Topological 5d ${\cal N}=2$ Gauge Theory: Novel Floer Homologies,

their Dualities, and an $A_{\infty}$-category of Three-Manifolds

Arif Er***E-mail<EMAIL_ADDRESS>Zhi-Cong Ong†††E-mail<EMAIL_ADDRESS>and Meng-Chwan Tan‡‡‡E-mail<EMAIL_ADDRESS>
Department of Physics
National University of Singapore
2 Science Drive 3, Singapore 117551

Abstract

We show how one can define novel gauge-theoretic Floer homologies of four,
three and two-manifolds from the physics of a certain topologically-twisted 5d
${\cal N}=2$ gauge theory via its supersymmetric quantum mechanics
interpretation. They are associated with Vafa-Witten, Hitchin and complexified
BF configurations on the four, three and two-manifolds, respectively. We also
show how one can define novel symplectic Floer homologies of Hitchin spaces,
which in turn will allow us to derive novel Atiyah-Floer correspondences that
relate our gauge-theoretic Floer homologies to symplectic intersection Floer
homologies of Higgs bundles. Furthermore, topological invariance and 5d
“S-duality” suggest a web of relations and a Langlands duality amongst these
novel Floer homologies and their loop/toroidal group generalizations. Last but
not least, via a 2d gauged Landau-Ginzburg model interpretation of the 5d
theory, we derive, from the soliton string theory that it defines and the 5d
partition function, a Fukaya-Seidel type $A_{\infty}$-category of Hitchin
configurations on three-manifolds and its Atiyah-Floer correspondence. Our
work therefore furnishes purely physical realizations and generalizations of
the mathematical conjectures and constructions of Haydys [1], Wang [2] and
Abouzaid-Manolescu [3], and more.

###### Contents

  1. 1 Introduction, Summary and Acknowledgements
  2. 2 A Topological 5d $\mathcal{N}=2$ Gauge Theory on $M_{4}\times\mathbb{R}$
    1. 2.1 Twisting along $M_{4}$
    2. 2.2 The Topological Action and the Haydys-Witten Equations
  3. 3 A Haydys-Witten Floer Homology of Four-Manifolds
    1. 3.1 5d Gauge Theory to SQM
    2. 3.2 An HW Floer Homology of $M_{4}$
  4. 4 A 4d-Haydys-Witten Floer Homology of Three-Manifolds
    1. 4.1 KK Reduction of HW Theory on $S^{1}$ and the Corresponding SQM
    2. 4.2 A 4d-HW Floer Homology of $M_{3}$
  5. 5 A 3d-Haydys-Witten Floer Homology of Two-Manifolds
    1. 5.1 KK Reduction of HW Theory on $S^{1}\times S^{1}$ and the Corresponding SQM
    2. 5.2 A 3d-HW Floer Homology of $M_{2}$
  6. 6 Symplectic Floer Homologies, Flat $G_{\mathbb{C}}$ Connections, and $\theta$-Hitchin Space
    1. 6.1 A 3d Sigma Model on $S^{1}\times S^{1}\times\mathbb{R}$
    2. 6.2 An SQM in the Double Loop Space of Flat $G_{C}$ Connections
    3. 6.3 A Symplectic Floer Homology in the Double Loop Space of Flat $G_{\mathbb{C}}$ Connections
    4. 6.4 An SQM in the Interval Space of the Loop Space of Flat $G_{\mathbb{C}}$ Connections
    5. 6.5 A Symplectic Intersection Floer Homology in the Loop Space of Flat $G_{\mathbb{C}}$ Connections
    6. 6.6 An SQM in the Interval Space of the Path Space of $\theta$-Hitchin Space
    7. 6.7 A Symplectic Intersection Floer Homology in the Path Space of $\theta$-Hitchin Space
  7. 7 Haydys-Witten Atiyah-Floer Correspondences
    1. 7.1 Splitting HW Theory on $M_{5}=M_{3}\times M_{1}\times\mathbb{R}$
    2. 7.2 An HW Atiyah-Floer Correspondence of $M_{3}\times S^{1}$
    3. 7.3 A 4d-HW Atiyah-Floer Correspondence of $M_{3}$
    4. 7.4 A 5d “Rotation” and a 4d-HW/VW Atiyah-Floer Correspondence Duality
  8. 8 Topological Invariance, 5d “S-duality”, and a Web of Relations
    1. 8.1 Topological Invariance of HW Theory and the Floer Homologies
    2. 8.2 5d “S-duality” of HW Theory as a Langlands Duality of Loop/Toroidal Group 4d/3d-HW Floer Homology
    3. 8.3 A Web of Relations Amongst the Floer Homologies
  9. 9 A Fukaya-Seidel type $A_{\infty}$-category of Three-Manifolds
    1. 9.1 HW Theory on $M_{3}\times\mathbb{R}^{2}$ as a 2d Model on $\mathbb{R}^{2}$ or SQM in Path Space
    2. 9.2 Non-constant Paths, Solitons, and Time-independent HW Configurations
    3. 9.3 The 2d Model on $\mathbb{R}^{2}$ and an Open String Theory in $\mathfrak{A}_{3}$
    4. 9.4 A Gauged LG Model, Soliton String Theory, the 5d-HW Partition Function, and an FS $A_{\infty}$-category of Hitchin Configurations on $M_{3}$
    5. 9.5 An Atiyah-Floer Correspondence for the FS $A_{\infty}$-category of Hitchin Configurations on $M_{3}$

##  Introduction, Summary and Acknowledgements

Introduction

In a visionary mathematical paper [1] by Haydys, he conjectured, with support
from some preliminary computations, novel invariants assigned to five, four
and three-manifolds from gauge theory. Specifically, he defined a certain
five-dimensional equation from which he could associate an integer to a five-
manifold, Floer-type homology groups to a four-manifold, and a Fukaya–Seidel-
type category to a three-manifold. This five-dimensional equation later
appeared in [4], too, albeit in a purely physical setting where the five-
manifold was of the form $M_{4}\times\mathbb{R}$, whence it was thereafter
named the Haydys-Witten equation.

In this paper, we aim, amongst other things, to physically realize the
aforementioned gauge invariants assigned to four and three-manifolds, and
derive their connection to symplectic invariants. To this end, we will study,
on various possibly decomposable five-manifolds in different topological
limits, the topologically-twisted 5d $\mathcal{N}=2$ gauge theory defined in
[5] whose BPS equations are exactly the Haydys-Witten equations. As an
offshoot, we would be able to derive a web of mathematically-novel relations
between these and a variety of other invariants, their Atiyah-Floer
correspondences, and more.

The computational techniques we employ are mainly those of standard Kaluza-
Klein reduction, topological reduction as pioneered in [6], recasting gauge
theories as supersymmetric quantum mechanics as pioneered in [7], and the
physical interpretation of Floer homology groups in terms of supersymmetric
quantum mechanics in infinite-dimensional space as elucidated in [8].

Let us now give a brief plan and summary of the paper.

A Brief Plan and Summary of the Paper

In $\S$2, we discuss general aspects of a topologically-twisted 5d
${\mathcal{N}}=2$ theory on $M_{5}=M_{4}\times\mathbb{R}$ resulting from the
Haydys-Witten (HW) twist, where the gauge group $G$ is taken to be a real,
simple, compact Lie group.

In $\S$3, we recast the 5d-HW theory as a 1d supersymmetric quantum mechanics
(SQM) in the space $\mathfrak{A}_{4}$ of irreducible $(A,B)$ fields on $M_{4}$
with action (3.6), where $A\in\Omega^{1}(M_{4})$ and $B\in\Omega^{2,+}(M_{4})$
are a gauge connection and self-dual two-form, respectively. This will in turn
allow us to express the partition function as (3.14):

$\boxed{\mathcal{Z}_{\text{HW},M_{4}\times\mathbb{R}}(G)=\sum_{k}{\cal
F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})=\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)=\mathcal{Z}^{\text{Floer}}_{\text{HW},M_{4}}(G)}$
(1.1)

where $\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)$ is a _novel_ Haydys-Witten
Floer homology class assigned to $M_{4}$ of degree $d_{k}$, defined by Floer
differentials described by the gradient flow equations (3.10):

$\boxed{\begin{aligned}
\frac{d{A}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{4}}\frac{\partial
V_{4}(A,B)}{\partial A^{b}}\\\
\frac{d{B}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{4}}\frac{\partial
V_{4}(A,B)}{\partial B^{b}}\end{aligned}}$ (1.2)

and Morse functional (3.12):

$\boxed{V_{4}(A,B)=\int_{M_{4}}\,\text{Tr}\,\Big{(}F^{+}\wedge\star
B+\frac{1}{3}B\wedge(\star_{3}(\mathscr{B}\wedge_{3}\mathscr{B}))\Big{)}}$
(1.3)

The chains of the HW Floer complex are generated by fixed critical points of
$V_{4}$, which correspond to _time-invariant Vafa-Witten (VW) configurations
on $M_{4}$_ given by time-independent solutions to the 4d equations (3.13):

$\boxed{\begin{aligned}
F^{+}+\star_{3}(\mathscr{B}\wedge_{3}\mathscr{B})&=0\\\ D\star
B&=0\end{aligned}}$ (1.4)

Note that $\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)$ was conjectured to exist by
Haydys in [1, §5]. We have therefore furnished a purely physical realization
of his mathematical conjecture.

In $\S$4, we let $M_{4}=M_{3}\times S^{1}$, and perform a Kaluza-Klein (KK)
dimensional reduction of 5d-HW theory by shrinking $S^{1}$ to be
infinitesimally small. We obtain the corresponding 1d SQM theory in the space
$\mathfrak{A}_{3}$ of irreducible fields $(A,B)\in\Omega^{1}(M_{3})$ on
$M_{3}$ with action (4.6), that is equivalent to the resulting 4d theory on
$M_{3}\times\mathbb{R}$. As before, this will allow us to express the
partition function as (4.8):

$\boxed{\mathcal{Z}_{\text{HW},M_{3}\times\mathbb{R}}(G)=\sum_{l}{\cal
F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})=\sum_{l}\text{HF}_{d_{l}}^{\text{HW}_{4}}(M_{3},G)=\mathcal{Z}^{\text{Floer}}_{\text{HW}_{4},M_{3}}(G)}$
(1.5)

where $\text{HF}_{d_{l}}^{\text{HW}_{4}}(M_{3},G)$ is a _novel_ 4d-Haydys-
Witten Floer homology class assigned to $M_{3}$ of degree $d_{l}$, defined by
Floer differentials described by the gradient flow equation (4.9):

$\boxed{\begin{aligned}
\frac{d{A}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{3}}\frac{\partial
V_{3}(A,B)}{\partial A^{b}}\\\
\frac{d{B}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{3}}\frac{\partial
V_{3}(A,B)}{\partial B^{b}}\end{aligned}}$ (1.6)

and Morse functional (4.10):

$\boxed{V_{3}(A,B)=\int_{M_{3}}\text{Tr}\,\bigg{(}F\wedge B-\frac{1}{3}B\wedge
B\wedge B\bigg{)}}$ (1.7)

The chains of the 4d-HW Floer complex are generated by fixed critical points
of $V_{3}$, which correspond to _time-invariant Hitchin configurations on
$M_{3}$_ given by time-independent solutions of the 3d equations (4.11):

$\boxed{\begin{aligned} F-B\wedge B=0\\\ DB=0\end{aligned}}$ (1.8)

In §5, we further specialize to the case where $M_{3}=M_{2}\times S^{1}$, and
perform a second KK dimensional reduction of 5d-HW theory by shrinking this
$S^{1}$ to be infinitesimally small. We obtain the corresponding 1d SQM theory
in the space $\mathfrak{A}_{2}$ of irreducible
$(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})$ fields on $M_{2}$ with action
(5.5), where $\mathcal{A},\bar{\mathcal{A}}\in\Omega^{1}(M_{2})$ and
$Z,\bar{Z}\in\Omega^{0}(M_{2})$ are (conjugate) complexified gauge connections
and complex scalar fields, respectively, that is equivalent to the resulting
3d theory on $M_{2}\times\mathbb{R}$. Again, this will allow us to express the
partition function as (5.7):

$\boxed{\mathcal{Z}_{\text{HW},M_{2}\times\mathbb{R}}(G)=\sum_{p}{\cal
F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})=\sum_{p}\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})=\mathcal{Z}^{\text{Floer}}_{\text{HW}_{3},M_{2}}(G_{\mathbb{C}})}$
(1.9)

where $\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})$ is a _novel_
3d-Haydys-Witten Floer homology class assigned to $M_{2}$ of degree $d_{p}$,
defined by Floer differentials described by the gradient flow equations (5.8):

$\boxed{\begin{aligned}
\frac{d\mathcal{A}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\mathcal{A}^{b}}\right)^{*}\qquad&\frac{d\bar{\mathcal{A}}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\bar{\mathcal{A}}^{b}}\right)^{*}\\\
\frac{d{Z}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial
Z^{b}}\right)^{*}\qquad&\frac{d\bar{Z}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\bar{Z}^{b}}\right)^{*}\end{aligned}}$
(1.10)

and Morse functional (5.9):

$\boxed{V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})=-i\int_{M_{2}}\,\text{Tr}\,\Big{(}Z\wedge\mathcal{F}-\bar{Z}\wedge\bar{\mathcal{F}}\Big{)}}$
(1.11)

The chains of the 3d-HW Floer complex are generated by fixed critical points
of $V_{2}$, which correspond to _time-invariant complexified BF configurations
on $M_{2}$_ given by time-independent solutions of the 2d equations (5.10):

$\boxed{\begin{aligned} \mathcal{F}&=0=\bar{\mathcal{F}}\\\
\mathcal{D}Z&=0=\bar{\mathcal{D}}\bar{Z}\end{aligned}}$ (1.12)

In $\S$6, we specialize to the case where $M_{5}=\Sigma\times
M_{2}\times\mathbb{R}$, and perform a topological reduction of 5d-HW theory
along a curved Riemann surface $M_{2}$ using the Bershadsky-Sadov-Johansen-
Vafa (BSJV) reduction method from [6], to obtain a 3d ${\mathcal{N}}=4$
topological sigma model on $\Sigma\times\mathbb{R}$. First, by considering
$\Sigma=S^{1}\times S^{1}$, we recast the 3d sigma model as a 1d SQM theory in
the double loop space $L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$
of the symplectic moduli space
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ of flat $G_{\mathbb{C}}$
connections on $M_{2}$ with action (6.13). This will allow us to express the
partition function as (6.15):

$\boxed{\mathcal{Z}_{\text{HW},S^{1}\times
S^{1}\times\mathbb{R}}(G)=\sum_{q}{\cal
F}^{q}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}=\sum_{q}\text{HSF}_{d_{q}}({L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})})=\mathcal{Z}^{\text{SympFloer}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}}$
(1.13)

where
$\text{HSF}_{d_{q}}^{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}$
is a _novel_ symplectic Floer homology class in
$L^{2}\mathcal{M}^{G}_{\text{flat}}(M_{2})$ of degree $d_{q}$ counted by Floer
differentials described by the gradient flow equations (6.16):

$\boxed{\begin{aligned}
\frac{dZ^{a}}{dt}&=-g^{a{\bar{b}}}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial
Z^{b}}\right)^{*}\qquad&\frac{d\bar{Z}^{a}}{dt}&=-g^{a{\bar{b}}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial\bar{Z}^{b}}\right)^{*}\\\
\frac{dZ^{\bar{a}}}{dt}&=-g^{{\bar{a}}b}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial
Z^{\bar{b}}}\right)^{*}\qquad&\frac{d\bar{Z}^{\bar{a}}}{dt}&=-g^{{\bar{a}}b}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial\bar{Z}^{\bar{b}}}\right)^{*}\end{aligned}}$
(1.14)

where $(Z^{a},Z^{\bar{a}},\bar{Z}^{a},\bar{Z}^{\bar{a}})$ are the coordinates
of $L^{2}\mathcal{M}^{G}_{\text{flat}}(M_{2})$, and

$\boxed{V_{\Sigma}(Z,\bar{Z})=\int_{S^{1}\times
S^{1}}drds\,g_{i{\bar{\jmath}}}\Big{(}Z^{i}\partial_{s}Z^{\bar{\jmath}}-iZ^{i}\partial_{r}\bar{Z}^{\bar{\jmath}}-\bar{Z}^{i}\partial_{s}\bar{Z}^{\bar{\jmath}}+i\bar{Z}^{i}\partial_{r}Z^{\bar{\jmath}}\Big{)}}$
(1.15)

is the Morse functional in (6.17). The chains of the symplectic Floer complex
are generated by fixed critical points of $V_{\Sigma}$, which correspond to
_time-invariant maps from $S^{1}\times S^{1}$ to
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$_ given by the time-
independent solutions of the 2d equations (6.18) on $S^{1}\times S^{1}$:

$\boxed{\begin{aligned}
i\partial_{r}Z^{i}+\partial_{s}\bar{Z}^{i}&=0=i\partial_{r}\bar{Z}^{i}+\partial_{s}Z^{i}\\\
i\partial_{r}\bar{Z}^{\bar{\imath}}-\partial_{s}{Z}^{\bar{\imath}}&=0=i\partial_{r}Z^{\bar{\imath}}-\partial_{s}\bar{Z}^{\bar{\imath}}\end{aligned}}$
(1.16)

Second, by considering $\Sigma=I\times S^{1}$, the 3d sigma model is first
recast as a 2d A-model on $I\times\mathbb{R}$, and then further recast as a 1d
SQM theory in the interval space
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$
of smooth trajectories between A-branes starting at $\mathscr{L}_{0}$ and
ending on $\mathscr{L}_{1}$ in the loop space
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ of maps from $S^{1}$ to
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ with action (6.22). This
will allow us to express the partition function as (6.26):

$\boxed{\mathcal{Z}_{\text{HW},\mathbb{R}\times I\rightarrow
S^{1}}(G)=\sum_{u}{\cal
F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}=\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}},\mathscr{L}_{0},\mathscr{L}_{1})=\mathcal{Z}^{\text{IntSympFloer}}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}}$
(1.17)

where
$\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}},\mathscr{L}_{0},\mathscr{L}_{1})$
is a symplectic intersection Floer homology class generated by the
intersection points of $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$ in
$L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}$, of degree $d_{u}$ counted
by the Floer differential realized as the flow lines of the SQM whose gradient
flow equations are the squared terms in (6.22) set to zero.

Third, by considering $\Sigma=I\times\mathbb{R}$, the 3d sigma model is first
recast as a 2d Aθ-model on $I\times\mathbb{R}$, and then further recast as a
1d SQM theory in the interval space
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$
of smooth trajectories between Aθ-branes starting at $\mathscr{P}_{0}(\theta)$
and ending on $\mathscr{P}_{1}(\theta)$ in the path space
$\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})$ of maps from
$\mathbb{R}$ to $\mathcal{M}^{G,M_{2}}_{H,\theta}$, the symplectic
$\theta$-Hitchin moduli space of $G$ on $M_{2}$, with action (6.38). This will
allow us to express the partition function as (6.42):

$\boxed{\mathcal{Z}_{\text{HW},\mathbb{R}\times
I\rightarrow\mathbb{R}}(G)=\sum_{v}{\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}=\sum_{v}\text{HSF}^{\text{Int}}_{d_{v}}(\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)},\mathscr{P}_{0},\mathscr{P}_{1})=\mathcal{Z}^{\text{IntSympFloer}}_{\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)}}}$
(1.18)

where
$\text{HSF}^{\text{Int}}_{d_{v}}(\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)},\mathscr{P}_{0},\mathscr{P}_{1})$
is a symplectic intersection Floer homology class generated by the
intersection points of $\mathscr{P}_{0}(\theta)$ and $\mathscr{P}_{1}(\theta)$
in $\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)}$, of
degree $d_{v}$ counted by the Floer differential realized as the flow lines of
the SQM whose gradient flow equations are the squared terms in (6.38) set to
zero.

