theoremdummy theorem lemmadummy lemma propositiondummy proposition
corollarydummy corollary conjecturedummy conjecture exampledummy example
definitiondummy definition remarkdummy remark questiondummy question
problemdummy problem assumptiondummy assumption

# Holomorphic Floer theory and the Fueter equation

Aleksander Doan, Semon Rezchikov

###### Abstract

We outline a proposal for a $2$-category $\mathrm{Fuet}_{M}$ associated to a
hyperkähler manifold $M$, which categorifies the subcategory of the Fukaya
category of $M$ generated by complex Lagrangians. Morphisms in this
$2$-category are formally the Fukaya–Seidel categories of holomorphic
symplectic action functionals. As such, $\mathrm{Fuet}_{M}$ is based on
counting maps to $M$ satisfying the Fueter equation with boundary values on
holomorphic Lagrangians. We make the first step towards constructing this
category by establishing some basic analytic results about Fueter maps, such
as the energy bound and maximum principle. When $M=T^{*}X$ is the cotangent
bundle of a Kähler manifold $X$ and $L_{0},L_{1}$ are the zero section and the
graph of the differential of a holomorphic function $F\colon X\to\mathbb{C}$,
we prove that all Fueter maps correspond to the complex gradient trajectories
of $F$ in $X$, which relates our proposal to the Fukaya–Seidel category of
$F$. This is a complexification of Floer’s theorem on pseudo-holomorphic
strips in cotangent bundles. Throughout the paper, we suggest problems and
research directions for analysts and geometers that may be interested in the
subject.

###### Contents

  1. 1 Introduction
  2. 2 Formal and categorical aspects
    1. 2.1 Topological field theories
    2. 2.2 The Fueter $2$-category
    3. 2.3 Complex Morse theory
    4. 2.4 The Fueter $2$-category from complex Morse theory
    5. 2.5 Decategorification I
    6. 2.6 Decategorification II
    7. 2.7 Combinatorics of Fueter domains
    8. 2.8 Cotangent bundles of Kähler manifolds
    9. 2.9 Examples from algebraic geometry
    10. 2.10 $3d$ mirror symmetry
    11. 2.11 Further analytic aspects
    12. 2.12 Further geometric aspects
  3. 3 The Fueter equation
    1. 3.1 Quaternion-antilinear maps
    2. 3.2 Fueter maps
    3. 3.3 Boundary conditions
    4. 3.4 Taming triples
    5. 3.5 Energy bounds
  4. 4 Convexity theory
    1. 4.1 Complex convexity and pseudo-holomorphic maps
    2. 4.2 Quaternionic convexity and Fueter maps
    3. 4.3 Conical completions
  5. 5 Cotangent bundles of symplectic manifolds
    1. 5.1 Almost complex and quaternionic structures
    2. 5.2 Convexity
    3. 5.3 Cauchy–Riemann operators
    4. 5.4 Complexified Floer correspondence
    5. 5.5 Remarks on the unperturbed equation
    6. 5.6 Interpretation of the perturbed equation
  6. A Notation and conventions
    1. A.1 Coordinates
    2. A.2 Symplectic geometry
    3. A.3 Lagrangian Floer theory
    4. A.4 Complex gradient flow
    5. A.5 Holomorphic Floer theory
  7. B Floer’s theorem revisited

##  1 Introduction

This paper studies a certain complexification of Lagrangian Floer homology.
Like its real counterpart, this theory arises from a $\sigma$-model: it
involves the study of maps

$U\colon E\to M$

from a three-manifold $E$, which we will take to be

$E=[0,1]\times\mathbb{R}^{2},$

to a manifold $M$ of real dimension $4n$ equipped with an almost quaternionic
structure $(I,J,K)$, that is: a triple of almost complex structures satisfying
the quaternionic relation $IJ=K$. The analog of the pseudo-holomorphic map
equation used in Lagrangian Floer theory,

$\begin{gathered}u\colon\mathbb{R}\times[0,1]\to M,\\\
\partial_{t}u+J(u)\partial_{\tau}u=0,\end{gathered}$ (1.1)

where $(t,\tau)$ are the coordinates on $\mathbb{R}\times[0,1]$, is the
_Fueter equation_ :

$\begin{gathered}U\colon[0,1]\times\mathbb{R}^{2}\to M,\\\
I(U)\partial_{\tau}U+J(U)\partial_{s}U+K(U)\partial_{t}U=0,\end{gathered}$
(1.2)

where $(\tau,s,t)$ are the coordinates on $[0,1]\times\mathbb{R}^{2}$. This
equation is in many ways the simplest higher-dimensional generalization of the
pseudo-holomorphic map equation which shares some of its good analytical
properties. As such, many analytical aspects of the Fueter equation are
tractable and can be developed systematically by analogy with pseudo-
holomorphic maps, although some core questions—most importantly, in
compactness theory—remain challenging [HNS09, Wal17b]. On the other hand, due
to the kinship of the Fueter equation with the pseudo-holomorphic map
equation, it conjecturally gives rise to a host of algebraic structures, some
of which we indicate in this paper, with rich connections to low-dimensional
topology, representation theory, and mathematical physics.

The Fueter equation has been previously studied from many different
perspectives. We list some of them below:

  * •

The linear variant of the Fueter equation, with $M$ a quaternionic vector
space, is simply the $3d$ massless Dirac equation. Solutions to the Dirac
equation on $\mathbb{R}^{3}$ are known to obey various quaternionic analogs of
phenomena in complex analysis [Fue35]. Indeed, finding a quaternionic analog
of holomorphic maps was the original motivation for introducing the Fueter
equation.111At least in mathematics: Dirac’s theory of the electron precedes
Fueter’s article by seven years.

  * •

Solutions to the Fueter equation whose domain $E$ is a closed, framed three-
manifold are critical points of a functional on the space of maps $E\to M$.
Gradient trajectories of this functional are solutions to the four-dimensional
version of the Fueter equation for maps $E\times\mathbb{R}\to M$. When $M$ is
a flat hyperkähler manifold, the moduli spaces of solutions to these equations
can be used to build an invariant of $M$ analogous to Hamiltonian Floer
homology [HNS09].

  * •

The Fueter equation arises naturally as a degeneration limit of various
equations in gauge theory, such as the instanton equation on Riemannian
manifolds with special holonomy Donaldson1998, Donaldson2011, Haydys2012,
Walpuski2017a, Walpuski2017b and the generalized Seiberg–Witten equations on
three- and four-dimensional manifolds [Tau13, Tau15, Tau16, HW15, WZ21]. In
both situations, $M$ is the hyperkähler quotient of a hyperkähler manifold by
an action of a compact Lie group, a typical example being the moduli space of
framed instantons on $\mathbb{R}^{4}$.

  * •

Maps satisfying the Fueter equation appear as a result of applying
supersymmetric localization to the physical 3d $N=4$ $\sigma$-model. The
corresponding result for the 4d N=2 theory, which reduces to the 3d N=4
theory, dates back at least to [AF94]. As such, Fueter maps are a mathematical
incarnation of the topological field theory underlying 3d mirror symmetry—an
active topic in mathematical physics and representation theory, which thus far
has been treated largely from an algebraic perspective.

Our motivations are closely aligned with the last point above, although we
learned this physical perspective after thinking about the results of this
paper and much of the algebraic structure indicated below from a purely
mathematical viewpoint. Heuristically, one can arrive at many expectations
regarding the Fueter equation by complexifying known results in symplectic
Floer theory, with the proviso that

_complexification categorifies symplectic invariants_.

Specifically, we study the analytic underpinnings of the complexification of a
classical theorem of Floer on cotangent bundles [Flo88a, Flo88], which we now
briefly review. Let $X$ be a closed manifold with a Morse function $f\colon
X\to\mathbb{R}$. The cotangent bundle $T^{*}X$ is a symplectic manifold and
the zero section and graph of ${\mathrm{d}}f$ are two Lagrangian submanifolds,
which we denote by $L_{0}$ and $L_{1}$. Floer proved that if the $C^{2}$ norm
of $f$ is sufficiently small (which can be always guaranteed by rescaling
$f$), then the Lagrangian Floer homology of $L_{0}$ and $L_{1}$ is canonically
identified with the Morse homology of $f$:

$HF_{*}(L_{0},L_{1})=HM_{*}(X,f).$ (1.3)

The two homology groups have the same generators, as the intersection points
of $L_{0}$ and $L_{1}$ correspond to the critical points of $f$, so the proof
consists in identifying the differentials. In Morse theory they are given by
counting gradient trajectories of $f$, while in Floer homology by counting
pseudo-holomorphic strips, that is: solutions to (1.1) with boundary on
$L_{0}$ and $L_{1}$ and asymptotic to $L_{0}\cap L_{1}$. Floer proves that
there is a time-dependent almost complex structure $J_{\tau}$ on $T^{*}X$
compatible with the symplectic structure and with the property that
$J_{\tau}$-holomorphic strips in $M=T^{*}X$ correspond to the gradient
trajectories of $f$ in $X$.

_We prove a complexified version of Floer’s correspondence._ On the Morse
theory side, we assume that $X$ is a Kähler manifold and $F\colon
X\to\mathbb{C}$ is a holomorphic function. Denote by $I_{X}$ the complex
structure on $X$. Gradient trajectories in Floer’s theorem are replaced by the
_complex gradient trajectories_ of $F$, that is: maps
$v\colon\mathbb{R}^{2}\to X$ satisfying

$\overline{\partial}_{I_{X}}(v)=i\nabla F(v)$

or, in coordinates $(s,t)$ on $\mathbb{R}^{2}$:

$\partial_{s}v+I_{X}(v)(\partial_{t}v-\nabla\mathrm{Re}(F)(v))=0.$ (1.4)

This equation is also known as the _Witten equation_ or _$\zeta$ -instanton
equation_ in the physics literature. Of course, it is a perturbation of the
$I_{X}$-holomorphic map equation by the Hamiltonian function $\mathrm{Im}(F)$,
as introduced by Floer. Solutions with $\partial_{s}v=0$ are simply the real
gradient trajectories of $\mathrm{Re}(F)$.

On the Lagrangian side, we consider, as before, the cotangent bundle
$M=T^{*}X$. The complex structure on $X$ induces one on $M$, and $M$ has a
canonical holomorphic symplectic form $\Omega$. Let $L_{0},L_{1}$ be the zero
section and the graph of $\mathrm{Re}({\mathrm{d}}F)$. They are complex
submanifolds of $M$ which are Lagrangian with respect to $\Omega$. Pseudo-
holomorphic strips (1.1) are replaced by _Fueter strips_ in $M$. By
definition, a Fueter strip is a map $U\colon[0,1]\times\mathbb{R}^{2}\to M$
satisfying the Fueter equation (1.2), with boundary on $L_{0}$ and $L_{1}$,
asymptotic to $L_{0}\cap L_{1}$ as $t\to\pm\infty$ and to pseudo-holomorphic
strips as $s\to\pm\infty$. These asymptotic boundary conditions are explained
in detail in Section 3.

######  Theorem .

Let $X$ be a Kähler manifold equipped and let $F\colon X\to\mathbb{C}$ be a
holomorphic function. If the $C^{2}$ norm of $F$ is sufficiently small, then
there is a $\tau$-dependent almost quaternionic structure
$(I_{\tau},J_{\tau},K_{\tau})$ on $M=T^{*}X$ such that Fueter strips

$\displaystyle U\colon[0,1]\times\mathbb{R}^{2}\to M,$ $\displaystyle
I_{\tau}(U)\partial_{\tau}U+J_{\tau}(U)\partial_{s}U+K_{\tau}(U)\partial_{t}U=0,$

with boundary on $L_{0}$ and $L_{1}$, the zero section and the graph of
$\mathrm{Re}({\mathrm{d}}F)$, correspond to the complex gradient trajectories
of $F$ in $X$.

When restricted to the asymptotic boundary $s\to\pm\infty$, this result
recovers Floer’s correspondence between the real gradient trajectories of
$\mathrm{Re}(F)$ and pseudo-holomorphic strips. We prove a similar theorem for
the cotangent bundles of almost Kähler manifolds, i.e. symplectic manifolds
equipped with a compatible almost complex structure and induced Riemannian
metric.

######  Theorem .

Let $X$ be an almost Kähler manifold and $F\colon X\to\mathbb{C}$ be pseudo-
holomorphic function. If the $C^{2}$ norm of $F$ is sufficiently small, then
there is an almost quaternionic triple on $M=T^{*}X$ and a perturbation of the
Fueter equation, see (5.6), whose all solutions with boundary on $L_{0}$ and
$L_{1}$ and asymptotic boundary values described above correspond to the
complex gradient trajectories of $F$ in $X$.

######  Remark .

The perturbation of the Fueter equation considered in Section 1 is by first
order terms which are proportional to the Nijenhuis tensor of $X$ and the
distance from the zero section. In particular, the equation is still elliptic
in a neigbhorhood of the zero section. For more details, see Section 5, where,
in addition, we discuss what is known for the unperturbed equation and give an
interpretation of Section 1 as an infinite-dimensional version of Floer’s
result.

Section 1 is a complexification of Floer’s result in the sense that a real
Morse function and its gradient trajectories have been replaced by their
complex analogs. At the same time, our theorem can be seen as the first steps
toward categorifying Floer’s isomorphism (1.3). This is best explained by the
following three general predictions:

  1. 1.

For a holomorphic Morse function $F\colon X\to\mathbb{C}$ on an Kähler
manifold (or, more generally, a symplectic Lefschetz fibration over
$\mathbb{C}$) there should be an associated $A_{\infty}$-category
$\mathrm{FS}(X,F)$ constructed using the complex gradient trajectories of
$F$.222We omit here various technical conditions that $X$ and $F$ should
satisfy, such as exactness of the symplectic form and some control on the
geometry of $X$ and the behavior of the holomorphic function at infinity. For
details, see [Hay15, Wan22], although it is likely that $\mathrm{FS}(X,F)$ can
be defined in greater generality. This category is conjecturally quasi-
isomorphic to the Fukaya–Seidel category of $F$ defined in [Sei08] by counting
pseudo-holomorphic polygons with boundaries on the Lefschetz thimbles of $F$.

Summarizing analogies with the real case:

Morse theory | Complex Morse theory
---|---
smooth manifold $X$ | Kähler manifold $X$
Morse function $f\colon X\to\mathbb{R}$ | holomorphic Morse function $F\colon X\to\mathbb{C}$
gradient trajectories of $f$ | complex gradient trajectories of $F$
Morse homology $HM_{*}(X,f)$ | Fukaya–Seidel category $\mathrm{FS}(X,F)$

  2. 2.

Let $(M,I,J,K)$ be a hyperkähler manifold (or, more generally, an almost
quaternionic manifold with three symplectic forms). For every pair of complex
Lagrangians $L_{0},L_{1}\subset M$, i.e. submanifolds which are holomorphic
with respect to $I$ and Lagrangian with respect to the symplectic forms
corresponding to $J$ and $K$, there should be an associated
$A_{\infty}$-category $\mathrm{Fuet}(L_{0},L_{1})$ constructed using Fueter
strips in $M$ with boundary on $L_{0}$ and $L_{1}$.

Comparing again to the real case:

Floer theory | Complex Floer theory
---|---
symplectic manifold $M$ | hyperkähler manifold $M$
Lagrangians $L_{0},L_{1}$ | complex Lagrangians $L_{0},L_{1}$
pseudo-holomorphic strips in $M$ | Fueter strips in $M$
Floer homology $HF_{*}(L_{0},L_{1})$ | Fueter category $\mathrm{Fuet}(L_{0},L_{1})$

  3. 3.

We expect these two categories to be related by the following conjecture,
which categorifies (1.3). We discuss it in detail in Section 2.8.

######  Conjecture .

Let $F\colon X\to\mathbb{C}$ be a holomorphic Morse function on a Kähler
manifold. If $M=T^{*}X$ and $L_{0}$, $L_{1}$ are the zero section and graph of
$\mathrm{Re}({\mathrm{d}}F)$, then there is an equivalence of
$A_{\infty}$-categories

$\mathrm{Fuet}(L_{0},L_{1})=\mathrm{FS}(X,F).$

The first prediction was made independently by Gaiotto–Moore–Witten [GMW15,
GMW17] and Haydys [Hay15], with some of the ideas going back to the seminal
work of Donaldson and Thomas on higher-dimensional gauge theory [DT98]. See
also [KKS16] for a discussion of the algebraic structures arising from the
Gaiotto–Moore–Witten proposal. The putative $A_{\infty}$ category
$\mathrm{FS}(X,F)$ associated with a holomorphic function $F\colon
X\to\mathbb{C}$ has:

  * •

Objects given by the critical points of $F$.

