-volume functional is convex along the geodesics.

There is also a formal GIT picture [57, section 6]: to some extent the space
$\mathcal{P}$ can be viewed as the infinitesimal complexification of
$Diff^{+}(L)$, with the space of orbits $\mathcal{T}$. The J-volume functional
is formally a Kähler potential on $\mathcal{P}$, and the $Diff^{+}(L)$-moment
map gives rise to critical points of the J-functional.

#### Limitations

From the viewpoint of special Lagrangian geometry, the limitations of the
Lotay-Pacini picture are:

  * •

The Lagrangians are largely relegated to the back stage. As clear from the
above, in the Calabi-Yau case the space of critical points is too enormous.
Their framework may be more useful in the negative Kähler-Einstein case, but
the lack of ellipticity is a severe obstacle.

  * •

There is no attempt to link up with Floer theory. As such, they lack
satisfactory existence criterions for the holomorphic curves, despite making
some limited progress in special cases. In fact, Lotay and Pacini’s geodesics
seem too oversimplified from the Floer theoretic viewpoint: one needs to
address transversality questions in general, and holomorphic disc breaking
should occur in moduli spaces of dimension $\geq 1$.

###  2.10 Analogy with tunneling effect

404040This section is meant to be purely inspirational, and its aim is to
present some analogies and comparisons with no claim to physical accuracy.

Notice the Thomas-Yau argument is a little mysterious from the following
classical perspective: how could two Lagrangians in possibly different
Hamiltonian isotopy classes communicate with each other? This situation seems
conceptually similar to the phenomenon of tunneling in quantum mechanics:
there may be several minima of the classical potential function separated by
potential wells, but there is a nontrivial quantum amplitude for the particles
to move in between, thereby removing the ground state degeneracy. In the
Thomas-Yau setup, we have two putative special Lagrangians, which are
analogous to the energy minima points, and the Thomas-Yau uniqueness statement
is similar to the removal of degeneracy, resulting in a unique ground state.

The physicists tell us that branes are dynamical objects and can fluctuate. If
so, it might make sense to ask about the amplitude for a brane to start with a
given configuration and end with another. Now if Lagrangian branes behave like
classical particles moving on an infinite dimensional space such as Solomon’s
space $\mathcal{O}$, one might expect Solomon’s formal picture to be relevant
for describing this amplitude, and the incompleteness of $\mathcal{O}$ would
suggest a nontrivial amplitude to tunnel outside $\mathcal{O}$ to another
Hamiltonian isotopy class. As a conflicting viewpoint, the use of Floer theory
in the Thomas-Yau argument suggests the Lagrangian branes communicate by the
strings stretched between them, so one might expect the amplitudes to be
computed in terms of worldsheet integrals. Lotay and Pacini’s geodesics fit
this viewpoint better, and have the major advantage of making sense outside a
given Hamiltonian isotopy class.

Tunneling effects made a famous appearance in Witten’s interpretation of Morse
theory [80] in terms of supersymmetric quantum mechanics, with an eye towards
applications in quantum field theory. In Witten’s context, the tunneling
amplitude between two critical points of a Morse function is to leading order
proportional to $e^{-S}$, where $S$ is proportional to the difference of the
two critical values. Now Solomon’s functional has some similarity with a Morse
function whose only critical points are minima. In sections 3.2, 3.7 we will
_unify_ the Solomon functional with the Lotay-Pacini geodesics, in the special
case of exact Lagrangians. Could the Solomon functional have any physical
interpretation in terms of the logarithm of the tunneling amplitudes? 414141As
a grain of salt, Witten’s interpretation concerns supersymmetric QFT, while
the Thomas-Yau picture concerns the dynamics of Lagrangian branes, which is a
target space perspective on string theory, which lacks a fundamental path
integral formulation. As such our suggested physcial interpretation is not
logically rigorous, but only a guess.

