# Non-equilibrium Photoionization and Hydrodynamic Simulations of Starburst-
driven Outflows

A Danehkar  M S Oey and W J Gray  Department of Astronomy, University of
Michigan, 1085 S. University Ave, Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>
###### Abstract

Starburst-driven galactic outflows in star-forming galaxies have been observed
to contain complex thermal structures and emission line features that are
difficult to explain by adiabatic fluid models and plasmas in photoionization
equilibrium (PIE) and collisional ionization equilibrium (CIE). We previously
performed hydrodynamic simulations of starburst-driven outflows using the
maihem module for non-equilibrium atomic chemistry and radiative cooling
functions in the hydrodynamics code flash, and calculated emission lines in
combined CIE and PIE conditions. In the present study, we consider time-
dependent non-equilibrium ionization (NEI) states produced by our maihem
simulations. Through extensive cloudy calculations made with the NEI states
from our hydrodynamic simulations, we predict the UV and optical emission line
profiles for starburst-driven outflows in time-evolving non-equilibrium
photoionization conditions. Our hydrodynamic results demonstrate applications
of non-equilibrium radiative cooling for H ii regions in starburst galaxies
hosting cool outflows.

##  1 Introduction

Starburst-driven galactic-scale outflows are typically seen in star-forming
galaxies [1, 2, 3, 4, 5, 6], which have complex multi-temperature structures
ranging from cold ($100$–1000 K) in radio and IR [7, 8] to warm ($\sim 10^{4}$
K) in optical [9, 10], and hot ($\sim 10^{7}$ K) in X-ray observations [11,
12]. Hydrodynamic properties of starburst-driven outflows have approximately
been modeled with adiabatic fluid models [13, 14, 15]. The adiabatic solution
of a free expanding wind can produce a hot bubble and a shell for an outflow
model surrounded by the ambient medium [16]. Moreover, the adiabatic solution
indicates that density and temperature of the free expanding wind decrease
with radius $r$ as $\rho\varpropto r^{-2}$ and $T\varpropto r^{-4/3}$ [14].
Thus, a starburst-driven outflow could have four regions: freely expanding
wind, hot bubble, shell, and ambient medium [13] (see Figure 1).

However, the adiabatic assumption seems to be inconsistent with strong cooling
and suppressed outflows seen in several star-forming galaxies [17, 18, 19].
Starburst-driven outflows have also been investigated using fluid models with
radiative cooling functions [20, 21], implying that cooling depends on the
metallicity, mass-loading, and wind terminal velocity. More recently, the
occurrence of strong radiative cooling has been investigated using radiative
cooling functions in non-equilibrium conditions [22], which depicted the
dependence of cooling on the metallicity, mass-loading, and wind velocity in
agreement with the previous numerical results derived from semianalytic models
[20]. Moreover, a large grid of hydrodynamic simulations indicate that
radiative cooling is stronger with higher mass-loading rate, and lower wind
terminal velocity [23]. However, emission lines predicted by this large grid
of models were calculated using photoionization equilibrium (PIE) and
collisional ionization equilibrium (CIE) that do not allow us to clearly
identify the parameter space with strong radiative cooling [23].

It has been found that emission lines calculated by non-equilibrium ionization
(NEI) could have a large departure from CIE [24, 25, 26] in plasma regions
($<10^{6}$ K) being in the transition from pure CIE to PIE [25]. Previously,
cooling functions and NEI chemistry networks were included in the package
maihem (Models of Agitated and Illuminated Hindering and Emitting Media) [27,
28, 22]. Some non-equilibrium ionization models built by maihem showed the
enhancements of O vi and C iv in non-equilibrium conditions [29].

Previoulsy, in Ref. [23], we obtained emission lines predicted by combined
CIE+PIE models of maihem hydrodynamic simulations for different outflow
parameters. In this recent work [30], we calculate emission lines for combined
NEI+PIE model using NEI states produced by our maihem hydrodynamic
simulations.

##  2 Numerical Methods

We consider a galactic-scale outflow driven by starburst feedback from a
spherically symmetric super star cluster (SSC) characterized with the SSC
radius $R_{\rm sc}$, mass deposition rate $\dot{M}$, wind terminal velocity
$V_{\infty}$, and stellar ionizing fields parameterized by ionizing luminosity
$L_{\rm ion}$ and spectral energy distribution (SED) surrounded by ambient
medium with density $\rho_{\rm amb}$ and temperature $T_{\rm amb}$. To
numerically simulate starburst-driven outflows with various parameters, we
model the fluid equations with appropriate initial and boundary conditions (§
2.1), ionizing radiation fields produced by SSCs (§ 2.2), radiative thermal
processes within outflows (§ 2.3), and non-equilibrium photoionization
processes (§ 2.4), as illustrated in Figure 2.

###  2.1 Hydrodynamic Simulations

We use the MUSCL method [31] and hybrid Riemann solver [32] in the
hydrodynamics code flash [33] to solve the fluid equations for mass
conservation, momentum conservation, and energy conservation coupled with
radiative cooling and heating functions of the NEI chemistry package maihem
[27, 28, 22] that are as follows:

$\frac{\partial\rho}{\partial t}+\vec{\nabla}\cdot\left(\rho\vec{u}\right)=0,$
(1) $\frac{\partial\vec{u}}{\partial
t}+(\vec{u}\cdot\vec{\nabla})\vec{u}+\frac{1}{\rho}\vec{\nabla}P=0,$ (2)
$\frac{\partial{E}}{\partial
t}+\vec{u}\cdot\vec{\nabla}E+\frac{P}{\rho}\vec{\nabla}\cdot\vec{u}=\frac{\Gamma-\Lambda}{\rho},$
(3)

where $\rho$, $u=|\vec{u}|$, $P=\left(\gamma-1\right)\rho\epsilon$, and
$E=\epsilon+\frac{1}{2}u^{2}$ is the fluid density, velocity, and thermal
pressure, and total energy per mass, respectively, $\epsilon$ the internal
energy per mass, $\gamma=5/3$ the heat capacity ratio, and $\Lambda$ and
$\Gamma$ the radiative cooling and heating rates per volume, respectively.

Figure 1: Schematic view of the starburst-driven outflows (left panel) with
the temperature profile (plotted by red line; right panel) and density profile
(blue line), showing the different regions (separated by dotted lines)
described by [13]: free wind, hot bubble, shell, and ambient medium.

To produce a starburst-driven outflow defined by $\dot{M}$ and $V_{\infty}$ at
the cluster boundary ($r=R_{\rm sc}$) surrounded by the ambient medium, we use
the analytic solutions of the steady-state outflow model in one-dimensional
spherical coordinates [14, 15, 20], and set density $\rho$, velocity $u$, and
temperature $T$ at $r=R_{\rm sc}$ with user-defined non-penetrating boundary
conditions as follows:

$\begin{array}[c]{ccc}\rho\Big{|}_{r=R_{\rm sc}}=\dfrac{\dot{M}}{4\pi R_{\rm
sc}^{2}u},\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode&u\Big{|}_{r=R_{\rm
sc}}=\dfrac{1}{2}V_{\infty},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
T\Big{|}_{r=R_{\rm sc}}=\dfrac{\mu}{\gamma k_{\rm B}}u^{2},\end{array}$ (4)

where $\mu$ is the mean mass per particle and $k_{\rm B}$ the Boltzmann
constant.

We also configure the initial values of the density, temperature, and velocity
at $t=0$ for the stationary ambient medium ($r>R_{\rm sc}$) with $\rho_{\rm
amb}$ and $T_{\rm amb}$ surrounding the SSC:

$\begin{array}[c]{ccc}\rho\Big{|}_{r>R_{\rm sc},\,t=0}=\rho_{\rm
amb},\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode\nobreak\leavevmode&u\Big{|}_{r>R_{\rm
sc},\,t=0}=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ T\Big{|}_{r>R_{\rm sc},\,t=0}=T_{\rm
amb}.\end{array}$ (5)

The outer boundary at $r=250$ pc sets zero-gradient boundary conditions that
allow the outflow to leave the simulation domain. We note for an ideal gas
$P=\rho k_{\rm B}T/\mu$.