  * •

Morphisms given by graded vector spaces generated by gradient flowlines
between the critical points of $F$. That is, for two critical points $p_{-}$
and $p_{+}$, let $\theta$ be the slope of the line connecting the
corresponding critical values. $\mathrm{Hom}(p_{-},p_{+})$ is the vector space
whose basis are the maps $\gamma\colon\mathbb{R}\to X$ satisfying

$\frac{{\mathrm{d}}\gamma}{{\mathrm{d}}t}=\nabla\mathrm{Re}(e^{-i\theta}F),\quad\lim_{t\to\pm\infty}\gamma(t)=p_{\pm}.$

  * •

The differential operation given by complex gradient trajectories (1.4) of
$e^{-i\theta}F$ asymptotic to the gradient flowlines of
$\mathrm{Re}(e^{-i\theta}F)$ as $s\to\pm\infty$ and to $p_{\pm}$ as
$t\to\pm\infty$. The higher $A_{\infty}$-operations are also defined by
counting complex gradient trajectories but with more complicated asymptotic
boundary conditions.

If the critical values of $F$ are in a convex position, this construction is
expected to recover the Fukaya–Seidel category of $F$. In general, one must
either perform a complex algebraic operation on the counts of the complex
gradient trajectories to define the $A_{\infty}$-operations Gaiotto2015,
Kapranov2016, or modify the construction to use curved gradient trajectories
[Hay15]. A rigorous construction of $\mathrm{FS}(X,F)$ involves solving a
number of interesting analytical problems concerning the moduli spaces of
complex gradient trajectories. Significant progress in this direction
following [Hay15] was made in [Wan22].

The second prediction—that we can associate an $A_{\infty}$-category
$\mathrm{Fuet}(L_{0},L_{1})$ with a pair of $I$-complex Lagrangians
$L_{0},L_{1}$ inside a hyperkähler manifold $(M,I,J,K)$— is an infinite-
dimensional version of the first one. To our knowledge, such a proposal has
not yet been described in the literature, although it is closely related to
the ongoing project of Kontsevich and Soibelman [KS20], as well as of
Bullimoore–Gaiotto–Dimofte–Hilburn [Bul+16, Appendix A], the upcoming work of
Bousseau [Bou] regarding relations to Donaldson–Thomas theory, and the
upcoming work of Khan [Kha] from a physical perspective.

Given $M$ and $L_{0},L_{1}$ as above, let $(\omega_{I},\omega_{J},\omega_{K})$
be the triple of Kähler forms corresponding to $(I,J,K)$. The starting point
of holomorphic Floer theory is the observation that the space of paths

$\mathcal{P}=\\{\gamma\colon[0,1]\to M\ |\ \gamma(0)\in L_{0},\ \gamma(1)\in
L_{1}\\}$

is an infinite-dimensional Kähler manifold with respect to the complex
structure and metric induced by $I$ and $\omega_{I}$. Moreover, the
_holomorphic symplectic action functional_

$\mathcal{A}_{\mathbb{C}}=\mathcal{A}_{J}+i\mathcal{A}_{K}\colon\mathcal{P}\to\mathbb{C}$
(1.5)

combining the symplectic action functionals with respect to $\omega_{J}$ and
$\omega_{K}$, is a holomorphic function on $\mathcal{P}$. (We ignore here that
$\mathcal{A}_{J}$ and $\mathcal{A}_{K}$ are actually multi-valued functions.
For example, we can assume that $\omega_{J},\omega_{K}$ are exact and
$L_{0},L_{1}$ are exact Lagrangians.) Therefore, we can formally apply the
construction the Fukaya–Seidel category, outlined above, to
$\mathcal{A}_{\mathbb{C}}\colon\mathcal{P}\to\mathbb{C}$. This is discussed in
Section 2. The result should be an $A_{\infty}$-category
$\mathrm{Fuet}(L_{0},L_{1})$ with

  * •

Objects given by the critical points of $\mathcal{A}_{\mathbb{C}}$, which
correspond to the intersection points of $L_{0}$ and $L_{1}$, as in Lagrangian
Floer theory.

  * •

Morphism spaces between two intersection points $p_{-}$ and $p_{+}$ generated
by the gradient trajectories of
$\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}})$ where $\theta$ is the
slope of the line connecting the corresponding critical values of
$\mathcal{A}_{\mathbb{C}}$. As in Lagrangian Floer theory, these trajectories
can be interpreted as $J_{\theta+\pi}$-holomorphic strips
$u\colon\mathbb{R}\times[0,1]\to M$ with boundary on $L_{0},L_{1}$, where

$J_{\theta}=(\cos\theta)J+(\sin\theta)K.$ (1.6)

  * •

The differential and $A_{\infty}$-operations defined by counting complex
gradient flowlines of $e^{-i\theta}\mathcal{A}_{\mathbb{C}}$. As we show in
Section 2, these correspond to the Fueter strips
$U\colon[0,1]\times\mathbb{R}^{2}\to M$ with respect to the rotated
quaternionic triple $(I,J_{\theta},K_{\theta})$, where
$K_{\theta}=IJ_{\theta}$, with boundary on $L_{0},L_{1}$, and asymptotic to
$J_{\theta}$-holomorphic strips as $s\to\pm\infty$ and to the intersection
points $p_{\pm}$ as $t\to\pm\infty$.

This brings us to the third prediction. If $F\colon X\to\mathbb{C}$ is a
holomorphic Morse function on a Kähler manifold, then $M=T^{*}X$ is a
holomorphic symplectic manifold, and the zero section and the graph of
${\mathrm{d}}(\mathrm{Re}F)$ are complex Lagrangians. While $M$ does not have
to carry a complete hyperkähler metric, we expect that
$\mathrm{Fuet}(L_{0},L_{1})$ is still defined by counting solutions to the
Fueter equation with respect to the time-dependent almost quaternionic triple
from Section 1. If this is the case, item 3 would be a direct consequence of
the correspondence between Fueter strips in $M$ and complex gradient flowlines
in $X$ established by Section 1.

In general, defining $\mathrm{Fuet}(L_{0},L_{1})$ is a challenging task, which
involves a number of interesting analytical questions. Some of these
foundational questions, such as transversality and compactness for the Fueter
equation, are considered in [HNS09, Wal17b, Esf]. Compactness theory, in
particular, requires further study, as it is currently unclear what role, if
any, is played by singular solutions to the Fueter equation. This is parallel
to the recent developments in gauge theory and calibrated geometry, which
suggest that such singular solutions are an unavoidable part of the theory
[Tau14, Tak18, DW21, Don21, Hay22]. In this paper we do not tackle these
foundational difficulties. We do, however, establish some basic analytic
results on Fueter strips, mimicking well-known theorems in Lagrangian Floer
theory.

######  Remark .

The proof of Section 1 excludes the possibility of the local sphere bubbling
or the formation of essential singularities described in [Wal17b], assuming
that $X$ is exact and $F$ is such that its complex gradient trajectories obey
an a priori $C^{0}$ bound. This is interesting as the corresponding metrics on
$T^{*}X$ may have complicated curvature, and thus existing methods for
excluding bubbling of Fueter maps do not apply to this setting. For more
details, see Section 5.4.

The first such result is an a priori energy bound for Fueter strips. Let
$(M,I,J,K)$ be an almost quaternionic manifold and $L_{0},L_{1}\subset M$ be
two submanifolds. The main examples are, of course, hyperkähler manifolds and
complex Lagrangians, but the next two theorems work in a greater generality.
Let $U\colon[0,1]\times\mathbb{R}^{2}\to M$ be a map with boundary on
$L_{0},L_{1}$. If $U$ is asymptotic to $J$-holomorphic strips as $s\to\infty$
in the sense explained earlier, then the integral of $|{\mathrm{d}}U|^{2}$
cannot be finite. Instead, by analogy with pseudo-holomorphic strips, we
introduce the following notion of energy:

$E(U)=\int_{[0,1]\times\mathbb{R}^{2}}|\partial_{s}U|^{2}+|\partial_{t}U-J(U)\partial_{\tau}U|^{2}.$
(1.7)

To state the energy identity, we need two more notions. First, let
$[U|_{\mathbb{R}^{2}}]$ be a homology class in $M$ relative to the limiting
$J$-holomorphic strips, represented by the restriction of $U$ to any slice
$\\{\tau\\}\times\mathbb{R}^{2}$; this is well-defined and independent of
$\tau$. Second, let us say that a complex two-form $\eta$ on $M$ is _$KI$
-taming_ if it is of type $(2,0)$ with respect to $J$ and $\mathrm{Re}(\eta)$
tames $K$ and $\mathrm{Im}(\eta)$ tames $I$. In particular, if $(M,I,J,K)$ is
a hyperkähler manifold and $(\omega_{I},\omega_{J},\omega_{K})$ is the triple
of Kähler forms corresponding to $(I,J,K)$, then $\eta=\omega_{K}+i\omega_{I}$
is a $KI$-taming two-form.

######  Theorem  (Energy identity).

Let $(M,I,J,K)$ be an almost quaternionic manifold. Suppose that there is a
closed $KI$-taming two-form $\eta$ on $M$ and $L_{0},L_{1}\subset M$ are
Lagrangian with respect to $\mathrm{Re}(\eta)$. There is a constant $C>0$ such
that for every Fueter strip $U\colon[0,1]\times\mathbb{R}^{2}\to M$ with
boundary on $L_{0},L_{1}$,

$E(U)\leq-C\langle[\mathrm{Im}(\eta)],[U|_{\mathbb{R}^{2}}]\rangle,$

where the right-hand side denotes the pairing between the second cohomology
and homology of $M$ relative to the limiting $J$-holomorphic strips.

The second result is a maximum principle for Fueter strips. In Section 4 we
introduce a notion of an almost quaternionic manifold with an $IJK$-conical
end. This is a quaternionic analog of pseudo-convexity known from complex
geometry. We, moreover, define what it means for submanifolds to be conical
with respect to two of the three almost complex structures. As an example, we
show that the cotangent bundle of a Stein manifold can be made into an almost
quaternionic manifolds with an $IJK$-conical end, with the zero section being
a $JK$-conical and $I$-holomorphic Lagrangian. With these notions, we prove
the following.

######  Theorem  (Maximum principle).

Let $(M,I,J,K)$ be an almost quaternionic manifold with an $IJK$-conical end
and let $L_{0},L_{1}\subset M$ be submanifolds which are either compact or
$JK$-conical with compact intersection $L_{0}\cap L_{1}$. There exists a
compact subset $M_{0}\subset M$ with the following property. If
$U\colon[0,1]\times\mathbb{R}^{2}\to M$ is a Fueter strip with boundary on
$L_{0},L_{1}$, then the image of $U$ is contained in $M_{0}$.

In symplectic geometry, pseudo-convex functions are used repeatedly to confine
pseudo-holomorphic curves. In particular, due to the the Weinstein tubular
neighborhood theorem, Floer’s result on pseudo-holomorphic strips in cotangent
bundles gives a description of local pseudo-holomorphic strips between two
Lagrangians in any symplectic manifold in terms of Morse-theoretic, or
topological data, and thus gives the first approximation to any invariant in
Lagrangian Floer theory. We expect that the convexity theory discussed in this
paper can be used similarly to compute the local Fueter trajectories between
complex Lagrangians in a general hyperkähler manifold.

This completes the summary of the results of this paper. These analytical
results are motivated by a rich set of expectations regarding the algebraic
and categorical structures that should arise from the study of the Fueter
equation. Indeed, Floer’s classical theorem mentioned earlier is the basic
input used to compute the Fukaya category of the cotangent bundle. The Fukaya
category is an $A_{\infty}$-category associated with a symplectic manifold.
From a physical perspective, it is related to the $A$-twist of the
$2$-dimensional $(2,2)$ superconformal $\sigma$-model with target $M$. Floer’s
isomorphism (1.3) is the simplest computation of the morphism group between
Lagrangians $L_{0}$, $L_{1}$, thought of as objects in the Fukaya category of
$M=T^{*}X$.

In the complex setting, all of this structure should be categorified. Namely,
we expect that that if $M$ is a hyperkähler manifold (or some weak version of
it, such as a holomorphic symplectic manifold or an almost quaternionic
manifold equipped with three symplectic forms), the $A_{\infty}$-categories
$\mathrm{Fuet}(L_{0},L_{1})$ for all pairs of complex Lagrangians assemble
into an $(A_{\infty},2)$-category $\mathrm{Fuet}_{M}$, in the sense of [BC18].
Its objects are given by complex Lagrangians, morphisms by intersection
points, and $2$-morphisms by pseudo-holomorphic strips between the
Lagrangians. This $2$-category should categorify the subcategory of the usual
Fukaya category of $(M,\omega_{I})$ whose objects are complex Lagrangians.
From a physical point of view, this $2$-category is the $2$-category of branes
in $M$ for the $A$-twist of the $3d$ $N=4$ $\sigma$-model. By analogy with the
real case, we expect that Section 1 will allow us to compute the _morphism
category_ of the Fueter $2$-category associated with the cotangent bundle
$M=T^{*}X$.

While a rigorous construction of $\mathrm{Fuet}_{M}$ involves many nontrivial
difficulties, many formal aspects are clear. Under the philosophy of $3d$
miror symmetry, the $2$-category $\mathrm{Fuet}_{M}$ is conjctured to be
equivalent to the Kapustin–Rozansky–Saulinas $2$-category of the mirror
hyperkähler manifold $M^{\vee}$. The motivation for this conjecture comes from
the physics literature, where it is postulated that $3d$ $N=4$ theories come
in pairs, and the $A$-twist of one model is equivalent with the $B$-twist of
the mirror model. Some initial work to model the Fueter $2$-category via
perverse sheaves of categories has been pursued in [GHM22]. In the case of
$M=T^{*}(\mathbb{C}^{*})^{n}$, the results presented in this paper seem to be
connected to an enhancement of homological mirror symmetry for toric
varieties.

We provide much additional context for the conjectural categorical structure
of the $3d$ topological field theory, as well for some of its connections to
other topics in mathematical physics, in Section 2. While Section 2 does not
contain any proofs, it is the core of the paper in the sense that it motivates
the results proved later. In Section 3 we proceed with the analytical study of
the Fueter equation; in particular, this section contains the proof of Section
1. In Section 4 we develop a quaternionic analog of pseudo-convexity theory
and prove Section 1. Finally, in Section 5 we apply the general theory to the
case of cotangent bundles, proving Section 1 and Section 1.

We hope that the differential-geometric study of the $A$-side of $3d$ mirror
symmetry gives rise to an active program of research. Throughout the paper, we
suggest problems and research directions for analyts and geometers that may be
interested in the subject.

#### Acknowledgements.

We thank Justin Hilburn and Ben Gammage for introducing us to the physical
aspects of $3d$ mirror symmetry, reviewed in Section 2.10, and for discussions
regarding algebraic aspects in Section 2.6 and Section 2.10; and Ahsan Khan
for pointing out the potential importance of twisted complexes in the
composition in the Fueter $2$-category. We are also grateful to Mohammed
Abouzaid, Denis Auroux, Andriy Haydys, Yan Soibelman, and Donghao Wang for
discussions which contributed to this paper. The first author is supported by
the National Science Foundation, Simons Society of Fellows, and Trinity
College, Cambridge. The second author is supported by the National Science
Foundation.

##  2 Formal and categorical aspects

As discussed in the introduction, we expect that the topological field theory
based on the Fueter equation assigns a $2$-category to a hyperkähler manifold
$X$. This section is a long, informal review of the underlying ideas, which
bridge mathematical disciplines. We hope that it will be useful to
mathematicians of different backgrounds who are interested in the differential
geometry and algebra of $3d$ mirror symmetry. Throughout this section, we
outline problems which we see as essential to the rigorous development of the
$3d$ $A$-model.

###  2.1 Topological field theories

This section briefly summarizes the basic philosophy of topological quantum
field theories (TQFT) and its connection to Floer homology. This material is
completely standard, but may be of interest to those who are not completely
familiar with the connection between partial differential equations and
algebraic structures.

#### Algebraic structure.

According to Atiyah [Ati88], a TQFT is a functor from a cobordism category of
manifolds to a monoidal category, e.g. the category of graded vector spaces
over a field $\mathbf{k}$. Concretely, the simplest notion of a
$2$-dimensional TQFT $F:\mathrm{Cob}_{2}\to\mathrm{Vect}_{\mathbf{k}}$ assigns
in a functorial way

  * •

a vector space $V=F(S^{1})$ to the circle $S^{1}$;

  * •

the vector space $V^{\otimes n}=F(\bigsqcup_{i=1}^{n}S^{1})$ to $n$ copies of
the circle $\bigsqcup_{i=1}^{n}S^{1}$;

  * •

a linear map $V^{\otimes n}\to V^{\otimes m}$ to any cobordism $\Sigma$ from
$\bigsqcup_{i=1}^{n}S^{1}$ to $\bigsqcup_{i=1}^{m}S^{1}$.

This gives us the following structure:

  * •

A commutative associative product $V\otimes V\to V$ and cocommutative
coassociative coproduct $V\to V\otimes V$ correspoding to the pair of pants
cobordism and its inverse, together with a unit $\mathbf{k}\to V$ for the
product and a counit $V\to\mathbf{k}$ for the coproduct.