##  3 Moduli space of holomorphic curves

The theme of this Chapter is the _bordism currents_ produced from the
$(n+1)$-dimensional universal family over the $(n-1)$-dimensional moduli
spaces of holomorphic curves. This theme unifies the search for the Floer
theoretic obstructions, and the problem of extending the Solomon functional to
the derived Fukaya category setting. One of our central techniques is
_integration over the moduli spaces_ , to prove both identities and
inequalities. The primary setting of this Chapter is exact graded compact
Lagrangians inside Calabi-Yau Stein manifolds, with occasional comments on the
compact Calabi-Yau case.

The _Floer theoretic obstructions_ are necessary conditions to the existence
of special Lagrangians in the exact and almost calibrated setting. We will
give a large number of a priori heuristic principles to place very stringent
constraints on what kind of obstructions we are looking for, then prove the
Floer theoretic obstructions under the extra hypotheses of automatic
transversality and a positivity condition (_cf._ section 3.5), and then
compare our picture to Joyce’s proposal on Bridgeland stability (_cf._ section
3.6). The converse direction, in search of _sufficient conditions_ for the
existence of special Lagrangians, will be laid out in Chapter 5.

We also present two perspectives on extending the Solomon functional to
unobstructed Lagrangians within the same $D^{b}Fuk(X)$ class. The more
elementary perspective (_cf._ section 3.2) works only in the exact setting,
and implies the first variation formula under Hamiltonian isotopies in a
rather straightforward manner. The second perspective (_cf._ section 3.7)
represents the Solomon functional as an integral over the moduli space of
holomorphic curves, which we hope can be generalized to the compact Calabi-Yau
setting. Section 3.8 contains more applications of the moduli integral
technique.

######  Remark 3.1.

The two _assumptions_ of automatic transversality (which roughly means that no
perturbation of almost complex structure is needed, _cf._ section 3.3) and the
positivity condition (_cf._ section 3.4) will feature prominently in the major
results of this Chapter. Correspondingly, we will not dwell on the detail of
the perturbation schemes aimed at overcoming the transversality issues, which
are painstakingly carried out in [7][81][82][14][15]. The expert readers may
convert to their favourite schemes as they prefer. For more on our clockwise
conventions such as gradings and signs, which differ from many symplectic
texts, see the Appendix.

###  3.1 Lotay-Pacini picture revisited

From a Floer theoretic perspective, the main insight of Lotay and Pacini
(_cf._ section 2.9) is that given two Lagrangians $L,L^{\prime}$ decorated
with suitable brane structures, one should look for a _family_ of holomorphic
curves with boundary on $L$ and $L^{\prime}$, which pass through any generic
point of $L$ and $L^{\prime}$. In their formal picture, such families are
called ‘geodesics’. What Lotay and Pacini did not provide is a good existence
criterion for their geodesics. Now, even though we will soon specialize to a
much simpler setting, we wish to explain how their geodesics fit into the
Thomas-Yau-Joyce picture.

In the setup of Bridgeland stability, the central charge is defined as a
homomorphism from the Grothendieck group of a triangulated category to
$\mathbb{C}$, which factorizes through a finitely generated lattice, viewed as
a numerical Grothendieck group. In view of the application to special
Lagrangians, the triangulated category is $D^{b}Fuk(X)$, and the numerical
Grothendieck group should be a subgroup of the homology group modulo torsion
$H_{n}(X,\mathbb{Z})/\text{tors}$. In particular,

  * •

If two Lagrangian branes $L,L^{\prime}$ define the same object in
$D^{b}Fuk(X)$, then this picture predicts them to lie in the same homology
class in $H_{n}(X)$.

  * •

If $L_{1}\to L\to L_{2}\to L_{1}[1]$ is a distinguished triangle, then
$[L_{1}]+[L_{2}]=[L]$ in $H_{n}(X)$.