We solve Eqs. (1)–(3) using the unsplit hydrodynamic solver [34, 35, 36]
included in flash with the 2nd-order MUSCL–Hancock scheme [31]. To have stable
solutions in regions with strong shocks, we utilize a hybrid Riemann solver
[32] that includes Roe [37] and HLLE solvers [38, 39]. Simulations are
performed on a grid with $N_{x}=512$ blocks and maximum refinement levels of
two, while the refinement criterion [40] is set to follow the fluid variables
$\rho$, $T$, $P$, and $u$. This provides a maximum resolution of 0.244 pc.

Figure 2: Schematic view of the connections between flash hydrodynamic
simulations, radiative cooling and heating functions in maihem used to produce
NEI states, ionizing SED model made by Starburst99, and non-equilibrium
photoionization calculations conducted by cloudy zone by zone using the
Starburst99 SED, physical conditions ($\rho$ and $T$), and NEI states
generated by non-equilibrium atomic chemistry functions in maihem.

###  2.2 Ionizing Radiation Source Modeling

Radiation fields produced by stars within the SSC result in photoionization
and radiative processes. To model the ionizing radiation source for our
hydrodynamic simulations and non-equilibrium photoionization calculations, we
generate a SED using the stellar population synthesis program Starburst99 [41,
42, 43] for a given total stellar mass $M_{\star}=2\times 10^{6}$ M⊙
corresponding to a mass-loading rate of $\dot{M}=10^{-2}$ M⊙ yr-1 at solar
metallicity and age of 1 Myr. For our Starburst99 model, we employ an IMF with
slope $\alpha=2.35$ in the mass range 0.5–150 M⊙, the Geneva rotational
stellar evolution models [44, 45, 46], and the extended Pauldrach/Hillier
stellar atmosphere [47, 48]. The ionizing luminosity and SED spectrum
calculated by Starburst99 are supplied to photo-heating functions of maihem
and non-equilibrium photoionization calculations performed by cloudy. To make
Starburst99 outputs usable for cloudy, we compile a binary format “mod” file
from the Starburst99 “spectrum” output file using the cloudy command compile
star. Moreover, we determine the ionizing luminosity ($L_{\rm ion}$) from the
total luminosity and H+ fraction at age 1 Myr read from the Starburst99
“quanta” output file.

###  2.3 Radiative Cooling and Heating Processes

To have radiative thermal effects in hydrodynamic simulations, it is necessary
to calculate radiative cooling and photo-heating rates. The cooling routine in
the maihem package implemented by Ref. [27] computes the radiative cooling
rate $\Lambda$ using the ion-by-ion cooling efficiencies $\Lambda_{i}$ for
different species [49] that depend on the metallicity $Z$ and temperature $T$
produced by hydrodynamic simulations. The heating routine in maihem added by
Ref. [22] calculates the photo-heating cooling rate $\Gamma$ using the
photoionization cross section [50, 51].

The radiative cooling $\Lambda$ and heating rates $\Gamma$ are calculated
using the ion number density $n_{i}$, and the electron number density $n_{e}$:

$\begin{array}[c]{cc}\Lambda=\displaystyle\sum_{i}n_{i}n_{e}\Lambda_{i},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \Gamma=\displaystyle\sum_{i}n_{i}\Gamma_{i},\end{array}$
(6)

where $\Lambda_{i}$ are the cooling efficiencies from [49], $\Gamma_{i}$ the
heating efficiencies estimated using the SED spectrum from Starburst99 and
photoionization cross sections from [50, 51].

Moreover, the maihem package as implemented by [27] also contains the atomic
chemistry functions that perform non-equilibrium ionization, collisional
ionization and recombination processes using collisional ionization rates
[52], recombination rates [53], and dielectronic recombination rates [27].
Thus, maihem supplies flash hydrodynamic simulations with radiative thermal
functions and non-equilibrium ionization that allow us to study time-dependent
radiative effects in starburst-driven outflows (see Figures 3 and 4; described
in § 3).

Figure 3: Various wind modes based on wind temperature profiles: adiabatic
bubble (AB), catastrophic cooling bubble (CB), and catastrophic cooling (CC),
described in [23]. Temperature profiles and their predicted adiabatic
solutions are shown by solid and dotted red lines, respectively. Radiative
cooling in free winds contributes to departures from the expected adiabatic
solutions. The boundaries of the bubble are also shown by dotted gray lines.

###  2.4 Non-equilibrium Photoionization Modeling

Non-equilibrium photoionization models are built by the program cloudy [54,
55, 56] using the density profile $\rho(r)$, temperature profile $T(r)$, and
NEI states $F_{\rm i}(r)$ of outflows as a function of $r$ predicted by
hydrodynamic simulations with maihem (see Figure 5; described in § 3),
together with the ionizing source made by Starburst99. The procedure is fully
described in [29] (non-CIE models). Previously, we also produced CIE+PIE
models using $\rho(r)$, $T(r)$, and SED, but without $F_{\rm i}(r)$ [23],
which incorporate hydrodynamic collisional ionization and photoionization
processes.

Modeling non-equilibrium photoionization requires performing multiple zone-by-
zone cloudy runs on physical conditions and NEI states with the radial
distance $r$ ranging from the cluster radius $R_{\rm sc}$ to the shell radius
$R_{\rm shell}$. For each zone ($r$, $r+\vartriangle r$), we run one cloudy
run for its density ($\rho$) specified with the cloudy command hden,
temperature ($T$) through the command constant temperature, and ionization
states ($F_{\rm i}$) using the cloudy command element name ionization.
Multiple cloudy runs compute the emissivities through iterations over all the
outflow zones, i.e. $r\in[R_{\rm sc},R_{\rm shell}]$. The SED (and ionizing
luminosity) made by Starburst99 is supplied to cloudy models (via the command
table star). As this procedure needs multiple cloudy runs for all the zones,
it is computationally expensive. This procedure continues until it reaches the
shell boundary $R_{\rm shell}$. Afterwards, we apply the emissivities obtained
from the CIE+PIE models in [23] to isothermal zones of the shell and the
ambient medium.

Figure 4: The wind temperature with radiative cooling relative to the
predicted adiabatic temperature, $f_{T}\equiv T_{\rm wind}/T_{\rm adi}$, in
the parameter space of the wind terminal velocity $V_{\infty}=250$, 500, and
1000 km s-1, ambient number density $n_{\rm amb}=1$, 10, and 100 cm-3, and
mass-loading rate $\dot{M}=10^{-3}$ and $10^{-2}$ M⊙ yr-1 for the models with
the solar metallicity, cluster radius $R_{\rm sc}=1$ pc, total stellar mass
$M_{\star}=2\times 10^{6}$ M⊙, and age $t=1$ Myr. Outflow models are
classified according to radiative/adiabatic cooling free winds and
presence/absence of bubbles in their temperature profiles, namely adiabatic
bubble (AB), catastrophic cooling (CC), catastrophic cooling bubble (CB), and
momentum-conserving (MC), see [23]. An optically-thick AB model is shown by
the bold font.

##  3 Results

Figure 1 (right) shows the temperature $T$ and density $\rho$ as a function of
radius $r$ predicted by our maihem simulation for an adiabatic example. It has
four different regions described in [13]: (a) freely expanding wind, (b) a hot
bubble, (c) a narrow dense shell, and (d) ambient medium. Ref. [23] classified
the the temperature profile of a freely expanding wind under the adiabatic
bubble (AB), catastrophic cooling bubble (CB), and catastrophic cooling (CC)
wind modes. The AB mode is the classic bubble model described by [13] and
shown in Figure 1 (right). In the CB mode, the temperature profile in the
region (a) has radiative cooling, but it still has a hot bubble. In CC mode,
the temperature profile of the free wind is also radiatively cooled, but it is
without any hot bubble. In Figure 3, we plot the temperature profiles for
three example models in the AB, CB, and CC modes with expected adiabatic
temperature profiles in dotted red lines.