  * •

By composing the cocounit with the product one gets a pairing
$\langle\cdot,\cdot\rangle:V\otimes V\to\mathbf{k}$, and dually, by composing
the unit and the product, a copairing $\Delta:\mathbf{k}\to V\otimes V$.
Decomposing $S^{1}\times I$ (corresponding to the identity $V\to V$) into the
composition of a pairing and a copairing implies that

$V\simeq V\otimes\mathbf{k}\xrightarrow{1\otimes\Delta}V\otimes V\otimes
V\xrightarrow{\langle\cdot,\cdot\rangle\otimes 1}\mathbf{k}\otimes V\simeq V$

is the identity map, and thus that the pairing is perfect. Writing the product
of $a,b\in V$ as $ab$, one verifies that $\langle ab,c\rangle=\langle
a,bc\rangle$.

  * •

The data of a perfect pairing together with a commutative associative unital
product satisfying the identity above is a _Frobenius algebra_ , and every
$2d$ TQFT determines and is determined by a Frobenius algebra [Koc03].

In summary, a TQFT is a functor from a cobordism category, which can be
specified by collection of algebraic operations satisfying certain identities.

#### Floer cohomology.

Certain examples of TQFTs are naturally associated with physical theories, and
indeed that was Atiyah’s original motivation for introducing the axioms pf
TQFT. Given $M$ in the cobordism category, $F(M)$ is meant to be the space of
quantum states of the field theory on $M$, while the map $F(M_{1})\to
F(M_{2})$ associated to a cobordism $N$ from $M_{1}$ to $M_{2}$ is meant to be
the Hamiltonian evolution of the quantum states through time. The cobordism
map is then typically defined by evaluating a path-integral over some
infinite-dimensional space of fields on $N$ [Fre12]. While the integration
over the infinte-dimensional space of fields is hard to make rigorous, in
certain cases, due to supersymmetry [SZ97] the path integral can be computed,
or rigorously defined, by counting solutions to some partial differential
equations. Many invariants of Floer-theoretic origin are of this form.

We now explain the general pattern by the example of Hamiltonian Floer
cohomology of a symplectic manifold. A Hamiltonian function $H:M\times
S^{1}\to\mathbb{R}$ on a symplectic manifold $M$ defines an action functional
$\mathcal{A}_{H}:\mathcal{L}M\to\mathbb{R}$ on the free loop space.
Hamiltonian Floer cohomology $HF^{*}(H)$, which is formally the Morse homology
of $\mathcal{A}_{H}$, computes the space of states of $2$-dimensional TQFT on
$M$ [Wit82, Flo89, Wit88]. Defining $HF^{*}(H)$ involves the study of the
gradient flow equation of $\mathcal{A}_{H}$, the Floer equation:

$\displaystyle u\colon\mathbb{R}\times S^{1}\to M,$
$\displaystyle\partial_{t}u+J(u)\partial_{\tau}u=\nabla H(u),$

with $(t,\tau)$ denoting the coordinates on $\mathbb{R}\times S^{1}$ and $J$
is an almost complex structure on $M$ compatible with the symplectic form.
This equation admits a natural generalization to surfaces, the inhomogeneous
pseudo-holomorpic map equation

$\begin{gathered}u:\Sigma\to M,\\\
({\mathrm{d}}u-\alpha(u))_{J}^{0,1}=0,\end{gathered}$

where $\alpha\in\Omega^{1}(\Sigma\times M,TM)$ is an inhomogenous term
generalizing $\nabla H$ above. Here $\Sigma=\hat{\Sigma}\setminus S$ is the
complement of a finite collection of points $S$ in a closed Riemann surface
$\hat{\Sigma}$. Counting solutions to (2.1) with asymptotic boundary
conditions at each point of $S$ defines TQFT operations corresponding to a
cobordism $\hat{\Sigma}\setminus U$, where $U$ is a small open neighborhood of
$S$ [Sei08].

Note that in the formulation of the inhomogeneous pseudo-holomorphic map
equation, one must fix boundary conditions _at infinity_ for $\Sigma$, even
though the corresponding cobordism is a compact manifold with boundary. In
general, there can be different kinds of boundary conditions for partial
differential equations, and these correspond to variations on the definition
of a TQFT. It may be that the surface $\Sigma$ needs to be decorated with
certain structures in order to make sense of the equations. Moreover, in order
to ensure compactness of the set of solutions to the equations, one may need
to restrict or modify the associated cobordism category further. As such, the
formal algebraic properties of the TQFT and the geometric and analytical
structure of the associated partial differential equation are closely
intertwined in a way that is challenging to fully axiomatize.

#### Category theory.

It is an old idea, formalized by the Baez–Dolan cobordism hypothesis [BD95],
that higher-dimensional TQFT are associated to structures in higher category
theory. A cobordism is simply a manifold with boundary, interpreted as a
morphism by dividing the boundary into two pieces. As such, one wishes to
interpret a cobordism-between-cobordisms, i.e. a manifold with corners modeled
on $\mathbb{R}_{>0}^{2}\times\mathbb{R}^{n-2}$ with a certain decomposition of
its boundary, as a morphism-between-morphisms in the cobordism _2-category_ ;
and more generally to define notions of cobordisms-between-cobordisms-between-
cobordisms in terms of general manifolds with corners, viewed as higher
morphisms in a cobordism $n$-category. An extended TQFT is then suppposed to
be a functor from a higher category of cobordisms to another higher category,
prototypically the $n$-category of $\mathbf{k}$-linear $(n-1)$-categories.

Figure 1: A two-dimensional cobordism between a pair of $1$-dimensional
cobordisms (in red). From the perspective of category theory this can be
interpreted as a morphism between morphisms.

While this general notion can be made precise [Lur09], when building a TQFT
out of solutions to partial differential equations it pays to be explicit.
Moreover, the analysis of the given equations may force us to modify the TQFT
formalism in minor ways. For example, while many aspects of the Fukaya
category can be formulated in a general extended TQFT framework, typically
symplectic topologists work in a formalism that is better described by the
language of an _open-closed $2d$ TQFT_ [MS06, And09]. In this formalism:

  * •

We have a category $\mathcal{F}$.

  * •

One assigns, to an interval with endpoints labeled by a pair of objects
$(L_{0},L_{1})$ of $\mathcal{F}$, the vector space given by the morphisms
$\mathcal{F}(L_{0},L_{1})$. There is also an assignment of a vector space
$\mathcal{F}(S^{1})$ to the circle.

  * •

To every surface with corners, with certain codimension one boundary
components labeled by objects, and the remaining codimension one boundary
components partitioned into incoming and outgoing pieces, one defines a
cobordism map. These maps compose functorially as we compose the corresponding
bordisms.

Figure 2: The open-closed field theory corresponding to the Fukaya category.
Composition in the Fukaya category arises from the cobordism in the bottom
left, and the open-closed map, discussed later in Section 2.3, from the
cobordism in the bottom right.

To a symplectic topologist, $\mathcal{F}$ is the Fukaya category of a
symplectic manifold $M$, for Lagrangian submanifolds $L_{0},L_{1}$ the vector
space $\mathcal{F}(L_{0},L_{1})$ is the Lagrangian Floer cohomology of
$L_{0},L_{1}$, and $\mathcal{F}(S^{1})$ is the Hamiltonian Floer cohomology of
$M$. One defines the cobordism map by counting the number of solutions to
(2.1) over all possible conformal structures on the domain of the cobordism.

In the next section we will give an outline of a similar formalism which
should encompass the basic algebraic structure associated to the Fueter TQFT.
The most complex object is the object assigned to a square with certain
labels, which will be the _Fueter $2$-category_. There are various notions of
$2$-categories with various levels of strictness in their composition; we
recall the easiest and strictest notion now. A strict $2$-category
$\mathcal{C}$ is a category enriched in the category of categories.
Concretely, this is

  * •

A set of objects $\mathrm{Ob}(\mathcal{C})$; we will write $L\in\mathcal{C}$
for $L\in\mathrm{Ob}(\mathcal{C})$.

  * •

For every pair of objects $(L_{0},L_{1})$, a _hom-category_
$\mathcal{C}(L_{0},L_{1})$. The objects the categories
$\mathcal{C}(L_{0},L_{1})$ are the $1$-morphisms in $\mathcal{C}$.

  * •

The morphisms between a pair of $1$-morphisms $p,q\in\mathcal{C}(L_{0},L_{1})$
are to be thought of as $2$-morphisms of $\mathcal{C}$. Thus, every
$2$-morphism has $1$-morphism as its source and its target, and given a triple
of morphisms $p,q,r\in\mathcal{C}(L_{0},L_{1})$ and a pair of $2$-morphisms
$u_{1}:p\to q$, $u_{2}:q\to r$, we can _vertically compose_ $(u_{1},u_{2})$ to
produce $u_{2}u_{1}:p\to r$.

  * •

We have composition bifunctors
$\mathcal{C}(L_{0},L_{1})\times\mathcal{C}(L_{1},L_{2})\to\mathcal{C}(L_{0},L_{2})$.
Thus, $1$-morphisms can be composed along objects, and a pair of $2$-morphisms
$u_{1}:p\to q$, $u_{2}:s\to t$ for $p,q\in\mathcal{C}(L_{0},L_{1})$ and
$s,t\in\mathcal{C}(L_{1},L_{2})$ _horizontally compose_ to a $2$-morphism
$u_{2}u_{1}:sp\to tq$ in $\mathcal{C}(L_{0},L_{2})$.

This data is graphically summarized in Figure 3. Note that for any
$L\in\mathcal{C}$, the category $\mathcal{C}(L,L)$ is necessarily _monoidal_ ,
with the monoidal structure given by composition of $1$-morphisms.

Figure 3: The data of a $2$-category. Top left: crosses represent objects,
black represent $1$-morphisms, red arrow represents a $2$-morphism. Top right:
Composition of $1$-morphisms. Bottom left: vertical composition of
$2$-morphisms. Bottom right: Horizontal composition of $2$-morphisms.

###  2.2 The Fueter $2$-category

We can now explain the formal structure of the Fueter TQFT. Let $M$ be a
hyperkähler manifold with complex structures $I,J,K$. We think of $M$ as a
complex manifold with distinguished complex structure $I$. The remaining part
of the hyperkähler structure is encoded in the Riemannian metric and
holomorphic symplectic form

$\Omega=\omega_{J}+i\omega_{K}$

where $\omega_{J},\omega_{K}$ are the Kähler forms corresponding to $J$, $K$.
Fix also a background angle $\hat{\theta}$; the final structure will not
depend on that choice up to isomorphism. To a very initial approximation, we
expect to define a $2$-category $\mathrm{Fuet}_{M}$ with the following
features.

  * •

The objects of $\mathrm{Fuet}_{M}$ are complex Lagrangians $L\subset M$, i.e.
submanifolds which are $I$-holomorphic and $\Omega$-Lagrangian.

  * •

A _hom-category_ $\mathrm{Fuet}_{M}(L_{0},L_{1})$ is, formally, the
Fukaya–Seidel category of the _holomorphic symplectic action functional_
$\mathcal{A}_{\mathbb{C}}$ on the space of paths
$\mathcal{P}=\mathcal{P}(M;L_{0},L_{1})$ from $L_{0}$ to $L_{1}$, given by
(1.5). Strictly speaking, what is well-defined is the differential of
$\mathcal{A}_{\mathbb{C}}$, which is a closed holomorphic $(1,0)$-form on
$\mathcal{P}$ given by

${\mathrm{d}}\mathcal{A}_{\mathbb{C}}(\gamma)v=\int_{0}^{1}\Omega\left(v(\tau),\frac{{\mathrm{d}}\gamma}{{\mathrm{d}}\tau}\right){\mathrm{d}}\tau$
(2.1)

for $\gamma\in\mathcal{P}$ and $v\in
T_{\gamma}\mathcal{P}=\Gamma([0,1],\gamma^{*}TM)$, with $\tau$ denoting the
coordinate on $[0,1]$. As usual [HS95], this one-form may or may not be the
differential of globally defined holomorphic function. We can pass to the
covering space $\widetilde{\mathcal{P}}\to\mathcal{P}$ with deck
transformation group given by

$\frac{\pi_{2}(M,L_{0}\cup
L_{1})}{\ker(\alpha\mapsto\langle[\Omega],\alpha\rangle)}$

so tat the pullback of ${\mathrm{d}}\mathcal{A}_{\mathbb{C}}$ to
$\widetilde{\mathcal{P}}$ is the differential of a function
$\mathcal{A}_{\mathbb{C}}$. The simplest setting is the _exact_ setting, in
which the deck transformation group is trivial. In this setting, we require
that there is a holomorphic one-form $\Lambda$ such that
$\Omega={\mathrm{d}}\Lambda$, and $\Lambda|_{L_{i}}={\mathrm{d}}H_{i}$ for
some holomorphic functions $H_{i}:L_{i}\to\mathbb{C}$. In that case,

$\mathcal{A}_{\mathbb{C}}(\gamma)=\int_{\gamma}\Lambda-
H_{1}(\gamma(1))+H_{0}(\gamma(0)).$ (2.2)

The usual definition of the Fukaya–Seidel category using Lefschetz thimbles is
not available in this infinite-dimensional setting. Instead, here
$\mathrm{FS}(\mathcal{A}_{\mathbb{C}})$ is meant to be defined via an
infinite-dimensional version of the complex Morse theory model of the Fukaya-
Seidel category, which we review in the next section.

  * •

In particular, the objects of $\mathrm{Fuet}_{M}(L_{0},L_{1})$, i.e. the
_1-morphisms_ of $\mathrm{Fuet}_{M}$, are the critical points of
$\mathcal{A}_{\mathbb{C}}$, which are in bijection with the constant paths
mapping to the intersection $L_{0}\cap L_{1}$. When $L_{0}$ and $L_{1}$
intersect transversely, these critical points are nondegenerate, which is a
requirement to make sense of the Fukaya–Seidel category. Heuristically, these
objects correspond to infinite-dimensional Lefschetz thimbles of the
holomorphic Morse function $\mathcal{A}_{\mathbb{C}}$.

  * •

The morphism cochain complex $\mathrm{Fuet}_{M}(L_{0},L_{1})(p_{-},p_{+})$,
for $p_{-},q_{+}\in L_{0}\cap L_{1}$ is generated by the gradient trajectories
of $\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}})$, where

$\theta=\arg(\mathcal{A}_{\mathbb{C}}(p_{-})-\mathcal{A}_{\mathbb{C}}(p_{+}))$

As in the standard Lagrangian Floer theory, they correspond to
$J_{\theta+\pi}$-holomorphic strips, or $J_{\theta}$-antiholomorphic strips,
with $J_{\theta}$ denoting the rotated complex structure (1.6). Explicitly,
the cochain complex $\mathrm{Fuet}_{M}(L_{0},L_{1})(p_{-},p_{+})$ is freely
generated over $\mathbb{Z}$ by the solutions to

$\displaystyle u:\mathbb{R}\times[0,1]\to M,$
$\displaystyle\partial_{t}u+J_{\theta+\pi}(u)\partial_{\tau}u=0,$ (2.3)

with boundary condition $u(t,i)\in L_{i}$ for $i=0,1$ and asymptotic boundary
conditions $\lim_{t\to\pm\infty}u(t,\tau)=p_{\pm}$. As usual, we only consider
solutions with finite Floer energy, which implies exponential converges with
derivatives a $t\to\pm\infty$.

  * •

The differential on $\mathrm{Fuet}_{M}(L_{0},L_{1})(p_{-},p_{+})$ counts
solutions to the complex gradient flow equation for
$\mathcal{A}_{\mathbb{C}}$. These, as we will see in Section 2.4, correspond
to solutions to the Fueter equation

$\begin{gathered}U:[0,1]\times\mathbb{R}^{2}\to M,\\\
I(U)\partial_{\tau}U+J_{\theta}(U)\partial_{s}U+K_{\theta}(U)\partial_{t}U=0,\\\
\end{gathered}$ (2.4)

and boundary conditions

$\displaystyle u(0,s,t)\in L_{0},\quad u(1,s,t)\in L_{1},$
$\displaystyle\lim_{s\to\pm\infty}U(\tau,s,t)=u_{\pm}(t,\tau),$
$\displaystyle\lim_{t\to\pm\infty}u(\tau,s,t)=p_{\pm},$

where again the convergence is assumed to be exponential. Here $J_{\theta}$ is
as before and $K_{\theta}=IJ_{\theta}$, and $u_{\pm}$ are solutions to (2.3)
corresponding to the gradient trajectories of
$\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}})$. Writing
$\mathcal{M}_{3}(u_{-},u_{+})$ for the moduli space of solutions to (2.4) and
$\mathcal{M}_{2}(p_{-},p_{+})$ for the moduli space of solutions to (2.3), we
have that the differential should formally be given by

${\mathrm{d}}u_{-}=\sum_{u_{+}\in\mathcal{M}_{2}(p_{-},p_{+})}\\#\mathcal{M}_{3}(u_{-},u_{+})u_{+},$
(2.5)

where $\\#$ denotes a signed count with respect to an orientation on
$\mathcal{M}_{3}(u_{-},u_{+})$ defined by means of index theory.