This means Floer theory must provide a bordism current between $L,L^{\prime}$
(resp. $L$ and $L_{1}\cup L_{2}$), namely an $(n+1)$-dimensional integration
current $\mathcal{C}$ with $\partial\mathcal{C}=L-L^{\prime}$ (resp.
$\partial\mathcal{C}=L-L_{1}-L_{2}$).424242The boundary of integration
currents do not detect contributions from supports of small enough dimension.
In this respect they are similar to pseudocycles, although when we generalize
to Lagrangians with very weak regularity later, the language of currents may
be more natural. In Floer theory, cycles are constructed from moduli spaces of
holomorphic curves and evaluation maps, so this current $\mathcal{C}$ should
come from families of holomorphic curves, possibly with highly sophisticated
perturbations and virtual techniques. This suggests the Lotay-Pacini
geodesics, construed in this homological sense, should be part of the Fukaya
category foundation necessary for the Thomas-Yau-Joyce program.

#### High brow viewpoint

The claim that _the $K$-theory of the derived Fukaya category of a symplectic
Calabi-Yau manifold (exact with suitable convexity at infinity, or compact)
factorizes through homology $H_{n}(X)$ modulo torsion_ seems well known to
symplectic topology experts, although a precise reference seems rather
difficult to find. We now sketch a high brow viewpoint explained to the author
by P. Seidel and S. Rezchikov, and will later explain in more detail a more
pedestrian approach in the exact setting. The claim is a formal consequence of
the existence of maps

$K_{0}(D^{b}Fuk(X))\xrightarrow{ch_{0}}HH_{0}(D^{b}Fuk(X))$
$HH_{0}(D^{b}Fuk(X))\xrightarrow{OC}QH^{n}(X;\Lambda_{nov})\simeq
H_{n}(X)\otimes\Lambda_{nov}.$

Here $HH_{0}(D^{b}Fuk(X))$ is the Hochschild homology in degree zero. The
first map sends the K-theory class of an unobstructed Lagrangian brane $L$
(compact, graded, oriented, with spin and bounding cochain structure) to the
unit $1_{L}\in HF^{0}(L,L)\to HH_{0}(D^{b}Fuk(X))$; the well definition of
this map is an essentially algebraic fact. Suppose $L,L^{\prime}$ are
isomorphic in $D^{b}Fuk(X)$, then there are closed morphisms $\alpha\in
CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ whose derived
category compositions are equal to $1_{L}\in HF^{0}(L,L)$ and
$1_{L^{\prime}}\in HF^{0}(L^{\prime},L^{\prime})$ in cohomology. The
Hochschild differential of $\beta\otimes\alpha$ exhibits
$1_{L}-1_{L^{\prime}}$ as a coboundary in the Hochschild chain complex, so
$1_{L}=1_{L^{\prime}}$ in $HH_{0}(D^{b}Fuk(X))$. Some additional calculation
shows the compatibility with distinguished triangles.

The second map is a special case of the open-closed string map, and in general
requires working over the Novikov field. One then needs the claim that $1_{L}$
is sent to the homology class $[L]\in H_{n}(X)$, without quantum correction.
The intuitive meaning of the open-closed string map is to consider holomorphic
discs with boundary on $L$ with an unconstrained boundary marked point, and
find the cycle in $X$ traced out by an interior marked point. The claim
amounts to saying that the only contribution comes from constant maps.
Unfortunately, the author is unable to locate a general reference. Granted
this claim, we would get by composition a map from $K_{0}(D^{b}Fuk(X))$ to
$H_{n}(X;\Lambda_{nov})$ which sends the K-theory class of $L$ to the homology
class $[L]$.