In $\S$7, we consider 5d-HW theory on $M_{5}=M_{3}\times
M_{1}\times\mathbb{R}$, and Heegaard split $M_{5}$ at $M_{3}$ with Heegaard
surface $\Sigma$. When $M_{1}=S^{1}$, via the topological invariance of HW
theory and the BSJV reduction procedure, we will obtain a _novel_ HW Atiyah-
Floer correspondence in (7.4):

$\boxed{\text{HF}^{\text{HW}}_{*}(M_{3}\times
S^{1},G)\cong\text{HSF}^{\text{Int}}_{*}({L\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},\mathscr{L}_{0},\mathscr{L}_{1})}$
(1.19)

where $\mathcal{M}^{G,\Sigma}_{\text{Higgs}}$ is the symplectic moduli space
of Higgs bundles of $G$ on $\Sigma$. In turn, this will also lead us to a
4d-HW Atiyah-Floer correspondence in (7.8):

$\boxed{\text{HF}^{\text{HW}_{4}}_{*}(M_{3},G)\cong\text{HSF}^{\text{Int}}_{*}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})}$
(1.20)

where
$\text{HSF}^{\text{Int}}_{*}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})$
is a symplectic intersection Floer homology class generated by the
intersection points of A-branes $L_{0}$ and $L_{1}$ in
$\mathcal{M}^{G,\Sigma}_{\text{Higgs}}$. It has a duality to a VW Atiyah-Floer
correspondence derived in an earlier work [8, eqn.(5.12)] via a topologically-
trivial “rotation” of the space orthogonal to $M_{3}$, as expressed in (7.10):

$\boxed{\text{HF}^{\text{HW}_{4}}_{*}(M_{3},G)\cong\text{HSF}^{\text{Int}}_{*}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\text{5d
``rotation}"}\text{HF}^{\text{inst}}_{*}(M_{3},G_{\mathbb{C}})\cong\text{HSF}^{\text{Int}}_{*}({\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}},{L}_{0}^{\prime},{L}_{1}^{\prime})}$
(1.21)

where $\text{HF}^{\text{inst}}_{*}(M_{3},G_{\mathbb{C}})$ is a
$G_{\mathbb{C}}$-instanton Floer homology class assigned to $M_{3}$.

In $\S$8, we first elucidate the implications of the topological invariance of
HW theory on the Floer homologies obtained hitherto. The results are given in
(8.1)-(8.7) and summarized as (I)

$\boxed{\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{M_{4}=M_{3}\times{\hat{S}}^{1}}\sum_{l}\text{HF}_{d_{l}}^{\text{HW}_{4}}(M_{3},G)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{M_{3}=M_{2}\times{\hat{S}}^{1}}\sum_{p}\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})}$
(1.22)

and

$\boxed{\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{M_{4}={\hat{M}}_{2}\times
S^{1}\times
S^{1}}\sum_{q}\text{HSF}_{d_{q}}({L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})})}$
(1.23)

where $S^{1}$ is a circle of fixed radius; ${\hat{S}}^{1}$ is a circle of
variable radius; $M_{2}$ is a Riemann surface of genus $g\geq 2$ with fixed
size; and $\hat{M}_{2}$ is a Riemann surface of genus $g\geq 2$ of variable
size. (II)

$\boxed{\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{M_{4}=M_{3}\times
S^{1}}\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},\mathscr{L}_{0},\mathscr{L}_{1})}$
(1.24)

and

$\boxed{\sum_{l}\text{HF}^{\text{HW}_{4}}_{d_{l}}(M_{3},G)=\sum_{w}\text{HSF}^{\text{Int}}_{d_{w}}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})}$
(1.25)

(III)

$\boxed{\begin{aligned}
\sum_{l}\text{HF}^{\text{HW}_{4}}_{d_{l}}(M_{3},G)&=\sum_{w}\text{HSF}^{\text{Int}}_{d_{w}}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})\\\
&\displaystyle\left\updownarrow\vphantom{\int}\right.\text{5d ``rotation}"\\\
\sum_{m}\text{HF}^{\text{inst}}_{d_{m}}(M_{3},G_{\mathbb{C}})&=\sum_{n}\text{HSF}^{\text{Int}}_{d_{n}}({\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}},{L}_{0}^{\prime},{L}_{1}^{\prime})\end{aligned}}$
(1.26)

(IV)

$\boxed{\sum_{p}\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})=\sum_{p}{\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{G_{\mathbb{C}}}_{\text{BF}}(M_{2})\big{)}}$
(1.27)

where
${\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{G_{\mathbb{C}}}_{\text{BF}}(M_{2})\big{)}$
is a de Rham class of zero-forms in the moduli space
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{BF}}(M_{2})$ of complexified BF
configurations on $M_{2}$, and the sum $p$ is a sum over all _time-invariant_
such configurations.

Second, we elucidate the implications of the 5d “S-duality” of HW theory, and
the results are given in (8.14), (8.18) and (8.21) as

$\boxed{\sum_{l}\text{HF}_{{d_{l}}}^{\text{HW}_{4}}(M_{3},L{G})\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\text{5d
``S-duality''}}\sum_{\tilde{l}}\text{HF}_{{d_{\tilde{l}}}}^{\text{HW}_{4}}(M_{3},(LG)^{\vee})}$
(1.28)
$\boxed{\sum_{q}\text{HF}_{{d_{q}}}^{\text{HW}_{3}}(M_{2},LL{G}_{\mathbb{C}})_{\scriptscriptstyle{Z,\bar{Z}=0}}\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{\text{5d
``S-duality''}}\sum_{\tilde{q}}\text{HF}_{{d_{\tilde{q}}}}^{\text{HW}_{3}}(M_{2},L(LG_{\mathbb{C}})^{\vee})_{\scriptscriptstyle{Z,\bar{Z}=0}}}$
(1.29)

where $G$/$G_{\mathbb{C}}$ are nonsimply-laced gauge groups with loop groups
$(LG)^{\vee}$/ $(LG_{\mathbb{C}})^{\vee}$ being Langlands dual to
$LG$/$LG_{\mathbb{C}}$ (at the level of their loop algebras); and

$\boxed{\sum_{q}\text{HF}_{d_{q}}^{\text{HW}_{3}}(M_{2},LLG_{\mathbb{C}})_{\scriptscriptstyle
Z,\bar{Z}=0}=\sum_{q}{\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{LLG_{\mathbb{C}}}_{\text{flat}}(M_{2})\big{)}}$
(1.30)

where
${\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{LLG_{\mathbb{C}}}_{\text{flat}}(M_{2})\big{)}$
is a de Rham class of zero-forms in the moduli space
$\mathcal{M}^{LLG_{\mathbb{C}}}_{\text{flat}}(M_{2})$ of flat toroidal
$G_{\mathbb{C}}$ connections on $M_{2}$, and the sum $q$ is a sum over all
_time-invariant_ such connections.

Lastly, we summarize the web of relations amongst all the Floer homologies,
and this is shown in Fig. 1, where the sizes of $\hat{S}^{1}$ and
$\hat{M}_{2}$ can be varied. In particular, from the web, we have, amongst
other additional relations,

$\boxed{\sum_{\tilde{l}}\text{HF}_{{d_{\tilde{l}}}}^{\text{HW}_{4}}(M_{3},(LG)^{\vee})\ext@arrow
9999{\arrowfill@\leftarrow\relbar\rightarrow}{\text{$M_{3}=M_{3}^{\prime}\bigcup
M_{3}^{\prime\prime}$}}{\text{$G$ nonsimply-
laced}}\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},\mathscr{L}_{0},\mathscr{L}_{1})}$
(1.31)

which is a Langlands duality in a 4d-HW Atiyah-Floer correspondence for loop
gauge groups!

$\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)$$\sum_{l}\text{HF}_{d_{l}}^{\text{HW}_{4}}(M_{3},G)$$\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},\mathscr{L}_{0},\mathscr{L}_{1})$$\sum_{p}\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})$$\sum_{q}\text{HSF}_{d_{q}}({L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})})$$\sum_{w}\text{HSF}^{\text{Int}}_{d_{w}}({\mathcal{M}^{G,\Sigma}_{\text{Higgs}}},{L}_{0},{L}_{1})$$\sum_{m}\text{HF}^{\text{inst}}_{d_{m}}(M_{3},G_{\mathbb{C}})$$\sum_{n}\text{HSF}^{\text{Int}}_{d_{n}}({\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}},{L}_{0}^{\prime},{L}_{1}^{\prime})$$\sum_{p}{\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{G_{\mathbb{C}}}_{\text{BF}}(M_{2})\big{)}$$\sum_{l}\text{HF}_{{d_{l}}}^{\text{HW}_{4}}(M_{3},L{G})$$\sum_{\tilde{l}}\text{HF}_{{d_{\tilde{l}}}}^{\text{HW}_{4}}(M_{3},(LG)^{\vee})$$\sum_{q}\text{HF}_{{d_{q}}}^{\text{HW}_{3}}(M_{2},LL{G}_{\mathbb{C}})_{\scriptscriptstyle{Z,\bar{Z}=0}}$$\sum_{\tilde{q}}\text{HF}_{{d_{\tilde{q}}}}^{\text{HW}_{3}}(M_{2},L(LG_{\mathbb{C}})^{\vee})_{\scriptscriptstyle{Z,\bar{Z}=0}}$$\sum_{q}{\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{LLG_{\mathbb{C}}}_{\text{flat}}(M_{2})\big{)}$$\sum_{\tilde{q}}{\text{H}}^{0}_{\text{dR}}\big{(}\mathcal{M}^{L{(LG_{\mathbb{C}})^{\vee}}}_{\text{flat}}(M_{2})\big{)}$pt$M_{4}=M_{3}\times\hat{S}^{1}$pt$M_{3}=M_{2}\times\hat{S}^{1}$$M_{4}=\hat{M}_{2}\times
S^{1}\times
S^{1}$$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$$M_{4}=M_{3}\times
S^{1}$$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$5d
“rotation”$M_{2}=\hat{M}_{2}$$M_{4}=M_{3}\times{S}^{1}$pt5d “S-duality”$G$
nonsimply-laced$M_{4}=M_{2}\times S^{1}\times S^{1}$pt5d
“S-duality”$G_{\mathbb{C}}$ nonsimply-
lacedpt$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$pt$M_{2}=\hat{M_{2}}$$G$
nonsimply-
laced$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$pt$M_{2}=\hat{M}_{2}$$G_{\mathbb{C}}$
nonsimply-laced

Figure 1: A web of relations amongst the Floer homologies.

In $\S$9, we consider the case where $M_{5}=M_{3}\times\mathbb{R}^{2}$, and
recast 5d-HW theory as a 2d gauged Landau-Ginzburg (LG) model on
$\mathbb{R}^{2}$ with target $\mathscr{A}_{3}$, the space of complexified
gauge connections on $M_{3}$. In turn, this 2d gauged LG model can be recast
as a 1d SQM theory in the path space
$\mathcal{M}(\mathbb{R},\mathscr{A}_{3})$. From the SQM and its critical
points that can be interpreted as solitons in the 2d gauged LG model, we
obtain (9.30):

$\boxed{\text{Hom}(\mathcal{A}_{I},\mathcal{A}_{J})_{\pm}\Longleftrightarrow\text{HF}^{G}_{d_{p}}(p^{IJ}_{\pm})}$
(1.32)

where $\text{HF}^{G}_{d_{p}}(p_{\pm}^{IJ})$ is a Floer homology class of
degree $d_{p}$ generated by $p_{\pm}^{IJ}$, the intersection points of left-
and right-thimbles, which represent the LG solitons that can be described as
the morphisms $\text{Hom}(\mathcal{A}_{I},\mathcal{A}_{J})_{\pm}$ whose
endpoints $\mathcal{A}_{I},\mathcal{A}_{J}$ correspond to $G$ Hitchin
configurations on $M_{3}$. Furthermore, via this equivalent description of
5d-HW theory as a 2d gauged LG model, we can interpret the 5d partition
function as a sum over tree-level scattering amplitudes of LG soliton strings
given by the maps of morphisms in (9.32):

$\boxed{\mu^{d}:\,\text{Hom}(\mathcal{A}_{I_{1}},\mathcal{A}_{I_{2}})_{-}\otimes\text{Hom}(\mathcal{A}_{I_{2}},\mathcal{A}_{I_{3}})_{-}\otimes\dots\otimes\text{Hom}(\mathcal{A}_{I_{d}},\mathcal{A}_{I_{d+1}})_{-}\longrightarrow\text{Hom}(\mathcal{A}_{I_{1}},\mathcal{A}_{I_{d+1}})_{+}}$
(1.33)

where $\text{Hom}(A_{*},A_{*})_{-}$ and $\text{Hom}(A_{*},A_{*})_{+}$
represent incoming and outgoing scattering soliton strings, as shown in Fig.
8.

Note that (1.32) and (1.33) underlie a Fukaya-Seidel type
$A_{\infty}$-category of $G$ Hitchin configurations on $M_{3}$ that was
conjectured to exist by Haydys in [1, $\S$5], and later rigorously constructed
by Wang in [2]. As such, we have a purely physical realization of Haydys’
mathematical conjecture and Wang’s mathematical construction. Also, Abouzaid-
Manolescu’s conjecture in [3, $\S$9.2] implies that $G_{\mathbb{C}}$-instanton
Floer homology of $M_{3}$, which is generated by $G$ Hitchin configurations on
$M_{3}$ (at $\theta=\pi/2$), can be categorified to give an
$A_{\infty}$-category of $M_{3}$. Thus, we also have a purely physical
realization and generalization of their mathematical conjecture.

Finally, via the results of $\S$7 with $M_{1}=\mathbb{R}$, we have, through
the one-to-one correspondence

$\boxed{\text{Hom}(\mathcal{A}_{I}(\theta),\mathcal{A}_{J}(\theta))_{\pm}\Longleftrightarrow\text{HSF}^{\text{Int}}_{d_{v}}(\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,\Sigma}_{H,\theta}\big{)},\mathscr{P}_{0},\mathscr{P}_{1})}$
(1.34)

in (9.37), an Atiyah-Floer type correspondence for the Fukaya-Seidel
$A_{\infty}$-category of $G$ Hitchin configurations on $M_{3}$!

Acknowledgements

We would like to thank M. Ashwinkumar, A. Haydys and R.P. Thomas for useful
discussions. This work is supported in part by the MOE AcRF Tier 1 grant
R-144-000-470-114.

##  A Topological 5d $\mathcal{N}=2$ Gauge Theory on $M_{4}\times\mathbb{R}$

In this section, we will study a certain topologically-twisted 5d
${\mathcal{N}}=2$ theory on $M_{4}\times\mathbb{R}$ with gauge group a real,
simple, compact Lie group $G$, where the BPS equations that its path integral
localizes onto are the Haydys-Witten equations [1, 4]. In turn, this will
allow us to get our desired results in later sections.

###  2.1 Twisting along $M_{4}$

The five-manifold of our 5d theory is $M_{4}\times\mathbb{R}$, where we have a
holonomy group $SO(4)_{E}$ for $M_{4}$. Given that

$SO(4)_{E}\cong SU(2)_{l}\otimes SU(2)_{r}\,,$ (2.1)

and the fact that $Spin(5)_{R}$, the $R$-symmetry group of this theory,
contains $Spin(4)_{R}\cong SU(2)_{a}\otimes SU(2)_{b}$ as a maximal subgroup,
the relevant topological twisting amounts to replacing $SU(2)_{r}$ with
$SU(2)_{r^{\prime}}$, the diagonal subgroup of $SU(2)_{r}\otimes SU(2)_{a}$.
As a $U(1)_{R}$ $R$-charge can be assigned to the fields under the Cartan
subgroup of the $SU(2)_{b}$ $R$-symmetry, upon twisting, the symmetry group of
the theory will become

$SU(2)_{l}\otimes SU(2)_{r}\otimes SU(2)_{a}\otimes U(1)_{R}\longrightarrow
SU(2)_{l}\otimes SU(2)_{r}^{\prime}\otimes U(1)_{R}\,.$ (2.2)

Noting that $\textbf{1}\otimes\textbf{2}=\textbf{2}$ and
$\textbf{2}\otimes\textbf{2}=\textbf{3}\oplus\textbf{1}$, one can compute that
the field content of the 5d $\mathcal{N}=2$ SYM theory will be modified as [5]

$\displaystyle A_{\mu}\>(\textbf{2,2,1})^{0}$ $\displaystyle\longrightarrow
A_{\mu}\>(\textbf{2,2})^{0}\>,$ (2.3) $\displaystyle
A_{t}\>(\textbf{1,1,1})^{0}$ $\displaystyle\longrightarrow
A_{t}\>(\textbf{1,1})^{0}\>,$
$\displaystyle\phi_{i}\>(\textbf{1,1,1})^{2}\oplus(\textbf{1,1,1})^{-2}\oplus(\textbf{1,1,3})^{0}$
$\displaystyle\longrightarrow\sigma\>(\textbf{1,1})^{2}\oplus\bar{\sigma}\>(\textbf{1,1})^{-2}\oplus
B_{\mu\nu}\>(\textbf{1,3})^{0}\>,$
$\displaystyle\lambda_{\alpha}\>(\textbf{2,1,2})^{1}\oplus(\textbf{2,1,2})^{-1}$
$\displaystyle\longrightarrow\psi_{\mu}\>(\textbf{2,2})^{1}\oplus\tilde{\psi}_{\mu}\>(\textbf{2,2})^{-1}\>,$
$\displaystyle\bar{\lambda}_{\dot{\alpha}}\>(\textbf{1,2,2})^{-1}\oplus(\textbf{1,2,2})^{1}$
$\displaystyle\longrightarrow\chi_{\mu\nu}\>(\textbf{1,3})^{-1}\>\oplus\eta\>(\textbf{1,1})^{-1}\oplus\tilde{\chi}_{\mu\nu}\>(\textbf{1,3})^{1}\>\oplus\tilde{\eta}\>(\textbf{1,1})^{1}\>.$

In (2.3), $\mu,\nu=0,1,2,3$ are indices along $M_{4}$, and $A_{t}$ is the
component of the 5d gauge field along $\mathbb{R}$. The indices
$\alpha,\dot{\alpha}$ are, respectively, the $SU(2)_{l}$ and $SU(2)_{r}$
indices along $M_{4}$. The index $i=1,2,\dots 5$ label the five scalar fields
in 5d $\mathcal{N}=2$ SYM theory. The dimensions of the fields under the
various $SU(2)$’s are denoted by boldfaced numbers while the $U(1)_{R}$
$R$-charges are denoted by the numbers in the superscript.

The resulting field content of this twist leaves, for the bosonic fields, the
gauge field $A_{\mu}\in\Omega^{1}(M_{4},ad(G))$ and
$A_{t}\in\Omega^{0}(M_{4},ad(G))$ unchanged, while the five original scalar
fields $\phi_{i}$ in the untwisted theory are now two scalars
$\sigma,\bar{\sigma}\in\Omega^{0}(M_{4},ad(G))$, and the remaining three now
form a self-dual two-form $B_{\mu\nu}\in\Omega^{2,+}(M_{4},ad(G))$. The
fermionic fields are now one-forms
$\psi_{\mu},\tilde{\psi_{\mu}}\in\Omega^{1}(M_{4},ad(G))$, self-dual two-forms
$\chi_{\mu\nu},\tilde{\chi}_{\mu\nu}\in\Omega^{2,+}(M_{4},ad(G))$, and scalars
$\eta,\tilde{\eta}\in\Omega^{0}(M_{4},ad(G))$. Here, $ad(G)$ is the adjoint
bundle of the underlying principal $G$-bundle.

Notice that only the spins along $M_{4}$ of the fields are modified, since the
twisting was performed only along $M_{4}$. Nonetheless, in doing so, we still
obtain a topological theory along the whole of $M_{5}$. This is to be
expected, since $\mathbb{R}$ is a flat direction. Furthermore, the
supersymmetry generators transform in the same representation as the fermion
fields, and since we now have two scalar fermion fields, we will also have two
nilpotent scalar supersymmetry generators, $\mathcal{Q}\>(\textbf{1,1})^{1}$
and $\bar{\mathcal{Q}}\>(\textbf{1,1})^{-1}$.

