$cd$, interchange of $ab$ with $cd$, and the algebraic Bianchi identity). This
fact is well-known [32], but not often mentioned in textbooks on relativity.

###  A.2 Special algebraic and differential operators

Now, suppose that we are working on an $n$-dimensional pseudo-Riemannian
manifold $(M,g)$, with Levi-Civita connection $\nabla$. As in Section 2.1, the
Young symmetrizes introduced above define vector bundles of Young symmetrized
covariant tensors $\mathrm{Y}^{d}T^{*}M\to M$, where $d$ stands for a Young
diagram. We define special linear algebraic and differential operators,
already briefly discussed in Section 2.2, between these Young symmetrized
tensor bundles occurring in the Calabi complex. Each of the corresponding
Young diagrams has at most two columns, where the first column usually has at
most $n$ cells and the second column has at most two cells. The operator
_trace_ ($\operatorname{tr}$) removes one row of cells, _metric exterior
product_ ($g\odot-$) adds one row of cells, _left_ or _right exterior
derivative_ ($\mathrm{d}_{L}$ and $\mathrm{d}_{R}$) adds one cell to the left
or right column respectively, and _left_ or _right divergence_ ($\delta_{L}$
and $\delta_{R}$) removes one cell from the left or right column respectively.
The name of each of these operators should be suggestive of their form, with
the main complication being to maintain appropriate Young symmetry.

In principle, the Littlewood-Richardson decomposition rules uniquely fix the
principal symbols of each of these operators up to a scalar multiple, with the
Levi-Civita operator canonically converting a first order principal symbol
into a first order operator. In practice, it takes a bit of work to find
explicit formulas for them, given that a naive application a Young symmetrizer
produces unmanageably large expressions. Moreover, the existence of the
intracolumn and intercolumn symmetrization identities introduces non-
uniqueness into possible explicit expressions. Below, we give explicit
formulas for these operators. In case of ambiguity, the choice was dictated by
practical convenience. Then, in Secs. A.3 and A.4 we show by explicit
calculation that they satisfy the required symmetrization identities and thus
carry the correct Young type.

$\displaystyle\operatorname{tr}[b]_{a_{1}\cdots a_{l}:b}$
$\displaystyle=b_{a_{1}\cdots a_{l}c:b}{}^{c},$ (104) $\displaystyle(g\odot
t)_{a_{1}\cdots a_{l}:bc}$ $\displaystyle=l(g_{b[a_{1}}t_{a_{2}\cdots
a_{l}]:c}-g_{c[a_{1}}t_{a_{2}\cdots a_{l}]:b}),$ (105)
$\displaystyle\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=l\nabla_{[a_{1}}b_{a_{2}\cdots a_{l}]:bc},$ (106)
$\displaystyle\delta_{L}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=\nabla^{a}b_{aa_{1}\cdots
a_{l}:bc}+2l^{-1}\nabla^{a}b_{[b|a_{1}\cdots a_{l}:|c]a},$ (107)
$\displaystyle\mathrm{d}_{R}[t]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=2\nabla_{[b}t_{|a_{1}\cdots
a_{l}:|c]}+2(l-1)^{-1}\nabla_{[\\{b\\}|}t_{\\{a_{1}\cdots a_{l}\\}:|c]},$
(108) $\displaystyle\delta_{R}[b]_{a_{1}\cdots a_{l}:b}$
$\displaystyle=\nabla^{c}b_{a_{1}\cdots a_{l}:bc}.$ (109)

Let us give an explicit example of (105) for $l=2$, which appears in the
formulas for the constant curvature Riemann tensor (123) and for the
linearized Riemann curvature operator (7):

$\displaystyle(g\odot h)_{a_{1}a_{2}:bc}$
$\displaystyle=g_{a_{1}b}h_{a_{2}c}-g_{a_{2}b}h_{a_{1}c}-g_{a_{1}c}h_{a_{2}b}+g_{a_{2}c}g_{a_{1}b},$
(110) $\displaystyle(g\odot g)_{a_{1}a_{2}:bc}$
$\displaystyle=2(g_{a_{1}b}g_{a_{2}c}-g_{a_{2}b}g_{a_{1}c}),$ (111)
$\displaystyle(\nabla\nabla\odot h)_{a_{1}a_{2}:bc}$
$\displaystyle=\nabla\nabla_{a_{1}b}h_{a_{2}c}-\nabla\nabla_{a_{2}b}h_{a_{1}c}-\nabla\nabla_{a_{1}c}h_{a_{2}b}+\nabla\nabla_{a_{2}c}g_{a_{1}b},$
(112)
$\displaystyle\qquad\text{where}~{}~{}\nabla\nabla_{ab}=\nabla_{(a}\nabla_{b)}=\frac{1}{2}(\nabla_{a}\nabla_{b}+\nabla_{b}\nabla_{a}).$