####  3.1.1 The exact embedded Lagrangian case

We specialize to the setting of Stein manifolds, and all Lagrangians are
assumed to be exact, graded and compact, and in particular carry an
orientation (_cf._ the Appendix for some basic Floer theory). The local
systems have holonomy in $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$. We
consider two transverse embedded Lagrangians $L,L^{\prime}$ in the _same
derived Fukaya category class_. By definition, we have closed morphisms
$\alpha\in CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$; we
sometimes view the Lagrangian intersections in $\beta$ as degree $n$ outputs.
Morever, in terms of the product structure on cohomology

$\begin{cases}HF^{0}(L,L^{\prime})\otimes HF^{0}(L^{\prime},L)\to
HF^{0}(L^{\prime},L^{\prime}),\\\ HF^{0}(L^{\prime},L)\otimes
HF^{0}(L,L^{\prime})\to HF^{0}(L,L),\end{cases}$

the composition $\alpha\circ\beta=1_{L^{\prime}}$ and
$\beta\circ\alpha=1_{L}$. These conditions completely characterize isomorphism
in $D^{b}Fuk(X)$. Our goal is to explain

######  Proposition 3.1.

There is a bordism current $\mathcal{C}$ such that
$\partial\mathcal{C}=L-L^{\prime}$ in the sense of currents.

The bordism current will be constructed from _universal families of
(perturbed) holomorphic strips_ with boundary on $L$ and $L^{\prime}$, and
with ends at $\alpha$ and $\beta$ (meaning that the ends of the strip converge
to intersection points of $L,L^{\prime}$ of degree $0$ and $n$ respectively,
and $\alpha,\beta$ encode the weighting factors to the contribution of these
intersection points). We will assume all the usual transversality assumptions
in Floer theory are satisfied, so the compactified moduli space
$\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$ of perturbed holomorphic
strips up to domain translation is a smooth manifold with boundary and
corners, of dimension $(n-1)$. The notation really stands for a formal sum of
many moduli spaces, coming from the summands of $\alpha,\beta$. The universal
family is fibred over this moduli space
$\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$, whose fibres are the
solutions to the Cauchy-Riemann equation (with domain dependent perturbations
of the almost complex structure), which we call perturbed holomorphic curves.
The fibres over the boundary of the moduli space are broken holomorphic
curves. The orientation on the universal family is induced from the complex
orientation on $\Sigma$ and the orientation on the moduli space, up to an
extra minus sign (_cf._ Example 6.2 for conventions). Upon evaluation to $X$
we obtain an $(n+1)$-dimensional current $\mathcal{C}$.

######  Remark 3.2.

If the Fukaya category is defined over $\mathbb{Z}$, then all the weighting
factors to the various universal families are all _integers_ , and
$\mathcal{C}$ is naturally an _integral current_. If we use Fukaya categories
over $\mathbb{Q}$ or $\mathbb{R}$ instead, then $\mathcal{C}$ is only
guaranteed to be a finite $\mathbb{Q}$ (resp. $\mathbb{R}$) linear combination
of integral currents.

Our main task is to understand the boundary of $\mathcal{C}$. There are two
sources of boundaries:

  * •

The holomorphic curves themselves have boundary along $L\cup L^{\prime}$. This
boundary contribution is always supported on $L\cup L^{\prime}$.

  * •

The compactified moduli spaces have boundary due to holomorphic strip
breaking.

In schematic notation, the boundary of the moduli space is described by

$\partial(\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}})=\bigcup_{r}\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}\times\overline{\mathcal{M}(r,\beta)/\mathbb{R}}.$
(16)

Here $r$ can range from all intersection points of degree between $1$ and
$n-1$. The notation $\mathcal{M}(\alpha,\beta)/\mathbb{R}$ stands for a
weighted sum of moduli spaces, with weighting coming from the holonomy factors
of the local systems.

Next comes a crucial observation. Although $2\leq\deg r\leq n-2$ give rise to
boundaries of the moduli space
$\overline{\mathcal{M}(\alpha,\beta)/\mathbb{R}}$, their contributions to
$\partial\mathcal{C}$ are contained in the universal families associated to
$\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ and
$\overline{\mathcal{M}(r,\beta)/\mathbb{R}}$. These smaller moduli spaces have
dimension at most $n-3$, and the corresponding universal families have
dimension at most $n-1$. By rectifiability considerations, the $n$-dimensional
current $\partial\mathcal{C}$ cannot receive contributions from at most
$(n-1)$-dimensional supports, so such disc breakings do not contribute to
$\partial\mathcal{C}$.