Figure 5: Temperature profiles $T(r)$ produced by the pure photoionization
(PIE) model and maihem hydrodynamic simulation, and density profile $n(r)$
generated by the hydrodynamic simulation (left panels) along with the
adiabatic predictions for hydrodynamic simulation (red dashed lines), and the
volume emissivities $\epsilon_{i}(r)$ of emission lines (right panels)
predicted by the pure photoionization (PIE), combined photoionization and
collisional ionization (CIE+PIE) from [23], combined non-equilibrium
ionization and photoionization (NEI+PIE) models from [30] The boundaries of
the bubble and the end of the shell are shown by dotted, dashed, and dash-
dotted, respectively. The model parameters are: $V_{\infty}=500$ km s-1,
$n_{\rm amb}=100$ cm-3, $\dot{M}=10^{-2}$ M⊙ yr-1, $R_{\rm sc}=1$ pc,
$M_{\star}=2\times 10^{6}$ M⊙, solar metallicity, and age $t=1$ Myr.

Figure 4 depicts the wind temperature predicted by radiative cooling over the
expected adiabatic temperature, $f_{T}\equiv T_{\rm wind}/T_{\rm adi}$, along
with the different wind modes in the parameter space of the wind terminal
velocity, ambient density, and mass-loading rate, where the cluster radius is
$R_{\rm sc}=1$ pc, total stellar mass $M_{\star}=2\times 10^{6}$ M⊙, solar
metallicity, and age $t=1$ Myr. The CC and CB modes are associated with values
of $f_{T}$ less than 0.75, whereas $0.75<f_{T}<1.25$ for the AB mode.
Additionally, we assign the momentum-conserving (MC) mode to those models
having totally suppressed wind due to catastrophic cooling (see [23] for
details). We also display the AB mode of an optically-thick model with the
bold font, whose ambient medium is not ionized. We see that higher mass-
loading rates and lower wind terminal velocities lead to strong cooling
effects in the free expanding wind.

Figure 5 (left panels) shows the temperature profiles $T(r)$ derived from one
example of our hydrodynamic simulations111Our results for all the hydrodynamic
simulations and photoionization calculations are presented as an interactive
figure in [23] and hosted on this URL: https://superwinds.astro.lsa.umich.edu/
with pure photoionization (PIE) and combined hydrodynamic and photoionization
thermal effects, the density profile $n(r)$ predicted by the hydrodynamic
simulation. Figure 5 (right panels) shows the emissivities obtained for one
hydrodynamic simulation with pure photoionization (PIE), combined hydrodynamic
collisional ionization and photoionization (CIE$+$PIE), and combined non-
equilibrium ionization and photoionization (NEI$+$PIE). The luminosities of
optical and UV emission lines are used to produce diagnostic diagrams based on
volume-integrated emissivities from CIE$+$PIE models [23] and NEI$+$PIE models
[30].

##  4 Conclusion

We have conducted hydrodynamic simulations of starburst-driven outflows using
the maihem module in flash to study the formation of radiatively cooling,
expanding winds, hot bubbles, and narrow dense shells (see Figure 1, right)
for various wind terminal velocity, ambient density, and mass-loading rate
(see Figure 4, also see [23]). Our modeling is implemented using the same
computational methods developed by [29] with boundary conditions based on the
semianalytic results found by [20]. Our findings confirm that radiative
cooling is increased by raising mass-loading rate and reducing wind terminal
velocity [23].

In the previous work [23], we predicted emissivities for the density and
temperature profiles generated by our hydrodynamic simulations in PIE and
CIE+PIE conditions, while we additionally calculate emissivities in NEI+PIE
conditions in this work (see Figure 5). Ionization states in PIE and CIE+PIE
conditions were calculated by cloudy (fully described in [23]), whereas in
NEI+PIE conditions we also provide cloudy with time-dependent NEI states
created by maihem [30]. As found by [29], C iv emissivity predicted by NEI+PIE
model is stronger than those made by PIE and CIE+PIE models (see Figure 5,
right), especially for strong cooling.

Our non-equilibrium photoionization models are conducted using a fixed typical
cluster radius (1 pc) and an ionizing source associated with a fixed total
stellar mass ($M_{\star}=2\times 10^{6}$ M⊙) and a final age of 1 Myr. We are
carrying out further hydrodynamic simulations to explore other parameters
contributing to radiative cooling in starburst-driven outflows such as cluster
radius, age, and time-dependent ionizing source.

flash was developed in part by DOE NNSA-ASC OASCR Flash Center at U. Chicago.

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$\displaystyle=\beta_{1}\wedge\beta_{2}+\sum_{s=0}^{k}\beta_{1}\wedge\psi_{s}\wedge\tau_{s}(v_{s})+(-1)^{|\beta_{2}|}\sum_{r=0}^{k}\phi_{r}\wedge\beta_{2}\wedge\tau_{r}(u_{r})$
$\displaystyle\qquad\qquad+\sum_{r,s=0}^{k}(-1)^{|\psi_{s}|}\phi_{r}\wedge\psi_{s}\wedge\tau_{r}(u_{r})\wedge\tau_{s}(v_{s})$
$\displaystyle=\tau(\beta_{1}\wedge\beta_{2},\sum_{s=0}^{k}\beta_{1}\wedge\psi_{s}\otimes
v_{s}+(-1)^{|\beta_{2}|}\sum_{r=0}^{k}\phi_{r}\wedge\beta_{2}\otimes u_{r})$
$\displaystyle\qquad\qquad+\sum_{r,s=0}^{k}(-1)^{|\psi_{s}|}\phi_{r}\wedge\psi_{s}\wedge\tau_{r}(u_{r})\wedge\tau_{s}(v_{s}).$

It thus remains to compute the formula for
$\tau_{s}(v_{s})\wedge\tau_{r}(u_{r})$. We can assume that $s<r$ (otherwise,
just flip the order at the cost of a sign). Computing this we get

$\displaystyle\tau_{s}(v_{s})\wedge\tau_{r}(u_{r})$
$\displaystyle=(v_{s,\operatorname{vert}}-\sum_{i=1}^{s-1}\eta_{i}\otimes
v_{s})\wedge(u_{r,\operatorname{vert}}-\sum_{j=1}^{r-1}\eta_{j}\otimes u_{r})$
$\displaystyle=\sum_{j=1}^{r-1}\eta_{j}\wedge(v_{s}\otimes
u_{r})_{\operatorname{vert}}-\sum_{i=1}^{s-1}\eta_{i}\wedge(v_{s}\otimes
u_{r})_{\operatorname{vert}}+\sum_{j=1}^{r-1}\sum_{i=1}^{s-1}\eta_{i}\wedge\eta_{j}\otimes(v_{s}\otimes
u_{r})$ $\displaystyle=\sum_{j=s}^{r-1}\eta_{j}\wedge(v_{s}\otimes
u_{r})_{\operatorname{vert}}-\sum_{j=s}^{r-1}\sum_{i=1}^{s-1}\eta_{j}\wedge\eta_{i}\otimes(v_{s}\otimes
u_{r})$ $\displaystyle=\sum_{j=s}^{r-1}\eta_{j}\wedge((v_{s}\otimes
u_{r})_{\operatorname{vert}}-\sum_{i=1}^{s+r-j-1}\eta_{i}\otimes(v_{s}\otimes
u_{r}))+\sum_{\begin{subarray}{c}i,j\geq s,\\\ i+j\leq
r+s-1\end{subarray}}\eta_{j}\wedge\eta_{i}\otimes(v_{s}\otimes u_{r})$
$\displaystyle=\tau(\sum_{j=s}^{r-1}\eta_{j}\otimes v_{s}\otimes u_{r}).$

Here we have used $\eta_{j}\otimes v_{s}\otimes u_{r}\in\Omega_{D}^{1}\otimes
L^{s+r-j}$ and the fact that symmetric sums of $\eta_{j}\wedge\eta_{i}$
vanish. ∎

###  7.4. A spectral sequence

Note that
$\Omega_{D}^{\bullet}(S_{k-1}(L))[-1]\subset\Omega_{D}^{\bullet}(S_{k}(L))[-1]$
is a subcomplex for all $k$. This gives rise to the following bounded
decreasing filtration $F^{\bullet}$ on $\Omega_{D}^{\bullet}(S_{k}(L))[-1]$:

$F^{p}\big{(}\Omega_{D}^{\bullet}(S_{k}(L))[-1]\big{)}=\begin{cases}\Omega_{D}^{\bullet}(S_{k}(L))[-1]&p\leq-k\\\
\Omega_{D}^{\bullet}(S_{-p}(L))[-1]&-k\leq p\leq 0\\\ 0&p>0\end{cases}$

######  Proposition 7.11.

The filtration $F^{\bullet}$ gives rise to a spectral sequence $E$ which
converges to the cohomology $H^{\bullet-1}_{D}(S_{k}(L))$. The $E_{1}$ page is
given by the cohomology of the local systems $L^{i}$:

$E_{1}^{p,q}=H^{p+q-1}(L^{-p}),\qquad p=0,...,-k.$

###### Proof.

Because the filtration is bounded, the spectral sequence converges to the
cohomology. The $E_{0}$ page of the spectral sequence is given by the
associated graded of the complex. Hence, for $p=0,...,-k$ we have

$E_{0}^{p,q}=\frac{F^{p}\big{(}\Omega_{D}^{p+q}(S_{k}(L))[-1]\big{)}}{F^{p+1}\big{(}\Omega_{D}^{p+q}(S_{k}(L))[-1]\big{)}}\cong\Omega_{D}^{p+q-1}(L^{-p}),$

and $E_{0}^{p,q}=0$ for all other values of $p$. The differential on the
complex is given by Equation 7.1. The portion that survives after taking the
associated graded is given by $d^{\nabla}$, the connection for the local
systems $L^{i}$. Hence, the $E_{1}$ page is given by

$E_{1}^{p,q}=H_{D}^{p+q-1}(L^{-p})$

for $p=0,...,-k$ and $E_{1}^{p,q}=0$ for all other values of $p$. ∎

###  7.5. The algebra in the case of the trivial line bundle

We study the special case where $L$ is the trivial bundle with trivial local
system. Let $t_{r}\in L^{r}$ be the unit section. Hence $t_{r}\otimes
t_{s}\cong t_{r+s}$. Now we prove a technical lemma that will be useful in
computing examples.

Let $(C^{\bullet},d)\subset(\Omega^{\bullet}_{D},d)$ be a quasi-isomorphic
sub-cdga. Assume that the Maurer-Cartan elements have the form
$\eta_{i}=\alpha_{i}\otimes t_{-i}$ for $\alpha_{i}\in C^{1}$. Define

$C^{\bullet}(\sigma)=\bigoplus_{i=0}^{k}(C^{\bullet}\otimes t_{i}[-1]),$

where we now view $t_{i}$ to have degree $+1$. It is straightforward to check
that $C^{\bullet}(\sigma)$ is a sub-cdga of
$\Omega^{\bullet}_{D}(S_{k}(L))[-1]$.

######  Lemma 7.12.

The cdga $C^{\bullet}(\sigma)$ is a quasi-isomorphic subalgebra of
$\Omega^{\bullet}_{D}(S_{k}(L))[-1]$.

###### Proof.

It suffices to show that the inclusion, denoted $i$, is a quasi-isomorphism.
Note first that the filtration $F^{\bullet}$ induces a filtration on
$C^{\bullet}(\sigma)$ and hence the inclusion induces a morphism between the
associated spectral sequences. The map on the $E_{1}$-page is given by

$i:H^{p+q-1}(C^{\bullet})\to H^{p+q-1}_{D}.$

This is an isomorphism because we have assumed that $C^{\bullet}$ is a quasi-
isomorphic subcomplex of $\Omega_{D}^{\bullet}$. Hence, by the Eilenberg-Moore
Comparison theorem [Wei94],
$i:C^{\bullet}(\sigma)\to\Omega^{\bullet}_{D}(S_{k}(L))[-1]$ is a quasi-
isomorphism. ∎

####  7.5.1. An Example

We continue with the example of Heisenberg group from Example 2.4. So recall
that $H(\mathbb{R})$ denotes the Heisenberg group, $H(\mathbb{Z})$ the
subgroup consisting of matrices with integer entries and
$X=H(\mathbb{R})/H(\mathbb{Z})$ the quotient. Let $a=dx$, $b=dy$ and $c=dz-
ydx$. These descend to $1$-forms in $\Omega^{1}(X)$ and satisfy $da=db=0$ and
$dc=a\wedge b$. Let

$C=\mathbb{R}[a,b,c]$

be the induced subalgebra of $\Omega^{\bullet}(X)$. Then by the Gysin
sequence, $C$ is quasi-isomorphic to $\Omega^{\bullet}(X)$. The cohomology is
given by

$H^{\bullet}(C)=\langle 1\rangle\oplus\langle a,b\rangle\oplus\langle
ac,bc\rangle\oplus\langle abc\rangle.$

Choose Maurer-Cartan elements given by

$\eta_{1}=a\otimes t_{-1},\qquad\eta_{2}=b\otimes
t_{-2},\qquad\eta_{3}=-c\otimes t_{-3}.$

This gives rise to the cdga

$C(\sigma)=\mathbb{R}[a,b,c]\langle t_{0},t_{1},t_{2},t_{3},t_{4}\rangle,$

with differentials

$dt_{0}=0,\ \ dt_{1}=0,\ \ dt_{2}=-at_{1},\ \ dt_{3}=-2at_{2}-bt_{1},\ \
dt_{4}=-3at_{3}-2bt_{2}+ct_{1},$

and products

$\displaystyle t_{0}t_{1}$ $\displaystyle=0,\ \ t_{0}t_{2}=at_{1},\ \
t_{0}t_{3}=at_{2}+bt_{1},\ \ t_{0}t_{4}=at_{3}+bt_{2}-ct_{1},\ \
t_{1}t_{2}=at_{2},$ $\displaystyle t_{1}t_{3}$ $\displaystyle=at_{3}+bt_{2},\
\ t_{1}t_{4}=at_{4}+bt_{3}-ct_{2},\ \ t_{2}t_{3}=bt_{3},\ \
t_{2}t_{4}=bt_{4}-ct_{3},\ \ t_{3}t_{4}=-ct_{4}.$

This algebra is quasi-isomorphic to $\Omega_{X}^{\bullet}(S_{4}(L))[-1]$ by
Lemma 7.12. Using the spectral sequence, we compute the cohomology algebra. It
is given by

$H^{\bullet}(C(\sigma))=H^{\bullet}(C)t_{0}\bigoplus\big{(}\langle
t_{1}\rangle\oplus\langle
bt_{1},bt_{2}+ct_{1},at_{4}+\frac{1}{2}bt_{3}-ct_{2}\rangle\oplus\langle
bct_{1},act_{3}+bct_{2},act_{4}+bct_{3}\rangle\oplus\langle
abct_{4}\rangle\big{)}.$

Computing the products of this algebra we find that only two of them are non-
vanishing. They are given by

$t_{1}\ast(at_{4}+\frac{1}{2}bt_{3}-ct_{2})=\frac{1}{6}bct_{1},\qquad(at_{4}+\frac{1}{2}bt_{3}-ct_{2})^{2}=abct_{4}.$

The dimensions of the cohomology of the associated $b^{5}$-algebroid are given
by

$\dim(H^{i}(A(\sigma)))=1,\ \ 4,\ \ 7,\ \ 6,\ \ 2.$

###  7.6. Universal algebra for foliations

Let $\mathfrak{g}_{k}=Lie(G_{k,1})$, which has basis
$z\partial_{z},z^{2}\partial_{z},...,z^{k}\partial_{z}$. Let
$x_{0},x_{1},...,x_{k-1}$ be the dual basis of $\mathfrak{g}_{k}^{*}$. Then
the Chevalley-Eilenberg algebra of this Lie algebra is given by

$CE(\mathfrak{g}_{k})=\mathbb{R}[x_{0},...,x_{k-1}],$

where each $x_{r}$ has degree $+1$ and the differential is given by

$\delta(x_{r})=\sum_{i=0}^{r}(i-r)x_{i}x_{r-i}.$

Now define

$S_{k}=\oplus_{r=0}^{k}CE(\mathfrak{g}_{k})t_{r}=\mathbb{R}[x_{0},...,x_{k-1}]\langle
t_{0},...,t_{k}\rangle,$

where each $t_{i}$ has degree $+1$. Define

$\delta(t_{r})=\sum_{i=0}^{r}(i-r)x_{i}t_{r-i},$

and

$t_{s}t_{r}=x_{s}t_{r}+x_{s+1}t_{r-1}+...+x_{r-1}t_{s+1},$

for $s<r$. Then $S_{k}$ is a cdga defined over $CE(\mathfrak{g}_{k})$.