The supersymmetry transformations of the twisted theory under
$\mathcal{Q},\bar{\mathcal{Q}}$ are presented in [5]. For our purposes in this
paper, we will consider only one of the two supercharges, i.e., $\mathcal{Q}$.
The transformations of the fields under $\mathcal{Q}$ are

$\displaystyle\delta A_{\mu}$ $\displaystyle=\,\psi_{\mu}\,,$ (2.4)
$\displaystyle\delta A_{t}$ $\displaystyle={\eta}\,,$ $\displaystyle\delta
B_{\mu\nu}$ $\displaystyle=\,{\chi}_{\mu\nu}\,,$ $\displaystyle\delta\sigma$
$\displaystyle=0\,,$ $\displaystyle\delta\bar{\sigma}$
$\displaystyle=-i\,\tilde{\eta},$ $\displaystyle\delta\psi_{\mu}$
$\displaystyle=-2iD_{\mu}\sigma\,,$ $\displaystyle\delta\tilde{\psi}_{\mu}$
$\displaystyle=-F_{t\mu}+D^{\nu}B_{\nu\mu}\,,$
$\displaystyle\delta\chi_{\mu\nu}$ $\displaystyle=-i[B_{\mu\nu},\sigma]\,,$
$\displaystyle\delta\tilde{\chi}_{\mu\nu}$
$\displaystyle=F^{+}_{\mu\nu}+\frac{1}{4}[B_{\mu\rho},B_{\nu}^{\rho}]+\frac{1}{2}D_{t}B_{\mu\nu}\,,$
$\displaystyle\delta\eta$ $\displaystyle=-2iD_{t}\sigma\,,$
$\displaystyle\delta\tilde{\eta}$ $\displaystyle=-2[\sigma,\bar{\sigma}]\,.$

Here,
$F^{+}_{\mu\nu}=\frac{1}{2}(F_{\mu\nu}+\frac{1}{2}\epsilon_{\mu\nu\rho\lambda}F^{\rho\lambda})$
is the self-dual part of the gauge field strength $F_{\mu\nu}$, and
$B_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\lambda}B^{\rho\lambda}$. The
supersymmetry variation $\mathcal{Q}$ is nilpotent up to gauge
transformations, where

$\displaystyle\delta^{2}A_{\mu}$ $\displaystyle\propto D_{\mu}\sigma\,,$ (2.5)
$\displaystyle\delta^{2}A_{t}$ $\displaystyle\propto D_{t}\sigma\,,$
$\displaystyle\delta^{2}\Phi$ $\displaystyle\propto[\phi,\sigma]\,.$

Here $\Phi$ represents all other fields that are not $A_{\mu}$ or $A_{t}$.
Since $\sigma$ is a generator of gauge transformations, we shall only consider
the case where it has no zero-modes such that the gauge connections are
irreducible whence the relevant moduli space is well-behaved.

###  2.2 The Topological Action and the Haydys-Witten Equations

The bosonic part of the action (involving only $A$ and $B$) can be written as

$S_{\text{bo}}=\frac{1}{e^{2}}\int_{M_{4}\times\mathbb{R}}d^{5}x\,\text{Tr}\big{(}|k_{\nu}|^{2}+|s_{\mu\nu}|^{2}\big{)}\,,$
(2.6)

where

$\displaystyle k_{\nu}$ $\displaystyle=F_{t\nu}-D^{\mu}B_{\mu\nu}\,,$ (2.7)
$\displaystyle s_{\mu\nu}$
$\displaystyle=F^{+}_{\mu\nu}+\frac{1}{4}[B_{\mu\rho},B^{\rho}_{\nu}]+\frac{1}{2}D_{t}B_{\mu\nu}\,.$

The $\mathcal{Q}$-exact topological action is [5]

$\displaystyle
S_{\text{HW}}=\frac{1}{e^{2}}\int_{M_{4}\times\mathbb{R}}d^{5}x$
$\displaystyle\,\text{Tr}\Big{(}|F_{t\nu}-D^{\mu}B_{\mu\nu}|^{2}+|F^{+}_{\mu\nu}+\frac{1}{4}[B_{\mu\rho},B^{\rho}_{\nu}]+\frac{1}{2}D_{t}B_{\mu\nu}|^{2}$
(2.8) $\displaystyle+2D_{\mu}\sigma D^{\mu}\bar{\sigma}+2D_{t}\sigma
D^{t}\bar{\sigma}-\frac{1}{2}[B_{\mu\nu},\sigma][B^{\mu\nu},\bar{\sigma}]+2[\sigma,\bar{\sigma}]^{2}$
$\displaystyle+\tilde{\eta}D_{\mu}\psi^{\mu}+\eta
D_{\mu}\tilde{\psi^{\mu}}+\tilde{\psi}_{\mu}D_{\nu}\chi^{\mu\nu}+\psi_{\mu}D_{\nu}\tilde{\chi}^{\mu\nu}-\tilde{\eta}D_{t}\eta-\tilde{\psi}_{\mu}D_{t}\psi^{\mu}-\tilde{\chi}_{\mu\nu}D_{t}\chi^{\mu\nu}$
$\displaystyle-\frac{1}{2}B_{\mu\nu}\\{\tilde{\eta},\chi^{\mu\nu}\\}+\frac{1}{2}B_{\mu\nu}\\{\eta,\tilde{\chi}^{\mu\nu}\\}+B_{\mu\nu}\\{\psi^{\mu},\tilde{\psi}^{\nu}\\}+B_{\mu\nu}\\{\tilde{\chi}^{\mu\rho},\chi^{\nu}_{\rho}\\}$
$\displaystyle+i\sigma\\{\tilde{\eta},\tilde{\eta}\\}+i\bar{\sigma}\\{\eta,\eta\\}-i\sigma\\{\tilde{\psi}_{\mu},\tilde{\psi}^{\mu}\\}-i\bar{\sigma}\\{\psi_{\mu},\psi^{\mu}\\}+i\bar{\sigma}\\{\chi_{\mu\nu},\chi^{\mu\nu}\\}+i\sigma\\{\tilde{\chi}_{\mu\nu},\tilde{\chi}^{\mu\nu}\\}\Big{)}\,.$

By setting the variation of the fermions in (2.4) to zero, we obtain the BPS
equations of the 5d theory as111Recall that the bosonic fields $\sigma$ and
$\bar{\sigma}$ have no zero-modes, so we can ignore the variation of fermions
involving them.

$\displaystyle F_{t\nu}-D^{\mu}B_{\mu\nu}$ $\displaystyle=0\,,$ (2.9)
$\displaystyle
F^{+}_{\mu\nu}+\frac{1}{4}[B_{\mu\rho},B^{\rho}_{\nu}]+\frac{1}{2}D_{t}B_{\mu\nu}$
$\displaystyle=0\,.$

These are the Haydys-Witten (HW) equations [1, 4], and configurations of
$(A_{\mu},B_{\mu\nu})$ satisfying (2.9) constitute a moduli space
$\mathcal{M}_{\text{HW}}$ that the path integral of the 5d theory would
localize onto, where $S_{\text{bo}}$ and thus $S_{\text{HW}}$ is indeed
minimized.

We shall henceforth denote the 5d theory with action $S_{\text{HW}}$ as HW
theory.

##  A Haydys-Witten Floer Homology of Four-Manifolds

In this section, we shall define, purely physically, a novel HW Floer homology
of $M_{4}$ via the $\mathcal{Q}$-cohomology of HW theory, through a
supersymmetric quantum mechanics (SQM) interpretation of the 5d gauge theory.

###  3.1 5d Gauge Theory to SQM

We would like to re-express the $\mathcal{N}=2$ gauge theory on
$M_{5}=M_{4}\times\mathbb{R}$ as an SQM model in $\mathfrak{A}_{4}$, the space
of irreducible $(A_{\mu},B_{\mu\nu})$ fields on $M_{4}$. To this end, we shall
employ the methods pioneered in [7].

We begin by subtracting from the action (2.8), the $\mathcal{Q}$-exact terms
$\delta(\eta D_{t}\bar{\sigma})$, $\delta(B_{\mu\nu}\tilde{\eta}B^{\mu\nu})$
and $\delta(\tilde{\eta}[\sigma,\bar{\sigma}])$. As we are ultimately
interested in the $\mathcal{Q}$-cohomology spectrum of HW theory, this
subtraction will have no relevant consequence. Using the self-duality
properties of $B_{\mu\nu}$, terms such as
$B_{\mu\nu}\\{\eta,\tilde{\chi}^{\mu\nu}\\}$ are topological, and can also be
ignored in (2.8), thereby leaving us with

$\displaystyle
S_{\text{HW}}=\frac{1}{e^{2}}\int_{M_{4}\times\mathbb{R}}d^{5}x$
$\displaystyle\,\text{Tr}\Big{(}|F_{t\nu}-D^{\mu}B_{\mu\nu}|^{2}+|F^{+}_{\mu\nu}+\frac{1}{4}[B_{\mu\rho},B^{\rho}_{\nu}]+\frac{1}{2}D_{t}B_{\mu\nu}|^{2}$
(3.1) $\displaystyle+\tilde{\eta}D_{\mu}\psi^{\mu}+\eta
D_{\mu}\tilde{\psi^{\mu}}+\tilde{\psi}_{\mu}D_{\nu}\chi^{\mu\nu}+\psi_{\mu}D_{\nu}\tilde{\chi}^{\mu\nu}-\tilde{\psi}_{\mu}D_{t}\psi^{\mu}-\tilde{\chi}_{\mu\nu}D_{t}\chi^{\mu\nu}$
$\displaystyle+2D_{\mu}\sigma
D^{\mu}\bar{\sigma}-i\sigma\\{\tilde{\psi}_{\mu},\tilde{\psi}^{\mu}\\}+i\sigma\\{\tilde{\chi}_{\mu\nu},\tilde{\chi}^{\mu\nu}\\}-i\bar{\sigma}\\{\psi_{\mu},\psi^{\mu}\\}+i\bar{\sigma}\\{\chi_{\mu\nu},\chi^{\mu\nu}\\}\Big{)}\,.$

The bosonic terms of the equivalent SQM action can be obtained by expanding
out the first line in (3.1) and collecting only those terms that do not
contain $A_{t}$ (as it can and will be integrated out). This gives

$S_{\text{HW}}=\frac{1}{e^{2}}\int_{\mathbb{R}}dt\,\text{Tr}\,\int_{M_{4}}d^{4}x\Big{|}\dot{A}_{\nu}-D^{\mu}B_{\mu\nu}\Big{|}^{2}+\frac{1}{4}\Big{|}\dot{B}_{\mu\nu}+(2F^{+}_{\mu\nu}+\frac{1}{2}[B_{\mu\rho},B^{\rho}_{\nu}])\Big{|}^{2}+\dots\,,$
(3.2)

where “$\dots$” represent the fermionic terms in the action, and
$\dot{A}_{\mu}$ represents the time-derivative of $A_{\mu}$, and similarly for
$\dot{B}_{\mu\nu}$.

The fermionic terms of the equivalent SQM action can be obtained by first
integrating $\eta,\tilde{\eta}$ out from (3.1) (since it is linear in
$\eta,\tilde{\eta}$), imposing the condition

$D_{\mu}\psi^{\mu}=D_{\mu}\tilde{\psi}^{\mu}=0\,.$ (3.3)

Furthermore, via integration by parts, (3.3) also removes from (3.1), the
terms $i\tilde{\psi}_{\mu}D_{\nu}\chi^{\mu\nu}$ and
$i\psi_{\mu}D_{\nu}\tilde{\chi}^{\mu\nu}$.

Using the equations of motion of $\sigma,\bar{\sigma}$ and plugging them back
into (3.1) yields a term

$\displaystyle\text{Tr}\int
d^{4}x\,\Big{(}\\{\tilde{\psi}^{\mu}(x),\tilde{\psi}_{\mu}(x)\\}-\\{\tilde{\chi}_{\mu\nu}(x),\tilde{\chi}^{\mu\nu}(x)\\}\Big{)}G(x-y)\Big{(}\\{\psi^{\mu}(y),\psi_{\mu}(y)\\}-\\{\chi_{\mu\nu}(y),\chi^{\mu\nu}(y)\\}\Big{)}\,,$
(3.4)

where $G$ is the Green’s function for the scalar Laplacian (on $M_{4}$).

Using the equations of motion for $A_{t}$ gives a term similar to (3.4), as
well as the term

$\text{Tr}\int
d^{4}x\,\Big{(}\\{\tilde{\psi}_{\mu}(x),\psi^{\mu}(x)\\}-\\{\tilde{\chi}_{\mu\nu}(x),\chi^{\mu\nu}(x)\\}\Big{)}G(x-y)\Big{(}D_{\mu}\dot{A}^{\mu}(y)+[\dot{B}^{\mu\nu}(y),B_{\mu\nu}(y)]\Big{)}\,.$
(3.5)

With regard to the equivalent SQM action, (3.4) will contribute towards the
four-fermi curvature term, while (3.5) will contribute towards the Christoffel
connection in the kinetic terms for the fermions.

After suitable rescalings, the equivalent SQM action can then be obtained from
(3.2) as

$\displaystyle S_{\text{SQM,HW}}=\frac{1}{e^{2}}\int_{\mathbb{R}}dt$
$\displaystyle\bigg{[}\Big{|}\dot{A}^{a}+g^{ab}_{\mathfrak{A}_{4}}\frac{\partial
V_{4}}{\partial
A^{b}}\Big{|}^{2}+\Big{|}\dot{B}^{a}+g^{ab}_{\mathfrak{A}_{4}}\frac{\partial
V_{4}}{\partial
B^{b}}\Big{|}^{2}+g_{\mathfrak{A}_{4}ab}\Big{(}\bar{\psi}^{a}\nabla_{t}\psi^{b}+\bar{\chi}^{a}\nabla_{t}\chi^{b}\Big{)}$
(3.6)
$\displaystyle+R^{ab}_{cd}\,\Big{(}\bar{\psi}_{a}\bar{\psi}_{b}\psi^{c}\psi^{d}-\bar{\psi}_{a}\bar{\psi}_{b}\chi^{c}\chi^{d}-\bar{\chi}_{a}\bar{\chi}_{b}\psi^{c}\psi^{d}+\bar{\chi}_{a}\bar{\chi}_{b}\chi^{c}\chi^{d}\Big{)}\bigg{]}\,,$

where $(A^{a},B^{a})$ and $a,b,c\dots$ are coordinates and indices of the
target $\mathfrak{A}_{4}$, respectively; $\psi^{a}$, $\chi^{a}$
$(\tilde{\psi}^{a}$, $\tilde{\chi}^{a})$ are tangent (co-tangent) vectors to
$A^{a}$, $B^{a}$ in $\mathfrak{A}_{4}$, respectively;
$\nabla_{t}\psi^{a}=\partial_{t}\psi^{a}+\Gamma^{a}_{bc}\big{(}\dot{x}^{b}+\dot{y}^{b}\big{)}\psi^{c}$;
$g_{\mathfrak{A}_{4}}$ and $R$ are the metric and Riemann tensor on
$\mathfrak{A}_{4}$, respectively; and $V_{4}(A,B)$ is the potential function.

###  3.2 An HW Floer Homology of $M_{4}$

In a TQFT, the Hamiltonian $H$ vanishes in the $\mathcal{Q}$-cohomology,
whence this means that for any state $|\mathcal{O}\rangle$ that is
nonvanishing in the $\mathcal{Q}$-cohomology, we have

$H|\mathcal{O}\rangle=\\{\mathcal{Q},\cdots\\}|\mathcal{O}\rangle=\mathcal{Q}(\cdots|\mathcal{O}\rangle)=\mathcal{Q}|\mathcal{O}^{\prime}\rangle=\\{\mathcal{Q},\mathcal{O}^{\prime}\\}|0\rangle=|\\{\mathcal{Q},\mathcal{O}^{\prime}\\}\rangle\sim
0\,.$ (3.7)

In other words, the $|\mathcal{O}\rangle$’s which span the relevant
$\mathcal{Q}$-cohomology of states in HW theory are actually ground states
that are therefore time-invariant. In particular, for HW theory on
$M_{5}=M_{4}\times\mathbb{R}$ with $\mathbb{R}$ as the time coordinate, its
relevant spectrum of states is associated only with $M_{4}$.

With $M_{5}=M_{4}\times\mathbb{R}$, $M_{4}$ is the far boundary of the five-
manifold and one needs to specify “boundary conditions” on $M_{4}$ to compute
the path integral. We can do this by first defining a restriction of the
fields to $M_{4}$, which we shall denote as $\Psi_{M_{4}}$, and then
specifying boundary values for these restrictions. Doing this is equivalent to
inserting in the path integral, an operator functional ${F}(\Psi_{M_{4}})$
that is nonvanishing in the ${\cal Q}$-cohomology (so that the path integral
will continue to be topological). This means that the corresponding partition
function of HW theory222The HW equations are elliptic, so the virtual
dimension of its moduli space $\mathcal{M}_{\text{HW}}$ will be zero [1],
whence just as in Vafa-Witten theory, it is a balanced TQFT, and one can
define the partition function.  can be computed as [9, eqn. (4.12)]

$\langle
1\rangle_{{F}(\Psi_{M_{4}})}=\int_{\mathcal{M}_{\text{HW}}}{F}(\Psi_{M_{4}})\,e^{-S_{\text{HW}}}\,.$
(3.8)

Since we demonstrated in the previous subsection that HW theory on
$M_{4}\times\mathbb{R}$ can be expressed as an SQM model in
$\mathfrak{A}_{4}$, we can write the partition function as

${\mathcal{Z}_{\text{HW},M_{4}\times\mathbb{R}}(G)=\langle
1\rangle_{{F}_{4}(\Psi_{M_{4}})}=\sum_{k}{\cal
F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})}\,.$ (3.9)

Here, ${\cal F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})$ in the
$\mathcal{Q}$-cohomology is the $k^{\text{th}}$ contribution to the partition
function that depends on the expression of ${{F}_{4}(\Psi_{M_{4}})}$ in the
bosonic fields on $M_{4}$ evaluated over the corresponding solutions of the HW
equations (2.9) restricted to $M_{4}$, and the summation in ‘$k$’ is over all
presumably isolated and non-degenerate configurations on $M_{4}$ in
$\mathfrak{A}_{4}$ that the equivalent SQM localizes onto.333This presumption
will be justified shortly.

Let us now ascertain what the ${\cal F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})$’s
correspond to. To this end, let us first determine the configurations that the
SQM localizes onto. These are configurations that minimize the SQM action
(3.6), i.e., they set the (zero modes of the) squared terms therein to zero.
They are therefore given by

$\boxed{\begin{aligned}
\frac{d{A}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{4}}\frac{\partial
V_{4}(A,B)}{\partial A^{b}}\\\
\frac{d{B}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{4}}\frac{\partial
V_{4}(A,B)}{\partial B^{b}}\end{aligned}}$ (3.10)

where the squaring argument [7] means that they have to obey the vanishing of
both the LHS and RHS _simultaneously_. In other words, the configurations that
the SQM localizes onto are fixed (i.e., time-invariant) critical points of the
potential $V_{4}$ in ${\mathfrak{A}}_{4}$.

What then should the explicit form of $V_{4}$ be, one might ask. To determine
this, note that the squared terms in (3.6) originate from the squared terms in
(3.2). Indeed, setting the squared terms in (3.2) to zero minimizes the
underlying 5d action, and this is consistent with setting the squared terms in
(3.6) to zero to minimize the equivalent SQM action. As such, one can deduce
the explicit form of $V_{4}$ by comparing (3.10) with the vanishing of the
squared terms in (3.2). Specifically, the vanishing of the squared terms in
(3.2) would give us

$\displaystyle\frac{d{A}_{\nu}}{dt}$ $\displaystyle=D^{\mu}B_{\mu\nu}\,,$
(3.11) $\displaystyle\frac{d{B}_{\mu\nu}}{dt}$
$\displaystyle=-2F^{+}_{\mu\nu}-\frac{1}{2}[B_{\mu\rho},B^{\rho}_{\nu}]\,,$

and by comparing this with (3.10), we find that

$\boxed{V_{4}(A,B)=\int_{M_{4}}\,\text{Tr}\,\Big{(}F^{+}\wedge\star
B+\frac{1}{3}B\wedge(\star_{3}(\mathscr{B}\wedge_{3}\mathscr{B}))\Big{)}}$
(3.12)

Here, $\mathscr{B}=\mathscr{B}_{i}d\gamma^{i}$ is a one-form on an auxiliary
$\mathbb{R}^{3}$ which corresponds to an interpretation of the two-form $B$ on
$M_{4}$ as a differential form that only has 3 independent components (because
of its self-duality),444Such a correspondence between $\mathscr{B}$ and $B$
has also been exploited by Haydys in [1, eqn. (37)]. where $\star_{3}$ and
$\wedge_{3}$ are the Hodge star operator and exterior product on this
auxiliary $\mathbb{R}^{3}$, respectively, such that the one-form
$\star_{3}(\mathscr{B}\wedge_{3}\mathscr{B})$ on $\mathbb{R}^{3}$ can be
interpreted as another two-form $B^{\prime}$ on $M_{4}$ whose components are
$B^{\prime}_{\mu\nu}\sim[B_{\mu\rho},B^{\rho}_{\nu}]$.