In the last equation we used the $\odot$ operation to define another
differential operator of definite Young type. This property follows directly
from that of (105).

###  A.3 Preservation of Young type

Each of the operators (104)–(109) maps tensors of one Young type into another
one, as is indicated by the index notation described in Sec. A.1. Below we
explicitly demonstrate that, by showing that the result of applying one of
these operators to a tensor of a given Young type always satisfies the
required intracolumn and intercolumn identities.

First, we list some key identities satisfied by the idempotent
antisymmetrization and column exchange operations. They follow from
straightforward, though possibly lengthy, application of their definitions.
Here, a tensor $t_{a_{1}\cdots a_{l}}$ is assumed to be fully antisymmetric.
Also, to simplify the notation for nested operations, we use the notation
$\\{\cdots\\}^{k}_{l}$, where the braces necessarily enclose the indices
$a_{k}$, $a_{k+1}$, …, $a_{l}$, though perhaps also others, to mean that we
apply the appropriate column exchange operation to these $a_{i}$ indices as if
they appeared in the order $\\{a_{k}\cdots a_{l}\\}$.

$\displaystyle(l+1)p_{[a}t_{a_{1}\cdots a_{l}]}$
$\displaystyle=p_{a}t_{a_{1}\cdots a_{l}}-p_{\\{a\\}}t_{\\{a_{1}\cdots
a_{l}\\}},$ (113) $\displaystyle p_{\\{b\\}}t_{\\{a_{1}\cdots a_{l}\\}}$
$\displaystyle=p_{a_{1}}t_{ba_{2}\cdots
a_{l}}+p_{\\{b\\}}t_{a_{1}\\{a_{2}\cdots a_{l}\\}},$ (114) $\displaystyle
p_{\\{a\\}}t_{b\\{a_{1}\cdots a_{l}\\}}$
$\displaystyle=-p_{\\{b\\}}t_{a\\{a_{1}\cdots a_{l}\\}},$ (115) $\displaystyle
p_{\\{b}q_{c\\}}t_{\\{a_{1}\cdots a_{l}\\}}$
$\displaystyle=p_{a_{1}}q_{\\{c\\}}t_{b\\{a_{2}\cdots
a_{l}\\}}+p_{\\{b}q_{c\\}}t_{a_{1}\\{a_{2}\cdots a_{l}\\}},$ (116)
$\displaystyle\left(p_{b^{\prime}}t_{\\{a_{1}\cdots
a_{l}\\}:c^{\prime}}-p_{c^{\prime}}t_{\\{a_{1}\cdots
a_{l}\\}:b^{\prime}}\right)\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}$
$\displaystyle=p_{\\{b\\}}t_{\\{a_{1}\cdots
a_{l}\\}:c}-p_{\\{c\\}}t_{\\{a_{1}\cdots a_{l}\\}:b},$ (117)
$\displaystyle\left(t_{b^{\prime}\\{a_{1}\cdots
a_{l}\\}:c^{\prime}a}-t_{b^{\prime}\\{a_{1}\cdots
a_{l}\\}:c^{\prime}a}\right)\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}$
$\displaystyle=-2(l-1)t_{[b|a_{1}\cdots a_{l}:|c]},$ (118) $\displaystyle
p_{\\{\\{b\\}\\}}t_{\\{\\{a_{1}\cdots a_{l}\\}\\}}$
$\displaystyle=lp_{b}t_{a_{1}\cdots a_{l}}-(l-1)p_{\\{b\\}}t_{\\{a_{1}\cdots
a_{l}\\}},$ (119) $\displaystyle p_{\\{b\\}}q_{\\{\\{c\\}}t_{\\{a_{1}\cdots
a_{l}\\}\\}^{1}_{l}}$ $\displaystyle=p_{\\{c\\}}q_{b}t_{\\{a_{1}\cdots
a_{l}\\}}+(p_{\\{b}q_{c\\}}-q_{\\{b}p_{c\\}})t_{\\{a_{1}\cdots a_{l}\\}},$
(120) $\displaystyle\left(p_{\\{\\{b^{\prime}\\}}t_{\\{a_{1}\cdots
a_{l}\\}\\}^{1}_{l}:c^{\prime}}-p_{\\{\\{c^{\prime}\\}}t_{\\{a_{1}\cdots
a_{l}\\}\\}^{1}_{l}:b^{\prime}}\right)\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}$
$\displaystyle=2(l-1)p_{[b|}t_{a_{1}\cdots
a_{l}:|c]}-2(l-2)p_{[\\{b\\}|}t_{\\{a_{1}\cdots a_{l}\\}:|c]},$ (121)
$\displaystyle p_{\\{\\{a\\}}t_{\\{a_{1}\cdots a_{l}\\}\\}^{1}_{l}:\\{bc\\}}$
$\displaystyle=p_{\\{a\\}}t_{\\{a_{1}\cdots a_{l}\\}:bc}$
$\displaystyle\quad{}+2p_{[\\{b\\}|}t_{\\{a_{1}\cdots
a_{l}\\}:|c]a}-2p_{[b|}t_{a_{1}\cdots a_{l}:|c]a}.$ (122)