Now let $A(\sigma)$ be a $b^{k+1}$-algebroid defined over $D\times\mathbb{R}$
with leaves $D$, $D\times\mathbb{R}^{+}$ and $D\times\mathbb{R}^{-}$. This is
specified by a Maurer-Cartan element
$\sigma\in\Omega_{D}^{1}(\mathfrak{g}_{k})$. Equivalently, this is defined by
the data of a morphisms of cdgas

$\psi_{\sigma}:CE(\mathfrak{g}_{k})\to\Omega_{D}^{\bullet},\qquad
x_{i}\mapsto\eta_{i}.$

This turns $\Omega_{D}^{\bullet}$ into an algebra over $CE(\mathfrak{g}_{k})$,
and this allows us to define a new cdga

$S_{k}\otimes_{CE(\mathfrak{g}_{k})}\Omega_{D}^{\bullet}.$

It is straightforward to check that this algebra is nothing but
$\Omega^{\bullet}_{D}(S_{k}(L))[-1]$. Hence, $S_{k}$ is the universal algebra
controlling the cohomology of $b^{k+1}$-algebroids.

###  7.7. Globalising the cohomology

Now let $D\subset M$ be a hypersurface in a manifold. The data $\sigma$
determines a $b^{k+1}$-algebroid $A(\sigma)$ on $M$. To make things concrete,
we may choose a tubular neighbourhood $U$ of $D$ which is isomorphic to
$L=\nu_{D}$, the normal bundle. By Lemma 7.6, we have an isomorphism of cdga’s

$\Omega_{U}^{\bullet}(\sigma)\cong\Omega_{U}^{\bullet}\oplus\Omega_{D}^{\bullet-1}(S_{k}(L))$

from which we obtain a surjective chain morphism
$\Omega_{U}^{\bullet}(\sigma)\to\Omega_{D}^{\bullet-1}(S_{k}(L))$. By
restricting forms we obtain a map

$\operatorname{Res}:\Omega_{M}^{\bullet}(\sigma)\to\Omega_{U}^{\bullet}(\sigma)\to\Omega_{D}^{\bullet-1}(S_{k}(L)).$

which we think of as a generalized residue map. Note that this map depends on
the choice of a tubular neighbourhood embedding. There is also an inclusion of
smooth forms $\Omega_{M}^{\bullet}\to\Omega_{M}^{\bullet}(\sigma)$ which is
also a chain map.

######  Remark 7.0.

The restricted algebroid $A(\sigma)|_{D}$ lives in the short exact sequence

$0\to L^{-k}\to A(\sigma)|_{D}\to TD\to 0,$

and this allows us to define a canonical higher residue map
$\operatorname{Res}^{k}:\Omega_{M}^{\bullet}(\sigma)\to\Omega^{\bullet-1}_{D}(L^{k})$
which extracts the component of an algebroid form that has a pole of order
$k+1$ along $D$. The kernel of this residue consists of algebroid forms for
the truncated algebroid $A(\sigma_{k-1})$. In other words, these are forms
with poles of order at most $k$. Hence, by making use of the splittings
$\tau_{r}$, we may iteratively extract the order $r$ residues of a form
$\alpha\in\Omega_{M}^{\bullet}(\sigma)$, for $0\leq r\leq k$. These residues
$\operatorname{Res}^{r}$ form the components of the map $\operatorname{Res}$
and they depend on both the algebroid $A(\sigma)$ as well as the choice of the
tubular neighbourhood embedding. For a $b^{k+1}$-algebroid determined by a
defining function for $D$, the $r^{th}$-residue $\operatorname{Res}^{r}$ just
extracts the coefficient of $\frac{dz}{z^{r+1}}$.

######  Lemma 7.13.

There is a short exact sequence of chain complexes

$0\to\Omega_{M}^{\bullet}\to\Omega_{M}^{\bullet}(\sigma)\to\Omega_{D}^{\bullet-1}(S_{k}(L))\to
0,$

which induces a long exact sequence in cohomology

$...\to H^{i}(M)\to H^{i}(A(\sigma))\to H^{i-1}(D,S_{k}(L))\to
H^{i+1}(M)\to...$

###### Proof.

The map $\Omega_{M}^{\bullet}\to\Omega_{M}^{\bullet}(\sigma)$ is an inclusion
and so is injective. Furthermore, since the map lands in smooth forms, these
have zero projection to $\Omega_{D}^{\bullet-1}(S_{k}(L))$. To see that
$\Omega_{M}^{\bullet}(\sigma)\to\Omega_{D}^{\bullet-1}(S_{k}(L))$ is
surjective, we choose a bump function $\psi_{U}$ which is supported in the
tubular neighbourhood and is equal to $1$ in a neighbourhood of $D$. Now let
$\alpha\in\Omega_{D}^{\bullet-1}(S_{k}(L))$. By Lemma 7.6, this comes from an
element $\tau(\alpha)\in\Omega_{U}^{\bullet}(\sigma)$, an algebroid form
defined on $U$. The form $\psi_{U}\tau(\alpha)$ can be extended by zero to a
global form in $\Omega_{M}^{\bullet}(\sigma)$. And since $\psi_{U}$ is the
identity on a neighbourhood of $D$, the singular part of
$\psi_{U}\tau(\alpha)$ is $\alpha$. Hence
$\operatorname{Res}(\psi_{U}\tau(\alpha))=\alpha$, establishing surjectivity.
Finally, to show exactness in the middle spot, note that a form $\omega$ with
$\operatorname{Res}(\omega)=0$ restricts to a form on $U$ with trivial
singular part. But this is nothing but a smooth form. ∎

######  Proposition 7.14.

The connecting homomorphism $\delta:H^{i-1}(D,S_{k}(L))\to H^{i+1}(M)$
vanishes. Therefore, we have isomorphisms

$H^{i}(A(\sigma))\cong H^{i}(M)\oplus H^{i-1}(D,S_{k}(L)),$

for all $i$.

###### Proof.

Let $\alpha\in\Omega_{D}^{i-1}(S_{k}(L))$ be a closed form and let
$\psi_{U}\tau(\alpha)$ be its extension to $M$. Then the connecting
homomorphism is given by $\delta(\alpha)=d(\psi_{U}\tau(\alpha)).$ By Lemma
7.9 we can write $\tau(\alpha)=\alpha_{0}\wedge\gamma+d\eta$, with
$\alpha_{0}$ and $\gamma$ smooth and closed forms pulled back from $D$ and
$\eta$ a ‘horizontal’ form with singularities along $D$. Hence

$\delta(\alpha)=d\psi_{U}\wedge\alpha_{0}\wedge\gamma+d\psi_{U}\wedge
d\eta=d(\psi_{U}\alpha_{0}\wedge\gamma-d\psi_{U}\wedge\eta).$

Note that $\psi_{U}\alpha_{0}\wedge\gamma$ extends by zero to a global form on
$M$. Second, $\eta$ is a form with singularities along $D$, but since
$\psi_{U}$ is identically equal to $1$ in a neighbourhood of $D$, its
derivative $d\psi_{U}$ vanishes in this neighbourhood, and hence
$d\psi_{U}\wedge\eta$ is smooth around $D$ and can be extended by zero to a
global smooth form on $M$. Let