Thus, the summation in ‘$k$’ in (3.9) is over all isolated and non-degenerate
critical points of (3.12) in $\mathfrak{A}_{4}$ that are also fixed.555As we
explain next, the aforementioned critical points correspond to Vafa-Witten
configurations on $M_{4}$. For them to be isolated, the actual dimension of
their moduli space must be zero. For an appropriate choice of $G$ and $M_{4}$
such that the actual dimension of the instanton sub-moduli space is zero, this
can be true. We shall assume such a choice of $G$ and $M_{4}$ henceforth;
specifically, we choose $G$ and $M_{4}$ such that
$\text{dim}(G)\,(1+b^{+}_{2})=4kh$, where $b^{+}_{2}$ is the positive second
betti number of $M_{4}$, $k$ is the instanton number on $M_{4}$, and $h$ is
the dual Coxeter number of $G$. As for their non-degeneracy, a suitable
perturbation of $V_{4}(A,B)$ which can be effected by introducing physically-
trivial $\mathcal{Q}$-exact terms to the action, would ensure this. We would
like to thank R.P. Thomas for discussions on this point.

Critical points of $V_{4}(A,B)$ are configurations in $\mathfrak{A}_{4}$ that
set the RHS of (3.10) to zero; they therefore correspond to configurations on
$M_{4}$ that set the RHS of (3.11) to zero. i.e., they are Vafa-Witten (VW)
configurations on $M_{4}$. In summary, the partition function (3.9) is an
algebraic sum of _fixed_ VW configurations on $M_{4}$ in $\mathfrak{A}_{4}$.

Notice that (3.10) are gradient flow equations, and they govern the classical
trajectory of the SQM model from one time-invariant VW configuration on
$M_{4}$ to another in $\mathfrak{A}_{4}$. Hence, just as in [7, 8], the
equivalent SQM model will physically define a novel gauge-theoretic Floer
homology theory.

Specifically, the _time-invariant VW configurations on $M_{4}$_ in
$\mathfrak{A}_{4}$, i.e., the time-independent solutions to the 4d equations

$\boxed{\begin{aligned}
F^{+}+\star_{3}(\mathscr{B}\wedge_{3}\mathscr{B})&=0\\\ D\star
B&=0\end{aligned}}$ (3.13)

will generate the chains of a Floer complex with _Morse functional
$V_{4}(A,B)$ in (3.12)_, where HW flow lines, described by time-varying
solutions to the _gradient flow equations ( 3.10)_, are the Floer
differentials such that the number of outgoing flow lines at each time-
invariant VW configuration on $M_{4}$ in $\mathfrak{A}_{4}$ is the degree
$d_{k}$ of the corresponding chain in the Floer complex.

In other words, we can also write (3.9) as

$\boxed{\mathcal{Z}_{\text{HW},M_{4}\times\mathbb{R}}(G)=\sum_{k}{\cal
F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})=\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)=\mathcal{Z}^{\text{Floer}}_{\text{HW},M_{4}}(G)}$
(3.14)

where each ${\cal F}^{G}_{\text{HW}}(\Psi_{M_{4}}^{k})$ can be identified with
a _novel_ class $\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)$ that we shall
henceforth name an HW Floer homology class assigned to $M_{4}$ defined by
(3.10), (3.12), (3.13) and the description above.

Note that $\text{HF}^{\text{HW}}_{d_{k}}(M_{4},G)$ was first conjectured to
exist by Haydys in [1, §5]. We have thus furnished a purely physical
realization of his mathematical conjecture.

##  A 4d-Haydys-Witten Floer Homology of Three-Manifolds

In this section, we specialize to the case where $M_{4}=M_{3}\times S^{1}$,
and perform a Kaluza-Klein (KK) dimensional reduction of HW theory by
shrinking $S^{1}$ to be infinitesimally small. This will allow us to
physically derive from its topologically-invariant $\mathcal{Q}$-cohomology, a
novel 4d-HW Floer homology of $M_{3}$.

###  4.1 KK Reduction of HW Theory on $S^{1}$ and the Corresponding SQM

Let the $S^{1}$-direction of $M_{4}=M_{3}\times S^{1}$ be along $x^{0}$, and
the ‘time’ direction be along $\mathbb{R}$ (recall it is the coordinate
labelled as $t$). $A_{0}$, the component of the gauge field along $S^{1}$, is
then interpreted as a scalar field on $M_{3}\times\mathbb{R}$ upon KK
reduction along $S^{1}$. From (2.4), the $\mathcal{Q}$-variation of $A_{\mu}$
(spanning the $x^{0}$ and $x^{i}$ directions, where $i=1,2,3$), and that of
$A_{t}$, are

$\displaystyle\delta C$ $\displaystyle=\zeta\,,\qquad\delta
A_{i}=\psi^{\prime}_{i}\,,$ (4.1) $\displaystyle\delta A_{4}$
$\displaystyle=\psi^{\prime}_{4}\,,$

where we have, for later convenience, relabeled $(A_{0},\psi_{0})$ as the
scalars $(C,\zeta)\in\Omega^{0}(M_{3}\times\mathbb{R})$; $A_{t}$ as $A_{4}$,
the ‘fourth’ component of the gauge field on $M_{3}\times\mathbb{R}$; and
$(\psi_{i},\eta)$ in (2.4) as $(\psi^{\prime}_{i},\psi^{\prime}_{4})$, which
now define a fermionic one-form
$\psi^{\prime}_{\mu^{\prime}}\in\Omega^{1}(M_{3}\times\mathbb{R})$, where
$\mu^{\prime}=1,2,3,4$.

Notice then from (2.4) that

$\delta^{2}A_{\mu^{\prime}}=2iD_{\mu^{\prime}}\sigma\,.$ (4.2)

That is, the nilpotency of $\mathcal{Q}$ up to gauge transformations by
$\sigma$ is maintained in the resulting 4d theory.

The other relevant 5d bosonic field which has components along $S^{1}$ is
$B_{\mu\nu}$. However, upon shrinking $S^{1}$,
$B_{\mu\nu}\in\Omega^{2,+}(M_{3}\times S^{1})$ can be interpreted as
$B_{i}\in\Omega^{1}(M_{3})$, as it really only has three independent
components along $M_{3}$ due to its self-duality properties. For later
convenience of explanation, let us continue to regard $B_{i}$ as
$B_{\mu^{\prime}\nu^{\prime}}\in\Omega^{2,+}(M_{3}\times\mathbb{R})$. In other
words, upon shrinking $S^{1}$, $B_{\mu\nu}$ becomes a self-dual two-form
$B_{\mu^{\prime}\nu^{\prime}}$ in the resulting 4d theory.

We are now ready to determine the conditions under which $S_{\text{bo}}$ in
(2.6) minimizes such that HW theory localizes when we KK reduce along $S^{1}$.
To this end, first note that (i) the non-vanishing components of $s_{\mu\nu}$
in (2.6) are $s_{0i}$ and $s_{ij}$; (ii) in a KK reduction along $S^{1}$, the
derivatives along $x^{0}$ must be set to zero; (iii) the zero-modes of $C$
should be set to zero (so as to avoid reducible connections in the moduli
space that the path integral localizes to). Then, in temporal gauge
$A_{4}=0$,666We can appeal to this gauge because we saw in $\S$3.1 that the
gauge field does not contribute along the ‘time’ direction in the end. we find
that the condition $k_{\mu}=0$ (on zero-modes) which minimizes
$S_{\text{bo}}$, would, from (2.7), correspond to

$\frac{d{A}^{i}}{dt}=-\epsilon^{ijk}\bigg{(}\partial_{j}B_{k}+[A_{j},B_{k}]\bigg{)}\,.$
(4.3)

Similarly, we find that the condition $s_{\mu\nu}=0$ (on zero modes) which
minimizes $S_{\text{bo}}$, would, from (2.7), correspond to

$\frac{d{B}^{i}}{dt}=-\frac{1}{2}\epsilon^{ijk}\bigg{(}F_{jk}-[B_{j},B_{k}]\bigg{)}\,.$
(4.4)

Here, we have used the fact that (i)
$B_{\mu^{\prime}\nu^{\prime}}\in\Omega^{2,+}(M_{3}\times\mathbb{R})$ can be
interpreted as a one-form on $M_{3}$ (as mentioned above), where its nonzero
independent components are $B_{4i}=B_{i}$, so
$A_{i},B_{i}\in\Omega^{1}(M_{3})$; (ii) from the self-duality condition
$B_{\mu^{\prime}\nu^{\prime}}=\frac{1}{2}\epsilon_{\mu^{\prime}\nu^{\prime}\rho^{\prime}\lambda^{\prime}}B^{\rho^{\prime}\lambda^{\prime}}$,
we can write $B_{i}=\epsilon_{ijk}B^{jk}$. Notice also that (4.3) and (4.4)
are effectively the KK reductions of (3.11) (with $A_{0}=C$ set to zero).

Just as in §3.1, this means that the 4d action can be written as

$S_{\text{HW}_{4}}=\frac{1}{e^{2}}\int
dt\,\text{Tr}\int_{M_{3}}d^{3}x\bigg{|}\dot{A}^{i}+\epsilon^{ijk}\bigg{(}\partial_{j}B_{k}+[A_{j},B_{k}]\bigg{)}\bigg{|}^{2}+\bigg{|}\dot{B}^{i}+\frac{1}{2}\epsilon^{ijk}\bigg{(}F_{jk}-[B_{j},B_{k}]\bigg{)}\bigg{|}^{2}+\dots\,,$
(4.5)

where “…” refers to fermionic terms. After suitable rescalings, the equivalent
SQM action can be obtained from (4.5) as

$S_{\text{SQM,H}}=\frac{1}{e^{2}}\int
dt\bigg{|}\dot{{A}}^{a}+g^{ab}_{\mathfrak{A}_{3}}\frac{\partial
V_{3}}{\partial{A}^{b}}\bigg{|}^{2}+\bigg{|}\dot{{B}}^{a}+g^{ab}_{\mathfrak{A}_{3}}\frac{\partial
V_{3}}{\partial{B}^{b}}\bigg{|}^{2}+\dots\,,$ (4.6)

where $({A}^{a},B^{a})$ and $a,b$ are coordinates and indices on the space
$\mathfrak{A}_{3}$ of irreducible $(A_{i},B_{i})$ fields on $M_{3}$,
respectively; $g^{ab}_{\mathfrak{A}_{3}}$ is the metric on $\mathfrak{A}_{3}$;
$V_{3}(A,B)$ is the potential function; and the choice of subscript ‘H’ will
be clear shortly.

By the squaring argument [7] applied to (4.5), the configurations that the
equivalent SQM localizes onto are those that set the LHS and RHS of
(4.3)-(4.4) _simultaneously_ to zero. In other words, the equivalent SQM
localizes onto time-invariant Hitchin configurations on $M_{3}$, where by
Hitchin configurations, we mean solutions to a 3d analog of Hitchin’s
equations defined as the vanishing of the RHS of (4.3) and (4.4).

###  4.2 A 4d-HW Floer Homology of $M_{3}$

Since the resulting 4d theory on $M_{3}\times\mathbb{R}$ can be interpreted as
an SQM model in $\mathfrak{A}_{3}$, its partition function can, like in (3.9),
be written as

${\mathcal{Z}_{\text{HW},M_{3}\times\mathbb{R}}(G)=\langle
1\rangle_{{F}_{3}(\Psi_{M_{3}})}=\sum_{l}{\cal
F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})}\,,$ (4.7)

where ${\cal F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})$ in the
$\mathcal{Q}$-cohomology is the $l^{\text{th}}$ contribution to the partition
function that depends on the expression of ${{F}_{3}(\Psi_{M_{3}})}$ in the
bosonic fields on $M_{3}$ evaluated over the corresponding solutions of (4.3)
and (4.4) restricted to $M_{3}$, and the summation in ‘$l$’ is over all
isolated and non-degenerate configurations on $M_{3}$ in $\mathfrak{A}_{3}$
that the equivalent SQM localizes onto.777This presumption that the
configurations will be isolated and non-degenerate is justified because (the
$\mathcal{Q}$-cohomology of) HW theory is topological in all directions and
therefore invariant when we shrink the $S^{1}$, so if $M_{3}$ (where
$M_{3}\times S^{1}=M_{4}$) and $G$ are chosen such as to satisfy the
conditions spelt out in footnote 5,
$\mathcal{Z}_{\text{HW},M_{3}\times\mathbb{R}}$ will be a discrete and non-
degenerate sum of contributions just like
$\mathcal{Z}_{\text{HW},M_{4}\times\mathbb{R}}$. We shall henceforth assume
such a choice of $M_{3}$ and $G$ whence the presumption would hold.

Let us now ascertain what the ${\cal
F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})$’s correspond to. Repeating here the
analysis in §3.2 with (4.6) as the action for the equivalent SQM model, we
find that we can also write (4.7) as

$\boxed{\mathcal{Z}_{\text{HW},M_{3}\times\mathbb{R}}(G)=\sum_{l}{\cal
F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})=\sum_{l}\text{HF}_{d_{l}}^{\text{HW}_{4}}(M_{3},G)=\mathcal{Z}^{\text{Floer}}_{\text{HW}_{4},M_{3}}(G)}$
(4.8)

where each ${\cal F}^{G}_{\text{HW}_{4}}(\Psi_{M_{3}}^{l})$ can be identified
with a _novel_ gauge-theoretic 4d-HW Floer homology class
$\text{HF}^{\text{HW}_{4}}_{d_{l}}(M_{3},G)$ of degree $d_{l}$ assigned to
$M_{3}$.

Specifically, the _time-invariant Hitchin configurations on $M_{3}$_ in
$\mathfrak{A}_{3}$ that obey the simultaneous vanishing of the LHS and RHS of
the _gradient flow equations_

$\boxed{\begin{aligned}
\frac{d{A}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{3}}\frac{\partial
V_{3}(A,B)}{\partial A^{b}}\\\
\frac{d{B}^{a}}{dt}&=-g^{ab}_{{\mathfrak{A}}_{3}}\frac{\partial
V_{3}(A,B)}{\partial B^{b}}\end{aligned}}$ (4.9)

will generate the chains of the 4d-HW Floer complex with _Morse functional_

$\boxed{V_{3}(A,B)=\int_{M_{3}}\text{Tr}\,\bigg{(}F\wedge B-\frac{1}{3}B\wedge
B\wedge B\bigg{)}}$ (4.10)

in $\mathfrak{A}_{3}$, where 4d-HW flow lines, described by time-varying
solutions to (4.9), are the Floer differentials such that the number of
outgoing flow lines at each time-invariant Hitchin configuration on $M_{3}$ in
$\mathfrak{A}_{3}$ which corresponds to a time-independent solution of the 3d
equations

$\boxed{\begin{aligned} F-B\wedge B=0\\\ DB=0\end{aligned}}$ (4.11)

is the degree $d_{l}$ of the corresponding chain in the 4d-HW Floer complex.

##  A 3d-Haydys-Witten Floer Homology of Two-Manifolds

In this section, we further specialize to the case where $M_{3}=M_{2}\times
S^{1}$, and perform a second KK dimensional reduction of HW theory by
shrinking this $S^{1}$ to be infinitesimally small. This will allow us to
physically derive from the topologically-invariant $\mathcal{Q}$-cohomology, a
novel 3d-HW Floer homology of $M_{2}$.

###  5.1 KK Reduction of HW Theory on $S^{1}\times S^{1}$ and the
Corresponding SQM

Let the $S^{1}$-direction of $M_{3}=M_{2}\times S^{1}$ be along $x^{1}$.
Further KK reduction along this $S^{1}$ amounts to setting $\partial_{1}\to
0$, $A_{1}=X$, $B_{1}=Y$, where $X,Y\in\Omega^{0}(M_{2})$ are scalars field on
$M_{2}$.

In particular, upon further KK reduction of HW theory along this $S^{1}$ in
$M_{3}$, (4.3) and (4.4) become

$\displaystyle\frac{d{A^{\alpha}}}{dt}$
$\displaystyle=-\epsilon^{\alpha\beta}\Big{(}D_{\beta}Y-[X,B_{\beta}]\Big{)}\,,$
(5.1) $\displaystyle\frac{dB^{\alpha}}{dt}$
$\displaystyle=-\epsilon^{\alpha\beta}\Big{(}D_{\beta}X+[Y,B_{\beta}]\Big{)}\,,$

and

$\displaystyle\frac{d{X}}{dt}$
$\displaystyle=-\epsilon^{\alpha\beta}D_{\alpha}B_{\beta}\,,$ (5.2)
$\displaystyle\frac{dY}{dt}$
$\displaystyle=-\frac{1}{2}\epsilon^{\alpha\beta}\Big{(}F_{\alpha\beta}-[B_{\alpha},B_{\beta}]\Big{)}\,,$

where $\alpha,\beta$ are indices on $M_{2}$. Defining a complexified
connection on $M_{2}$ as $\mathcal{A}=A+iB\in\Omega^{1}(M_{2},\mathbb{C})$, a
complex scalar field on $M_{2}$ as $Z=X+iY\in\Omega^{0}(M_{2},\mathbb{C})$,
(5.1) and (5.2) can be written as

$\displaystyle\dot{\mathcal{A}}^{\alpha}+i\epsilon^{\alpha\beta}\bar{\mathcal{D}}_{\beta}\bar{Z}$
$\displaystyle=0=\dot{\bar{\mathcal{A}}}^{\alpha}-i\epsilon^{\alpha\beta}\mathcal{D}_{\beta}Z\,,$
(5.3)
$\displaystyle\dot{Z}+\frac{i}{2}\epsilon^{\alpha\beta}\bar{\mathcal{F}}_{\alpha\beta}$
$\displaystyle=0=\dot{\bar{Z}}-\frac{i}{2}\epsilon^{\alpha\beta}\mathcal{F}_{\alpha\beta}\,,$

where
$\mathcal{D}_{\alpha}\equiv\partial_{\alpha}+[\mathcal{A}_{\alpha},\cdot]$ and
$\bar{\mathcal{D}}_{\alpha}\equiv\partial_{\alpha}+[\bar{\mathcal{A}}_{\alpha},\cdot]$.
In other words, 4d-HW theory, upon a second KK reduction along this $S^{1}$ to
a 3d theory on $M_{2}\times\mathbb{R}$, localizes onto configurations that
satisfy (5.3).

Just as in §3.1, this means that the 3d action can be written as

$\displaystyle S_{\text{HW}_{3}}=\frac{1}{e^{2}}\int
dt\int_{M_{2}}\text{Tr}\Bigg{(}$
$\displaystyle\big{|}\dot{\mathcal{A}}^{\alpha}+i\epsilon^{\alpha\beta}\bar{\mathcal{D}}_{\beta}\bar{Z}\big{|}^{2}+\big{|}\dot{Z}+\frac{i}{2}\epsilon^{\alpha\beta}\bar{\mathcal{F}}_{\alpha\beta}\big{|}^{2}$
(5.4)
$\displaystyle+\bigg{|}\dot{\bar{\mathcal{A}}}^{\alpha}-i\epsilon^{\alpha\beta}\mathcal{D}_{\beta}Z\bigg{|}^{2}+\bigg{|}\dot{\bar{Z}}-\frac{i}{2}\epsilon^{\alpha\beta}\mathcal{F}_{\alpha\beta}\bigg{|}^{2}+\dots\Bigg{)}\,,$

where “$\dots$” refers to fermionic terms. After suitable rescalings, the
equivalent SQM action can be obtained from (5.4) as

$\displaystyle S_{\text{SQM,BF}}=\frac{1}{e^{2}}\int dt$
$\displaystyle\bigg{|}\dot{\mathcal{A}}^{a}+g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}}{\partial\mathcal{A}^{b}}\right)^{*}\bigg{|}^{2}+\bigg{|}\dot{Z}^{a}+g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}}{\partial Z^{b}}\right)^{*}\bigg{|}^{2}$ (5.5)
$\displaystyle+\bigg{|}\dot{\bar{\mathcal{A}}}^{a}+g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}}{\partial\bar{\mathcal{A}}^{b}}\right)^{*}\bigg{|}^{2}+\bigg{|}\dot{\bar{Z}}^{a}+g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}}{\partial\bar{Z}^{b}}\right)^{*}\bigg{|}^{2}+\dots\,,$

where $(\mathcal{A}^{a},Z^{a},\bar{\mathcal{A}}^{a},{\bar{Z}}^{a})$ and $a,b$
are coordinates and indices on the space $\mathfrak{A}_{2}$ of irreducible
$(\mathcal{A}^{\alpha},Z,\bar{\mathcal{A}}^{\alpha},{\bar{Z}})$ fields on
$M_{2}$, respectively; $g^{ab}_{\mathfrak{A}_{2}}$ is the metric on
$\mathfrak{A}_{2}$; $V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})$ is the
potential function; and the choice of the subscript ‘BF’ will be clear
shortly.