Next, we show how the above key identities can be used to explicitly
demonstrate that the required symmetrization identities are satisfied. We try
to indicate which of the key identities are used and where, while also
silently making use of the symmetrization properties of the Young type tensors
on which the operations are being performed.

For the _trace_ (104), the intracolumn identities are obvious, so there is
only one intercolumn identity to check:

$\displaystyle\operatorname{tr}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}}$
$\displaystyle=b_{\\{a_{1}\cdots
a_{l}\\}c:\\{b\\}}{}^{c}~{}~{}\eqref{eq:id-c1}$
$\displaystyle=b_{\\{a_{1}\cdots a_{l}c\\}:\\{b\\}}{}^{c}-b_{\\{a_{1}\cdots
a_{l}b\\}:c}{}^{c}=b_{a_{1}\cdots a_{l}c:b}{}^{c}$
$\displaystyle=\operatorname{tr}[b]_{a_{1}\cdots a_{l}:b}.$

For the _metric exterior product_ (105), the intracolumn identities are
obvious, so there are two intercolumn identities to check:

$\displaystyle(g\odot t)_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}c}$
$\displaystyle=l(g_{\\{b\\}\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:c}-g_{c\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:\\{b\\}})~{}~{}\eqref{eq:id-asym}$
$\displaystyle=-l(l+1)(g_{[b[a_{1}}t_{a_{2}\cdots
a_{l}]]:c}-g_{c[[a_{1}}t_{a_{2}\cdots a_{l}]:b]})$
$\displaystyle\quad{}+l(g_{b[a_{1}}t_{a_{2}\cdots
a_{l}]:c}-g_{c[a_{1}}t_{a_{2}\cdots a_{l}]:b})$ $\displaystyle=(g\odot
t)_{a_{1}\cdots a_{l}:bc},$ $\displaystyle(g\odot t)_{\\{a_{1}\cdots
a_{l}\\}:\\{bc\\}}$ $\displaystyle=l(g_{b^{\prime}\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:c^{\prime}}-g_{c^{\prime}\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:b^{\prime}})\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}~{}~{}\eqref{eq:id-c2m}$
$\displaystyle=l(g_{\\{b\\}\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:c}-g_{\\{c\\}\\{[a_{1}}t_{a_{2}\cdots
a_{l}]\\}:b})~{}~{}\eqref{eq:id-asym}$
$\displaystyle=-l(l+1)(g_{[b[a_{1}}t_{a_{2}\cdots
a_{l}]]:c}-g_{[c[a_{1}}t_{a_{2}\cdots a_{l}]]:b})$
$\displaystyle\quad{}+l(g_{b[a_{1}}t_{a_{2}\cdots
a_{l}]:c}-g_{c[a_{1}}t_{a_{2}\cdots a_{l}]:b})$ $\displaystyle=(g\odot
t)_{a_{1}\cdots a_{l}:bc}.$

The double anti-symmetrizations vanished because of the identities
$g_{[ab]}=0$ and $t_{[a_{2}\cdots a_{l}:a_{1}]}=0$, with the latter following
from a combination of (113) and an intercolumn identity. Also, we have used
the fact that $p_{[a_{1}}t_{a_{2}\cdots a_{l}]:b}$ is a tensor of the
corresponding Young type, which follows from the identities in the paragraph
below.