$\chi(\alpha,\psi_{U},h)=\psi_{U}\alpha_{0}\wedge\gamma-d\psi_{U}\wedge\eta,$

a smooth global form which depends on the choices of $\alpha$, a metric $h$ on
$L$, a tubular neighbourhood, and the bump function $\psi_{U}$. Then
$\delta(\alpha)=d\chi$, so that the connecting homomorphism vanishes. ∎

Let’s take a closer look at the residue map $\operatorname{Res}$. It factors
as $\operatorname{Res}=\pi_{2}\circ r_{1}$, where

$r_{1}:\Omega_{M}^{\bullet}(\sigma)\to\Omega_{U}^{\bullet}(\sigma)\cong\Omega_{U}^{\bullet}\oplus\Omega_{D}^{\bullet}(S_{k}(L))[-1],$

and the second map $\pi_{2}$ is the projection onto the second factor. The
first map is a morphism of cdga’s and induces on cohomology the algebra map

$H(r_{1}):H^{\bullet}(A(\sigma))\to H^{\bullet}(D)\oplus
H^{\bullet}(D,S_{k}(L))[-1].$

Now $H^{\bullet}(M)\subset H^{\bullet}(A(\sigma))$ is a subalgebra and if we
restrict the map $H(r_{1})$ to $\beta\in H^{\bullet}(M)$ we get

$H(r_{1})(\beta)=(\beta|_{D},0)\in H^{\bullet}(D)\oplus
H^{\bullet}(D,S_{k}(L))[-1].$

Note that the restriction map $H^{\bullet}(M)\to H^{\bullet}(D)$ is a
homomorphism and in this way any $H^{\bullet}(D)$-module becomes a
$H^{\bullet}(M)$-module. In particular, $H^{\bullet}(D,S_{k}(L))[-1]$ is in
this way an $H^{\bullet}(M)$-module.

######  Proposition 7.15.

The sequence

$0\to H^{\bullet}(M)\to H^{\bullet}(A(\sigma))\to
H^{\bullet}(D,S_{k}(L))[-1]\to 0$

is a short exact sequence of $H^{\bullet}(M)$-modules.

###### Proof.

The sequence is exact by Proposition 7.14. Note first that $H^{\bullet}(M)\to
H^{\bullet}(A(\sigma))$ is a morphism of algebras and hence in particular, a
morphism of $H^{\bullet}(M)$-modules. To see that $H(\operatorname{Res})$ is a
module morphism, we compute, for $\beta\in H^{\bullet}(M)$ and $\omega\in
H^{\bullet}(A(\sigma))$ :

$\displaystyle H(\operatorname{Res})(\beta\wedge\omega)$
$\displaystyle=\pi_{2}(H(r_{1})(\beta\wedge\omega))$
$\displaystyle=\pi_{2}(H(r_{1})(\beta)\wedge H(r_{1})(\omega))$
$\displaystyle=\pi_{2}((\beta|_{D},0)\wedge(\tilde{\omega},H(\operatorname{Res})(\omega)))$
$\displaystyle=\pi_{2}(\beta|_{D}\wedge\tilde{\omega},\beta|_{D}\wedge
H(\operatorname{Res})(\omega))$ $\displaystyle=\beta|_{D}\wedge
H(\operatorname{Res})(\omega)$ $\displaystyle=\beta\ast
H(\operatorname{Res})(\omega).$

The first equality uses the factorization, the second uses the fact that
$H(r_{1})$ is an algebra homomorphism, the third uses the above calculation of
the map, the fourth uses Proposition 7.10, and the last equality uses the
definition of the action of $H^{\bullet}(M)$ on $H^{\bullet}(D,S_{k}(L))[-1]$.
∎

######  Remark 7.0.

One may expect that the decomposition
$H^{\bullet}(A(\sigma))=H^{\bullet}(M)\oplus H^{\bullet-1}(D,S_{k}(L))$ can be
made compatibly with the algebra structures, generalizing Proposition 7.10. At
present we do not know whether this is true.

###  7.8. Restriction to the boundary

Recall that the restricted algebroid $A(\sigma)|_{D}$ sits in the following
short exact sequence

$0\to L^{-k}\to A(\sigma)|_{D}\to TD\to 0.$

Dualizing and taking wedge powers gives the following short exact sequence of
complexes

(7.2)
$0\to\Omega_{D}^{\bullet}\to\Omega^{\bullet}_{A(\sigma)|_{D}}\to\Omega^{\bullet-1}_{D}\otimes
L^{k}\to 0.$

It can be useful to express the restriction morphism
$\mathcal{R}:\Omega_{M}^{\bullet}(\sigma)\to\Omega^{\bullet}_{A(\sigma)|_{D}}$
in terms of the decomposition of Lemma 7.6. First, the subcomplex
$\Omega_{M}^{\bullet}$ maps to the subcomplex
$\Omega_{D}^{\bullet}\subset\Omega^{\bullet}_{A(\sigma)|_{D}}$ via the
restriction map.

Next, we focus on the subcomplex $\Omega_{D}^{\bullet-1}(S_{k}(L))$. Note that
$S_{k-1}(L)\subset S_{k}(L)$ is a subbundle which is preserved by the
connection. Hence, we get a short exact sequence of complexes

$0\to\Omega_{D}^{\bullet-1}(S_{k-1}(L))\to\Omega_{D}^{\bullet-1}(S_{k}(L))\to\Omega_{D}^{\bullet-1}\otimes
L^{k}\to 0.$

######  Lemma 7.16.

The restriction map restricts to the following map

$\mathcal{R}:\Omega_{D}^{\bullet-1}(S_{k-1}(L))\to\Omega_{D}^{\bullet},\qquad\sum_{r=0}^{k-1}\alpha_{r}\mapsto\sum_{r=0}^{k-1}\alpha_{r}\wedge\eta_{r}.$

Therefore, we obtain a map of short exact sequences

$0$$\Omega_{D}^{\bullet-1}(S_{k-1}(L))$$\Omega_{D}^{\bullet-1}(S_{k}(L))$$\Omega_{D}^{\bullet-1}\otimes
L^{k}$$0$$0$$\Omega_{D}^{\bullet}$$\Omega^{\bullet}_{A(\sigma)|_{D}}$$\Omega_{D}^{\bullet-1}\otimes
L^{k}$$0$$\mathcal{R}$$\mathcal{R}$$=$

As a result, we obtain a Gysin-like long exact sequence computing the
cohomology of $A(\sigma)|_{D}$.

######  Proposition 7.17.

There is a long exact sequence of cohomology groups

$...\to H^{i}_{D}\to H^{i}(A(\sigma)|_{D})\to H^{i-1}_{D}(L^{k})\to
H^{i+1}_{D}\to...$

with connecting homomorphism given by the wedge product with the extension
class $c(\sigma)\in H^{2}_{D}(L^{-k})$:

$\delta=(-1)^{i}c(\sigma)\wedge\ :H^{i-1}_{D}(L^{k})\to H^{i+1}_{D}.$

###### Proof.

By Lemma 7.16 we may compute the connecting homomorphism as
$\mathcal{R}\circ\delta$, where $\delta$ is the connecting homomorphism of the
short exact sequence for $\Omega_{D}^{\bullet}(S_{k}(L))$. This later sequence
is split as a graded vector space. Hence, given $\omega\in
H^{i-1}_{D}(L^{k})$, the connecting homomorphism is computed as

$\displaystyle\mathcal{R}\circ\delta(\omega)$
$\displaystyle=\mathcal{R}(d\omega)=\mathcal{R}(\sum_{j=1}^{k-1}(j-r)\eta_{j}\wedge\omega)$
$\displaystyle=(-1)^{i-1}(\sum_{j=1}^{k-1}(j-r)\eta_{j}\wedge\eta_{k-j})\wedge\omega$
$\displaystyle=(-1)^{i}c(\sigma)\wedge\omega.$

In the second equality we used the expression for the differential from
Equation 7.1, using the fact that $d^{\nabla}\omega=0$. In the last equality
we used the expression for $c(\sigma)$ from Theorem 2.2. ∎

Given a symplectic form $\omega\in\Omega^{2}(A(\sigma))$, we can use the above
result to decompose the cohomology class $[\omega|_{D}]$. This allows us to
study the geometry induced on the hypersurface $D$, as in Proposition 2.7 and
Proposition 2.8.