Now for $\deg r=1$, the moduli spaces
$\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ are zero dimensional, so their
presence merely amounts to some counting factors. The condition for $\alpha$
to be _closed_ in $CF^{0}(L,L^{\prime})$ is equivalent to the weighted count
of $\overline{\mathcal{M}(\alpha,r)/\mathbb{R}}$ being zero. This weighted sum
appears as the coefficient of the $n$-dimensional current defined by the
universal family over $\overline{\mathcal{M}(r,\beta)/\mathbb{R}}$. Thus we
see that $r\in CF^{1}(L,L^{\prime})$ does not contribute to
$\partial\mathcal{C}$. Similarly, the condition for $\beta$ to be closed in
$CF^{0}(L^{\prime},L)$ implies that $r\in CF^{1}(L^{\prime},L)$ (alternatively
viewed as degree $n-1$ intersections from $L$ to $L^{\prime}$) does not
contribute to $\partial\mathcal{C}$. In summary, $\partial\mathcal{C}$ must be
an integration cycle supported on $L\cup L^{\prime}$.

Since $\partial\mathcal{C}$ is itself the boundary of a current, it must be
closed. This explains why $\partial\mathcal{C}$ is a constant linear
combination of the integration cycle of $L$ and $L^{\prime}$, instead of some
nontrivial function times these cycles. The constant coefficients can be
pinned down by counting the number of holomorphic strips passing through a
given generic point on $L$ (resp. $L^{\prime}$), and the choice of the generic
point does not matter. Such counts are precisely the geometric interpretation
of the Floer product $HF^{0}(L,L^{\prime})\otimes HF^{0}(L^{\prime},L)\to
HF^{0}(L^{\prime},L^{\prime}),$ and $HF^{0}(L^{\prime},L)\otimes
HF^{0}(L,L^{\prime})\to HF^{0}(L,L)$ (_cf._ Example 6.1). When the moduli
space orientations are taken into account, we obtain
$\partial\mathcal{C}=L-L^{\prime}$ (_cf._ Example 6.2 for an exposition on
signs).

######  Remark 3.3.

(Homological uniqueness of the bordism current) Some auxiliary perturbation
data goes into the construction of $\mathcal{C}$ due to the need to ensure
transversality. If we fix $L,L^{\prime}$, but change the domain dependent
almost complex structures, then the difference of two bordism currents
$\mathcal{C}-\mathcal{C}^{\prime}$ has zero boundary in the sense of currents.
Recall that Stein manifolds have the homotopy type of a CW complex of
dimension $\leq n$, and thus $H_{k}(X)=0$ for $k\geq n+1$, so
$\mathcal{C}-\mathcal{C}^{\prime}$ must be the boundary of an
$(n+2)$-dimensional current. For an alternative viewpoint, this
$(n+2)$-dimensional current can be concretely provided by parametrized
families of pseudoholomorphic curves (_cf._ Remark 3.5).

####  3.1.2 Immersed case

Still working in the exact setting, we now allow
$(L,b),(L^{\prime},b^{\prime})$ to be unobstructed immersed Lagrangians with
transverse self intersections. Assume they intersect transversally, and define
isomorphic objects in $D^{b}Fuk(X)$. We now wish to explain why Proposition
3.1 should continue to hold even in the immersed setting, without delving too
deep into the specifics of the perturbation schemes and transversality issues.
For some background on the immersed Floer theory, see the Appendix 6.2.