###  5.2 A 3d-HW Floer Homology of $M_{2}$

Since the resulting 3d theory on $M_{2}\times\mathbb{R}$ can be interpreted as
an SQM model of complexified fields in $\mathfrak{A}_{2}$, its partition
function can, like in (3.9), be written as

${\mathcal{Z}_{\text{HW},M_{2}\times\mathbb{R}}(G)=\langle
1\rangle_{{F}_{2}(\Psi_{M_{2}})}=\sum_{p}{\cal
F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})}\,,$ (5.6)

where ${\cal F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})$ in the
$\mathcal{Q}$-cohomology is the $p^{\text{th}}$ contribution to the partition
function that depends on the expression of ${{F}_{2}(\Psi_{M_{2}})}$ in the
bosonic fields on $M_{2}$ evaluated over the corresponding solutions of
(5.1)-(5.2) restricted to $M_{2}$, and the summation in ‘$p$’ is over all
isolated and non-degenerate configurations on $M_{2}$ in $\mathfrak{A}_{2}$
that the equivalent SQM localizes onto.888This presumption that the
configurations will be isolated and non-degenerate is justified because (the
$\mathcal{Q}$-cohomology of) HW theory is topological in all directions and
therefore invariant when we further shrink this second $S^{1}$, so if $M_{2}$
(where $M_{2}\times S^{1}\times S^{1}=M_{4}$) and $G$ are chosen such as to
satisfy the conditions spelt out in footnote 5,
$\mathcal{Z}_{\text{HW},M_{2}\times\mathbb{R}}$ will be a discrete and non-
degenerate sum of contributions just like
$\mathcal{Z}_{\text{HW},M_{4}\times\mathbb{R}}$. We shall henceforth assume
such a choice of $M_{2}$ and $G$ whence the presumption would hold.

Let us now ascertain what the ${\cal
F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})$’s correspond to. Repeating here the
analysis in §3.2 with (5.5) as the action for the equivalent SQM model, we
find that we can also write (5.6) as

$\boxed{\mathcal{Z}_{\text{HW},M_{2}\times\mathbb{R}}(G)=\sum_{p}{\cal
F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})=\sum_{p}\text{HF}_{d_{p}}^{\text{HW}_{3}}(M_{2},G_{\mathbb{C}})=\mathcal{Z}^{\text{Floer}}_{\text{HW}_{3},M_{2}}(G_{\mathbb{C}})}$
(5.7)

where each ${\cal F}^{G}_{\text{HW}_{3}}(\Psi_{M_{2}}^{p})$ can be identified
with a _novel_ gauge-theoretic 3d-HW Floer homology class
$\text{HF}^{\text{HW}_{3}}_{d_{p}}(M_{2},G_{\mathbb{C}})$ of degree $d_{p}$
assigned to $M_{2}$.

Specifically, the _time-invariant complexified BF configurations on $M_{2}$_
in $\mathfrak{A}_{2}$ that obey the simultaneous vanishing of the LHS and RHS
of the _gradient flow equations_

$\boxed{\begin{aligned}
\frac{d{\mathcal{A}^{a}}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\mathcal{A}^{b}}\right)^{*}\qquad&\frac{d{\bar{\mathcal{A}}^{a}}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\bar{\mathcal{A}}^{b}}\right)^{*}\\\
\frac{d{Z}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial
Z^{b}}\right)^{*}\qquad&\frac{d{\bar{Z}}^{a}}{dt}&=-g^{ab}_{\mathfrak{A}_{2}}\left(\frac{\partial
V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})}{\partial\bar{Z}^{b}}\right)^{*}\end{aligned}}$
(5.8)

will generate the chains of the 3d-HW Floer complex with _Morse functional_

$\boxed{V_{2}(\mathcal{A},Z,\bar{\mathcal{A}},\bar{Z})=-i\int_{M_{2}}\,\text{Tr}\,\Big{(}Z\wedge\mathcal{F}-\bar{Z}\wedge\bar{\mathcal{F}}\Big{)}}$
(5.9)

in $\mathfrak{A}_{2}$, where 3d-HW flow lines, described by time-varying
solutions to (5.8), are the Floer differentials such that the number of
outgoing flow lines at each time-invariant complexified BF configuration on
$M_{2}$ in $\mathfrak{A}_{2}$ which corresponds to a time-independent solution
of the 2d equations

$\boxed{\begin{aligned} \mathcal{F}&=0=\bar{\mathcal{F}}\\\
\mathcal{D}Z&=0=\bar{\mathcal{D}}\bar{Z}\end{aligned}}$ (5.10)

is the degree $d_{p}$ of the corresponding chain in the 3d-HW Floer complex.

##  Symplectic Floer Homologies, Flat $G_{\mathbb{C}}$ Connections, and
$\theta$-Hitchin Space

In this section, we will specialize to the case where $M_{5}=\Sigma\times
M_{2}\times\mathbb{R}$, and perform a topological reduction of HW theory along
$M_{2}$, a Riemann surface of genus $g\geq 2$, using the Bershadsky-Sadov-
Johansen-Vafa reduction method [6], to obtain a 3d ${\mathcal{N}}=4$
topological sigma model on $\Sigma\times\mathbb{R}$, where $\Sigma=S^{1}\times
S^{1}$, $I\times S^{1}$ and $I\times\mathbb{R}$. By recasting this 3d sigma
model as an SQM in the appropriate space, we will be able to derive, from the
topologically-invariant $\mathcal{Q}$-cohomology, a novel symplectic Floer
homology in the double loop space of flat $G_{\mathbb{C}}$ connections on
$M_{2}$, and a novel symplectic intersection Floer homology in the loop space
of flat $G_{\mathbb{C}}$ connections on $M_{2}$ and the path space of
$\theta$-Hitchin space of $M_{2}$.

###  6.1 A 3d Sigma Model on $S^{1}\times S^{1}\times\mathbb{R}$

In $M_{5}=S^{1}\times S^{1}\times M_{2}\times\mathbb{R}$, denote by $x^{M}$
with $M\in\\{0,1\\}$ the coordinates of $S^{1}\times S^{1}$, $x^{m}$ with
$m\in\\{2,3\\}$ the coordinates of $M_{2}$, and $t$ the coordinate of
$\mathbb{R}$. topological reduction of the action along $M_{2}$ can be
achieved by scaling the metric along $M_{2}$ to the vanishing limit, as was
done in [6].

By writing in (3.1) the metric of $M_{2}$ as $g_{M_{2}}\rightarrow\epsilon
g_{M_{2}}$, we can ignore the terms with positive powers of $\epsilon$ since
they will not survive when we take the limit $\epsilon\rightarrow 0$. Only
those terms with zero power of $\epsilon$ are of interest since they will
survive when we take the limit $\epsilon\rightarrow 0$. These terms can be
identified as those with a single contraction in $M_{2}$ in (3.1), whence the
relevant 5d action is

$\displaystyle S_{\epsilon^{0}}=\frac{1}{e^{2}}\int
d^{4}xdt\,\text{Tr}\Big{(}$
$\displaystyle|F_{tm}-D^{M}B_{Mm}|^{2}+2\left|F^{+}_{Mm}+\frac{1}{4}[B_{Mn},B_{m}^{n}]+\frac{1}{4}[B_{MN},B_{m}^{N}]+\frac{1}{2}D_{t}B_{Mm}\right|^{2}$
(6.1) $\displaystyle-2D_{m}\sigma
D^{m}\bar{\sigma}+\tilde{\eta}D_{m}\psi^{m}+\eta
D_{m}\tilde{\psi^{m}}+\tilde{\psi}_{m}D_{t}\psi^{m}+2\tilde{\chi}_{mM}D_{t}\chi^{mM}$
$\displaystyle+(\tilde{\psi}_{m}D_{M}-\tilde{\psi}_{M}D_{m})\chi^{mM}+(\psi_{m}D_{M}-\psi_{M}D_{m})\tilde{\chi}^{mM}$
$\displaystyle+i\sigma(2\\{\tilde{\chi}_{mM},\tilde{\chi}^{mM}\\}-\\{\tilde{\psi}_{m},\tilde{\psi}^{m}\\})+i\bar{\sigma}(2\\{\chi_{mM},\chi^{mM}\\}-\\{\psi_{m},\psi^{m}\\})\Big{)}\,.$

That said, there are also terms with negative powers of $\epsilon$ which will
blow up when we take the limit $\epsilon\rightarrow 0$, and they are those
with more than one contraction along $M_{2}$. As such, these terms need to be
set to $0$ to ensure the finiteness of the reduction process, whence we will
have the conditions

$\displaystyle
F^{+}_{mn}+\frac{1}{4}[B_{mM},B_{n}^{M}]+\frac{1}{2}D_{t}B_{mn}=D^{m}B_{mM}$
$\displaystyle=0\,,$ (6.2a) $\displaystyle D^{n}B_{nm}$ $\displaystyle=0\,.$
(6.2b)

Re-expressing in (6.1), the coordinates of $M_{2}$ as complex coordinates
defined by $z=x^{2}+ix^{3}$, the final expression for the relevant 5d action
that will survive when we take the limit $\epsilon\rightarrow 0$ is999To
arrive at the following expression, we have used the fact that $B_{\mu\nu}$ is
a self-dual 2-form in $\Omega^{2,+}(M_{4},ad(G))$, and that the terms
$B_{MN}=\pm B_{01}=\pm B_{23}=B_{mn}$ in (6.1), vanish. This fact can be
understood as follows. Firstly, one can exploit the self-duality of
$B_{\mu\nu}$ to define a section $\phi=B_{0z}dx^{0}\wedge dz-
iB_{1z}dx^{1}\wedge dz+B_{0{\bar{z}}}dx^{0}\wedge
d{\bar{z}}+iB_{1{\bar{z}}}dx^{1}\wedge d{\bar{z}}$ of $\Omega^{1}(S^{1}\times
S^{1})\otimes\Omega^{1}(M_{2})$, where the remaining independent component
$B_{01}$ can then be interpreted as a scalar 0-form w.r.t rotations on
$M_{5}$. Secondly, one can understand from (6.2b) that $B_{01}$ generates
infinitesimal gauge transformations whilst keeping $A_{z}$ and $A_{{\bar{z}}}$
constant. Thus, to ensure that gauge connections are irreducible whence the
moduli space of (6.2) (which will play a relevant role in our resulting 3d
theory) is well-behaved, we shall consider the case that $B_{01}=0$.

$\displaystyle S_{\epsilon^{0}}=\frac{1}{e^{2}}\int
d^{2}x|dz|^{2}dt\,{\textrm{Tr}\;}\Big{(}$
$\displaystyle\,D_{t}A_{z}D_{t}A^{z}+D^{M}B_{Mz}D_{N}B^{Nz}$ (6.3)
$\displaystyle\,+D^{M}A_{z}D_{M}A^{z}+\frac{1}{2}D_{t}B_{Mz}D_{t}B^{Mz}+\dots\Big{)}\,,$

where “$\dots$” refers to fermionic terms and terms with the scalars $\sigma$
and $\bar{\sigma}$ in (6.1). Note that the terms with covariant derivatives in
$M_{2}$ ($D_{z}$ or $D_{\bar{z}}$) have been omitted as they will not
contribute when we take the limit $\epsilon\rightarrow 0$.101010When $M_{2}$
is shrunken away in the vanishing limit of $\epsilon\to 0$, it is clear that
the partial derivatives $\partial_{z,{\bar{z}}}$ will not contribute. The
remaining terms of the covariant derivatives will be paired to auxiliary
fields, which will be integrated out in a later step. Therefore, we have
chosen not to present them in the current expression.

The finiteness conditions in (6.2) can now be expressed as

$\displaystyle F_{z{\bar{z}}}-\frac{1}{2}[B_{Mz},B^{M}_{\bar{z}}]$
$\displaystyle=0\,,$ (6.4) $\displaystyle D^{z}B_{Mz}$ $\displaystyle=0\,.$

Let us denote by $\mathbf{V}\equiv\partial_{0}+\partial_{1}$ a unit vector
along $S^{1}\times S^{1}$, $F_{z{\bar{z}}}=F$ the field strength two-form on
$M_{2}$, and
$\varphi\equiv\iota_{\mathbf{V}}\phi=(B_{0z}-iB_{1z})dz+(B_{0{\bar{z}}}+iB_{1{\bar{z}}})d{\bar{z}}$
a section of $\Omega^{0}(S^{1}\times
S^{1}\times\mathbb{R})\otimes\Omega^{1}(M_{2})$ obtained by performing an
interior product of the section $\phi$ with the unit vector $\mathbf{V}$ (see
definition of $\phi$ in footnote 9). Then, with an appropriate scaling of
$\varphi$, we can also express (6.4) as

$\boxed{\begin{aligned} F-\varphi\wedge\varphi&=0\\\
D\varphi&=0\end{aligned}}$ (6.5)

This is just Hitchin’s equations for $G$ on $M_{2}$!

Recall that in (6.5), $F_{z{\bar{z}}}=D_{z}A_{\bar{z}}$,
$F_{{\bar{z}}z}=D_{\bar{z}}A_{z}$,
$\varphi=\varphi_{z}dz+\varphi_{\bar{z}}d{\bar{z}}$, where
$\varphi_{z}=B_{0z}-iB_{1z}$ and
$\varphi_{\bar{z}}=B_{0{\bar{z}}}+iB_{1{\bar{z}}}$. These $A$ and $B$
components are exactly those that appear in $S_{\epsilon^{0}}$ in (6.3). What
this means is that when we take the limit $\epsilon\to 0$ and shrink $M_{2}$
away, $S_{\epsilon^{0}}$ in (6.3) becomes an action of a 3d theory on
$S^{1}\times S^{1}\times\mathbb{R}$ with the bosonic $A$’s and $B$’s as
scalars and one-forms, respectively, which can be interpreted as coordinates
of the space of solutions $(A_{M_{2}},\varphi)$ to (6.5) (modulo gauge
transformations). In other words, we will have a 3d $\mathcal{N}=4$
topological sigma model with symplectic target $\mathcal{M}^{G}_{H}(M_{2})$,
the Hitchin moduli space of $G$ on $M_{2}$. For $M_{2}$ a Riemann surface of
genus $g\geq 2$, $\mathcal{M}^{G}_{H}(M_{2})$ is well-behaved (see also
footnote 9). We shall henceforth assume this to be the case.

Let us now determine the sigma model action. Via the cotangent bases
$(\alpha_{iz},\alpha_{{\bar{\imath}}{\bar{z}}})$ and
$(\beta_{iz},\beta_{{\bar{\imath}}{\bar{z}}})$ that can be defined for the
base and fiber of the hyperKähler target space $\mathcal{M}^{G}_{H}(M_{2})$,
we can write the bosons in (6.3) as

$\displaystyle A_{z}$ $\displaystyle=X^{i}\alpha_{iz}\,,\qquad$
$\displaystyle\qquad A_{\bar{z}}$
$\displaystyle=X^{{\bar{\imath}}}\alpha_{{\bar{\imath}}{\bar{z}}}\,,$ (6.6)
$\displaystyle B_{Mz}$ $\displaystyle=Y^{i}_{M}\beta_{iz}\,,\qquad$
$\displaystyle\qquad B_{M{\bar{z}}}$
$\displaystyle=Y^{\bar{\imath}}_{M}\beta_{{\bar{\imath}}{\bar{z}}}\,,$

where $(X,Y)$ are coordinates of the base and fiber of
$\mathcal{M}^{G}_{H}(M_{2})$, respectively.

What about the fermions in (6.3)? Well, they can be interpreted as tangent
vectors of $\mathcal{M}^{G}_{H}(M_{2})$, and can thus be written as

$\displaystyle\psi_{z}$ $\displaystyle=\lambda^{i}_{1}\alpha_{iz}\,,\qquad$
$\displaystyle\qquad\psi_{\bar{z}}$
$\displaystyle=\lambda^{{\bar{\imath}}}_{1}\alpha_{{\bar{\imath}}{\bar{z}}}\,,$
(6.7) $\displaystyle\tilde{\psi}_{z}$
$\displaystyle=\lambda^{i}_{2}\alpha_{iz}\,,\qquad$
$\displaystyle\qquad\tilde{\psi}_{\bar{z}}$
$\displaystyle=\lambda^{{\bar{\imath}}}_{2}\alpha_{{\bar{\imath}}{\bar{z}}}\,,$
$\displaystyle\chi_{Mz}$ $\displaystyle=\rho^{i}_{1M}\beta_{iz}\,,\qquad$
$\displaystyle\qquad\chi_{M{\bar{z}}}$
$\displaystyle=\rho^{{\bar{\imath}}}_{1M}\beta_{{\bar{\imath}}{\bar{z}}}\,,$
$\displaystyle\tilde{\chi}_{Mz}$
$\displaystyle=\rho^{i}_{2M}\beta_{iz}\,,\qquad$
$\displaystyle\qquad\tilde{\chi}_{M{\bar{z}}}$
$\displaystyle=\rho^{{\bar{\imath}}}_{2M}\beta_{{\bar{\imath}}{\bar{z}}}\,,$

where $(\lambda,\rho)$ are tangent vectors of the base and fiber of
$\mathcal{M}^{G}_{H}(M_{2})$, respectively.

Finally, let us topologically reduce (6.3) along $M_{2}$ by taking the limit
$\epsilon\rightarrow 0$. Noting that the fields $\sigma$, $\bar{\sigma}$,
$A_{t}$, $A_{M}$, $\eta$, $\tilde{\eta}$, $\psi_{M}$, and $\tilde{\psi}_{M}$
can be integrated out to get the curvature and Christoffel connection terms
for the 3d fermions (as was done in the context of an SQM in equations (3.4)
and (3.5)), the resultant action is111111To arrive at the following
expression, we have done three things. Firstly, the indices of the worldvolume
were lowered via the fact that the metric of $S^{1}\times S^{1}$ is flat, i.e.
$\eta^{01}=\eta^{10}=0$ and $\eta^{00}=\eta^{11}=1$. Secondly, the $Y$-,
$\rho_{1}$-, and $\rho_{2}$-fields were written only in their $M=0$ index –
this is because from the self-duality of $B$, $\chi$, $\tilde{\chi}$, we have,
in the field correspondences (6.6) and (6.7), a duality between different $M$
indices of the $Y$-, $\rho_{1}$-, and $\rho_{2}$-fields, respectively, i.e.,
$Y^{i}_{0}=-iY^{i}_{1}$, $Y^{\bar{\imath}}_{0}=iY^{\bar{\imath}}_{1}$,
$\rho^{i}_{\alpha 0}=-i\rho^{i}_{\alpha 1}$, and $\rho^{\bar{\imath}}_{\alpha
0}=i\rho^{\bar{\imath}}_{\alpha 1}$, for $\alpha\in\\{1,2\\}$. Thirdly, the
single $M=0$ index is omitted for the aforementioned fields as it is
effectively the only index, i.e., its specification is redundant.

