Holomorphic Floer theory and the Fueter equation Aleksander Doan Semon Rezchikov August 23 2023

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Holomorphic Floer theory and the Fueter equation
Aleksander Doan, Semon Rezchikov
August 23, 2023
Abstract
We outline a proposal for a 2-category FuetMassociated to a hyperk¨ahler man-
ifold M, which categorifies the subcategory of the Fukaya category of Mgenerated
by complex Lagrangian submanifolds. Morphisms in this 2-category are formally the
Fukaya–Seidel categories of holomorphic symplectic action functionals. As such, FuetM
is based on counting maps [0,1]×R2Msatisfying the Fueter equation with boundary
values on complex Lagrangians.
We make the first step towards constructing this category by establishing some
basic analytic results about Fueter maps, such as an energy bound and a maximum
principle. When M=TXis the cotangent bundle of a K¨ahler manifold Xand
L0, L1are the zero section and the graph of the differential of a holomorphic function
F:XC, we prove that Fueter maps correspond to the complex gradient trajectories
of Fin 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 Introduction 3
2 Formal aspects 10
2.1 The Fueter 2-category ............................ 10
2.2 Complex Morse theory ............................ 13
2.3 The Fueter 2-category from complex Morse theory ............ 20
2.4 Decategorifications of the Fueter 2-category ................ 22
2.5 Cotangent bundles of K¨ahler manifolds ................... 24
2.6 3dmirror symmetry ............................. 25
3 The Fueter equation 27
3.1 Quaternion-antilinear maps ......................... 27
3.2 Fueter maps .................................. 28
3.3 Boundary conditions ............................. 30
3.4 Taming triples ................................ 31
3.5 Energy bounds ................................ 34
1
arXiv:2210.12047v2 [math.SG] 21 Aug 2023
4 Convexity theory 37
4.1 Complex convexity and pseudo-holomorphic maps ............ 37
4.2 Quaternionic convexity and Fueter maps .................. 39
4.3 Conical completions ............................. 40
5 Cotangent bundles of almost complex manifolds 42
5.1 Almost complex and quaternionic structures ................ 42
5.2 Convexity ................................... 42
5.3 Cauchy–Riemann operators ......................... 45
5.4 Complexified Floer correspondence ..................... 46
5.5 Remarks on the modified equation ..................... 48
A Topological field theories 51
B Problems and conjectures 55
B.1 More on the Fueter 2-category ....................... 55
B.2 Fueter Category and Holomorphic Floer Theory .............. 56
B.3 Combinatorics of Fueter domains ...................... 59
B.4 Examples from algebraic geometry ..................... 61
B.5 Further analytic aspects ........................... 63
B.6 Further geometric aspects .......................... 67
C Floer’s theorem revisited 69
D Notation and conventions 71
D.1 Coordinates .................................. 71
D.2 Symplectic geometry ............................. 71
D.3 Lagrangian Floer theory ........................... 72
D.4 Complex gradient flow ............................ 72
D.5 Holomorphic Floer theory .......................... 73
2
1 Introduction
This paper studies a complexification of Lagrangian Floer homology. Like its real
counterpart, this theory arises from a σ-model: it involves the study of maps
U:EM
from a three-manifold E, which we will take to be
E= [0,1] ×R2,
to a manifold Mof real dimension 4nequipped with an almost quaternionic structure
(I, J, K), that is: a triple of almost complex structures satisfying the quaternionic rela-
tion IJ =K. The analog of the pseudo-holomorphic map equation used in Lagrangian
Floer theory,
u:R×[0,1] M,
tu+J(u)τu= 0,(1.1)
where (t, τ) are the coordinates on R×[0,1], is the Fueter equation:
U: [0,1] ×R2M,
I(U)τU+J(U)sU+K(U)tU= 0,(1.2)
where (τ, s, t) are the coordinates on [0,1] ×R2. This equation is perhaps the sim-
plest higher-dimensional generalization of the pseudo-holomorphic map equation which
shares 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;Wal17c]. 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 generalizing the well-known structures appearing
in symplectic topology, with rich connections to low-dimensional topology, representa-
tion theory, and mathematical physics. The goal of this paper is to explore some of
these ideas and connections.
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 Ma quaternionic vector space,
is the 3dmassless Dirac equation. Solutions to the Dirac equation on R3are
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 Fueter’s paper.
Solutions to the Fueter equation whose domain Eis a closed, framed three-
manifold are critical points of a functional on the space of maps EM. Gradient
trajectories of this functional are solutions to the four-dimensional version of the
Fueter equation for maps E×RM. When Mis a flat hyperk¨ahler manifold,
the moduli spaces of solutions to these equations can be used to build an invariant
of Manalogous to Hamiltonian Floer homology [HNS09].
The Fueter equation arises naturally as a degeneration limit of various equa-
tions in gauge theory, such as the instanton equation on Riemannian manifolds
with special holonomy [DT98;DS11;Hay12;Wal17a;Wal17b] and the general-
ized Seiberg–Witten equations on three- and four-dimensional manifolds [Tau13;
Tau15;Tau16;HW15;WZ21]. In both situations, Mis the hyperk¨ahler quotient
of a hyperk¨ahler manifold by an action of a compact Lie group, a typical exam-
ple being the moduli space of framed instantons on R4. There is also a parallel
picture for calibrated submanifolds of special holonomy manifolds.
3
Maps satisfying the Fueter equation appear as a result of applying supersymmetric