  * •

The compositions between $1$-morphisms

$\mathrm{Ob}(\mathrm{Fuet}_{M}(L_{0},L_{1}))\otimes\mathrm{Ob}(\mathrm{Fuet}_{M}(L_{1},L_{2}))\to\mathrm{Ob}(\mathrm{Fuet}_{M}(L_{0},L_{2}))$

are defined by counting $J_{\hat{\theta}}$-holomorphic triangles in Figure 4.
Recall that $\hat{\theta}$ is the fixed background angle and does not depend
on $L_{0},L_{1},L_{2}$; the role of $\hat{\theta}$ is discussed in the next
section.

Figure 4: The Fueter $2$-category. Left: Objects are holomorpic Lagrangians,
$1$-morpisms are intersection points, $2$-morphisms are
$J_{\theta+\pi}$-holomorphic strips. Right: Composition of $1$-morphisms is
given by the count of $J_{\hat{\theta}}$-holomorphic triangles.

  * •

The horizontal compositions and vertical compositions are given by counts of
solutions to the Fueter equations with boundary conditions as in Figure 5 and
Figure 6, respectively.

Figure 5: Vertical compostion in the Fueter $2$-category. Domain is
$\mathbb{C}\times[0,1]$, boundary conditions as on the right. Grey denotes
Lagrangian boundary conditions, blue denotes asymptotic pseudoholomorphic
strips.  Figure 6: Horizontal composition in the Fueter $2$-category. Domain
is $\mathbb{R}\times(\mathbb{D}^{2}\setminus\\{3\text{ points}\\})$. As
before, Lagrangians are boundary conditions while pseudo-holomorphic triangles
and strips are asymptotic conditions.

The most basic problems are then to develop the elementary theory of solutions
to the Fueter equation with these types of boundary conditions. One expects
the linearization of (2.4) to be a Fredholm operator admitting an index
formula.

######  Problem .

Develop the basic Fredholm theory of the Fueter equation 2.4 with the above
boundary conditions. Characterize algebro-topological data needed to assign
orientations to the moduli spaces of solutions and gradings to $1$-morphisms
in $\mathrm{Fuet}_{M}$, such that the index of the linearization of (2.4) at a
solution $U$ is the difference of the gradings of $u_{-}$ and $u_{+}$.

######  Problem .

As usual in Floer homology, the above counts of solutions are only correct if
the linearizations of the defining equations (2.3), (2.4) are surjective on
solutions, and only if the resulting solutions are isolated. This may not
occur for a fixed hyperkähler triple, so a perturbation scheme, either virtual
or geometric, is required to make sense of this prescription in general.
Develop such a perturbation scheme.

To define the compositions of $1$-morphisms as in Figure 5 and Figure 6, one
must carefully specify the boundary conditions of the Fueter equation, and one
must specify how the angle $\theta$ should vary over the domain because it is
different for different strips. More generally, to show that composition is
associative, one should compute the counts of solutions to the Fueter equation
on domains with boundary conditions labeled by points in the associahedron, so
that $\mathrm{Fuet}_{M}(L_{0},L_{1})$ is the cohomology category of an
$A_{\infty}$-category. This category is meant to be the complex Morse theory
model of the Fukaya-Seidel category of $\mathcal{A}_{\mathbb{C}}$. The complex
Morse theory model is necessary to make sense of the Fukaya-Seidel category in
the infinite-dimensional setting. As such, the problem of the precise
definition of the Fueter equation to be solved to define
$\mathrm{Fuet}_{M}(L_{0},L_{1})$ is to be taken by analogy to a better-
understood finite-dimensional construction, which we review below.

###  2.3 Complex Morse theory

#### Fukaya–Seidel category.

Let $F\colon X\to\mathbb{C}$ be a holomorphic Morse function on a Kähler
manifold. We assume that $X$ is an exact symplectic manifold with a pseudo-
convex end, and that $F$ has a controlled behavior at infinity. For example,
$X$ could be a Stein manifold. Seidel [Sei08] defined an $A_{\infty}$-category
$\mathrm{FS}(X,F)$, the Fukaya–Seidel category of the pair $(X,F)$. This
category enlarges the Fukaya category of closed Lagrangian submanifolds of $X$
by adding in certain non-compact Lagrangians: the _Lefschetz thimbles_ $L_{p}$
associated to critical points $p$ of $F$. Up to exact Lagrangian isotopy, the
thimbles are given by the unstable manifolds of the critical points of $F$ for
the gradient flow of $\mathrm{Re}(F)$. It is a remarkable insight of Seidel
that this enlargement of the Fukaya category of $X$ is in fact _simpler_ than
the original category, in the sense that it can be computed entirely from the
Floer cohomology of the thimbles themselves [Sei01].

Figure 7: The Fukaya-Seidel category. Objects are closed Lagrangians and non-
compact Lagrangians with prescribed asymptotic behavior. All objects can be
computed from the distinguished objects corresponding to the Lefschetz
thimbles.

#### Morse theory construction.

It has been proposed, simultaneously in the physics [GMW15a] and mathematics
[Hay15] literature, to push this observation of Seidel further in order to
construct an alternative model of $\mathrm{FS}(X,F)$, which we call the
_complex Morse theory model_ , that never refers to any Lagrangian
submanifolds of $X$. The motivation for this problem is the study of the
Fukaya–Seidel categories of infinite-dimensional holomorphic functionals,
especially the holomorphic Chern–Simons functional [Wit10].

The basic problem with applying any of the usual models of the Fukaya–Seidel
category to the infinite-dimensional setting is that working with infinite-
dimensional Lagrangian submanifolds as boundary conditions for certain partial
differential equations is typically not amenable to standard analytic methods.
Instead, in the complex Morse theory model, one is expected to define the
category $\mathrm{FS}(X,F)$ in terms of the gradient flows of
$\mathrm{Re}(e^{-i\theta}F)$ and the solutions to the _complex gradient
equation_

$\begin{gathered}u:\mathbb{R}^{2}\to X,\\\
\partial_{s}u+I_{X}(u)\partial_{t}u=\nabla\mathrm{Im}(e^{-i\theta}F)(u),\end{gathered}$
(2.6)

where $(s,t)$ are the coordinates on $\mathbb{R}^{2}$, $I_{X}$ is the complex
structure on $X$, and we are interested in solutions which converge
exponentially to the real gradient trajectories of
$\mathrm{Re}(e^{-i\theta}F)$ as $s\to\pm\infty$. Studying the complex gradient
equation when $X$ is a certain space of connections then reproduces previous
elliptic differential equations of interest, such as the Kapustin–Witten
equation [Hay15].

The basic idea of the complex Morse theory model is that Lefschetz thimbles
for $F$ are in bijection with the critical points of $F$. To define the
thimble $L_{p}$ associated to a critical point $p\in Crit(W)$, one fixes a
_background angle_ $\hat{\theta}\in S^{1}$ which is not the difference of
arguments of any pair of critical values of $F$. For simplicity, we will take
$\hat{\theta}=\pi$. One then defines the thimble $L_{p}$ to be the unstable
manifold of the gradient flow of $\mathrm{Re}(e^{-i\hat{\theta}}F)$. As such,
a pair of Lefschetz thimbles $(L_{p},L_{q})$ will not intersect, as each
thimble will project to parallel lines in the $F$ plane. Instead, one defines
the intersections of $L_{p}$ and $L_{q}$ by rotating the angle at which
$L_{p}$ flows out of $p$ slightly, or alternatively, flowing $L_{p}$ by the
the Hamiltonian flow of a function which is zero on the interior of $X$ and
agrees with $\mathrm{Re}(e^{-i\hat{\theta}}F)$ on the ends of $X$. As usual,
in the Fukaya category these intersections are the generators of the cochain
complex $\mathrm{FS}(X,F)(L_{p},L_{q})$.

Figure 8: The complex Morse theory model $\mathrm{FS}(X,F)$ of the Fukaya-
Seidel category does not involve Lagrangian submanifolds of $X$. The
generating objects are taken to be critical points of $F$, and the morphisms
are generated by gradient flows of $\mathrm{Re}(e^{-i\theta}F)$ between these
critical points. In simple cases, these are naturally in bijection with
intersection points between the thimbles, as shown above.

In the complex Morse theory model, one requires a definition this complex
which does not rely on intersecting Lagrangian submanifolds. To accomplish
this, one uses the fact that given two critical points $p,q$, with no critical
points $r$ such that

$\mathrm{Im}(F(p))<\mathrm{Im}(F(r))<\mathrm{Im}(F(q)),$

the intersection points between the Lefschetz thimbles $L_{p},L_{q}$ of $p$
and $q$ are in bijection with the gradient flow lines of
$-\mathrm{Re}(e^{-i\theta}F)$ from $p$ to $q$, where $\theta=\arg(F(p)-F(q))$;
one sets these to be the generators of $\mathrm{FS}(X,F)(p,q)$ in the complex
Morse theory model.

######  Remark .

The picture to keep in mind is when $F(p)$ and $F(q)$ are on a vertical line
in $\mathbb{C}=\mathbb{R}^{2}$, with $F(q)$ above $F(p)$. In that case, the
gradient flow of $-\mathrm{Re}(e^{-i\theta}F)$ agrees with the Hamiltonian
flow of $\mathrm{Re}(e^{-i\hat{\theta}}F)$ where the background angle
$\hat{\theta}=\pi$ in our convention. See Figure 10.

In the Fukaya–Seidel category, the differential on
$\mathrm{FS}(X,F)(L_{p_{-}},L_{p_{+}})$ is given by counting holomorphic
strips. These intersections are the critical points of the symplectic action
functional
$\mathcal{A}_{L_{p_{-}},L_{p_{+}}}:\mathcal{P}_{L_{p_{-}},L_{p_{+}}}\to\mathbb{R}$
on the space of paths from $L_{p_{-}}$ to $L_{p_{+}}$, and the holomorphic
strips are the gradient flows of this action functional. To write a
corresponding holomorphic curve equation for the differential in the complex
Morse theory model, one uses the observation that because $F$ is holomorphic,
the gradient flows of $-\mathrm{Re}(e^{-i\theta}F)$ from ${p_{-}}$ to
${p_{+}}$ also Hamiltonian flows of $-\mathrm{Im}(e^{-i\theta}F)$. Thus,
instead of the space of paths $[0,1]\to X$ from $L_{p_{-}}$ to $L_{p_{+}}$,
one studies the space of _infinite paths_ with exponential decay condition
near the critical points:

$\mathcal{P}_{p_{-},p_{+}}=\\{\gamma:\mathbb{R}\to
X;\lim_{t\to\pm\infty}\gamma(t)=p_{\pm}\text{ exponentially}\\}$ (2.7)

and defines the symplectic action functional $\mathcal{A}_{p_{-},p_{+}}$
perturbed by this Hamiltonian:

$\mathcal{A}_{p_{-},p_{+}}(\gamma)=\int_{\gamma}\lambda+\int_{\mathbb{R}}\mathrm{Im}(e^{-i\theta}(F(\gamma(t))-F(p_{\pm}))){\mathrm{d}}t.$
(2.8)

where $\lambda$ is a primitive for the symplectic form $\omega$. Equation
(2.6) is then the formal gradient flow of $\mathcal{A}_{p_{-},p_{+}}$, if we
impose the conditions

$\lim_{t\to\pm\infty}u(s,t)=p_{\pm},\quad\text{and}\quad\lim_{s\to\pm\infty}=\gamma_{\pm}$
(2.9)

where $\gamma_{\pm}$ are gradient trajectories of $-\mathrm{Re}(e^{-i\theta
F})$ from $p_{-}$ to $p_{+}$. The differential on
$\mathrm{FS}(X,F)(p_{-},p_{+})$ is then given by counting solutions to (2.6)
with conditions (2.9); writing $\mathcal{M}(p_{-},p_{+})$ for this space of
solutions the differential is formally given by

${\mathrm{d}}\gamma_{+}=\sum_{\gamma_{-}\in\mathrm{Crit}(\mathcal{A}_{p_{-},p_{+}})}\\#\mathcal{M}(\gamma_{-},\gamma_{+})\gamma_{-}.$
(2.10)

######  Remark .

The equation that we have called the complex gradient flow equation recurs
repeatedly in the literature under different names, as the Witten equation
[Wit93, FJR13], the $\zeta$-instanton equation [GMW15a], and, of course, it is
the inhomogeneous pseudo-holomorphic map equation introduced by Floer. We
suggest the moniker “complex gradient flow” for (2.6), and the _complex Morse
theory model_ for its application to Fukaya–Seidel categories, in order not to
privilege any particular application of this basic differential equation. We
also note that this terminology goes back to [DT98].

Continuing onwards, the Fukaya–Seidel category has compositions given by
counts of pseudo-holomorphic triangles, and more generally
$A_{\infty}$-operations given by counts of pseudo-holomorphic polygons, with
boundaries on the thimbles and corners on the intersections of the thimbles,
see Figure 9.

Figure 9: Composition in the Fukaya–Seidel category is given by counts of
pseudoholomorphic triangles.

The basic corresponding ansatz in the complex Morse theory model is to count
solutions to the complex gradient equation with asymptotics given by critical
points and gradient flows of $F$, as in Figure 10.

Figure 10: Composition in the complex Morse theory model of the Fukaya-Seidel
category is given by solutions to the complex gradient equation (2.6) with
prescribed asymptotics. Formally, the above complex gradient trajectory
corresponds to the pseudo-holomorphic triangle of Figure 9. Figure 11: Left:
Morphisms in the Fukaya–Seidel category cannot be identified with gradient
flows between corresponding critical points when an intervening critical point
exists. As such there is no natural identification between objects in the
complex Morse model, defined naively, and thimbles in the Fukaya–Seidel
category. Right: When critical points are in convex position, one can
straight-line gradient flows with intersection points of thimbles which are
not given by the standard unstable manifolds of
$\mathrm{Re}(e^{i\hat{\theta}}F)$.

Beyond this, the existing approaches become less precise. The essential
challenge is that while there is a natural bijection between the gradient flow
lines from $p_{-}$ to $p_{+}$ and the intersection of the corresponding
thimbles $(L_{p_{-}},L_{p_{+}})$ when there are no critical values in between
the images of the thimbles, this identification fails for general pairs of
critical points. Instead, the number of gradient trajectories from $p_{-}$ to
$p_{+}$ is in bijection with the intersection points of a pair of $L_{p_{-}}$
and a different Lagrangian $\tilde{L}_{p_{+}}$, which is related to
$\tilde{L}_{p_{+}}$ by a series of _Hurwitz moves_ ; see Figure 11.
Unfortunately, this Lagrangian $\tilde{L}_{p_{+}}$ depends on the choice of
the background angle $\hat{\theta}$ in a strong sense: these Lagrangians are
not isomorphic in $\mathrm{FS}(X,F)$ as one varies $\hat{\theta}$. Thus, the
naive definition of the morphisms in the complex Morse theory model cannot be
correct, as this precludes any natural map from the complex Morse theory model
to the usual model of the Fukaya–Seidel category, which must send the critical
points to well-defined objects of $\mathrm{FS}(X,F)$.

There are two proposed approaches to resolve this problem, which are
schematically compared in Figure 12. The first, proposed by Haydys [Hay15]
suggests to count broken Morse trajectories and broken complex gradient
trajectories which first flow out from $p_{-}$ to a point with
$-\mathrm{Re}(e^{-i\hat{\theta}}F)$ very large and then back to $p_{-}$, in
order to define morphisms in the complex Morse theory model. As such, these
have are always in bijection with the intersection points of the corresponding
thimbles. This approach has recently been developed further by [Wan22], and
shown to produce a category equivalent to the Fukaya–Seidel category.
Crucially, the combinatorics of the $A_{\infty}$ operations are the familiar
ones from the Fukaya category. However, this approach has the downside of
introducing many nontrivial perturbations to the complex and real gradient
flow equation, which should make computations more difficult, especially when
applying the complex Morse theory model to the infinite-dimensional setting;
see Section 2.4.

The work of Gaiotto-Moore-Witten [GMW15a] takes a different approach, and
provides a combinatorial formula for defining an $A_{\infty}$-category in
terms of standard, straight-line gradient trajectories between critical points
together with solutions to the unperturbed complex gradient flow equation
(2.6). The price of this advantageous geometric setup is a different proposed
compactification of solutions to the complex gradient flow equation, which is
not based on $A_{\infty}$ combinatorics but instead produces the $A_{\infty}$
category $\mathrm{FS}(X,F)$ via a highly involved process. A description of
this formula in more traditional mathematical language was given in [KKS16].

Figure 12: Two natural solutions to the problems posed by intervening critical
points in Figure 11). Left: Proposal of Haydys [Hay15], made rigorous by
[Wan22]. Instead of straight-line gradient flows, one can use time-dependent
gradient flows of $\mathrm{Re}(e^{-i\theta(t)}F)$. This simplifies many issues
by may make computations in the infinite-dimensional context of Section 2.2
significantly more difficult. Right: Proposal of Gaiotto–Moore–Witten
[GMW15a]. An alternative compactification of the moduli space of solutions to
the complex gradient flow equation should make it possible to use straight-
line gradient flows, at the cost of novel analysis and novel algebraic
constructions.