$S_{\text{3d}}=\frac{1}{e^{2}}\int d^{2}x\,dt\,L_{1}+L_{2}\,,$ (6.8)
$\displaystyle L_{1}=g_{i{\bar{\jmath}}}\Big{(}$
$\displaystyle\partial_{t}X^{i}\partial_{t}X^{\bar{\jmath}}+\partial_{t}Y^{i}\partial_{t}Y^{\bar{\jmath}}+\partial_{0}X^{i}\partial_{0}X^{\bar{\jmath}}+\partial_{1}X^{i}\partial_{1}X^{\bar{\jmath}}+\partial_{0}Y^{i}\partial_{0}Y^{\bar{\jmath}}+\partial_{1}Y^{i}\partial_{1}Y^{\bar{\jmath}}$
$\displaystyle+\lambda^{i}_{2}\nabla_{t}\lambda^{\bar{\jmath}}_{1}+4\rho^{i}_{2}\nabla_{t}\rho^{\bar{\jmath}}_{1}-\lambda^{i}_{2}\nabla_{0}\rho^{\bar{\jmath}}_{1}+i\lambda^{i}_{2}\nabla_{1}\rho^{\bar{\jmath}}_{1}-\lambda^{i}_{1}\nabla_{0}\rho^{\bar{\jmath}}_{2}+i\lambda^{i}_{1}\nabla_{1}\rho^{\bar{\jmath}}_{2}\Big{)}\,,$
$\displaystyle
L_{2}=\Omega_{i{\bar{\jmath}}k{\bar{l}}}\Big{(}8\rho^{i}_{1}\rho^{{\bar{\jmath}}}_{1}\rho^{k}_{2}\rho^{{\bar{l}}}_{2}+\frac{1}{2}\lambda^{i}_{1}\lambda^{\bar{\jmath}}_{1}\lambda^{k}_{2}\lambda^{\bar{l}}_{2}-2\lambda^{i}_{1}\lambda^{\bar{\jmath}}_{1}\rho^{k}_{2}\rho^{{\bar{l}}}_{2}-2\rho^{i}_{1}\rho^{{\bar{\jmath}}}_{1}\lambda^{k}_{2}\lambda^{{\bar{l}}}_{2}\Big{)}\,,$

where $\Omega_{i{\bar{\jmath}}k{\bar{l}}}$ is the Riemann curvature tensor of
the target space $\mathcal{M}^{G}_{H}(M_{2})$.

The supersymmetric variations descend from (2.4) for the fields that survive
the topological reduction from $M_{5}$ to $S^{1}\times S^{1}\times\mathbb{R}$,
and they are given by

$\displaystyle\delta X^{i}$ $\displaystyle=\lambda^{i}_{1}\,,\qquad$
$\displaystyle\delta X^{\bar{\imath}}$
$\displaystyle=\lambda^{\bar{\imath}}_{1}\,,$ (6.9) $\displaystyle\delta
Y^{i}$ $\displaystyle=\rho^{i}_{1}\,,\qquad$ $\displaystyle\delta
Y^{\bar{\imath}}$ $\displaystyle=\rho^{\bar{\imath}}_{1}\,,$
$\displaystyle\delta\lambda^{i}_{1}$ $\displaystyle=0\,,\qquad$
$\displaystyle\delta\lambda^{\bar{\imath}}_{1}$ $\displaystyle=0\,,$
$\displaystyle\delta\rho^{i}_{1}$ $\displaystyle=0\,,\qquad$
$\displaystyle\delta\rho^{{\bar{\imath}}}_{1}$ $\displaystyle=0\,,$
$\displaystyle\delta\tilde{\lambda}^{i}_{2}$
$\displaystyle=-\partial_{t}X^{i}+\partial_{0}Y^{i}+i\partial_{1}Y^{i}\,,\qquad$
$\displaystyle\delta\tilde{\lambda}^{\bar{\imath}}_{2}$
$\displaystyle=-\partial_{t}X^{\bar{\imath}}+\partial_{0}Y^{\bar{\imath}}-i\partial_{1}Y^{\bar{\imath}}\,,$
$\displaystyle\delta\tilde{\rho}^{i}_{2}$
$\displaystyle=\partial_{t}Y^{i}+\partial_{0}X^{i}-i\partial_{1}X^{i}\,,\qquad$
$\displaystyle\delta\tilde{\rho}^{\bar{\imath}}_{2}$
$\displaystyle=\partial_{t}Y^{\bar{\imath}}+\partial_{0}X^{\bar{\imath}}+i\partial_{1}X^{\bar{\imath}}.$

###  6.2 An SQM in the Double Loop Space of Flat $G_{C}$ Connections

Relabelling the coordinates $(x^{0},x^{1})$ as $(r,s)$ for convenience, we get
the BPS equations of our 3d $\mathcal{N}=4$ topological sigma model with
target $\mathcal{M}^{G}_{H}(M_{2})$ as

$\displaystyle\partial_{t}X^{i}$
$\displaystyle=\partial_{r}Y^{i}+i\partial_{s}Y^{i}\,,$
$\displaystyle\quad\partial_{t}X^{\bar{\imath}}$
$\displaystyle=\partial_{r}Y^{\bar{\imath}}-i\partial_{s}Y^{\bar{\imath}}\,,$
(6.10) $\displaystyle\partial_{t}Y^{\bar{\imath}}$
$\displaystyle=-\partial_{r}X^{\bar{\imath}}-i\partial_{s}X^{\bar{\imath}}\,,$
$\displaystyle\quad\partial_{t}Y^{i}$
$\displaystyle=-\partial_{r}X^{i}+i\partial_{s}X^{i}\,.$

In fact, it can be determined from the (6.10) that
$\mathcal{M}^{G}_{H}(M_{2})=\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$,
the moduli space of flat $G_{\mathbb{C}}$ connections on $M_{2}$.121212 By
performing KK reductions of (6.10) along either of the $S^{1}$’s, the BPS
equations of the resulting 2d sigma model will tell us that the holomorphic
coordinates of $\mathcal{M}^{G}_{H}(M_{2})$ are (complex) linear combinations
of $X$ and $Y$, i.e. the 2d sigma model sees $\mathcal{M}^{G}_{H}(M_{2})$ in
either complex structure $J$ or $K$, depending on the choice of $S^{1}$ that
is being reduced. As the KK reduction is a topological process on the 3d
worldvolume that will not alter the target space, it can be inferred that the
3d sigma model will also see $\mathcal{M}^{G}_{H}(M_{2})$ in either complex
structure $J$ or $K$, whence
$\mathcal{M}^{G}_{H}(M_{2})=\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.

Therefore, it is more appropriate to re-express the BPS equations in (6.10) as

$\displaystyle\partial_{t}Z^{i}$
$\displaystyle=-i\partial_{r}Z^{i}-\partial_{s}\bar{Z}^{i}\,,$
$\displaystyle\quad\partial_{t}Z^{\bar{\imath}}$
$\displaystyle=-i\partial_{r}Z^{\bar{\imath}}+\partial_{s}\bar{Z}^{\bar{\imath}}\,,$
(6.11) $\displaystyle\partial_{t}\bar{Z}^{\bar{\imath}}$
$\displaystyle=i\partial_{r}\bar{Z}^{\bar{\imath}}-\partial_{s}{Z}^{\bar{\imath}}\,,$
$\displaystyle\quad\partial_{t}\bar{Z}^{i}$
$\displaystyle=i\partial_{r}\bar{Z}^{i}+\partial_{s}Z^{i}\,,$

where $Z^{i}\equiv X^{i}+iY^{i}$,
$Z^{{\bar{\imath}}}\equiv{X}^{{\bar{\imath}}}+i{Y}^{{\bar{\imath}}}$,
$\bar{Z}^{i}\equiv X^{i}-iY^{i}$, and
$\bar{Z}^{{\bar{\imath}}}\equiv{X}^{{\bar{\imath}}}-i{Y}^{{\bar{\imath}}}$,
are the coordinates of $\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.

Just as in §3.1, we can now rewrite the 3d action in (6.8) as

$\displaystyle S_{\text{3d}}=\frac{1}{e^{2}}\int dt\int_{S^{1}\times S^{1}}$
$\displaystyle|\dot{Z}^{i}+i\partial_{r}Z^{i}+\partial_{s}\bar{Z}^{i}|^{2}+|\dot{Z}^{\bar{\imath}}+i\partial_{r}Z^{\bar{\imath}}-\partial_{s}\bar{Z}^{\bar{\imath}}|^{2}$
(6.12)
$\displaystyle+|\dot{\bar{Z}}^{\bar{\imath}}-i\partial_{r}\bar{Z}^{\bar{\imath}}+\partial_{s}\bar{Z}^{\bar{\imath}}|^{2}+|\dot{\bar{Z}}^{i}-i\partial_{r}\bar{Z}^{i}-\partial_{s}Z^{i}|^{2}+\dots\,,$

where “…” refer to fermionic terms in (6.8). After suitable rescalings, the
equivalent SQM action can be obtained from (6.12) as

$\displaystyle S_{\text{SQM},\Sigma}=\frac{1}{e^{2}}\int dt$
$\displaystyle\left|\dot{Z}^{a}+g^{a{\bar{b}}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}\left(\frac{\partial
V_{\Sigma}}{\partial
Z^{b}}\right)^{*}\right|^{2}+\left|\dot{Z}^{\bar{a}}+g^{{\bar{a}}b}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}\left(\frac{\partial
V_{\Sigma}}{\partial Z^{\bar{b}}}\right)^{*}\right|^{2}$ (6.13)
$\displaystyle+\left|\dot{\bar{Z}}^{\bar{a}}+g^{{\bar{a}}b}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}\left(\frac{\partial
V_{\Sigma}}{\partial\bar{Z}^{\bar{b}}}\right)^{*}\right|^{2}+\left|\dot{\bar{Z}}^{a}+g^{a{\bar{b}}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}\left(\frac{\partial
V_{\Sigma}}{\partial\bar{Z}^{b}}\right)^{*}\right|^{2}+\dots\,,$

where $(Z^{a},Z^{\bar{a}},\bar{Z}^{a},\bar{Z}^{\bar{a}})$ and
$g^{a{\bar{b}}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}$ are
the coordinates and the metric on the double loop space
$L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ of maps from
$S^{1}\times S^{1}$ to the symplectic manifold
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, respectively; and
$V_{\Sigma}(Z,\bar{Z})$ is the potential function. In other words, we
equivalently have an SQM in
$L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.

###  6.3 A Symplectic Floer Homology in the Double Loop Space of Flat
$G_{\mathbb{C}}$ Connections

Since the resulting 3d theory on $S^{1}\times S^{1}\times\mathbb{R}$ can be
interpreted as an SQM model in
$L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, its partition
function can be written as

$\mathcal{Z}_{\text{HW},S^{1}\times S^{1}\times\mathbb{R}}(G)=\sum_{q}{\cal
F}^{q}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}\,,$ (6.14)

where the $q^{\text{th}}$ contribution ${\cal
F}^{q}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}$ to the
partition function, is in the $\mathcal{Q}$-cohomology, and the summation in
‘$q$’ is over all isolated and non-degenerate configurations on $S^{1}\times
S^{1}$ in $L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ that the
equivalent SQM localizes onto.131313That the configurations will be isolated
and non-degenerate will be clear shortly.

Via a similar analysis to that in §3.2 with (6.13) as the action for the
equivalent SQM model, we find that we can also write (6.14) as

$\boxed{\mathcal{Z}_{\text{HW},S^{1}\times
S^{1}\times\mathbb{R}}(G)=\sum_{q}{\cal
F}^{q}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}=\sum_{q}\text{HSF}_{d_{q}}({L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})})=\mathcal{Z}^{\text{SympFloer}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}}$
(6.15)

where each ${\cal
F}^{q}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})}$ can be
identified with a _novel_ symplectic Floer homology class
$\text{HSF}_{d_{q}}({L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})})$
of degree $d_{q}$ in $L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.

Specifically, the time-invariant maps in
$L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ that obey the
simultaneous vanishing of the LHS and RHS of the gradient flow equations

$\boxed{\begin{aligned}
\frac{dZ^{a}}{dt}&=-g^{a{\bar{b}}}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial
Z^{b}}\right)^{*}\qquad&\frac{d\bar{Z}^{a}}{dt}&=-g^{a{\bar{b}}}_{L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial\bar{Z}^{b}}\right)^{*}\\\
\frac{dZ^{\bar{a}}}{dt}&=-g^{{\bar{a}}b}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial
Z^{\bar{b}}}\right)^{*}\qquad&\frac{d\bar{Z}^{\bar{a}}}{dt}&=-g^{{\bar{a}}b}_{L^{2}{\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}}}\left(\frac{\partial
V_{\Sigma}(Z,\bar{Z})}{\partial\bar{Z}^{\bar{b}}}\right)^{*}\end{aligned}}$
(6.16)

will generate the chains of the symplectic Floer complex with _Morse
functional_

$\boxed{V_{\Sigma}(Z,\bar{Z})=\int_{S^{1}\times
S^{1}}drds\,g_{i{\bar{\jmath}}}\Big{(}Z^{i}\partial_{s}Z^{\bar{\jmath}}-iZ^{i}\partial_{r}\bar{Z}^{\bar{\jmath}}-\bar{Z}^{i}\partial_{s}\bar{Z}^{\bar{\jmath}}+i\bar{Z}^{i}\partial_{r}Z^{\bar{\jmath}}\Big{)}}$
(6.17)

such that flow lines, described by time-varying solutions to (6.16), are the
Floer differentials whereby the number of outgoing flow lines at each time-
invariant map in $L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$
which corresponds to a time-independent solution of the 2d equations

$\boxed{\begin{aligned}
i\partial_{r}Z^{i}+\partial_{s}\bar{Z}^{i}&=0=i\partial_{r}\bar{Z}^{i}+\partial_{s}Z^{i}\\\
i\partial_{r}\bar{Z}^{\bar{\imath}}-\partial_{s}{Z}^{\bar{\imath}}&=0=i\partial_{r}Z^{\bar{\imath}}-\partial_{s}\bar{Z}^{\bar{\imath}}\end{aligned}}$
(6.18)

on $S^{1}\times S^{1}$, is the degree $d_{q}$ of the corresponding chain in
the symplectic Floer complex.

Notice that we can equivalently combine the terms in (6.18) to get
$\partial_{\bar{w}}Y^{i}=0=\partial_{w}\bar{Y}^{\bar{\imath}}$ and
$\partial_{\bar{w}}\bar{X}^{\bar{\imath}}=0=\partial_{w}X^{i}$, which
describes simultaneous holomorphic and antiholomorphic maps from $S^{1}\times
S^{1}$ to the fiber and base of $\mathcal{M}^{G}_{H}(M_{2})$, respectively, so
the time-invariant maps in
$L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ that they correspond
to, are indeed isolated and non-degenerate.141414The dimension $d$ of the
moduli space of (anti)holomorphic maps from a genus $g$ Riemann surface to a
target space $\mathcal{M}$ is given by
$d=c_{1}(\mathcal{M})\cdot\beta+\text{dim}\mathcal{M}(1-g)$, where $c_{1}$ is
the first Chern class and $\beta$ is the degree of the map. In our case, $g=1$
and $c_{1}(\mathcal{M})=0$, so $d=0$. Thus, the critical point set of
$V_{\Sigma}$ in $L^{2}\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ is,
in our case, zero-dimensional, and is therefore made up of isolated points
which, one can, up to the addition of physically-inconsequential
$\mathcal{Q}$-exact terms in the action to deform $V_{\Sigma}$, assume to be
non-degenerate.

###  6.4 An SQM in the Interval Space of the Loop Space of Flat
$G_{\mathbb{C}}$ Connections

Replacing the $S^{1}$ along the $s$-direction with an interval $I$, the action
(6.8) can be expressed as

$\displaystyle S_{\text{3d}}=\frac{1}{e^{2}}\int dt\int_{I\times S^{1}}drds$
$\displaystyle|\dot{Z}^{i}+i\partial_{r}Z^{i}+\partial_{s}\bar{Z}^{i}|^{2}+|\dot{Z}^{\bar{\imath}}+i\partial_{r}Z^{\bar{\imath}}-\partial_{s}\bar{Z}^{\bar{\imath}}|^{2}$
(6.19)
$\displaystyle+|\dot{\bar{Z}}^{\bar{\imath}}-i\partial_{r}\bar{Z}^{\bar{\imath}}+\partial_{s}Z^{\bar{\imath}}|^{2}+|\dot{\bar{Z}}^{i}-i\partial_{r}\bar{Z}^{i}-\partial_{s}Z^{i}|^{2}+\dots\,,$

where “$\dots$” are boundary and fermionic terms.

A 2d Sigma Model on $\mathbb{R}\times I$ with Target the Loop Space of Flat
$G_{\mathbb{C}}$ Connections

After suitable rescalings, it can be recast as151515Here, we have made use of
Stokes’ theorem and the fact that $S^{1}$ has no boundary to note that
$\partial_{s}X$, $\partial_{s}Y$ should vanish in their integration over
$S^{1}$.

$\displaystyle S_{\text{2d},I\rightarrow S^{1}}=\frac{1}{e^{2}}\int
dt\int_{I}dr$
$\displaystyle|\dot{Z}^{A}+i\partial_{r}Z^{A}|^{2}+|\dot{Z}^{\bar{A}}+i\partial_{r}Z^{\bar{A}}|^{2}+|\dot{\bar{Z}}^{\bar{A}}-i\partial_{r}\bar{Z}^{\bar{A}}|^{2}+|\dot{\bar{Z}}^{A}-i\partial_{r}\bar{Z}^{A}|^{2}+\dots\,,$
(6.20)

where $(Z^{A},Z^{\bar{A}},\bar{Z}^{A},\bar{Z}^{\bar{A}})$ are the coordinates
on the loop space $L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ of maps
from $S^{1}$ to the symplectic manifold
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$. In other words, the 3d
sigma model on $\mathbb{R}\times I\times S^{1}$ with target
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ is equivalent to a 2d
sigma model on $\mathbb{R}\times I$ with target
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.

A 2d A-model with Branes $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$

The 2d sigma model describes open strings with worldsheet $\mathbb{R}\times I$
propagating in $L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, ending on
branes $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$. Let $w=\frac{1}{2}(t+ir)$.
Then, its BPS equations can be read off from (6.20) and expressed as

$\displaystyle\partial_{\bar{w}}Z^{A}$ $\displaystyle=0\,,$
$\displaystyle\partial_{\bar{w}}Z^{\bar{A}}$ $\displaystyle=0\,,$ (6.21)
$\displaystyle\partial_{w}\bar{Z}^{\bar{A}}$ $\displaystyle=0\,,$
$\displaystyle\partial_{w}\bar{Z}^{A}$ $\displaystyle=0\,.$

In other words, the 2d sigma model is characterized by holomorphic maps of
$\mathbb{R}\times I$ into
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$. As such, our 2d sigma
model is an A-model, and $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$ are A-branes
in $L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, where
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ can also be regarded as
$L\mathcal{M}^{G}_{H}(M_{2})$ in complex structure $J$.161616To understand
this claim, first, note that the subspace of constant maps in
$L\mathcal{M}^{G}_{H}(M_{2})$ is actually $\mathcal{M}^{G}_{H}(M_{2})$, so
$L\mathcal{M}^{G}_{H}(M_{2})$ would necessarily inherit the hyperKähler
structure of $\mathcal{M}^{G}_{H}(M_{2})$, i.e., $L\mathcal{M}^{G}_{H}(M_{2})$
can also be in complex structure $I$, $J$, or $K$. Second,
$\mathcal{M}^{G}_{H}(M_{2})$ in complex structure $J$ or $K$ is
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, and in our case where the
holomorphic coordinates of $\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$
are the $Z$’s, it would mean that
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ results from
$\mathcal{M}^{G}_{H}(M_{2})$ being in complex structure $J$. Altogether, this
means that we can, in our case, regard
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ as
$L\mathcal{M}^{G}_{H}(M_{2})$ in complex structure $J$.  $\mathscr{L}_{0}$ and
$\mathscr{L}_{1}$ are therefore necessarily of type $(*,\text{A},*)$ in
complex structure $(I,J,K)$.