The isomorphism condition gives us closed morphisms $\alpha\in
CF^{0}(L,L^{\prime})$ and $\beta\in CF^{0}(L^{\prime},L)$ whose cohomological
compositions give the identities. At the chain level,

$m_{2}^{b,b^{\prime}}(\beta,\alpha)=1_{L}-m_{1}^{b}(\gamma),\quad
m_{2}^{b^{\prime},b}(\alpha,\beta)=1_{L^{\prime}}+m_{1}^{b^{\prime}}(\gamma^{\prime})$

where $1_{L},1_{L^{\prime}}$ stand for the geometric units (represented by a
sum of local maximum points of Hamiltonian functions on $L,L^{\prime}$
respectively), and $\gamma,\gamma^{\prime}$ are elements in $CF^{-1}(L,L)$,
$CF^{-1}(L^{\prime},L^{\prime})$ respectively. Notice in the almost calibrated
case, $\gamma,\gamma^{\prime}$ would be both zero, since there are no self
intersections of degree $-1$.

As before, the bordism current shall be constructed from the universal family
of (perturbed) holomorphic curves with boundary on $L$ and $L^{\prime}$. But
instead of working only with holomorphic strips, we need holomorphic polygons
with corners not only at intersection points in $\alpha,\beta$, but also at
points in $b,b^{\prime}$. In addition to the holomorphic strip moduli space
$\mathcal{M}(\alpha,\beta)/\mathbb{R}$, we also need the moduli space of
polygons $\mathcal{M}(b,\ldots b,\alpha,b^{\prime},\ldots b^{\prime},\beta)$,
and $\mathcal{M}(b,\ldots,b,\gamma)$, $\mathcal{M}(b^{\prime},\ldots
b^{\prime},\gamma^{\prime})$. The notation here is a shorthand for a weighted
sum of many moduli spaces of polygons. Since the bounding cochain elements
have Floer degrees one, these moduli spaces all have dimension $n-1$. The
energy of the polygons satisfies the topological formula (66), so by the
Novikov positivity requirement of bounding cochains, there is a uniform a
priori energy bound once $\alpha,\beta,\gamma,\gamma^{\prime}$ are given,
whence there are in fact only finitely many moduli spaces involved. Each
moduli space provides a universal family of holomorphic curves, and the sum of
all the contributions defines an $(n+1)$-dimensional current $\mathcal{C}$.
For sign conventions, see the Appendix 6.2, and Example 6.2.

The boundary of $\mathcal{C}$ comes from two sources: the boundary of the
individual holomorphic curves which lie on $L\cup L^{\prime}$, and the
boundary of the compactified moduli spaces. In the exact setting, there are no
sphere bubbles. As in the embedded case, for support dimension reasons, the
boundaries of the compactified moduli space that can contribute to
$\partial\mathcal{C}$, is caused by curve breaking into two pieces arising in
$0$ and $n-2$ dimensional moduli spaces. The cancellation of these
contributions is very similar to the standard argument for the Floer
differential to square to zero (_cf._ the Appendix 6.2):

  * •

For breakings at a nodal point mapping to $CF^{1}(L,L^{\prime})$ (resp.
$CF^{1}(L^{\prime},L)$), the contributions vanish due to the closedness
condition $m_{1}^{b,b^{\prime}}(\alpha)=0$ (resp. the closedness of $\beta$).

  * •

For breakings at degree 2 self intersection point on $L$ (resp. $L^{\prime}$),
the contributions vanish due to the Mauer-Cartan equation on $b$ (resp.
$b^{\prime}$).

  * •

A new way of disc breaking/splitting, is at a degree zero self intersection
$r$ on $L$ (the $L^{\prime}$ case being entirely similar). The discs of
$\mathcal{M}(b,\ldots b,\alpha,b^{\prime},\ldots b^{\prime},\beta)$ can break
into a virtual dimension zero disc with input at
$b,\ldots,\alpha,b^{\prime},\ldots\beta,b,\ldots b$ and output at $r$, and a
virtual dimension $n-1$ disc with input corners at $b,\ldots b,r$. On the
other hand, the discs of $\mathcal{M}(b,\ldots,b,\gamma)$ can break into a
virtual dimension zero disc with input at $b,\ldots,\gamma,b,\ldots b$ and
output at $r$, and a virtual dimension $n-1$ disc with input corners at
$b,\ldots b,r$. These two effects cancel out.