Most aspects of the analytical setup underlying the Gaiotto–Moore–Witten (GMW)
approach remain entirely open. In particular, the basic compactness result
conjectured in [GMW15a] has not been established.

######  Problem .

Define an analytical setup for the complex gradient flow equation and prove
the correctness of the combinatorial compactifications of the spaces of
solutions to (2.6) as proposed in [GMW15a].

In order to make sense of this problem, the precise conditions and
perturbation scheme for the complex gradient equation in the
Gaiotto–Moore–Witten approach remain to be specified.

######  Remark .

Another nontrivial aspect of the compactness theory for the complex gradient
flow equation is that the the energy identities of Haydys [Hay15, Section 2.4]
(and the corresponding formulae in [KKS16]) are difficult to make sense of, as
the values of the action functional, as written in these references, are
_infinite_. For the purpose of defining the differential, this can be fixed by
renormalizing the action functional as in (2.8). For defining higher
$A_{\infty}$-operations, a nontrivial renormalization procedure must be
performed in order to extract a finite energy identity controlling the $L^{2}$
energy of solutions to the complex gradient equation [Wan22]).

######  Remark .

In the case where the critical values of $F$ are the vertices of a convex
polytope, there is a natural identification between thimbles and critical
points such that the intersection points of the thimbles are in bijection with
the counts of gradient flows; see Figure 11. It is a nontrivial fact that the
combinatorics of Gaiotto–Moore–Witten then simplify dramatically in this
context and reduce to the typical $A_{\infty}$ combinatorics of the Fukaya
category; this is a consequence of the fact that the secondary polytope of a
convex polytope is an associahedron, and can be extracted from [KKS16].

#### Hochschild homology.

There is a natural way to produce a vector space from an $A_{\infty}$-category
$\mathcal{F}$, namely, its _Hochschild homology_ $HH_{*}(\mathcal{F})$. We
will not go into this construction in detail, except to say that there is an
explicit chain complex computing $HH_{*}(\mathcal{F})$ written in terms of the
$A_{\infty}$-operations on $\mathcal{F}$ called the _cyclic bar complex_ , and
that when $\mathcal{F}$ is a Fukaya category of a symplectic manifold $X$,
there is an _open-closed map_ [Sei06]

$\mathcal{OC}:HH_{*}(\mathcal{F})\to HF^{*}(X,H)$ (2.11)

for some appropriate Hamiltonian $H:X\to\mathbb{R}$, which depends on the
variant of the Fukaya category being considered. The Hamiltonian Floer
cohomology group $HF^{*}(X,H)$ is object assigned to a circle by the open-
closed field theory Section 2.1) which assigns $\mathcal{F}(L_{0},L_{1})$ to
the interval with endpoints labeled by $(L_{0},L_{1})$ , and the open-closed
map is a chain-level version of all the cobordism maps associated to genus-
zero cobordisms with one circle boundary and many interval boundaries. In many
cases, including the case of the Fukaya–Seidel category, the open-closed map
is known to be an isomorphism [Abo10, GPS19].

Recall that to define the thimbles $L_{p}$ generating the Fukaya–Seidel
category, we had to choose a background angle $\hat{\theta}$. Denote by
$\mathrm{FS}_{\hat{\theta}}(X,F)$ the category defined for $\hat{\theta}$. The
appropriate Hamiltonian to take for
$\mathcal{F}=\mathrm{FS}_{\hat{\theta}}(X,F)$ is
$H=-\mathrm{Re}(e^{-i\hat{\theta}}F)$, or rather a small real rescaling of
this Hamiltonian. As we take this Hamiltonian to zero (on some compact
subdomain of $X$ containing all critical points, which arises in the technical
setup for the Fukaya–Seidel category), $HF^{*}(X,H)$ is identified with the
Morse homology group $HM_{*}(-\mathrm{Re}^{-i\hat{\theta}}F)$. This, in turn,
agrees with the homology of $X$ relative to the subset $\\{x\in
X|-\mathrm{Re}^{-i\hat{\theta}}F(x)\ll 0)$. For the conventional choice
$\hat{\theta}=\pi$, this is simply $H_{*}(X,\\{\mathrm{Re}(F)\gg 0\\})$.

In the construction of $\mathrm{FS}_{\hat{\theta}}(X,F)$ of $F$, which a
priori depends on the choice of $\hat{\theta}$, one uses $\hat{\theta}$ to
define an ordering $p_{1}<\ldots<p_{r}$ on the critical points
$\\{p_{1},\ldots,p_{r}\\}$ of $F$. This is used to define first the _directed
Fukaya–Seidel category_ $\mathrm{FS}^{\to}_{\hat{\theta}}(X,F)$, which only
has morphisms from $L_{p_{j}}$ to $L_{p_{k}}$ if $j\leq k$, with the only
endomorphism of $L_{p}$ being the identity map. The full category
$\mathrm{FS}_{\hat{\theta}}(X,F)$ is then the category of _twisted complexes_
on $\mathrm{FS}^{\to}_{\hat{\theta}}(X,F)$. By algebraic properties of
Hochschild homology, this implies that $\mathrm{FS}_{\hat{\theta}}(X,F)$ has
the same Hochschild homology as $\mathrm{FS}_{\hat{\theta}}^{\to}(X,F)$,
which, because of the directedness, is easily seen to be a vector space freely
generated by $\\{L_{p_{1}},\ldots,L_{p_{r}}\\}$. Geometrically, these cycles
give a basis for the relative homology group $H_{*}(X,\\{\mathrm{Re}(F)\gg
0\\})$ (with corresponding modification when $\hat{\theta}\neq\pi$). When $F$
is a Morsification of a non-Morse holomorphic function $F_{0}$ with a single
singular fiber, this would be the group of _vanishing cycles_ of the singular
fiber.

#### Background angle and monodromy.

As we see in the description above, the Fukaya–Seidel category
$\mathrm{FS}_{\hat{\theta}}(X,F)$ a priori depends on the choice of an angle
$\hat{\theta}$, and the directed subcategory
$\mathrm{FS}^{\to}_{\hat{\theta}}(X,F)$ manifestly depends on $\hat{\theta}$
because $\hat{\theta}$ determines the ordering on the critical points.

However, using the topological interpretation of the Hochschild homology group
$HH_{*}(\mathrm{FS}_{\hat{\theta}}(X,F))$ as
$H_{*}(X,\\{\mathrm{Re}(e^{-i\hat{\theta}}F)\gg 0\\})$, one sees that these
groups form a local system of groups over the space of $\hat{\theta}\in
S^{1}$. On the topological level, the dependence of the categories on
$\hat{\theta}$ corresponds to the fact that the thimbles $L_{p_{i}}$, which
implicitly depend on $\hat{\theta}$, do not define a trivialization of the
local system of relative homology groups. Now, as $\hat{\theta}$ varies
through angles avoiding the exceptional values

$\hat{\theta}^{*}=\arg(F(p))-\arg(F(q))$

for pairs $p,q\in\mathrm{Crit}(F)$, the ordering on the critical points does
not change, the thimbles $L_{p}$ stays the same up to exact Lagrangian
isotopy, and the bases of $HH_{*}(\mathrm{FS}_{\hat{\theta}}(X,F))$ given by
the open-closed map form a trivialization of the local system.

At the exceptional angles, a theorem of Seidel [Sei01] tell us that the
difference between the directed subcategories
$\mathrm{FS}^{\to}_{\hat{\theta}^{*}-\epsilon}(X,F)$ and
$\mathrm{FS}^{\to}_{\hat{\theta}^{*}+\epsilon}(X,F)$ for a small $\epsilon>0$
is given by an algebraic operation called a _mutation_ , inducing an
equivalence

$F_{\hat{\theta}^{*}}:\mathrm{FS}_{\hat{\theta}^{*}-\epsilon}(X,F)\to\mathrm{FS}_{\hat{\theta}^{*}+\epsilon}(X,F).$
(2.12)

Thus, in the end, the full Fukaya–Seidel category
$\mathrm{FS}_{\hat{\theta}}(X,F)$ up to equivalence does not depend on the
choice of background angle $\hat{\theta}$. However, it is more precise to
think of the categories $\mathrm{FS}_{\hat{\theta}}(X,F)$ as forming a _local
system of categories_ over $S^{1}$. Indeed, the content of Picard–Lefschetz
theory is that the bases of $H_{*}(X,\\{\mathrm{Re}(e^{-i\hat{\theta}}F)\gg
0\\})$ given by the thimbles $L_{p_{i}}$ are related at the exceptional angles
$\hat{\theta}^{*}$ by an elementary integral change-of-basis matrix
$\tau_{\hat{\theta}^{*}}$ referred to as a Picard–Lefschetz transformation. It
turns out that the equivalences $F_{\hat{\theta}^{*}}$ categorify the
Picard–Lefschetz transformations on relative homology in the sense that

$\mathcal{OC}(1_{L_{p}^{\hat{\theta}^{*}+\epsilon}})=\tau_{\hat{\theta}^{*}}(\mathcal{OC}(1_{L_{p}^{\hat{\theta}^{*}-\epsilon}})).$
(2.13)

where $1_{O}$ denotes the unit of the object $O$ and
$L_{p}^{\hat{\theta}}\in\mathrm{FS}_{\hat{\theta}}$ denotes the thimble
defined using the background angle $\hat{\theta}$. In other words, as we
change the background angles, when the homology classes of thimbles change by
a Picard-Lefshetz transformations, the corresponding objects change by a
mutation – the object associated to the new thimble is algebraically a _cone_
on a morphism from the previous thimble to a thimble associated to the end of
an exceptional gradient trajectory at angle $\hat{\theta}^{*}$. For many
natural choices of function $F$, the total monodromy of the local system
around $S^{1}$ is not trivial. On the level of Fukaya–Seidel categories, this
corresponds to the unpublished theorem [AG15] conjectured by Seidel [Sei06]
that the composition of the functors $F_{\hat{\theta}}$ is the Serre functor
of $\mathrm{FS}_{\hat{\theta}}(X,F)$, which tends to be nontrivial as well.

###  2.4 The Fueter $2$-category from complex Morse theory

As explained in Section 2.2, the hom-category $\mathrm{Fuet}_{M}(L_{0},L_{1})$
in the Fueter $2$-category is meant to be based off of the complex Morse
theory model of the holomorphic symplectic action functional
$\mathcal{A}_{\mathbb{C}}$. We now explain the correspondence, focusing on the
exact setting as in (2.2). By (2.1) and the nondegeneracy of $\Omega$, the
critical points of $\mathcal{A}_{\mathbb{C}}$ are the constant paths with
values in $L_{0}\cap L_{1}$. We can view $M$ as a real symplectic manifold
with respect to any of the symplectic forms

$\omega_{\theta}=\mathrm{Re}(e^{-i\theta}\Omega).$ (2.14)

For each $\theta\in S^{1}$, we have that $\omega_{\theta}$ is compatible with
$J_{\theta}=\cos\theta J+\sin\theta K$. In the exact setting these forms have
primitives given by $\lambda_{\theta}=\mathrm{Re}(e^{-i\theta}\Lambda)$, and
$\lambda_{\theta}|_{L_{i}}$ has a primitive given by
$H_{i}^{\theta}=\mathrm{Re}(e^{-i\theta}H_{i})$. The holomorphic symplectic
action functional $\mathcal{A}_{\mathbb{C}}$ thus gives rise to a family of
real symplectic action functionals

$\mathcal{A}_{\theta}(\gamma):=\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}}(\gamma))=\int_{\gamma}\lambda_{\theta}-H_{1}^{\theta}(\gamma(1))+H_{0}^{\theta}(\gamma(0)),$
(2.15)

which have differentials

${\mathrm{d}}\mathcal{A}_{\theta}(\gamma)v=Re(e^{-i\theta}{\mathrm{d}}\mathcal{A}_{\mathbb{C}}(\gamma)v)=\int_{0}^{1}\omega_{\theta}\left(v(\tau),\frac{{\mathrm{d}}\gamma}{{\mathrm{d}}\tau}\right){\mathrm{d}}\tau.$
(2.16)

In particular, $\mathcal{A}_{J}:=\mathcal{A}_{0}$ is the symplectic action
functional when $M$ is thought of as symplectic with respect to $\omega_{J}$,
and $\mathcal{A}_{K}:=\mathcal{A}_{\pi/2}$ is the symplectic action functional
when $M$ is thought of as symplectic with respect to $\omega_{K}$. In these
conventions, we can write
$\mathcal{A}_{\mathbb{C}}=\mathcal{A}_{J}+i\mathcal{A}_{K}$.

By the standard computation of the Hessian of the real symplectic action
functional [Flo89], we see that the critical points of
$\mathcal{A}^{\mathbb{C}}$ are nondegenerate exactly when $L_{0}$ intersects
$L_{1}$ transversely. Moreover, in our conventions, summarized in Section A.3,
the gradient flow equation for $\mathcal{A}_{\theta}$ is the
$J_{\theta}$-holomorphic curve equation

$\begin{gathered}u:\mathbb{R}\to\mathcal{P},\\\
\partial_{t}u=\nabla\mathcal{A}_{\theta}(u(t))=-J_{\theta}(u)\partial_{\tau}u.\end{gathered}$
(2.17)

In our conventions for the complex Morse theory model, the morphisms from $p$
to $q$ should be generated by gradient flows of
$-\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}})$ where $\theta=\arg(p-q)$,
i.e. gradient flows of $-\mathcal{A}_{\theta}=\mathcal{A}_{\theta+\pi}$, that
is: $J_{\theta+\pi}$-holomorphic strips.

Finally, the differential, and more generally the $A_{\infty}$-operations, in
the complex Morse theory model should be given by solutions to the complex
gradient flow equation (2.6). Plugging in $F=\mathcal{A}_{\mathbb{C}}$ into
(2.6) and using (2.17) to compute the gradient of the action functional, we
get that complex gradient trajectories $U:\mathbb{C}\to\mathcal{P}$ are
equivalent to solutions of

$\displaystyle U:\mathbb{R}^{2}\times[0,1]\to M,$
$\displaystyle\partial_{s}U+I(U)\partial_{t}U=\nabla\mathrm{Re}(e^{-i(\theta+\pi/2)}\mathcal{A}_{\mathbb{C}})=-J_{\theta+\pi/2}(U)\partial_{\tau}U=-K_{\theta}(U)\partial_{\tau}U,$

where

$K_{\theta}=IJ_{\theta}=\cos\theta K-\sin\theta J$ (2.18)

is the complex structure such that $(I,J_{\theta},K_{\theta})$ is a
quaternionic triple. Moving all terms to the left hand side, and multiplying
the equation by $J_{\theta}$ we get that the above solutions solve the Fueter-
type equation

$I(U)\partial_{\tau}U+J_{\theta}(U)\partial_{s}U-K_{\theta}(U)\partial_{t}U=0.$
(2.19)

In order to recover the more standard Fueter equation (1.2), we can instead
define $\mathrm{FS}(\mathcal{P},\mathcal{A}_{\mathbb{C}})$ via solutions to
the _anti-complex gradient flow equation_

$\displaystyle U:\mathbb{C}\to\mathcal{P},$
$\displaystyle\partial_{s}U-I(U)\partial_{t}U=\nabla\mathrm{Im}(-e^{-i\theta}\mathcal{A}_{\mathbb{C}}).$

Solutions to the above equation are in bijection with solutions to (2.4) as
desired. The $s$-invariant asymptotics are the $J_{\theta}$-_anti_ -holomophic
strips, i.e. the $J_{\theta+\pi}$-holomorphic strips, that give generate the
morphism complexes $\mathrm{Fuet}_{M}(L_{0},L_{1})(p,q)$. The coordinate
change $(s,t)\leftrightarrow(-s,t)$ induces a bijection between complex
gradient trajectories and complex anti-gradient trajectories, and this
explains why the opposing conventions in the differentials (2.5) and (2.10).
We have thus reproduced the description of the Fueter hom-category of Section
2.2.

#### Complex linearity and anti-linearity.

Upon first glance, one may find the appearance of the anti-complex gradient
flow in this derivation strange. However, it is actually rather natural, as
the Fueter equation is the condition that the differential of the map is
_anti_ -quaternion-linear. We discuss the underlying linear algebra further in
Section 3.1. Given this observation, is no surprise that the dimensional
reductions of the Fueter equations (2.4) are $J_{\theta}$-_anti_ -holomorphic
strips. Moreover, unlike in dimension two, where holomorphic and anti-
holomorphic maps are related by a conjugation of the underlying domain, the
analog of the Fueter equation where we ask for the differential to be
quaternion-linear has essentially no solutions [Fue35], so if we wish to
preserve the symmetric form of the Fueter equations (1.2), we must incorporate
the corresponding preference for anti-gradient trajectories and anti-
holomorphic strips into our conventions. Luckily, via the substitution
$J_{\theta}\to J_{\theta+\pi}$ and the reordering of inputs in
$A_{\infty}$-operations, one can set the conventions up such that anti-
holomorphic strips never explicitly arise in the Fueter $2$-category, as we
saw in s Section 2.2.