An SQM in the Interval Space of the Loop Space of Flat $G_{\mathbb{C}}$
Connections

Just as in §3.1, the equivalent SQM action can be obtained from (6.20) as

$\displaystyle S_{\text{SQM},I\rightarrow S^{1}}=\frac{1}{e^{2}}\int dt$
$\displaystyle\left|\dot{Z}^{\alpha}+g^{\alpha\bar{\beta}}_{I\rightarrow
S^{1}}\left(\frac{\partial V_{I\rightarrow S^{1}}}{\partial
Z^{\beta}}\right)^{*}\right|^{2}+\left|\dot{Z}^{\bar{\alpha}}+g^{\bar{\alpha}\beta}_{I\rightarrow
S^{1}}\left(\frac{\partial V_{I\rightarrow S^{1}}}{\partial
Z^{\bar{\beta}}}\right)^{*}\right|^{2}$ (6.22)
$\displaystyle+\left|\dot{\bar{Z}}^{\bar{\alpha}}+g^{\bar{\alpha}\beta}_{I\rightarrow
S^{1}}\left(\frac{\partial V_{I\rightarrow
S^{1}}}{\partial\bar{Z}^{\bar{\beta}}}\right)^{*}\right|^{2}+\left|\dot{\bar{Z}}^{\alpha}+g^{\alpha\bar{\beta}}_{I\rightarrow
S^{1}}\left(\frac{\partial V_{I\rightarrow
S^{1}}}{\partial\bar{Z}^{\beta}}\right)^{*}\right|^{2}+\dots\,,$

where $(Z^{\alpha},Z^{\bar{\alpha}},\bar{Z}^{\alpha},\bar{Z}^{\bar{\alpha}})$
and $g^{\alpha\bar{\beta}}_{I\rightarrow S^{1}}$ are the coordinates and
metric on the interval space
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$
of smooth trajectories from $\mathscr{L}_{0}$ to $\mathscr{L}_{1}$ in
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, respectively; and
$V_{I\rightarrow S^{1}}(Z,\bar{Z})$ is the potential function. In other words,
we equivalently have an SQM in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$.

###  6.5 A Symplectic Intersection Floer Homology in the Loop Space of Flat
$G_{\mathbb{C}}$ Connections

Since the resulting 2d A-model on $\mathbb{R}\times I$ with target
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$ and action (6.20), can be
interpreted as an SQM in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$
with action (6.22), its partition function can be expressed as

$\mathcal{Z}_{\text{HW},\mathbb{R}\times I\rightarrow S^{1}}(G)=\sum_{u}{\cal
F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}\,,$ (6.23)

where the $u^{\text{th}}$ contribution ${\cal
F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$ to the partition
function, is in the $\mathcal{Q}$-cohomology, and the summation in ‘$u$’ is
over all isolated and non-degenerate configurations on $I$ in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$
that the equivalent SQM localizes onto.171717That the configurations on $I$
will be isolated and non-degenerate can be justified as follows. As we will
show, the aforementioned configurations on $I$ are constant paths which
correspond to the intersection points of ${\mathscr{L}}_{0}$ and
${\mathscr{L}}_{1}$. These points can always be made isolated and non-
degenerate by adding physically-inconsequential $\mathcal{Q}$-exact terms to
the SQM action which will (1) correspond to a deformation of the 2d sigma
model worldsheet such that the ${\mathscr{L}}_{0}$ and ${\mathscr{L}}_{1}$
branes can be moved to intersect only at isolated points, and (2) deform the
SQM potential accordingly such that its critical points will be non-
degenerate. Let us now determine what ${\cal
F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$ is.

To this end, first note that the BPS equations of the 2d A-model in (6.21) can
also be written as

$\displaystyle\frac{\partial Z^{A}}{\partial t}+i\frac{\partial
Z^{A}}{\partial r}$ $\displaystyle=0\,,\qquad$ $\displaystyle\frac{\partial
Z^{\bar{A}}}{\partial t}+i\frac{\partial Z^{\bar{A}}}{\partial r}$
$\displaystyle=0\,,$ (6.24)
$\displaystyle\frac{\partial\bar{Z}^{\bar{A}}}{\partial
t}-i\frac{\partial\bar{Z}^{\bar{A}}}{\partial r}$ $\displaystyle=0\,,\qquad$
$\displaystyle\frac{\partial\bar{Z}^{A}}{\partial
t}-i\frac{\partial\bar{Z}^{A}}{\partial r}$ $\displaystyle=0\,.$

By the squaring argument [7] applied to (6.22), we find that its equivalent
SQM will localize onto configurations that set the LHS and RHS of the squared
terms therein _simultaneously_ to zero. In other words, its equivalent SQM
will localize onto time-invariant critical points of $V_{I\rightarrow S^{1}}$
that correspond to 2d A-model configurations that obey

$\displaystyle\frac{\partial Z^{A}}{\partial r}$ $\displaystyle=0\,,\qquad$
$\displaystyle\frac{\partial Z^{\bar{A}}}{\partial r}$ $\displaystyle=0\,,$
(6.25) $\displaystyle\frac{\partial\bar{Z}^{\bar{A}}}{\partial r}$
$\displaystyle=0\,,\qquad$ $\displaystyle\frac{\partial\bar{Z}^{A}}{\partial
r}$ $\displaystyle=0\,.$

These are constant intervals, which correspond to stationary trajectories in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$,
i.e., intersection points of $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$.

Second, via a similar analysis to that in §3.2, with (6.22) as the action for
the equivalent SQM model, we thus find that we can also express (6.23) as

$\boxed{\mathcal{Z}_{\text{HW},\mathbb{R}\times I\rightarrow
S^{1}}(G)=\sum_{u}{\cal
F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}=\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}},\mathscr{L}_{0},\mathscr{L}_{1})=\mathcal{Z}^{\text{IntSympFloer}}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}}$
(6.26)

where each ${\cal F}^{u}_{L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}}$
can be identified with
$\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G_{\mathbb{C}},M_{2}}_{\text{flat}}},\mathscr{L}_{0},\mathscr{L}_{1})$,
a symplectic intersection Floer homology class, that is generated by the
intersection points of $\mathscr{L}_{0}$ and $\mathscr{L}_{1}$ in
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$, where its degree $d_{u}$
is counted by the Floer differential realized as the flow lines of the SQM
whose gradient flow equations are the squared terms in (6.22) set to zero.

###  6.6 An SQM in the Interval Space of the Path Space of $\theta$-Hitchin
Space

$\theta$-deformed 3d Sigma Model BPS Equations

Let us further relabel the coordinate $r$ as $\tau$. By replacing the $S^{1}$
along the $\tau$-direction with $\mathbb{R}$, and the remaining $S^{1}$ along
the $s$-direction with $I$, we will have a 3d sigma model on $\mathbb{R}\times
I\times\mathbb{R}$, i.e., $I\times\mathbb{R}^{2}$. For later convenience, we
shall take $\tau$ to be the temporal direction (instead of $t$).

Notice that we can make use of the rotational symmetry of $\mathbb{R}^{2}$ to
rotate the 3d sigma model about $I$ by an angle $\theta$. The BPS equations
(6.10) (which continue to apply here even though we have replaced $S^{1}\times
S^{1}$ with $\mathbb{R}\times I$) then become

$\displaystyle\partial_{\tau}X^{i}$
$\displaystyle=-\partial_{t}Y^{i}+i\partial_{s}\big{(}X^{i}\cos(\theta)+Y^{i}\sin(\theta)\big{)}\,,$
(6.27) $\displaystyle\partial_{\tau}Y^{i}$
$\displaystyle=\partial_{t}X^{i}-i\partial_{s}\big{(}Y^{i}\cos(\theta)-X^{i}\sin(\theta)\big{)}\,,$
$\displaystyle\partial_{\tau}{X}^{\bar{\imath}}$
$\displaystyle=-\partial_{t}{Y}^{\bar{\imath}}-i\partial_{s}\big{(}{X}^{\bar{\imath}}\cos(\theta)+{Y}^{\bar{\imath}}\sin(\theta)\big{)}\,,$
$\displaystyle\partial_{\tau}{Y}^{\bar{\imath}}$
$\displaystyle=\partial_{t}{X}^{\bar{\imath}}+i\partial_{s}\big{(}\bar{Y}^{\bar{\imath}}\cos(\theta)-{X}^{\bar{\imath}}\sin(\theta)\big{)}\,,$

which can be viewed as $\theta$-deformed 3d sigma model BPS equations.

$\theta$, and Flat $G_{\mathbb{C}}$ Connections or Higgs Bundles on $M_{2}$

To determine what role $\theta$ plays, let us take a few specific values of
$\theta$ and discuss the consequences. When $\theta=\pi/2$, (6.27) becomes

$\displaystyle\partial_{\tau}X^{i}$
$\displaystyle=-\partial_{t}Y^{i}+i\partial_{s}Y^{i}\,,$
$\displaystyle\quad\partial_{\tau}Y^{i}$
$\displaystyle=\partial_{t}X^{i}+i\partial_{s}X^{i}\,,$ (6.28)
$\displaystyle\partial_{\tau}{X}^{\bar{\imath}}$
$\displaystyle=-\partial_{t}{Y}^{\bar{\imath}}-i\partial_{s}{Y}^{\bar{\imath}}\,,$
$\displaystyle\quad\partial_{\tau}{Y}^{\bar{\imath}}$
$\displaystyle=\partial_{t}{X}^{\bar{\imath}}-i\partial_{s}{X}^{\bar{\imath}}\,,$

which is similar in form to (6.10) (where the time-derivative of the $(X,Y)$
fields is equal to the spatial derivatives of the $(Y,X)$ fields), whence it
can be determined that the target space of the 3d sigma model is once again
$\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})$.181818The same reasoning
in footnote 12 applies here.

When $\theta=\pi$, (6.27) becomes

$\displaystyle\partial_{\tau}X^{i}$
$\displaystyle=-\partial_{t}Y^{i}-i\partial_{s}X^{i}\,,$
$\displaystyle\quad\partial_{\tau}Y^{i}$
$\displaystyle=\partial_{t}X^{i}+i\partial_{s}Y^{i}\,,$ (6.29)
$\displaystyle\partial_{\tau}{X}^{\bar{\imath}}$
$\displaystyle=-\partial_{t}{Y}^{\bar{\imath}}+i\partial_{s}{X}^{\bar{\imath}}\,,$
$\displaystyle\quad\partial_{\tau}{Y}^{\bar{\imath}}$
$\displaystyle=\partial_{t}X^{i}-i\partial_{s}{Y}^{\bar{\imath}}\,,$

for which it can be determined that the symplectic target space of the 3d
sigma model is $\mathcal{M}^{G}_{\text{Higgs}}(M_{2})$, the moduli space of
Higgs bundles of $G$ on $M_{2}$.191919To see this, we can adapt the reasoning
employed in footnote 12, where the BPS equations of the resulting 2d sigma
model will tell us that the holomorphic coordinates of
$\mathcal{M}^{G}_{H}(M_{2})$ are independently $X$ and $Y$, i.e., the 2d sigma
model now sees $\mathcal{M}^{G}_{H}(M_{2})$ in complex structure $I$. Thus, it
can be inferred that the 3d sigma model will also see
$\mathcal{M}^{G}_{H}(M_{2})$ in complex structure $I$, whence
$\mathcal{M}^{G}_{H}(M_{2})=\mathcal{M}^{G}_{\text{Higgs}}(M_{2})$.

This tells us that the complex structure of the target space
$\mathcal{M}^{G}_{H}(M_{2})$ varies as we vary $\theta$. Therefore, let us
henceforth denote the target space in complex structure $\mathcal{J}(\theta)$,
where $\mathcal{J}(\theta)$ is a linear combination of the complex structures
$I$, $J$, $K$ that depends on $\theta$, as
$\mathcal{M}^{G}_{H,\theta}(M_{2})$, the $\theta$-Hitchin moduli space of $G$
on $M_{2}$.

The $\theta$-rotated 3d Sigma Model Action

The action of the $\theta$-rotated 3d sigma model is

$\displaystyle S_{\text{3d}}(\theta)=\frac{1}{e^{2}}\int d\tau
ds\int_{\mathbb{R}}dt$
$\displaystyle|\partial_{\tau}{X}^{i}+p^{i}(\theta)|^{2}+|\partial_{\tau}{Y}^{i}-q^{i}(\theta)|^{2}$
(6.30)
$\displaystyle+|\partial_{\tau}{{X}}^{\bar{\imath}}+{\bar{p}}^{\bar{\imath}}(\theta)|^{2}+|\partial_{\tau}{{Y}}^{\bar{\imath}}-{\bar{q}}^{\bar{\imath}}(\theta)|^{2}+\dots\,,$

where

$\displaystyle p^{i}(\theta)$
$\displaystyle=\partial_{t}Y^{i}-i\partial_{s}\big{(}X^{i}\cos(\theta)+Y^{i}\sin(\theta)\big{)}\,,$
(6.31) $\displaystyle q^{i}(\theta)$
$\displaystyle=\partial_{t}X^{i}-i\partial_{s}\big{(}Y^{i}\cos(\theta)-X^{i}\sin(\theta)\big{)}\,,$
$\displaystyle{\bar{p}}^{\bar{\imath}}(\theta)$
$\displaystyle=\partial_{t}{Y}^{\bar{\imath}}+i\partial_{s}\big{(}{X}^{\bar{\imath}}\cos(\theta)+{Y}^{\bar{\imath}}\sin(\theta)\big{)}\,,$
$\displaystyle{\bar{q}}^{\bar{\imath}}(\theta)$
$\displaystyle=\partial_{t}{X}^{\bar{\imath}}+i\partial_{s}\big{(}{Y}^{\bar{\imath}}\cos(\theta)-{X}^{\bar{\imath}}\sin(\theta)\big{)}\,,$

and “$\dots$” refer to boundary and fermionic terms.

A 2d Sigma Model on $\mathbb{R}\times I$ with Target the Path Space of
$\theta$-Hitchin Space

As before, we can recast the $\theta$-rotated 3d sigma model on
$\mathbb{R}\times I\times\mathbb{R}$ as a 2d sigma model on $\mathbb{R}\times
I$ with target the path space
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2})\big{)}$ of
maps from $\mathbb{R}$ to $\mathcal{M}^{G}_{H,\theta}(M_{2})$, whereby the
action is

$\displaystyle
S_{\text{2d},I\rightarrow\mathbb{R}}(\theta)=\frac{1}{e^{2}}\int
d\tau\int_{I}ds$
$\displaystyle|\partial_{\tau}{X}^{C}+p^{C}(\theta)|^{2}+|\partial_{\tau}{Y}^{C}-q^{C}(\theta)|^{2}$
(6.32)
$\displaystyle+|\partial_{\tau}{{X}}^{\bar{C}}+{\bar{p}}^{\bar{C}}(\theta)|^{2}+|\partial_{\tau}{{Y}}^{\bar{C}}-{\bar{q}}^{\bar{C}}(\theta)|^{2}+\dots\,,$

where $(X^{C},Y^{C},{X}^{\bar{C}},{Y}^{\bar{C}})$ are the coordinates on
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2})\big{)}$, and

$\displaystyle p^{C}$
$\displaystyle=-i\partial_{s}\big{(}X^{C}\cos(\theta)+Y^{C}\sin(\theta)\big{)}\,,$
(6.33) $\displaystyle q^{C}$
$\displaystyle=-i\partial_{s}\big{(}Y^{C}\cos(\theta)-X^{C}\sin(\theta)\big{)}\,,$
$\displaystyle{\bar{p}}^{\bar{C}}$
$\displaystyle=i\partial_{s}\big{(}{X}^{\bar{C}}\cos(\theta)+{Y}^{\bar{C}}\sin(\theta)\big{)}\,,$
$\displaystyle{\bar{q}}^{\bar{C}}$
$\displaystyle=i\partial_{s}\big{(}{Y}^{\bar{C}}\cos(\theta)-{X}^{\bar{C}}\sin(\theta)\big{)}\,.$

A 2d $\text{A}_{\theta}$-model with Branes $\mathscr{P}_{0}(\theta)$ and
$\mathscr{P}_{1}(\theta)$

We thus have a 2d sigma model that describes open strings with worldsheet
$\mathbb{R}\times I$ propagating in
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2})\big{)}$,
ending on branes $\mathscr{P}_{0}(\theta)$ and $\mathscr{P}_{1}(\theta)$. Let
$w(\theta)$ and ${\bar{w}}(\theta)$ be generalized coordinates on the
worldsheet such that
$\partial_{w(\theta)}=\partial_{\tau}-ie^{i\theta}\partial_{s}$, and
$\partial_{{\bar{w}}(\theta)}=\partial_{\tau}+ie^{-i\theta}\partial_{s}$.
Then, its BPS equations can be read off from (6.32) and expressed as

$\displaystyle\partial_{{\bar{w}}(-\theta)}\bar{Z}^{\bar{C}}+\partial_{{\bar{w}}(\theta)}Z^{\bar{C}}$
$\displaystyle=0\,,$
$\displaystyle\quad\partial_{{\bar{w}}(\theta)}Z^{C}-\partial_{{\bar{w}}(-\theta)}\bar{Z}^{C}$
$\displaystyle=0\,,$ (6.34)
$\displaystyle\partial_{w(-\theta)}Z^{C}+\partial_{w(\theta)}\bar{Z}^{C}$
$\displaystyle=0\,,$
$\displaystyle\quad\partial_{w(\theta)}\bar{Z}^{\bar{C}}-\partial_{w(-\theta)}Z^{\bar{C}}$
$\displaystyle=0\,,$

where $Z^{C}\equiv X^{C}+iY^{C}$, $Z^{\bar{C}}\equiv
X^{\bar{C}}+iY^{\bar{C}}$, $\bar{Z}^{C}\equiv X^{C}-iY^{C}$, and
${\bar{Z}}^{\bar{C}}\equiv X^{\bar{C}}-iY^{\bar{C}}$.

When $\theta=0$, (6.34) can be written as

$\displaystyle\partial_{{\bar{w}}}{X}^{\bar{C}}$ $\displaystyle=0\,,$
$\displaystyle\quad\partial_{{\bar{w}}}Y^{C}$ $\displaystyle=0\,,$ (6.35)
$\displaystyle\partial_{w}X^{C}$ $\displaystyle=0\,,$
$\displaystyle\quad\partial_{w}{Y}^{\bar{C}}$ $\displaystyle=0\,,$

where $w=w(0)=\frac{1}{2}(\tau+is)$ and
${\bar{w}}={\bar{w}}(0)=\frac{1}{2}(\tau-is)$. In other words, the 2d sigma
model is characterised by holomorphic/antiholomorphic maps of
$\mathbb{R}\times I$ into
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{\text{Higgs}}(M_{2})\big{)}$.
That is, it is a standard A-model.

When $\theta=\pi$, (6.34) can be written as

$\displaystyle\partial_{{\bar{w}}}X^{C}$ $\displaystyle=0\,,$
$\displaystyle\quad\partial_{{\bar{w}}}Y^{\bar{C}}$ $\displaystyle=0\,,$
(6.36) $\displaystyle\partial_{w}X^{\bar{C}}$ $\displaystyle=0\,,$
$\displaystyle\quad\partial_{w}Y^{C}$ $\displaystyle=0\,,$

where ${\bar{w}}=w(\pi)=\frac{1}{2}(\tau-is)$ and
$w={\bar{w}}(\pi)=\frac{1}{2}(\tau+is)$. In other words, the 2d sigma model is
characterised by holomorphic/antiholomorphic maps of $\mathbb{R}\times I$ into
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{\text{Higgs}}(M_{2})\big{)}$.
That is, it is a standard A-model.

When $\theta=\pi/2$, (6.34) can be written as

$\displaystyle\partial_{{\bar{w}}}(X^{\bar{C}}+Y^{\bar{C}})$
$\displaystyle=0\,,$ $\displaystyle\quad\partial_{{\bar{w}}}(X^{C}-Y^{C})$
$\displaystyle=0\,,$ (6.37) $\displaystyle\partial_{w}(X^{C}+Y^{C})$
$\displaystyle=0\,,$ $\displaystyle\quad\partial_{w}(X^{\bar{C}}-Y^{\bar{C}})$
$\displaystyle=0\,.$

In other words, the 2d sigma model is characterised by
holomorphic/antiholomorphic maps of $\mathbb{R}\times I$ into
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(M_{2})\big{)}$.
That is, it is a standard A-model.

As such, our 2d sigma model is a $\theta$-generalized A-model with
$\theta$-generalized branes. We will henceforth call it an
$\text{A}_{\theta}$-model with $\text{A}_{\theta}$-branes
$\mathscr{P}_{0}(\theta)$ and $\mathscr{P}_{1}(\theta)$ in
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2})\big{)}$, where
if $\theta=0$, $\pi$ and $\pi/2$, it is just a standard A-model with standard
A-branes. From (6.35)-(6.37), one can see that the standard A-branes at
$\theta=0$, $\pi$ and $\pi/2$ are necessarily of type $(\text{A},*,*)$,
$(\text{A},*,*)$ and $(*,*,\text{A})$ in complex structure $(I,J,K)$,
respectively.