##  Appendix A Group and Lie algebroid extensions

In this section we recall some general theory for the extension of groups, Lie
algebroids and Lie groupoids. For a detailed account on Lie algebroid
extensions see [Bra10], or [Cra03].

###  A.1. Group extensions

We consider group extensions of the form

$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1,$

with $A$ Abelian. Let $\sigma:C\rightarrow B$ be any set-theoretic splitting,
then we may define a $C$-module structure:

$c\cdot a:=\sigma(c)\cdot a\cdot\sigma(c)^{-1}$

This structure does not depend on the choice of splitting as $A$ is abelian.
Consequently, one may consider the complex of group-cochains with values in
$A$: $\Omega^{k}(C;A)$.

Given any splitting $\sigma:C\rightarrow B$ we may consider

$C_{\sigma}(c_{1},c_{2})=\sigma(c_{1}c_{2})^{-1}\sigma(c_{1})\sigma(c_{2}).$

It then follows that using $\sigma$, $B$ is isomorphic to
$C\ltimes_{C_{\sigma}}A$ which is $C\times A$ as a set and has group structure
given by

$(c_{1},a_{1})\cdot(c_{2},a_{2})=(c_{1}\cdot c_{2},c_{1}^{-1}\cdot a_{1}\cdot
a_{2}\cdot(c_{1}c_{2})^{-1}\cdot C_{\sigma}(c_{1},c_{2})).$

The cohomology class $[C_{\sigma}]\in H^{2}(C;A)$ is independent on the choice
of splitting, and vanishes precisely when the extension is split, that is $B$
is isomorphic to $A\rtimes C$ as groups.

###  A.2. Lie algebroid extensions

The following result is well-known, but for completeness-sake we will still
provide a proof:

######  Proposition A.1.

Suppose that we have a short exact sequence of Lie algebroids:

$0\rightarrow B\rightarrow A\rightarrow C\rightarrow 0,$

with $B$ contained in $\ker\rho(A)$. Then:

  1. (1)

Any vector bundle splitting $\sigma:C\rightarrow A$ induces a $C$-connection
on $B$, given by $\nabla_{c}(b):=[\sigma(c),b]$.

  2. (2)

If $\sigma$ is a bracket preserving splitting $\Omega^{\bullet}(C;B)$ obtains
the structure of a dgla. If $\tilde{\sigma}$ is any other bracket preserving
splitting then $\sigma-\tilde{\sigma}\in{\rm MC}(\Omega^{\bullet}(C,B))$.

  3. (3)

When $B$ is abelian, the representation $\nabla$ does not depend on the choice
of splitting $\sigma$.

###### Proof.

1) We have that

$\nabla_{c}(fb)=[\sigma(c),fb]=f[\sigma(c),b]+\mathcal{L}_{\rho_{A}(\sigma(c))}(b)=f[\sigma(c),b]+\mathcal{L}_{\rho_{C}(c)}(b),$

and

$\nabla_{fc}(b)=[f\sigma(c),b]=f[\sigma(c),b]-\mathcal{L}_{\rho_{B}(b)}(f)c=f[\sigma(c),b],$

as $B$ has trivial anchor.
2) We have to show that

$d^{\nabla}(\sigma-\tilde{\sigma})+\frac{1}{2}[\sigma-\tilde{\sigma},\sigma-\tilde{\sigma}]=0.$

We have

$\displaystyle d^{\nabla}(\sigma-\tilde{\sigma})(X,Y)$
$\displaystyle=\nabla_{X}(\sigma(Y)-\tilde{\sigma}(Y))-\nabla_{Y}(\sigma(X)-\tilde{\sigma}(Y))-\sigma([X,Y])+\tilde{\sigma}([X,Y])$
$\displaystyle=[\sigma(X),\sigma(Y)-\tilde{\sigma}(Y)]-[\sigma(Y),\sigma(X)-\tilde{\sigma}(X)]-\sigma([X,Y])+\tilde{\sigma}([X,Y])$
$\displaystyle=[\sigma(X),\sigma(Y)]-[\sigma(X),\tilde{\sigma}(Y)]-[\sigma(Y),\sigma(X)]+[\sigma(Y),\tilde{\sigma}(X)]-\sigma([X,Y])+\tilde{\sigma}([X,Y])$
$\displaystyle=C_{\sigma}(X,Y)+[\sigma(X),\sigma(Y)]-[\sigma(X),\tilde{\sigma}(Y)]+[\sigma(Y),\tilde{\sigma}(X)]+\tilde{\sigma}([X,Y]),$

whereas

$\displaystyle\frac{1}{2}[\sigma-\tilde{\sigma},\sigma-\tilde{\sigma}]$
$\displaystyle=[\sigma(X)-\tilde{\sigma}(X),\sigma(Y)-\tilde{\sigma}(Y)]$
$\displaystyle=[\sigma(X),\sigma(Y)]-[\sigma(X),\tilde{\sigma}(Y)]-[\tilde{\sigma}(X),\sigma(Y)]+[\tilde{\sigma}(X),\tilde{\sigma}(Y)].$

From this we conclude that

$d^{\nabla}(\sigma-\tilde{\sigma})-\frac{1}{2}[\sigma-\tilde{\sigma},\sigma-\tilde{\sigma}]=C_{\sigma}+C_{\tilde{\sigma}}.$

3) As $\sigma(c)-\tilde{\sigma}(c)$ lies in $B$ as the sequence is exact it
follows that $[\sigma(c)-\tilde{\sigma}(c),b]=0$ for all $b\in B$.
Consequently $\nabla_{c}(b)$ does not depend on the choice of splitting. ∎

Given a short exact sequence of Lie algebroids, and a vector bundle splitting
$\sigma:C\rightarrow A$, we may define the curvature of $\sigma$ by:

$C_{\sigma}\in\Omega^{2}(C;B),\quad
C_{\sigma}(X,Y):=[\sigma(X),\sigma(Y)]-\sigma([X,Y]).$

######  Proposition A.2.

Given a vector bundle splitting $\sigma:C\rightarrow A$ of a short exact
sequence of Lie algebroids, the curvature $C_{\sigma}$ defines a cohomology
class in $H^{2}(C;B)$, where the representation is induced by the above
proposition.

  * •

$C:=[C_{\sigma}]\in H^{2}(C;B)$ is independent from the choice of splitting.

##  Appendix B Characteristic classes of (symplectic) foliations

We recall some theory of characteristic classes of (symplectic) foliations.

Let $\Omega^{\bullet}_{\mathcal{F}}(M)$ denote the kernel of the restriction
map $\iota^{*}:\Omega^{\bullet}(M)\rightarrow\Omega^{\bullet}(\mathcal{F})$.
We thus obtain a short exact sequence of cochain complexes:

(B.1)
$0\rightarrow\Omega_{\mathcal{F}}^{\bullet}(M)\rightarrow\Omega^{\bullet}(M)\rightarrow\Omega^{\bullet}(\mathcal{F})\rightarrow
0.$

We also consider $\Omega^{\bullet}(\mathcal{F},\nu^{*})$, endowed with the
differential induced by the Bott-connection. We have several maps relating
these complexes, the first is:

(B.2) $\displaystyle p:\Omega^{k+1}_{\mathcal{F}}(M)$
$\displaystyle\rightarrow\Omega^{k}(\mathcal{F},\nu^{*}),$ $\displaystyle
p(\alpha)(X_{1},\ldots,X_{k})(\bar{N})$
$\displaystyle=\alpha(X_{1},\ldots,X_{n},N),$

with $X_{1},\ldots,X_{n}\in\Gamma(T\mathcal{F})$ and where for
$N\in\mathfrak{X}(M)$, $\bar{N}$ denotes the corresponding section of the
zero-section. Another map, which we denote simply by the differential, is
given by

$\displaystyle d:H^{k}(\mathcal{F})$ $\displaystyle\rightarrow
H^{k+1}_{\mathcal{F}}(M),$ $\displaystyle[\alpha]$
$\displaystyle\mapsto[d\tilde{\alpha}],$

where $\tilde{\alpha}\in\Omega^{k}(M)$ is any extension of $\alpha$. We may
combine both these maps to obtain the variation map $d_{\nu}=p\circ
d:H^{k}(\mathcal{F})\rightarrow H^{k}(\mathcal{F},\nu^{*})$.