#### Background angles.

In the complex Morse theory model for the Fukaya–Seidel category, a choice of
background angle $\hat{\theta}$ is used all existing proposals to put an
ordering on the critical points of the complex Morse function, as in the more
standard construction of the Fukaya-Seidel category. In particular, the
complex Morse theory model of the Fukaya–Seidel category
$\mathrm{FS}_{\hat{\theta}}^{\zeta}(X,F)$ also contains a directed subcategory
$\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(X,F)$, see [Wan22], and the categories
$\mathrm{FS}_{\hat{\theta}}^{\zeta}(X,F)$ also form a local system of
categories over $S^{1}$ as the paragraph on background angles in Section 2.3.

To define $\mathrm{Fuet}_{M}$, we should also fix a single background angle
$\hat{\theta}$ and then formally define
$\mathrm{Fuet}_{M}(L_{0},L_{1})=\mathrm{FS}^{\zeta}_{\hat{\theta}}(\mathcal{P},\mathcal{A}_{\mathbb{C}})$,
using the same $\hat{\theta}$ for all pairs $L_{0},L_{1}$. The horizontal
composition of morphisms
$\mathrm{Fuet}_{M}(L_{0},L_{1})\otimes\mathrm{Fuet}_{M}(L_{1},L_{2})\to\mathrm{Fuet}_{M}(L_{0},L_{2})$
should be given in terms of a count of reguklar $J_{\hat{\theta}}$-holomorphic
triangles. In between any two intersection points of exact Lagrangians $L_{0}$
and $L_{1}$, there are $J_{\theta}$-holomorphic strips only for isolated
values of $\theta$. If we choose a different background angle
$\hat{\theta}^{\prime}$ to define $\mathrm{Fuet}_{M}$, there should be an
equivalence between the $2$-categories built using these choices via analogs
of the functors (2.12) above. We explore this structure further in Section 2.5
below.

######  Remark .

In a formal definition of the $A_{\infty}$-category
$\mathrm{Fuet}_{M}(L_{0},L_{1})$, one must take either the
Gaiotto–Moore–Witten approach or the Haydys approach to the complex Morse
theory model. It is probable that the Haydys approach is more straightforward
in the setting of $J_{\theta}$-holomorphic curves and Fueter maps, as it does
not involve solving Section 2.3 as a first step. However, this has the
significant downside that the morphism complexes in the Haydys model for
$\mathrm{Fuet}_{M}(L_{0},L_{1})$ will be generated by solutions to a
perturbation of the $J_{\theta(t)}$-holomorphic strip equation, where
$\theta(t)$ is a very particular family of angles $\theta:\mathbb{R}\to
S^{1}$. Finding such $J_{\theta}(t)$-holomorphic strips in a situation with a
concrete geometry may be significantly harder than finding
$J_{\theta}$-holomorphic strips for fixed $\theta$, and thus may make it
challenging to compute Fueter hom-categories in any situation where a formal
method for computation may not be available. In certain cases, such as the
setting of Section 2.8 below, the Haydys model should suffice to provide a
computable definition; but we see both approaches as valid and necessary for
the development of $3d$ mirror symmetry.

#### Novikov rings.

In many cases, the holomorphic action functional $\mathcal{A}^{\mathbb{C}}$
defined by its differential (2.1) is multivalued on the path space, and one
must pass to the Novikov cover to make it single-valued. As such, defining
$\mathrm{Fuet}_{M}(L_{0},L_{1})=\mathrm{FS}(\mathcal{P},\mathcal{A}_{\mathbb{C}})$
would then give rise to an infinite number of objects, somehow related by
deck-transformations. Even in the finite-dimensional case, when
${\mathrm{d}}F\in\Omega^{1,0}(X)$ is replaced by a closed holomorphic one-form
on $X$, there is yet no standard formalism for studying the Fukaya–Seidel
categories of the pullback of the one-form to the appropriate Novikov cover of
$X$. While it is not clear to the authors if there is any essential challenge
in adapting the existing definitions of the Fukaya–Seidel category to this
setting, it is important for the foundations of the Fueter $2$-category to
give a precise exposition of such a construction.

######  Problem .

Define the Fukaya–Seidel category of a closed holomorphic one-form on a Kähler
manifold, both with a more classical symplectic approach and in the complex
Morse theory model.

It is advantageous and nontrivial to give a definition where the one-form is
not required to be holomorphic. Indeed, Fukaya–Seidel categories make sense
for symplectic Lefschetz fibrations, which have a significant amount of
flexibility. However, it is not immediate what is the best definition of a
correspondingly flexible symplectic of a holomorphic one-form; the essential
challenges in defining this notion are about establishing energy bounds for
the relevant pseudo-holomorphic curves or corresponding complex Morse
trajectories. The flexibility of such a definition would be helpful in the
infinite-dimensional setting, where one wishes to perturb the holomorphic
symplectic action functional or to work in some _weak_ or _almost_ hyperkähler
setup, cf. Section 2.11.

###  2.5 Decategorification I

The Fukaya category of a symplectic manifold $M$ categorifies the topological
invariants of $M$, such as the intersection form. In the same way, we expect
that if $M$ is hyperkähler, the Fueter $2$-category of $M$ categorifies the
symplectic invariants of $M$, including the Fukaya category itself.

As should be clear from Section 2.2, the Fueter $2$-category takes into
account the many nontrivial $J_{\theta}$-holomorphic strips stretching between
pairs of complex Lagrangians. By analogy with the finite-dimensional case,
discussed in Section 2.3, the Fueter category defined using background angle
$\hat{\theta}$ should satisfy

$HH_{*}(\mathrm{Fuet}^{\hat{\theta}}_{M}(L_{0},L_{1})))\simeq
HM_{*}(\mathcal{A}_{\hat{\theta}+\pi})=HF^{*}(L_{0},L_{1};\omega_{\hat{\theta}},J_{\hat{\theta}}).$
(2.20)

On other words, _the Fueter $2$-category should categorify the subcategory of
the Fukaya category containing complex Lagrangians_. In particular, applying
Hochschild homology to the $A_{\infty}$-bifunctors (see [Sei08] for algebraic
background),

$\mathrm{Fuet}_{M}(L_{0},L_{1})\otimes\mathrm{Fuet}_{M}(L_{1},L_{2})\to\mathrm{Fuet}_{M}(L_{0},L_{2})$
(2.21)

should recover the composition in the Fukaya category. This identify follows
from the definition of composition of $1$-morphisms in Section 2.2. We
symbolically write this naive decategorification of the Fukaya category as

$HH_{*}(\mathrm{Fuet}_{M}^{\hat{\theta}})\simeq\mathrm{Fuk}^{h}(M,\omega_{\theta}),$
(2.22)

where
$\mathrm{Fuk}^{h}(M,\omega_{\theta})\subset\mathrm{Fuk}(M,\omega_{\theta})$ is
the full subcategory with objects the complex Lagrangian submanifolds.

There has been a significant amount of previous work which gives insight into
the Floer cohomology of complex Lagrangians. The basic observation is that if
two complex Lagrangian submanifolds intersect transversely, then the gradings
of their intersection points are all the same. As such, one has that

$HF^{*}(L_{0},L_{1})\simeq\mathbb{Z}(L_{0}\cap L_{1})\text{ when }L_{0}\text{
is transverse to }L_{1}.$ (2.23)

In the case of non-transverse intersections, Solomon–Verbitsky have proven

######  Proposition  ([SV19]).

For generic values of $\theta\in S^{1}$, all $J_{\theta}$-holomorphic polygons
with boundary on a finite collection of complex Lagrangians are constant.

The basic reason for both of these phenomena is explained by the fact that the
Floer cohomology of $L_{0},L_{1}$ is the gradient flow of the real part of a
holomorphic Morse function. Indeed, when we compute
$HF^{*}(L_{0},L_{1},\omega_{\theta})$, we are allowed to use any complex
structure _tamed_ [MS12, Chapter 1] by $\omega_{\theta}$, which is an open
condition. Thus, instead of using $J_{\theta}$ to define the holomorphic curve
equation, we can use $J_{\theta+\epsilon}$ for a small generic $\epsilon$ to
compute $HF^{*}(L_{0},L_{1};\omega_{\theta})$. The image of a
$J_{\theta+\epsilon}$-holomorphic curve under $\mathcal{A}_{\mathbb{C}}$ will
project to a straight line with argument $\theta+\epsilon$ in the complex
plane; as such, for generic $\epsilon$, there will be no such lines between a
pair of critical values, and thus no such non-constant holomorphic curves
between intersection points, and in the case of transverse intersection, and
no differential on the resulting Floer complex.

######  Remark .

In examples of importance to mathematical physics and representation theory,
one may hope that there exist a collection of complex Lagrangians which
_generate_ the Fukaya category, and thus we have
$\mathrm{Fuk}^{h}(M,\omega_{\theta})\simeq\mathrm{Fuk}(M,\omega_{\theta})$.
Specializing to $M$ of representation-theoretic interest, this should give
geometric characterizations of various representation-theoretic invariants. We
give several potential examples below.

######  Remark .

To recover the Fukaya category upon taking Hochschild homology, composition of
$1$-morphisms in $\mathrm{Fuet}_{M}$ should be given by the usual counts of
$J_{\hat{\theta}}$-holomorphic triangles, as opposed to any kind of
parameterized counts over $S^{1}$. As one varies $J_{\hat{\theta}}$, by
bifurcation theory, the counts of $J_{\hat{\theta}}$-holomorphic triangles
should jump exactly when a $J_{\hat{\theta}}$-holomorphic strip appears,
_which should couple precisely with the nontrivial mutation isomorphisms_
(2.12) from
$\mathrm{Fuet}_{M,\hat{\theta}^{*}-\epsilon}(L_{i},L_{j})\to\mathrm{Fuet}_{M,\hat{\theta}^{*}+\epsilon}(L_{i},L_{j})$
as $\hat{\theta}$ varies through an exceptional angle $\hat{\theta}^{*}$, in
order given an equivalence of two categories
$\mathrm{Fuet}_{M,\hat{\theta}^{*}-\epsilon}\simeq\mathrm{Fuet}_{M,\hat{\theta}^{*}+\epsilon}$
making the choice of background angle _irrelevant_.

There are several existing works which study the enhancements of the Floer
theory of complex Lagrangians that arise from the ambient holomorphic
symplectic geometry. We argue that all of them can viewed as simplifications
of the data associated to the hom-categories of the Fueter $2$-category.

#### Connections to perverse sheaves.

As a consequence of Section 2.5, one might expect that $HF^{*}(L_{0},L_{1})$
is computable in terms of classical topological invariants, even when $L_{0}$
is not transverse to $L_{1}$. A proposed definition of the correct topological
invariant was given by Brav–Bussi–Dupont–Joyce–Szendroi [Bra+15], who defined
a _perverse sheaf of vanishing cycles_ $P_{L_{0},L_{1}}$ supported on
$L_{0}\cap L_{1}$. Heuristically, one defines this sheaf locally by writing
$L_{1}$ as the graph of the differential of a holomorphic function
$F:L_{0}\to\mathbb{C}$, and then defines $P_{L_{0},L_{1}}$ as the _sheaf of
vanishing cycles_ [Mas99] of $F$. An isolated intersection of $(L_{0},L_{1})$
corresponds to an isolated critical point of $F$, and hypercohomology of the
summand of $P_{L_{0},L_{1}}$ supported at that intersection point is simply
the number of critical points of a Morsification $\tilde{F}$ of $F$. It is a
well-known conjecture that
$HF^{*}(L_{0},L_{1})\simeq\mathbf{H}(P_{L_{0},L_{1}})$, which generalizes
(2.23).

The sheaf of vanishing cycles $P_{L_{0},L_{1}}$ admits a _monodromy
automorphism_ $h:P_{L_{0},L_{1}}\to P_{L_{0},L_{1}}$. In the case where
$L_{1}$ is the graph of a differential of a function $F:L_{0}\to\mathbb{C}$ in
$T^{*}L_{0}$ and $F$ has a single isolated critical point, this corresponds to
considering the monodromy automorphism of $\tilde{F}$. We call the induced
automorphism $h:\mathbb{H}(P_{L_{0},L_{1}})\to\mathbb{H}(P_{L_{0},L_{1}})$ the
_local monodromy_.

On the side of the Lagrangian Floer cohomology group, one can consider the
Floer-theoretic continuation map

$HF^{*}(L_{0},L_{1},\omega_{\hat{\theta}})\to
HF^{*}(L_{0},L_{1},\omega_{\hat{\theta}})$ (2.24)

given by counting solutions to

$\begin{gathered}u:\mathbb{R}\times[0,1]\to M,\\\
\partial_{t}u+J_{\theta(t)}(u)\partial_{\tau}u=0,\\\
\lim_{s\to\pm\infty}u(s,t)\in L_{0}\cap L_{1}\end{gathered}$ (2.25)

where $\theta:\mathbb{R}\to S^{1}$ is equal to $\hat{\theta}$ for $|t|\gg 0$,
and such that $\theta$ winds once around the $S^{1}$ as $t$ increases. We
argue that this monodromy map is the map induced by the Serre functor of
$\mathrm{Fuet}_{M}(L_{0},L_{1})$ under (2.20). Indeed, in the finite-
dimensional case, the Picard-Lefschetz transformations of (2.13) are simply
the changes of basis induced by the bifurcation analysis of the family of
Morse functions $Re(e^{-i\theta}W)$. It is standard [Flo88] that instead of
constructing isomorphisms of Morse-Floer homology groups using bifurcation
analysis, one can construct the same maps using a continuation equation; the
continuation equation for the family
$\mathrm{Re}(e^{-i\theta}\mathcal{A}_{\mathbb{C}})$ is the equation (2.25)
above. Thus, the global monodromy map, which is an object of real symplectic
topology, is the simplest shadow of nontrivial structures associated to the
Fueter $2$-category.

#### Connections to resurgence.

It is expected, following Kontsevich–Soibelman [KS20], that the difference
between the local and the global monodromy can be viewed as a _Stokes datum_.
Indeed, the formal path integral

$\int_{\mathcal{P}_{L_{0},L_{1}}}e^{i\hbar\mathcal{A}_{\mathbb{C}}(\gamma)}\mathcal{D}\gamma$
(2.26)

should satisfy an irregular differential equation in $\hbar$ by analogy to the
finite-dimensional case of exponential integrals. The irregular
Riemann–Hilbert correspondence, in the corresponding finite dimensional case

$\int_{\Gamma}e^{i\hbar F(x)}\mathrm{vol}$ (2.27)

where $F:X\to\mathbb{C}$ is a finite dimensional holomorphic Morse function
and $\mathrm{vol}\in\Omega^{top}(X)$ is a holomorphic volume form, then allows
one to reconstruct the differential equation satisfied by functions of the
form (2.27). We will not exposit the theory here; for details, see for example
[PS03]. One may summarize [KS20] by saying that the exceptional
$J_{\theta}$-holomorphic strips between complex Lagrangians may be counted to
enhance the $S^{1}$ family of Fukaya categories
$\mathrm{Fuk}(M,\omega_{\theta})$ into a holomorphic family of categories over
over $\mathbb{P}^{1}$ where morphisms between complex Lagrangians are
$D$-modules with irregular singularities at $0$ and $\infty$, which formally
correspond to the path integrals (2.26).

In the case of $(L_{0},L_{1})$ being a pair of complex Lagrangian submanifolds
of a complex cotangent bundle $T^{*}X$, the Stokes data associated to the two
integrals (2.26), (2.27) should be identified as a consequence of Floer’s
theorem (1.3). The categorification of this statement is our Section 2.8
below. Thus, we see the structure of the Fueter $2$-category as a
categorification of the approach to holomorphic Floer theory suggested in
[KS20].

The following problems, while not strictly related to the Fueter $2$-category,
would go a long way to giving foundations to the Fueter $2$-category in the
setting where the complex Lagrangians involved have bad intersections.

######  Problem .

Make sense of the global monodromy operator
$HF^{*}(L_{0},L_{1};\omega_{\hat{\theta}})\to
HF^{*}(L_{0},L_{1};\omega_{\hat{\theta}})$ in cases of interest when $L_{0}$
intersects $L_{1}$ in a non-compact set. This involve finding appropriate
perturbations of $L_{1}$ to a family of non-holomorphic submanifolds
$\tilde{L}^{\theta}_{1}$, depending on $\theta$, which are Lagrangian for
$\omega_{\theta}$ and are all transverse to $L_{0}$. The core of the problem
is to understand the appropriate constraints on the perturbations near the
ends at infinity of $L_{0}\cap L_{1}$, in order to make the cohomology groups
$HF^{*}(L_{0},\tilde{L}^{\theta}_{1};\omega_{\theta})$ isomorphic to
$\mathbb{H}(P_{L_{0},L_{1}})$ for every $\theta$.

######  Problem .

Define a $D$-module associated to a pair of complex Lagrangians
$(L_{0},L_{1})$ which intersect non-transversely in a non-compact locus.

######  Remark .

The authors do not know of any cases where the global monodromy operator is
known to differ from the local monodromy operator except for the cases of
complex Lagrangians in complex cotangent bundles relevant to the work of
[KS20]. Explicit computations of this operator for interesting complex
Lagrangians would be significant progress towards novel computations of the
Fueter $2$-category.