An SQM in the Interval Space of the Path Space of $\theta$-Hitchin Space

Once again, we can obtain the equivalent SQM action as

$\displaystyle S_{\text{SQM},I\rightarrow\mathbb{R}}=\frac{1}{e^{2}}\int
d\tau$
$\displaystyle\left|\partial_{\tau}{X}^{\gamma}+g^{\gamma\bar{\delta}}_{I\rightarrow\mathbb{R}}\left(\frac{\partial
V_{I\rightarrow\mathbb{R}}}{\partial
X^{\delta}}\right)^{*}\right|^{2}+\left|\partial_{\tau}{\bar{X}}^{\bar{\gamma}}+g^{\bar{\gamma}\delta}_{I\rightarrow\mathbb{R}}\left(\frac{\partial
V_{I\rightarrow\mathbb{R}}}{\partial\bar{X}^{\bar{\delta}}}\right)^{*}\right|^{2}$
(6.38)
$\displaystyle+\left|\partial_{\tau}{Y}^{\gamma}+g^{\gamma\bar{\delta}}_{I\rightarrow\mathbb{R}}\left(\frac{\partial
V_{I\rightarrow\mathbb{R}}}{\partial
Y^{\delta}}\right)^{*}\right|^{2}+\left|\partial_{\tau}{\bar{Y}}^{\bar{\gamma}}+g^{\bar{\gamma}\delta}_{I\rightarrow\mathbb{R}}\left(\frac{\partial
V_{I\rightarrow\mathbb{R}}}{\partial\bar{Y}^{\bar{\delta}}}\right)^{*}\right|^{2}+\dots\,,$

where $(X^{\gamma},Y^{\gamma},\bar{X}^{\bar{\gamma}},\bar{Y}^{\bar{\gamma}})$
and $g^{\gamma\bar{\delta}}_{I\rightarrow\mathbb{R}}$ are, respectively, the
coordinates and metric on the interval space
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$
of smooth trajectories from $\mathscr{P}_{0}(\theta)$ to
$\mathscr{P}_{1}(\theta)$ in
$\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2})\big{)}$; and
$V_{I\rightarrow\mathbb{R}}(X^{\gamma},Y^{\gamma},\bar{X}^{\bar{\gamma}},\bar{Y}^{\bar{\gamma}},\theta)$
is the potential function.

In other words, we equivalently have an SQM in
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$.

###  6.7 A Symplectic Intersection Floer Homology in the Path Space of
$\theta$-Hitchin Space

Likewise, since the 2d Aθ-model on $\mathbb{R}\times I$ with target
$\mathcal{M}(\mathbb{R},\mathcal{M}^{G}_{H.\theta}(M_{2}))$ can be interpreted
as an SQM in
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$
with action (6.38), its partition function can be expressed as

$\mathcal{Z}_{\text{HW}^{\theta},\mathbb{R}\times
I\rightarrow\mathbb{R}}(G)=\sum_{v}{\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}\,,$ (6.39)

where the $v^{\text{th}}$ contribution ${\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$ to the
partition function, is in the $\mathcal{Q}$-cohomology, and the summation in
‘$v$’ is over all isolated and non-degenerate configurations on $I$ in
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$
that the equivalent SQM localizes onto.202020The same reasoning as footnote 17
applies here. Let us determine what ${\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$ is.

To this end, first note that the BPS equations of the 2d Aθ-model in (6.34)
can also be written as

$\displaystyle\frac{\partial X^{C}}{\partial\tau}-i\frac{\partial}{\partial
s}\big{(}X^{C}\cos(\theta)+Y^{C}\sin(\theta)\big{)}$ $\displaystyle=0\,,\quad$
$\displaystyle\frac{\partial Y^{C}}{\partial\tau}+i\frac{\partial}{\partial
s}\big{(}Y^{C}\cos(\theta)-X^{C}\sin(\theta)\big{)}$ $\displaystyle=0\,,$
(6.40)
$\displaystyle\frac{\partial\bar{X}^{\bar{C}}}{\partial\tau}+i\frac{\partial}{\partial
s}\big{(}\bar{X}^{\bar{C}}\cos(\theta)+\bar{Y}^{\bar{C}}\sin(\theta)\big{)}$
$\displaystyle=0\,,\quad$
$\displaystyle\frac{\partial\bar{Y}^{\bar{C}}}{\partial\tau}-i\frac{\partial}{\partial
s}\big{(}\bar{Y}^{\bar{C}}\cos(\theta)-\bar{X}^{\bar{C}}\sin(\theta)\big{)}$
$\displaystyle=0\,.$

By the squaring argument [7] applied to (6.38), we find that its equivalent
SQM will localize onto configurations that set the LHS and RHS of the squared
terms therein _simultaneously_ to zero. In other words, its equivalent SQM
will localize onto time-invariant critical points of
$V_{I\rightarrow\mathbb{R}}$ that correspond to 2d Aθ-model configurations
that obey

$\displaystyle\frac{\partial}{\partial
s}\big{(}X^{C}\cos(\theta)+Y^{C}\sin(\theta)\big{)}$ $\displaystyle=0\,,\quad$
$\displaystyle\frac{\partial}{\partial
s}\big{(}Y^{C}\cos(\theta)-X^{C}\sin(\theta)\big{)}$ $\displaystyle=0\,,$
(6.41) $\displaystyle\frac{\partial}{\partial
s}\big{(}\bar{X}^{\bar{C}}\cos(\theta)+\bar{Y}^{\bar{C}}\sin(\theta)\big{)}$
$\displaystyle=0\,,\quad$ $\displaystyle\frac{\partial}{\partial
s}\big{(}\bar{Y}^{\bar{C}}\cos(\theta)-\bar{X}^{\bar{C}}\sin(\theta)\big{)}$
$\displaystyle=0\,.$

These are constant intervals, which correspond to stationary trajectories in
$\mathcal{M}(\mathscr{P}_{0},\mathscr{P}_{1})_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$,
i.e., intersection points of $\mathscr{P}_{0}(\theta)$ and
$\mathscr{P}_{1}(\theta)$.

Second, via a similar analysis to that in §3.2, with (6.38) as the action for
the equivalent SQM model, we thus find that we can also express (6.39) as

$\boxed{\mathcal{Z}_{\text{HW}^{\theta},\mathbb{R}\times
I\rightarrow\mathbb{R}}(G)=\sum_{v}{\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}=\sum_{v}\text{HSF}^{\text{Int}}_{d_{v}}(\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)},\mathscr{P}_{0},\mathscr{P}_{1})=\mathcal{Z}^{\text{IntSympFloer}}_{\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)}}}$
(6.42)

where each ${\cal
F}^{v}_{\mathcal{M}(\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta})}$ can be
identified with a symplectic intersection Floer homology class
$\text{HSF}^{\text{Int}}_{d_{v}}(\mathcal{M}\big{(}\mathbb{R},\mathcal{M}^{G,M_{2}}_{H,\theta}\big{)},\mathscr{P}_{0},\mathscr{P}_{1})$
that is generated by the intersection points of $\mathscr{P}_{0}(\theta)$ and
$\mathscr{P}_{1}(\theta)$ in
$\mathcal{M}(\mathbb{R},\mathcal{M}^{G}_{H,\theta}(M_{2}))$, where its degree
$d_{v}$ is counted by the Floer differential realized as the flow lines of the
SQM whose gradient flow equations are the squared terms in (6.38) set to zero.

##  Haydys-Witten Atiyah-Floer Correspondences

In this section, we consider HW theory on $M_{5}=M_{3}\times
S^{1}\times\mathbb{R}$. We decompose $M_{5}$ into two pieces by performing a
Heegaard split of $M_{3}$ along a Riemann surface $\Sigma$. Then, by the
topological invariance of HW theory under a shrinking of $\Sigma$, and the
results of the previous section, we will be able to derive an HW Atiyah-Floer
correspondence between HW Floer homology of $M_{3}\times S^{1}$ and symplectic
intersection Floer homology of the loop space of Higgs bundles, which, in
turn, implies its 4d version between 4d-HW Floer homology of $M_{3}$ and
symplectic intersection Floer homology of the space of Higgs bundles. We will
also derive a duality between our 4d-HW Atiyah-Floer correspondence and the VW
Atiyah-Floer correspondence in [8].

###  7.1 Splitting HW Theory on $M_{5}=M_{3}\times M_{1}\times\mathbb{R}$

Splitting HW Theory on $M_{5}=M_{3}\times M_{1}\times\mathbb{R}$ via Heegaard
Splitting $M_{3}$

Let us perform a Heegaard split of
$M_{3}=M_{3}^{\prime}\bigcup_{\Sigma}M_{3}^{\prime\prime}$ along $\Sigma$, as
shown in Fig. 2(a), whence we can view $M_{3}^{\prime}$ and
$M_{3}^{\prime\prime}$ as nontrivial fibrations of $\Sigma$ over intervals
$I^{\prime}$ and $I^{\prime\prime}$, where $\Sigma$ goes to zero size at one
end of each of the intervals. The metric on $M_{3}^{\prime}$ and
$M_{3}^{\prime\prime}$ can then be written as

$ds^{2}_{\text{3d}}=(dx^{B})^{2}+f(x^{B})(g_{\Sigma})_{ab}dx^{a}dx^{b}\,,$
(7.1)

where $x^{a,b}$ are coordinates on the Riemann surface $\Sigma$; $x^{B}$ is a
coordinate on $I^{\prime}$ and $I^{\prime\prime}$; and $f(x^{B})$ is a scalar
function along $I^{\prime}$ and $I^{\prime\prime}$. Therefore,
$M_{5}=M_{5}^{\prime}\bigcup_{\mathscr{B}}M_{5}^{\prime\prime}$, where
$M_{5}^{\prime}=M_{3}^{\prime}\times M_{1}\times\mathbb{R}$,
$M_{5}^{\prime\prime}=M_{3}^{\prime\prime}\times M_{1}\times\mathbb{R}$, and
$\mathscr{B}$ is their common boundary. This is illustrated in Fig. 2(b).

$\Sigma$$M_{3}^{\prime}$$M_{3}^{\prime\prime}$ (a) $M_{3}$ as a connected sum
of three-manifolds $M_{3}^{\prime}$ and $M_{3}^{\prime\prime}$ along the
Riemann surface $\Sigma$.

$\mathscr{B}$$M_{5}^{\prime}$$M_{5}^{\prime\prime}$ (b) $M_{5}$ splits into
five-manifolds $M_{5}^{\prime}$ and $M_{5}^{\prime\prime}$ along their common
boundary $\mathscr{B}$.

Figure 2: Heegaard splits of $M_{3}$ and $M_{5}$.212121This diagram is adapted
from [8, Fig. 2].

Because of the topological invariance of HW theory, we are free to perform a
Weyl rescaling of the corresponding Heegaard split metrics on $M_{5}^{\prime}$
and $M_{5}^{\prime\prime}$ to

$ds^{2}_{\text{5d}}=\frac{1}{f(x^{B})}\left[(dx^{A})^{2}+(ds)^{2}+(dx^{B})^{2}\right]+(g_{\Sigma})_{ab}dx^{a}dx^{b}\,,$
(7.2)

where $x^{A}$ and $s$ are the coordinates on $\mathbb{R}$ and $M_{1}$,
respectively. The prefactor of $f(x^{B})^{-1}$ is effectively a scaling factor
on $\mathbb{R}\times M_{1}\times I^{\prime}$ and $\mathbb{R}\times M_{1}\times
I^{\prime\prime}$, whence their topologies are left unchanged. We can thus
regard
$M_{5}=M_{5}^{\prime}\bigcup_{\mathscr{B}}M_{5}^{\prime\prime}=(\mathbb{R}\times
I^{\prime}\times M_{1}\times\Sigma)\bigcup_{\mathscr{B}}(\mathbb{R}\times
I^{\prime\prime}\times M_{1}\times\Sigma)$, where $\mathscr{B}=\Sigma\times
M_{1}\times\mathbb{R}$.

Hence, HW theory on $M_{5}$ can be regarded as a union of an HW theory on
$\mathbb{R}\times I^{\prime}\times M_{1}\times\Sigma$ and an HW theory on
$\mathbb{R}\times I^{\prime\prime}\times M_{1}\times\Sigma$ along their common
boundary $\mathscr{B}=\Sigma\times M_{1}\times\mathbb{R}$.

###  7.2 An HW Atiyah-Floer Correspondence of $M_{3}\times S^{1}$

HW Theory on $M_{5}^{\prime}$($M_{5}^{\prime\prime}$) as a 2d A-model on
$\mathbb{R}\times I^{\prime}$($I^{\prime\prime}$) with Branes
$\mathscr{L}_{0},\mathscr{L}_{1}^{\prime}$
($\mathscr{L}_{0}^{\prime\prime},\mathscr{L}_{1}$) in
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(\Sigma)$

Let $M_{1}=S^{1}$. Notice that topological HW theory on
$M_{5}^{\prime}=\mathbb{R}\times I^{\prime}\times S^{1}\times\Sigma$ and
$M_{5}^{\prime\prime}=\mathbb{R}\times I^{\prime\prime}\times
S^{1}\times\Sigma$, when $\Sigma$ is shrunken, can be interpreted as a 3d
sigma model on $\mathbb{R}\times I^{\prime}\times S^{1}$ and $\mathbb{R}\times
I^{\prime\prime}\times S^{1}$, respectively, whose action is (6.19). In turn,
these 3d sigma models can be interpreted as 2d sigma models with action
(6.20).

That is, HW theory on $M_{5}^{\prime}=\mathbb{R}\times I^{\prime}\times
S^{1}\times\Sigma$ and $M_{5}^{\prime\prime}=\mathbb{R}\times
I^{\prime\prime}\times S^{1}\times\Sigma$ can also be interpreted as a 2d
A-model on $\mathbb{R}\times I^{\prime}$ and $\mathbb{R}\times
I^{\prime\prime}$ with branes $(\mathscr{L}_{0},\mathscr{L}_{1}^{\prime})$ and
($\mathscr{L}_{0}^{\prime\prime},\mathscr{L}_{1}$) in
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(\Sigma)$, respectively, whose
action is (6.20).

An HW Atiyah-Floer Correspondence of $M_{3}\times S^{1}$

Therefore, the union of an HW theory on $M_{5}^{\prime}$ and an HW theory on
$M_{5}^{\prime\prime}$ along their common boundary $\mathscr{B}$ to get an HW
theory on $M_{5}$, can be interpreted as the union of a 2d A-model on
$\mathbb{R}\times I^{\prime}$ with branes
$(\mathscr{L}_{0},\mathscr{L}_{1}^{\prime})$ and a 2d A-model on
$\mathbb{R}\times I^{\prime\prime}$ with branes
$(\mathscr{L}_{0}^{\prime\prime},\mathscr{L}_{1})$ along their common boundary
$\mathscr{R}=\mathscr{L}_{1}^{\prime}=\mathscr{L}_{0}^{\prime\prime}$ to get a
2d A-model on $\mathbb{R}\times I$ with branes
$(\mathscr{L}_{0},\mathscr{L}_{1})$. This is illustrated in Fig. 3.

$\mathscr{L}_{0}$ $\mathscr{L}_{1}^{\prime}$ $\mathscr{L}_{0}^{\prime\prime}$
$\mathscr{L}_{1}$ $\mathscr{R}$$\mathbb{R}\times I^{\prime}$$\mathbb{R}\times
I^{\prime\prime}$ $\mathscr{L}_{0}$ $\mathscr{L}_{1}$ $\mathbb{R}\times I$
Figure 3: Union of 2d A-models along their common boundary $\mathscr{R}$.

In other words, $G$-type HW theory on $M_{3}\times S^{1}\times\mathbb{R}$ can
be interpreted as a 2d A-model on $\mathbb{R}\times I$ with branes
$(\mathscr{L}_{0},\mathscr{L}_{1})$ in
$L\mathcal{M}^{G_{\mathbb{C}}}_{\text{flat}}(\Sigma)$ whose action is (6.20).
This means that we can equate their partition functions in (3.14) and (6.26),
respectively, to get

${\sum_{k}\text{HF}^{\text{HW}}_{d_{k}}(M_{3}\times
S^{1},G)=\sum_{u}\text{HSF}^{\text{Int}}_{d_{u}}({L\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}},\mathscr{L}_{0},\mathscr{L}_{1})}\,,$
(7.3)

where $\mathscr{L}_{0},\mathscr{L}_{1}$ are necessarily isotropic-coisotropic
of $(\text{A},\text{A},\text{B})$ type in complex structure $(I,J,K)$.222222In
$\S$6.4, we saw that the branes $\mathscr{L}_{0,1}$ must be of type
$(*,\text{A},*)$ in complex structure $(I,J,K)$; this means they can either be
$(\text{A},\text{A},\text{B})$ or $(\text{B},\text{A},\text{A})$ branes (as
$(\text{B},\text{A},\text{B})$ and $(\text{A},\text{A},\text{A})$ branes are
not possible [10, footnote 24]). Since in this case, we require that
$\mathscr{L}_{0,1}$, when restricted to
$\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}$, ought to correspond to
branes that wrap $\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}$ where
flat $G_{\mathbb{C}}$ connections on $\Sigma$ are extendable to $M_{3}$, i.e.,
the branes must be Lagrangian [11] and therefore isotropic-coisotropic [12],
it would mean that $\mathscr{L}_{0,1}$ must also be isotropic-coisotropic.
Although $(\text{A},\text{A},\text{B})$ and $(\text{B},\text{A},\text{A})$
branes are both isotropic-coisotropic, we choose $\mathscr{L}_{0,1}$ to be
$(\text{A},\text{A},\text{B})$ branes for later convenience.

We would now like to ascertain if there is a one-to-one correspondence between
$(k,d_{k})$ and $(u,d_{u})$, which is to ask if there is a degree-to-degree
isomorphism between the HW Floer homology and the symplectic intersection
Floer homology.

To ascertain if there is a one-to-one correspondence between ‘$k$’ and ‘$u$’,
first, note that each ‘$k$’ refers to a time-invariant critical point of
$V_{4}$ in $\mathfrak{A}_{4}$ which corresponds to a time-invariant solution
of the 5d BPS equations (2.9) on $M_{3}\times S^{1}\times\mathbb{R}$. Second,
note that each ‘$u$’ refers to a time-invariant critical point of $V_{I\to
S^{1}}$ in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{{L\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}}}$
which corresponds to a time-invariant solution of the 3d BPS equations given
by the squared terms in (6.19) set to zero. Third, note that the 3d BPS
equations are a direct topological reduction of the 5d BPS equations along
$\Sigma\subset M_{3}$ (which we can regard as $I\times\Sigma$ in our context),
whence there is a one-to-one correspondence between time-invariant solutions
of the latter and former. Altogether, this means that there is a one-to-one
correspondence between ‘$k$’ and ‘$u$’.

To ascertain if there is a one-to-one correspondence between ‘$d_{k}$’ and
‘$d_{u}$’, first, note that the flow lines between time-invariant critical
points of $V_{4}$ in $\mathfrak{A}_{4}$ that realize the Floer differential of
$\text{HF}^{\text{HW}}_{*}$ which counts ‘$d_{k}$’, correspond to solutions of
the gradient flow equations (3.10) which, in turn, correspond to solutions of
the 5d BPS equations (2.9) on $M_{3}\times S^{1}\times\mathbb{R}$. Second,
note that the flow lines between time-invariant critical points of $V_{I\to
S^{1}}$ in
$\mathcal{M}(\mathscr{L}_{0},\mathscr{L}_{1})_{{L\mathcal{M}^{G_{\mathbb{C}},\Sigma}_{\text{flat}}}}$
that realize the Floer differential of $\text{HSF}^{\text{Int}}_{*}$ which
counts ‘$d_{u}$’, correspond to solutions of the gradient flow equations given
by the squared terms in (6.22) set to zero which, in turn, correspond to
solutions of the 3d BPS equations given by the squared terms in (6.19) set to
zero. Third, note again that the 3d BPS equations are a direct topological
reduction of the 5d BPS equations along $\Sigma\subset M_{3}$ (which we can
regard as $I\times\Sigma$ in our context), whence there is a one-to-one
correspondence between solutions of the latter and former. Altogether, this
means that there is a one-to-one correspondence between ‘$d_{k}$’ and
‘$d_{u}$’.