######  Definition B.1.

The symplectic variation of a symplectic foliation
$(\mathcal{F},\omega_{\mathcal{F}})$ is the class ${\rm
var}_{\omega_{\mathcal{F}}}:=d_{\nu}[\omega_{\mathcal{F}}]\in
H^{2}(\mathcal{F},\nu^{*})$.

Let $\theta\in\Omega^{1}(M;\nu)$ denote the quotient map arising from the
sequence

$0\rightarrow T\mathcal{F}\rightarrow TM\rightarrow\nu\rightarrow 0.$

One may phrase the involutivity condition for $\mathcal{F}$ purely in terms of
$\theta$ as follows:

######  Lemma B.2.

Let $\theta\in\Omega^{1}(M;\nu)$ and a flat connection $\nabla$ on $\nu$ be
given. Then $\ker\theta$ is involutive if and only if

$d^{\nabla}\theta=\beta\wedge\theta,$

for some $\beta\in\Omega^{1}(M)$.

######  Lemma B.3.

Let $\mathcal{F}$ be a foliation on $M$, and $\nabla$ a flat connection on the
normal bundle $\nu$ up to gauge equivalence. Then

$m(\mathcal{F},\nabla):=[\beta|_{\mathcal{F}}]\in H^{1}(\mathcal{F}),$

is a well-defined cohomology class, called the modular class of
$(\mathcal{F},\nabla)$.

Let

$\delta:H^{1}(\mathcal{F})\rightarrow H^{2}_{\mathcal{F}}(D)\rightarrow
H^{1}(\mathcal{F},\nu^{*})$

be the connection morphism of (B.1) combined with the map $p$ of Equation B.2.

######  Definition B.4.

The variation of $\mathcal{F}$ is defined as
$\operatorname{Var}_{\mathcal{F}}:=\delta(m(\mathcal{F},\nabla))$.

######  Lemma B.5.

The variation $\operatorname{Var}_{\mathcal{F}}$ does not depend on the choice
of connection.

###### Proof.

Changing to another flat connection means

$\tilde{\nabla}=\nabla+\gamma,$

for $\gamma\in\Omega^{1,cl}_{X}$. Then

$d^{\tilde{\nabla}}\theta=(\beta+\gamma)\wedge\theta.$

Hence, $m(F,\tilde{\nabla})=(\beta+\gamma)|_{F}=m(F,\nabla)+\gamma|_{F}$.
Therefore,

$\delta(m(F,\tilde{\nabla}))=\delta(m(F,\nabla))+d\gamma=\delta(m(F,\nabla)).$

∎

######  Proposition B.6.

Consider a pair $(F,\nabla)$, let $\theta\in\Omega^{1}_{X}\otimes\nu$, and let
$m\in H^{1}_{F}$ be the modular class. Then the following are equivalent

  1. (1)

$\mathrm{var}(F)=\delta(m)=0$,

  2. (2)

there exists a flat connection $\tilde{\nabla}$ on $\nu$ such that
$d^{\tilde{\nabla}}\theta=0$,

  3. (3)

there exists a flat extension of the Bott connection on $\nu$.

Furthermore, if these conditions are equivalent, the flat extension of the
Bott connection is unique up to an element of $H^{0}_{F}(\nu^{*})$. In other
words, it is unique up to a flat section of the Bott connection on $\nu^{*}$.

###### Proof.

First from $1$ to $2$. Assume that $\delta(m)=0$. This means that the class
$[d\beta]\in H^{1}_{F}(\nu^{*})$ is $0$, implying that $d\beta$ is in fact
exact in $\Omega^{\bullet}_{F}\otimes\nu^{*}$. This means that there is some
$\eta\in\nu^{*}$ with $d\eta=d\beta$. Therefore,
$\tilde{\beta}=\beta-\eta\in\Omega_{X}^{1}$ is closed. But

$\tilde{\beta}\wedge\theta=\beta\wedge\theta-\eta\wedge\theta=\beta\wedge\theta.$

Now since $\tilde{\beta}$ is closed, we may define a new flat connection
$\tilde{\nabla}=\nabla-\tilde{\beta}$ on $\nu$. With respect to this
connection

$d^{\tilde{\nabla}}\theta=d^{\nabla}\theta-\tilde{\beta}\wedge\theta=\beta\wedge\theta-\beta\wedge\theta=0.$

Now we show $2$ implies $1$. So let $\tilde{\nabla}$ be a flat connection such
that $d^{\tilde{\nabla}}\theta=0$. It will differ from $\nabla$ by a closed
$1$-form: $\tilde{\nabla}=\nabla-\gamma$, for $\gamma\in\Omega^{1,cl}_{X}$.
Hence

$0=d^{\tilde{\nabla}}\theta=d^{\nabla}\theta-\gamma\wedge\theta=\beta\wedge\theta-\gamma\wedge\theta.$

Therefore, $\beta\wedge\theta=\gamma\wedge\theta$. This means that
$\beta|_{F}=\gamma|_{F}$ and so

$\delta(m)=[d\gamma]=0.$

To show that $2$ and $3$ are equivalent we prove that a connection $\nabla$ on
$\nu$ extends the Bott connection if and only if $d^{\nabla}\theta=0$. Recall
that the Bott connection is a $TF$ connection $\nabla^{B}$ on $\nu$ defined as
follows

$\nabla^{B}_{V}(W)=\theta([V,\tilde{W}]),$

where $V\in TF$ and $W\in\nu$ which has an extension $\tilde{W}\in TX$.
Therefore, a flat $TX$-connection $\nabla$ on $\nu$ extends the Bott
connection if and only if

$\nabla_{V}(\theta(U))=\theta([V,U])$

for $V\in F$ and $U\in TX$. But since $\theta(V)=0$, this equation may be
rewritten as follows:

$d^{\nabla}\theta(V,U)=\nabla_{V}(\theta(U))-\nabla_{U}(\theta(V))-\theta([V,U])=0.$

Hence, $\nabla$ extends Bott if and only if $d^{\nabla}\theta$ vanishes when
one input lies in $F$. But since $\nu$ has rank $1$, this is precisely the
condition that $d^{\nabla}\theta=0$ ∎

######  Definition B.7.

A corank $1$ foliation $F\subset TX$ on $X$ has trivial variation if
$\mathrm{var}(F)=0$.

Now let $F$ be a foliation with trivial variation. Let
$\theta\in\Omega_{X}^{1}(\nu)$ be the defining $1$-form and let $\nabla$ be a
flat connection such that $d^{\nabla}\theta=0$. Now let $\alpha\in
H^{0}_{F}(\nu^{*})$. Namely, $\alpha\in\nu^{*}\subset\Omega_{X}^{1}$ such that
$d\alpha=0$. Then $\alpha\wedge\theta=0$ and hence
$d^{\tilde{\nabla}}(\theta)=0$, where $\tilde{\nabla}=\nabla+\alpha$.

Let $t_{\alpha}\in\nu^{*}$ be the section $\alpha$ but viewed as a section of
the line bundle rather than a $1$-form. Then

$\alpha=\langle t_{\alpha},\theta\rangle,$

and

$0=d\alpha=\langle d^{\nabla}t_{\alpha},\theta\rangle+\langle
t_{\alpha},d^{\nabla}\theta\rangle=\langle
d^{\nabla}t_{\alpha},\theta\rangle.$

This means that $d^{\nabla}t_{\alpha}\in\nu^{*}\otimes\nu^{*}$. Hence,

$d^{\nabla}t_{\alpha}=\langle u_{\alpha},\theta\rangle,$

with $u_{\alpha}\in\nu^{-2}$.

Now suppose that $[\theta]=0\in H^{1}_{X}(\nu,\nabla)$. This means that there
is some section $s\in\nu$ such that

$\nabla(s)=\theta.$

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