###  2.6 Decategorification II

In Section 2.5 we discussed the Fukaya category as a certain
decategorification of the Fueter $2$-category. However, there are multiple
ways to decategorify a $2$-category. If a $2$-category arises from a TQFT, one
form of such a decategorification can be obtained by studying the field theory
on closed manifolds, as opposed to manifolds with boundary or corners. For
example, as we saw in Section 2.1, an open-closed 2d TQFT assigns a vector
space not just to an interval but also to a circle. In the case of the Fukaya
category, this vector space is the Hamiltonian Floer cohomology of $M$, and is
defined by computing the Morse homology of the action functional on the _loop
space_ instead of on the space of paths. Algebraically, the Hamiltonian Floer
cohomology of $M$ is meant to be recovered by taking the Hochschild homology
of the Fukaya category.

In the setting of the Fueter equation, we can formally do the same, and
formally define a $1$-category

$\mathrm{Fuet}_{M}(S^{1})=\mathrm{FS}(\mathcal{A}^{\mathbb{C}}_{S^{1}})$

where $\mathcal{A}^{\mathbb{C}}_{S^{1}}$ is the holomorphic symplectic action
functional on the loop space

$\begin{gathered}\mathcal{A}^{\mathbb{C}}_{S^{1}}:LX\to\mathbb{C},\\\
{\mathrm{d}}\mathcal{A}^{\mathbb{C}}_{S^{1}}(\gamma)v=\int_{S^{1}}\Omega\left(v(\tau),\frac{{\mathrm{d}}\gamma}{{\mathrm{d}}\tau}\right){\mathrm{d}}\tau.\end{gathered}$
(2.28)

On a formal level, $\mathrm{Fuet}_{M}(S^{1})$ has objects given by critical
points of $\mathcal{A}^{\mathbb{C}}_{S^{1}}$, which are the constant loops
into $X$. The $1$-morphisms would then be given by gradient flows of
$\mathrm{Re}(e^{i\theta}\mathcal{A}^{\mathbb{C}}_{S^{1}})$ between these
points, which are the _$J_{\theta}$ -holomorphic spheres in $X$_, for all
$\theta\in S^{1}$. Correspondingly, the the differential on the morphism
complexes in this category should be given by counts of Fueter maps

$\begin{gathered}U:\mathbb{R}\times S^{2}\to M,\\\
\partial_{s}U-I(U)(\bar{\partial}^{J_{\theta}}_{z}U)=0,\end{gathered}$ (2.29)

with $s$ denoting the coordinate on $\mathbb{R}$, $z$ denoting the holomorphic
coordinate on $S^{2}$, and with asymptotics given by $J_{\theta}$-holomorphic
spheres in $M$. Similarly, the higher $A_{\infty}$-operations should count
solutions Fueter maps from non-compact $3$-manifolds without boundary with
ends modeled on $\mathbb{R}\times S^{2}$.

Unfortunately, $\mathcal{A}^{\mathbb{C}}_{S^{1}}$ is only Morse–Bott rather
than Morse, which leads to the infinitude of objects of
$\mathrm{Fuet}_{M}(S^{1})$ in the initial formal approach to this category
outline above. As such, the complex Morse theory model is not available, and
it is a basic problem to find a general perturbation scheme to make sense of
$\mathrm{FS}(\mathcal{A}^{\mathbb{C}}_{S^{1}}$), or to make sense of the
Morse–Bott version of the complex Morse theory model; see Section 2.11 for
more on this problem.

In any case, applying the finite dimensional relation between Hochschild
homology and Morse homology again, we should expect that

$HH_{*}(\mathrm{Fuet}_{M}(S^{1}))\simeq
HM_{*}(\mathrm{Re}(e^{-i\hat{\theta}}\mathcal{A}^{\mathbb{C}}_{S^{1}}))=HF^{*}(X,\omega_{\hat{\theta}}),$
(2.30)

with the right-hand side denoting the Hamiltonian Floer cohomology of
$(X,\omega_{\hat{\theta}})$. We expect that there is a categorification of the
isomorphism above to a map

$\overline{HH}_{*}(\mathrm{Fuet}_{M})\to\mathrm{Fuet}_{M}(S^{1}).$ (2.31)

which (assuming that this map is an isomorphism) recovers (2.30) by taking
$HH_{*}$ of both sides _again_. Here, $\overline{HH}_{*}$ is an invariant of a
$2$-category, which is distinct from the $1$-category (2.22) produced by
taking Hochschild homology of all hom-categories. Recall that the Fukaya
category $\mathrm{Fuk}(X)$ is really an $A_{\infty}$-category, which, from the
perspective of homotopy theory, is a particular model for a stable
$(\infty,1)$-category [Lur17]. A stable $(\infty,1)$-category has a monoidal
$(\infty,1)$-category of endomorphisms $\mathrm{End}(\mathcal{F}(X))$, and a
unit object represented by the identity functor
$\mathbf{1}:\mathcal{F}(X)\to\mathcal{F}(X)$. The Hochschild homology
$HH_{*}(\mathcal{F}(X))$ is then simply the homotopy groups of the two-sided
self-tensor product of the identity $\mathbf{1}$, thought of as an object of
$\mathrm{End}(\mathcal{F}(X))$.

Correspondingly, the Fueter $2$-category described in Section 2.2 should be
the cohomology category of a sort of $(A_{\infty},2)$-category defined by
counting Fueter maps from various domains. It is not clear what is the precise
algebraic model of an $(A_{\infty},2)$-category is appropriate for this
setting; we discuss this problem in the next section. Nonetheless, whatever
the appropriate model is, a basic desideratum is that the data of the Fueter
$2$-category should give rise, from the perspective of homotopy theory, to an
$(\infty,2)$-category, which we will also denote by $\mathrm{Fuet}_{M}$ for
convenience. An $(\infty,2)$-category is meant to be an object where where the
morphisms between two objects form an $(\infty,1)$-category. Certain
foundations for $(\infty,2)$-categories have been laid by Gaitsgory–Rozenblym
[GR17]. In particular, given an $(\infty,2)$-category $\mathcal{C}$ such as
the one associated to the Fueter $2$-category, there is an
$(\infty,2)$-category of endomorphisms $\mathrm{End}(\mathcal{C})$, and a unit
object $\mathbf{1}_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}$ corresponding to
the identity functor. Using this machinery, we get a rather inexplicit
definition of categorical Hochschild homology as

$\overline{HH}_{*}(\mathrm{Fuet}_{M}):=\mathbf{1}_{\mathrm{Fuet}_{M}}\otimes_{(Fuet_{M}\otimes
Fuet_{M}^{op})}\mathbf{1}_{\mathrm{Fuet}_{M}}.$ (2.32)

We can take a different perspective to link up with a topic of interest in
representation theory. Assume that we have a collection of generating objects
$L_{0},\ldots,L_{k}$ of $\mathrm{Fuet}_{M}$, meaning that $\mathrm{Fuet}_{M}$
embeds faithfully into the category of modules over the category
$\mathrm{End}(L_{0}\oplus\ldots\oplus L_{k})$ via the Yoneda embedding. Now
$End(L_{0}\oplus\ldots\oplus L_{k})$ is going to be a _monoidal_ $A_{\infty}$
category, where the monoidal structure arises from composition of
$1$-morphisms as in Section 2.2. We will then have that
$\overline{HH}_{*}(\mathrm{Fuet}_{M})$ is the _categorical trace_ of the
identity functor on this monoidal $1$-category [Gai+19, Chapter 3]. Thus, one
can use ansatzes motivated by representation theory to make guesses as to the
nature of $\mathrm{Fuet}_{M}(S^{1})$ for $M$ of representation-theoretic
interest.

###  2.7 Combinatorics of Fueter domains

While abstract foundations for $(\infty,2)$-category have been systematically
explored, they are not sufficient for development of the Fueter $2$-category.
Indeed, symplectic topology depends significantly on the identification of the
associahedron with a connected component the real locus in the moduli space of
stable spheres, which underlies the combinatorics of $A_{\infty}$-categories
and makes it possible to write down many basic operations such as the open-
closed map.

A higher-dimensional generalization of the $A_{\infty}$ combinatorics is
required for the full formulation of the Fueter $2$-category. One basic
combinatorial model for an $(\infty,2)$-category which generalizes is given by
the $2$-associahedra, which underly the theory of $(A_{\infty},2)$-categories
[BC18]. These were developed with the aim of piecing together the Fukaya
$A_{\infty}$ categories of all sympletic manifolds into an
$(\infty,2)$-categorical object. It is an interesting question to the authors
whether the $2$-associahedra can be arranged to control the boundary
conditions in such a way that one can give a formal construction of the
$(\infty,2)$ category $\mathrm{Fuet}_{M}$ given a plausible compactification
theory for the Fueter equations.

A concrete manifestation of this problem is a consequence of the fact that the
counts of holomorphic triangles defining composition of $1$-morphisms in the
Fueter $2$-category may give a _sum_ of $1$-morphisms of $\mathrm{Fuet}_{M}$
as an output. In particular, it is possible that the composition of
$1$-morphisms may only be defined on the total Fukaya Seidel categories

$tw\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{0},L_{1}})\otimes
tw\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{1},L_{2}})\to
tw\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{0},L_{2}}),$
(2.33)

where we recall that
$tw\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{i},L_{j}})=\mathrm{FS}^{\zeta}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{i},L_{j}})=\mathrm{Fuet}_{M}(L_{i},L_{j})$
by definition, and may not be induced by a functor on the directed Fukaya-
Seidel categories

$\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{0},L_{1}})\otimes\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{1},L_{2}})\to\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}^{\mathbb{C}}_{L_{0},L_{2}}).$
(2.34)

Indeed, the correct definition of composition may take as input a pair $p\in
L_{0}\cap L_{1}$, $q\in L_{1}\cap L_{2}$, and produce a _twisted complex of
intersection points_ (see [Sei08] for algebraic background)

$m_{2}(p,q)\in
tw\mathrm{FS}^{\zeta,\to}_{\hat{\theta}}(\mathcal{A}_{L_{0},L_{2}}^{\mathbb{C}})=\mathrm{Fuet}_{M}(L_{0},L_{2}).$

This possibility was pointed out to us by Ahsan Khan [Kha], and is explored in
that forthcoming work. See Figure 13 for a diagram of the domain of a Fueter
map producing the Maurer–Cartan datum defining the twisted complex
$m_{2}(p,q)$, as well as comments as to why such putative Maurer–Cartan data
may in fact be forced to be zero. Regardless, clarifying the combinatorial and
higher-categorical operadic structures associated to the Fueter equations is
an exciting problem, which in large part belongs to topology, geometry, and
algebra rather than to geometric analysis, and thus may be interesting and
accessible to mathematicians working in other disciplines.

Figure 13: This diagram was explained to the authors by Ahsan Khan [Kha]. It
shows asymptotic conditions drawn on $S^{2}=\partial B^{3}$ for a Fueter map
which may give rise to a Maurer–Cartan datum given by a count of $2$-morphisms
$u\in Fuet(L_{0},L_{2})(r,r^{\prime})$ for the twisted complex of intersection
points $r\in L_{0}\cap L_{2}$ corresponding to the composition of the
$1$-morphisms $p\in Fuet(L_{0},L_{1})$ and $q^{\prime}\in Fuet(L_{1},L_{2})$.
The counts of holomorphic triangles $C_{pq^{\prime}}^{r}$, as per Section 2.2,
define the untwisted object $\mathcal{O}=\oplus_{r}C_{pq^{\prime}}^{r}r$, and
the above domains define morphisms $r\to r^{\prime}$ which can be used to
deform the differential on $\mathcal{O}$ appropriately. Note however that if
each of $(L_{0},L_{1},L_{2})$ intersect transversely then all holomorphic
curves (blue) must be constant by Solomon–Verbitsky [SV19], and then
$r=r^{\prime}$ and the Maurer–Cartan datum is zero. Whether the Maurer–Cartan
data associated to such domains are ever nonzero remains to be be clarified in
general and in examples.

######  Problem .

Explain the precise cominatorics of surfaces and boundary conditions needed to
make sense of the horizontal compositions of $2$-morphisms in the Fueter
$2$-category. This likely involves the introduction of certain novel operads
which are $2$-categorical counterparts to the $A_{\infty}$ operad. These
combinatorics should be different in Gaiotto–Moore–Witten and Haydys models
for the Fueter $2$-category. Can one provide a concrete, combinatorial
definition of $(\infty,2)$-categories by following this perspective?

######  Problem .

Make sense of the cyclic bar complex of a $2$-category. Define the open-closed
map (2.31) and use it to compute $\mathrm{Fuet}_{M}$ for interesting cases.

###  2.8 Cotangent bundles of Kähler manifolds

As the previous discussions show, there are significant analytical and
combinatorial challenges that arise in the definition of the Fueter
$2$-category. We now come to the first examples where we have explicit
expectations as to the behavior of the Fueter $2$-category. These examples are
cotangent bundles of Kähler manifolds. As we explain in the introduction, the
following conjecture is a quaternionic version of Floer’s result.

######  Conjecture .

Let $X$ be a Kähler manifold with an exact symplectic form and pseudo-convex
end (for example, a Stein manifold). Let $F:X\to\mathbb{C}$ be a holomorphic
Morse function. Then, given certain geometric assumptions on $F$ near the ends
of $X$,

$\mathrm{Fuet}_{T^{*}X}(L_{0},L_{1})\simeq\mathrm{FS}(X,F),$ (2.35)

where $L_{0}$ is the zero-section of $T^{*}X$ and
$L_{1}=\Gamma({\mathrm{d}}F)$ is the graph of ${\mathrm{d}}F$.

This conjecture is supported by our Section 1. As a definition of the complex
Morse theory model of $\mathrm{FS}(X,F)$ in the Haydys version is now
available [Wan22], this conjecture should be accessible by adapting that model
to the infinite-dimensional setting and generalizing the analysis performed in
the current paper.

This conjecture, together with the existence of the Fueter $2$-category, would
have remarkable implications. Indeed, just as Lagrangian Floer cohomology
groups are invariant under Hamiltonian flows, the Fueter equation and the
associated categorical invariants enjoy natural invariance properties under
the flows of _holomorphic Hamiltonians_. We explore this phenomenon further in
Section 2.11; for now, it suffices to note that a holomorphic function

$H:M\to\mathbb{C}$

determines a _holomorphic Hamiltonian vector field_ $X_{H}$ on $M$ by

$\iota_{X_{H}}\Omega=-dH,$ (2.36)

such that $X_{H}$ is the Hamiltonian vector field of
$\mathrm{Re}(e^{-i\theta}H)$ with respect to $\omega_{\theta}$ for all
$\theta\in S^{1}$ simultaneously. As such, given a pair of holomorphic
functions $F_{1},F_{2}:X\to\mathbb{C}$, we get a triple of Lagrangians
$L_{1}=\Gamma({\mathrm{d}}F_{1})$, $L_{2}=\Gamma({\mathrm{d}}F_{2})$, and
$L_{12}=\Gamma({\mathrm{d}}(F_{1}+F_{2}))$. Writing $\pi:T^{*}X\to X$ for the
projection, the time one holomorphic Hamiltonian flow of $H=-\pi^{*}F_{1}$
takes $L_{1}$ to the zero section $L_{0}$, and $L_{12}$ to $L_{2}$. Thus, the
existence of of the composition operation

$\mathrm{Fuet}_{T^{*}X}(L_{0},L_{1})\otimes\mathrm{Fuet}_{T^{*}X}(L_{1},L_{12})\to\mathrm{Fuet}_{T^{*}X}(L_{0},L_{12})$
(2.37)

implies, after applying the time one flow of $X_{H}$ to the second hom-
category above and using Section 2.8, the surprising

######  Conjecture .

Let $X$ be as above and let $F_{1},F_{2}:X\to\mathbb{C}$ be holomorphic Morse
functions. Then, given certain geometric assumptions on $F_{1}$ and $F_{2}$
near the ends of $X$, there exists a monoidal functor

$\mathrm{FS}(X,F_{1})\otimes\mathrm{FS}(X,W_{2})\to\mathrm{FS}(X,F_{1}+F_{2}).$
(2.38)

Given a third holomorphic Morse function $F_{3}$, under appropriate
assumptions on the geometry at infinity, the monoidal products for the pairs
$(F_{1},F_{2})$, $(F_{2},F_{3})$, $(F_{1}+F_{2},F_{3})$, and
$(F_{1},F_{2}+F_{3})$ should commute.

This is surprising, as there is no straightforward way to compose Lefschetz
thimbles associated to multiple symplectic Lefschetz fibrations on the same
symplectic manifold using purely symplectic methods. Searching for the
definitions of such functors using symplectic geometry is a significant
problem; we caution that the functors of Section 2.8 should only exist as
stated under yet-uncertain tameness assumptions on the geometry of the $F_{i}$
at infinity, together with similarly uncertain geometric conditions ruling out
the contribution of essential singularities of Fueter maps to the categorical
structures. The most likely setting where such functors may exist are when $X$
is flat, as in that case all bubbling of Fueter maps can be ruled out a
priori. It is possible that in the general case, the bubbling of Fueter maps
contributes in an interesting way to the above algebraic structure. We explore
a possible interpretation of Section 2.8 in Section 2.9 below.