After the cancellation of all moduli space boundaries, the only contributions
$\partial\mathcal{C}$ are supported in $L\cup L^{\prime}$. As in the embedded
case, $\partial\mathcal{C}$ is locally a constant multiple of the underlying
cycles of $L,L^{\prime}$. The interpretation of the geometric unit pins down
$\partial\mathcal{C}=L-L^{\prime}$ as in the embedded case.

######  Example 3.2.

If $L$ and $L^{\prime}$ are disjoint Lagrangian branes which both define the
zero object in $D^{b}Fuk(X)$, then $\alpha,\beta$ are both zero, and
$\mathcal{C}$ comes entirely from the $\gamma,\gamma^{\prime}$ contributions.
Of course, zero Lagrangian objects have zero homology class, which cannot
happen in the almost calibrated case.

######  Remark 3.4.

If there are degree $-2$ self intersections of $L$, then the choice of
$\gamma$ is only unique up to $m_{1}^{b}$ of some element in $CF^{-2}(L,L)$.
The corresponding choice of $\mathcal{C}$ would be ambiguous by the boundary
of an $(n+2)$-dimensional integration current. As a closely related issue, our
conditions on $\alpha,\beta$ are merely cohomological, so in general we can
adjust $\alpha$ and $\beta$ by coboundary terms, which would affect
$\mathcal{C}$ also by the boundary of an $(n+2)$ dimensional current. If we
impose $L,L^{\prime}$ to be almost calibrated, then there are no
$CF^{-1}(L,L^{\prime})$ elements to begin with, and these phenomena do not
happen.

On the other hand, $\mathcal{C}$ still depends on the choice of local systems
and bounding cochains, which may contribute nontrivial holonomy factors. Gauge
equivalent choices affect $\mathcal{C}$ by the boundary of an
$(n+2)$-dimensional current. One may naturally ask:

######  Question 3.

Up to gauge equivalence of bounding cochains and local systems, is there an
optimal representative of $\mathcal{C}$?

######  Question 4.

Given an exact isotopy with surgery between $L$ and $L^{\prime}$ among
unobstructed Lagrangians, is there a preferred choice of $\mathcal{C}$ (_cf._
Question 2)?

#### Distinguished triangles

In our convention, an immersed Lagrangian can be made up of _several connected
components_. A prototypical situation is when $L^{\prime}$ is the union of two
immersed Lagrangians $L_{1}$ and $L_{2}$, with some degree one intersections
in $CF^{1}(L_{2},L_{1})$ arising as part of the bounding cochain data of
$L^{\prime}$. When the brane structure is taken into account, we can view
$L^{\prime}$ as a twisted complex built from $L_{1},L_{2}$ (with bounding
cochains $b_{1},b_{2}$ suppressed in the notation) and a closed morphism
$\gamma\in CF^{1}(L_{2},L_{1})=CF^{0}(L_{2}[-1],L_{1})$. Inside $D^{b}Fuk(X)$,

$L\simeq L^{\prime}\simeq\left(\begin{matrix}(L_{2},b_{2})&\\\
\gamma&(L_{1},b_{1})\end{matrix}\right).$

We have a distinguished triangle

$L_{2}[-1]\xrightarrow{\gamma}L_{1}\to\text{Cone}(\gamma)\xrightarrow{}L_{2},$

and $L\simeq L^{\prime}\simeq\text{Cone}(\gamma)$. Rotating the triangles, we
get another distinguished triangle

$L_{1}\to L\to L_{2}\xrightarrow{\gamma}L_{1}[1].$

The bordism current between $L$ and $L^{\prime}$ is an $(n+1)$-dimensional
integration current, with $\partial\mathcal{C}=L-L_{1}-L_{2}.$ In particular,
this explains that the Grothendieck group of $D^{b}Fuk(X)$ should factorize