localization to the physical 3d N= 4 σ-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. Heuristi-
cally, one can arrive at many expectations regarding the Fueter equation by complexi-
fying 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 clas-
sical theorem of Floer on cotangent bundles [Flo88b;Flo88a], which we now briefly
review. Let Xbe a closed manifold with a Morse function f:XR. The cotangent
bundle TXis a symplectic manifold and the zero section and graph of dfare two
Lagrangian submanifolds, which we denote by L0and L1. Floer proved that if the C2
norm of fis sufficiently small (which can be always guaranteed by rescaling f), then
the Lagrangian Floer homology of L0and L1is canonically identified with the Morse
homology of f:
HF(L0, L1) = HM(X, f ).(1.3)
The two homology groups have the same generators, as the intersection points of L0
and L1correspond 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 L0and L1and asymptotic to L0L1. Floer proved (1.3) by
constructing a time-dependent almost complex structure Jτon TXcompatible with
the symplectic structure and with the property that Jτ-holomorphic strips in M=TX
correspond to the gradient trajectories of fin X.
We prove a complexified version of Floer’s correspondence. On the Morse theory
side, we assume that Xis a K¨ahler manifold and F:XCis a holomorphic function.
Denote by Ithe complex structure on X. Gradient trajectories in Floer’s theorem are
replaced by the complex gradient trajectories of F, that is: maps v:R2Xsatisfying
I(v) = iF(v)
or, in coordinates (s, t) on R2:
sv+I(v)(tv− ∇Re(F)(v)) = 0.(1.4)
This equation is also known as the Witten equation or ζ-instanton equation in the
physics literature. Of course, it is the same as the non-homogenous perturbation of the
I-holomorphic map equation by the Hamiltonian function Im(F), introduced by Floer.
Solutions with sv= 0 are simply the real gradient trajectories of Re(F).
On the Lagrangian side, we consider, as before, the cotangent bundle M=TX.
The complex structure on Xinduces one on M, and Mhas a canonical holomorphic
symplectic form Ω. Let L0, L1be the zero section and the graph of Re(dF). They
are complex submanifolds of Mwhich are Lagrangian with respect to Ω. Pseudo-
holomorphic strips (1.1) are replaced by Fueter strips in M. By definition, a Fueter
strip is a map U: [0,1] ×R2Msatisfying the Fueter equation (1.2), with boundary
on L0and L1, asymptotic to L0L1as t→ ±∞ and to pseudo-holomorphic strips as
s→ ±∞. These asymptotic boundary conditions are explained in detail in Section 3.
4
Theorem 1.1. Let Xbe a compact K¨ahler manifold with boundary and let F:X
Cbe a holomorphic function. If Fis C2small, then there is a τ-dependent almost
quaternionic structure (Iτ, Jτ, Kτ)on M=TXsuch that all Fueter strips
U: [0,1] ×R2M,
Iτ(U)τU+Jτ(U)sU+Kτ(U)tU= 0,
with boundary on L0and L1, the zero section and the graph of Re(dF), correspond to
the complex gradient trajectories of Fin X.
Remark 1.2. When restricted to the asymptotic boundary s→ ±∞, this result
recovers Floer’s correspondence between the real gradient trajectories of Re(F) and
pseudo-holomorphic strips.
Remark 1.3. While Theorem 1.1 concerns compact manifolds with boundary and
arbitrary holomorphic functions, typically one considers the complex gradient equation
for a holomorphic Morse function F:b
XCon a noncompact, complete K¨ahler
manifolds (or, more generally, a symplectic Lefschetz fibration). Under appropriate
assumptions on the growth of Fand the geometry of b
Xat infinity, all critical points
and complex gradient trajectories of Flie in a fixed compact subset Xb
X[Wan22],
[FJY18]. There is an analogous maximum principle for Fueter strips discussed below.
Remark 1.4. This result is a special case of Theorem 5.5 about the cotangent bundles
of almost complex manifolds. In the general case, the correspondence holds for a
perturbation of the Fueter equation by first order terms which are proportional to the
distance from the zero section and the torsion of a natural connection on the almost
complex manifold. This perturbation term vanishes if and only if the manifold is
ahler.
Theorem 1.1 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. For a holomorphic Morse function F:XCon an K¨ahler manifold or, more
generally, a symplectic Lefschetz fibration, there should be an associated A-
category FS(X, F ) constructed using the complex gradient trajectories of F. 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 the analogies with the real case:
Morse theory Complex Morse theory
smooth manifold Xahler manifold X
Morse function f:XRholomorphic Morse function F:XC
gradient trajectories of fcomplex gradient trajectories of F
Morse homology HM(X, f ) Fukaya–Seidel category FS(X, F )
2. Let (M, I, J, K) be a hyperk¨ahler manifold (or, more generally, an almost quater-
nionic manifold with three symplectic forms). For every pair of complex La-
grangians L0, L1M, i.e. submanifolds which are holomorphic with respect to I
and Lagrangian with respect to the symplectic forms corresponding to Jand K,
there should be an associated A-category Fuet(L0, L1) constructed using Fueter
strips in Mwith boundary on L0and L1. Comparing again to the real case:
Floer theory Complex Floer theory
symplectic manifold Mhyperk¨ahler manifold M
Lagrangians L0, L1complex Lagrangians L0, L1
pseudo-holomorphic strips in MFueter strips in M
Floer homology HF(L0, L1) Fueter category Fuet(L0, L1)
5
摘要:

HolomorphicFloertheoryandtheFueterequationAleksanderDoan,SemonRezchikovAugust23,2023AbstractWeoutlineaproposalfora2-categoryFuetMassociatedtoahyperk¨ahlerman-ifoldM,whichcategorifiesthesubcategoryoftheFukayacategoryofMgeneratedbycomplexLagrangiansubmanifolds.Morphismsinthis2-categoryareformallytheFu...

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