###  2.9 Examples from algebraic geometry

Using the conjectures of Section 2.8 together with the existing results on
$2d$ mirror symmetry, one can derive predictions regarding existing invariants
which may be surprising without the perspective of the Fueter TQFT. We thank
Denis Auroux for initial inspiration regarding Section 2.9 and Justin Hilburn
and Benjamin Gammage for further discussion regarding the relevant toric
geometry.

For example, let us specialize Section 2.8 to $X=(\mathbb{C}^{*})^{n}$. In
this setting, $X$ and $T^{*}X$ admit flat metrics, and thus bubbling of Fueter
maps is a priori excluded [HNS09], and thus the conjecutres are more likely to
hold without modification. The study of the Fukaya–Seidel categories of
Laurent polynomials $F:(\mathbb{C}^{*})^{n}\to\mathbb{C}$ is part of the
subject of homological mirror symmetry for toric varieties, which identify
these categories with categories of coherent sheaves on certain toric
varieties. While there are at this point a variety of approaches to the
subject [Abo09, Fan+08, HH22, She21], we state an early result below:

######  Theorem  (Abouzaid [Abo09]).

Let $Y$ be a smooth projective toric variety, which is determined by a
simplicial rational polyhedrial fan $\Delta$ with $|\Delta|=\mathbb{R}^{n}$
and such that all the maximal cones of $\Delta$ are nonsingular [Ful93]. Then
$D^{b}Coh(Y))\simeq\mathrm{FS}(Y,F_{Y})$, where
$F:(\mathbb{C}^{*})^{n}\to\mathbb{C}$ is any generic Laurent polynomial which
consists of monomials with exponents given by the primitive vertices of the
$1$-cones of $\Delta$.

One can ask what operation on toric varieties corresponds to taking the sum
the sum of two such Laurent polynomials $(F_{Y_{1}},F_{Y_{2}})$. We argue that
this corresponds to superimposing the associated fans. Explicitly, let us take
the product of the toric varieties $Y_{1}\times Y_{2}$ and then taking a GIT
quotient with respect to the diagonal torus action
$(\mathbb{C}^{*})^{n}\subset(\mathbb{C}^{*})^{n}\times(\mathbb{C}*)^{n}\curvearrowright
Y_{1}\times Y_{2}$. We take this with respect to an equivariant ample line
bundle $\xi=\xi_{1}\boxtimes\xi_{2}$ where $\xi_{i}$ is an equivariant ample
line bundle on $Y_{i}$. The GIT quotient $Y=Y_{1}\times
Y_{2}//_{\xi}(\mathbb{C}^{*})^{n}$ will will have a fan given by the cones
through zero on the faces of the convex hull of the union of each of the
Newton polytopes associated to $F_{Y_{1}}$ and $F_{Y_{2}}$. Indeed, this
follows from the fact that the dual polytope $(C_{1}\cap C_{2})^{*}$ of an
intersection of convex polytopes $C_{1},C_{2}$ is the convex hull of
$C_{1}^{*}\cup C_{2}^{*}$, together with the bijection between ample
equivariant bundles on a toric variety $Y_{\Delta}$ and polyhedra with normal
fan $\Delta$, and the description of the GIT quotient of a toric variety by a
subtorus [Tha96].

Thus, given a pair of smooth proper toric varieties $(Y_{1},Y_{2})$ with
associated Laurent polynomials $(F_{Y_{1}},F_{Y_{2}})$, we have a diagram

${Y:=Y_{1}\times
Y_{2}//_{\xi}(\mathbb{C}^{*})^{n}}$${Y_{1}}$${Y_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\pi_{2}}$
(2.39)

We conjecture that this interesting operation may be mirror to addition of
Laurent polynomials in the following sense:

######  Conjecture .

Suppose that the toric variety $Y$ as above is smooth and proper. Then the
functor of Section 2.8, under homological mirror symmetry for
$(Y_{1},Y_{2},Y)$, becomes the functor

$\begin{gathered}D^{b}Coh(Y_{1})\otimes D^{b}Coh(Y_{2})\to D^{b}Coh(Y)\\\
(\mathcal{L}_{1},\mathcal{L}_{2})\mapsto\pi_{1}^{*}\mathcal{L}_{1}\otimes\pi_{2}^{*}\mathcal{L}_{2}.\end{gathered}$
(2.40)

Figure 14: A composition functor on toric varieties inspired by Conjecture
2.8. Here, the sum of $(x+y+1/xy)$ and $x+y+1/x+1/y$ is mirror to the del
Pezzo surface given by the $2$-point blow-up of $\mathbb{P}^{2}$, which is
equal to the $1$-point blow-up of $\mathbb{P}^{1}\times\mathbb{P}^{1}$.
Addition of Laurent polynomials corresponds to the convex hull of the union of
their Newton polytopes.

It is a basic test of our understanding of the Fueter $2$-category to fully
understand the composition operation on hom-categories in the simplest
setting. One should be able to generalize this conjecture to cases where $Y$
is not smooth by taking a resolution of $Y$. It is interesting to consider the
case where the $Y_{i}$ are not proper; unfortunately, while
$\mathrm{FS}(Y_{i},F_{Y_{i}})$ can be defined via sheaf-theoretic methods
[GPS18], there is currently no known complex Morse theory model of
Fukaya–Seidel categories where the wrapping is not fully stopped. To get a
handle on the functor of (2.40), recall that equivariant line bundles on
$Y_{i}$ correspond to piecewise integral functions on the fan which are linear
on each cone, and to forget the equivariant structure one considers such
piecewise-linear functions up to globally linear functions [Kly90]. Thus, the
map (2.40) simply corresponds to addition of piecewise-linear functions; the
sum of two piecewise-linear functions is piecewise linear on the superimposed
cones! Moreover, these line bundles are mirror to explicit Lagrangian
submanifolds [Abo09]. This example can be worked out completely in the
simplest nontrivial case, as in Figure 14.

#### Singularity categories.

There are a number of other examples in 2-dimensional mirror symmetry beyond
the case of toric mirror symmetry of Section 2.9 where multiple functions the
same manifold are known the be mirror to certain objects of algebraic
geometry. A natural case is the setting of $X=\mathbb{C}^{n}$, which is also
flat; this includes the cases of mirror symmetry for $ADE$ singularities
[KS01, Ebe16] and Berglund–Hubsch mirror symmetry [BH93]. It is interesting to
explore the consequences of Section 2.8 on the $B$-side in these cases and to
identify constructions for the putative monoidal operations via the methods of
algebraic geometry. This may help clarify under what geometric conditions the
composition operations of Section 2.8 make sense, which should suggest the
appropriate way to handle the differential geometry of non-compact complex
Lagrangians in the Fueter $2$-category.

###  2.10 $3d$ mirror symmetry

This section is strongly influenced by our discussions with Justin Hilburn.
Thus far, we have discussed the Fueter TQFT from a purely mathematical
perspective, by complexifying and categorifying the Fukaya category of
symplectic topology. A different way to arrive at the same story is through
the perspective of quantum field theory. Indeed, the Fueter equation can be
seen to arise by by studying the $3d\;N=4\;\sigma$-model, which is a certain
supersymmetric field theory where the classical fields are maps from
$3$-manifolds to a hyperkähler $4n$-manifold, augmented with supersymmetric
data. This model is the dimensional reduction of the $4d\;N=2$ $\sigma$-model,
which is a corresponding theory for maps from a $4$-manifold with a
hyperkähler target. The identification of the $4$-dimensional Fueter equation
as the partial differential equation governing the $A$-type twist of the
$4d\;N=2$ model dates back to [AF94]; one can derive from that work that the
$A$-type twist of the $3d\;N=4$ $\sigma$-model is governed by the Fueter
equation (1.2) , using the fact that the dimensional reduction of the
$4d\;N=2$ model is the $3d\;N=4$ model. Dimensionally reducing further, we
have that the reduction of the $3d\;N=4$ model to $2d$ is the
$2d\;N=(2,2)$-model, the $\sigma$-model responsible for mirror symmetry of
Calabi-Yau manifolds; this corresponds to the fact that the dimensional
reduction of the Fueter equations is the pseudo-holomorphic curve equation.

The $3d\;N=4$ and $4d\;N=2$ theories are objects of active study. Most
commonly in mathematics, they arise in their _gauged variants_. Namely, there
are natural partial differential equations on maps from a $3$ or $4$-manifold
to a hyperkähler manifold $M$ equipped with a hyperhamiltonian action [Hit87]
of a compact Lie group $G$. We do not write down these equations in full
generality, but we note that in the case of $M=T^{*}V$ for $V$ a complex
representation of a compact Lie group, these equations reproduce many standard
equations of gauge theory; for example, taking $V=\mathbb{C}$ and $M=U(1)$
recovers, after a minor modification, the usual Seiberg-Witten equations
[Nak15, Remark 6.2]. Thus, gauge theorists should think of this paper as
studying the _nonlinear, un-gauged variant of the Seiberg-Witten equations_.

#### The $B$-type twist.

All of the field theories mentioned above, besides admitting an $A$-type
twist, also admit a $B$-type twist. Just as the $A$-type twist of the $3d$
$N=4$ $\sigma$-model has a $2$-category of boundary conditions
$\mathrm{Fuet}_{M}$, which we explore in this paper, the $B$-type twist also
has a $2$-category of boundary conditions, for which the initial attempts at a
definition were given by Kapustin–Rozansky–Saulinas [KRS09, KR09]. We thus
call this $2$-category of boundary conditions for the $B$-type twist the _KRS_
$2$-category $KRS_{M}$. This category, just like the category $Coh_{X}$ of
boundary conditions of the $B$-type twist of the $2D$ $N=(2,2)$ $\sigma$-model
into $X$, is of purely algebraic nature. The original papers propose that for
a cotangent bundle $T^{*}X$, the objects $2$-category $KRS_{T^{*}X}$ should
contain the complex Lagrangians in $X$, just like the Fueter $2$-category. The
model case for morphisms in the KRS $2$-category is the analog of Section 2.8
for the $B$ model: when $L_{0}\subset T^{*}X$ is the zero section and
$L_{1}\subset T^{*}X$ is the graph of the differential of $F:X\to\mathbb{C}$,
one has

$KRS_{T^{*}X}(L_{0},L_{1})\simeq D_{\mathbb{Z}/2}(X,F)$ (2.41)

where $D_{\mathbb{Z}/2}(X,F)$ is a small modification of the category
$\mathrm{MF}(F)$ of _matrix factorizations_ [Orl04], the $B$-twist analog of
the Fukaya-Seidel category. . Similarly, in the KRS $2$-category, the analog
of Section 2.8 holds with Fukaya-Seidel categories replaced by the categories
$D_{\mathbb{Z}/2}(X,F)$.

More generally, $KRS_{M}$ should make sense for any complex symplectic $M$,
and should contain complex Lagrangians as objects. The morphism categories
between complex Lagrangians $KRS_{M}(L_{0},L_{1})$ should the categories of
global sections of a sheaf of categories supported on $L_{0}\cap L_{1}$ which
near a point $p\in L_{0}\cap L_{1}$ is meant to be a shift of the category
$D_{\mathbb{Z}/2}(U,F)$ for some holomorphic $F:L_{0}\supset U\to\mathbb{C}$,
where $L_{1}$ is viewed as the graph of the differential of a function $F$
defined on a neighborhood $U$ of $p$. Beyond this, Kapustin–Rozansky–Saulinas
propose that in some sense, there should be more objects of the KRS
$2$-category beyond the complex Lagrangians; in particular, a holomorphic
generating function should specify an object in $KRS(T^{*}X)$, and more
generally, certain sheaves of categories should define objects in the $KRS$
$2$-category. Partial foundational work for this idea can be found in [Gai13,
Ari20].

#### 3d mirror symmetry.

Most commonly, the $3d\;N=4$ theory is studied in its gauged linear variant
via the geometry and representation theory associated to its _Higgs and
Coloumb branches_ , which are certain non-compact complete hyperkähler
manifolds [Huy11]. The $3d\;N=4$ theories are supposed to come in pairs, and
the correspondence between paired theories goes by the name of _3d mirror
symmetry_ [IS96]. Given a pair of dual theories
$\mathcal{X},\mathcal{X}^{\vee}$, the correspondence is expected to pair

  * •

The Coloumb branch of $\mathcal{X}$ with the Higgs branch of
$\mathcal{X}^{\vee}$

  * •

The $A$-type twist of $\mathcal{X}$ with the $B$-type twist of
$\mathcal{X}^{\vee}$. In particular, there should be an equivalence of
$2$-categories of boundary conditions.

In certain cases, dual pairs of $3d$ $N=4$ theories are associated to pairs of
hyperkähler manifolds, or stacks, $X,X^{\vee}$, and one expects that the
Fueter $2$-category of $X$ is exchanged with the KRS $2$-category of
$X^{\vee}$. This is the statement of 2-categorical 3d mirror symmetry. Some
initial progress towards this idea has been achieved in [GHM22], who develop a
conjectural sheaf theoretic model, by analogy to [GPS19], of a _wrapped_
variant of the Fueter $2$-category and relate this to a KRS $2$-category of a
3d mirror geometry. We discuss the problem of wrapping Fueter maps below.

###  2.11 Further analytic aspects

To establish the existence categorical structures described above, a number of
essential analytic aspects of the Fueter equation remain to be clarified.
Excluding the well-known open problems regarding the compactness theory of the
Fueter equation, we outline what we see as the basic conceptual challenges
suggested by this categorical framework. These are the problems of
transversality of the critical locus and Fueter trajectories, of wrapping
Fueter maps, and the most general problem of the domain of definition of the
Fueter $2$-category.

#### Morse–Bott theory.

An essential challenge in setting up the Fueter $2$-category is that the
holomorphic action functional $\mathcal{A}^{\mathbb{C}}_{L,L}$ is not Morse,
but is instead _Morse-Bott_ , with critical locus equal to $L$ viewed as the
set of constant paths from $L$ to $L$. As such, one does not have access to
the basic structures of the complex Morse theory model of
$\mathrm{FS}(\mathcal{A}^{\mathbb{C}}_{L,L})$, which require that the
holomorphic function is Morse, and a Morse–Bott analog requires significant
new ideas

######  Problem .

Make sense of the monoidal endomorphism category $\mathrm{Fuet}_{M}(L,L)$
where $L$ is a complex Lagrangian.

In the finite dimensional case, one way to define the Fukaya–Seidel category
of a Morse–Bott symplectic Lefschetz fibration $F:X\to\mathbb{C}$ by
perturbing $F$ through maps to a nearby map $\tilde{F}$ which is Morse (if
this is possible), and defining $\mathrm{FS}(X,F):=\mathrm{FS}(X,\tilde{F})$.
It is crucial, to control the behavior of the perturbation
$F\rightsquigarrow\tilde{F}$ at infinity, since otherwise this definition of
$\mathrm{FS}(X,F)$ may be highly dependent on perturbation. For example, if
the critical locus $\mathrm{Crit}(F)$ of $F$ is compact, one should use a
perturbation of $F$ which is constant near infinity. If $\mathrm{Crit}(F)$ is
non-compact, then there may be no preferred perturbation, and there may be
multiple inequivalent ways of defining the complex Morse theory of $F$.
Indeed, in the real case of the Floer cohomology of a non-compact Lagrangian
(which amounts to the real Morse theory of a real Morse–Bott action functional
with non-compact critical locus), there are essentially different variants of
Lagrangian Floer cohomology, such as the wrapped [AS05] or infinitesimally
wrapped [NZ09] variants. We note that Section 1 and the associated Section 2.8
suggest that in the case of non-compact complex Lagrangians $L$, perturbing
$L$ through nearby exact complex Lagrangians can drastically change the
behavior of the Fueter hom-category, essentially because different holomorphic
functions can have very different asymptotic geometry.

Unfortunately, even the idea of perturbing the Morse–Bott functional
$\mathcal{A}^{\mathbb{C}}_{L,L}$ to a Morse action functional is non-trivial.
The basic challenge is the control of the energy (1.7) of the associated
Fueter trajectories, which are needed to have any hope of compactness of
resulting moduli spaces.

######  Problem .

Find a way of perturbing $\mathcal{A}^{\mathbb{C}}_{L,L}$ through action
functionals to one with nondegenerate critical points such there still exist
energy bounds for the associated perturbations of Fueter trajectories.

In the setting of Lagrangian Floer cohomology, when $L$ is compact, one can
perturb $\mathcal{A}_{L,L}$ to a Morse action functional by adding a
Hamiltonian term to the symplectic action, or equivalently by perturbing $L$
via an exact deformation to a new Lagrangian $\tilde{L}$ which intersects $L$
transversely. Unfortunately, while it is easy to prove energy bounds for the
corresponding perturbations of $\mathcal{A}^{\mathbb{C}}_{L,L}$ in the