For the _left exterior derivative_ (106), the intracolumn identities are
obvious, so there are two intercolumn identities to check:

$\displaystyle\mathrm{d}_{L}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}c}$
$\displaystyle=l\nabla_{\\{[a_{1}}b_{a_{2}\cdots
a_{l}]\\}:\\{b\\}c}~{}~{}\eqref{eq:id-asym}$
$\displaystyle=-l(l+1)\nabla_{[[a_{1}}b_{a_{2}\cdots
a_{l}]:b]c}+l\nabla_{[a_{1}}b_{a_{2}\cdots
a_{l}]:bc}=\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc},$
$\displaystyle\mathrm{d}_{L}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{bc\\}}$
$\displaystyle=l\nabla_{\\{[a_{1}}b_{a_{2}\cdots
a_{l}]\\}:\\{bc\\}}~{}~{}\eqref{eq:id-asym}$
$\displaystyle=\nabla_{\\{a_{1}}b_{a_{2}\cdots
a_{l}\\}:\\{bc\\}}-\nabla_{\\{\\{a_{1}\\}}b_{\\{a_{2}\cdots
a_{l}\\}\\}^{1}_{l}:\\{bc\\}}~{}~{}\eqref{eq:id-c2}$
$\displaystyle=\nabla_{a_{1}}b_{\\{a_{2}\cdots
a_{l}\\}:\\{bc\\}}+\nabla_{b}b_{\\{a_{2}\cdots a_{l}\\}:a_{1}\\{c\\}}$
$\displaystyle\quad{}-\nabla_{\\{\\{a_{1}\\}}b_{\\{a_{2}\cdots
a_{l}\\}\\}^{2}_{l}:\\{bc\\}}-\nabla_{\\{\\{b\\}}b_{\\{a_{2}\cdots
a_{l}\\}\\}^{2}_{l}:a_{1}\\{c\\}}~{}~{}\eqref{eq:id-c1i2m},\eqref{eq:id-c1j}$
$\displaystyle=\nabla_{a_{1}}b_{a_{2}\cdots a_{l}:bc}+\nabla_{b}b_{a_{2}\cdots
a_{l}:a_{1}c}-\nabla_{c}b_{\\{a_{2}\cdots a_{l}\\}:a_{1}\\{b\\}}$
$\displaystyle\quad{}-\nabla_{\\{a_{1}\\}}b_{\\{a_{2}\cdots
a_{l}\\}:bc}-2\nabla_{[\\{b\\}|}b_{\\{a_{2}\cdots
a_{l}\\}:|c]a_{1}}+2\nabla_{[b|}b_{a_{2}\cdots a_{l}:|c]a_{1}}$
$\displaystyle\quad{}-(\nabla_{b^{\prime}}b_{\\{a_{2}\cdots
a_{l}\\}:a_{1}c^{\prime}}-\nabla_{c^{\prime}}b_{\\{a_{2}\cdots
a_{l}\\}:a_{1}b^{\prime}})\delta^{c^{\prime}}_{\\{c}\delta^{b^{\prime}}_{b\\}}~{}~{}\eqref{eq:id-c2m}$
$\displaystyle=l\nabla_{[a_{1}}b_{a_{2}\cdots
a_{l}]:bc}-2\nabla_{[\\{b\\}|}b_{\\{a_{2}\cdots a_{l}\\}:|c]a_{1}}$
$\displaystyle\quad{}+2\nabla_{[\\{c\\}|}b_{\\{a_{2}\cdots a_{l}\\}:a_{1}|b]}$
$\displaystyle=l\nabla_{[a_{1}}b_{a_{2}\cdots
a_{l}]:bc}=\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc}.$

For the _left divergence_ (107), the intracolumn identities are obvious. It
remains to check the two intercolumn identities:

$\displaystyle\delta_{L}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}c}$
$\displaystyle=\nabla^{a}(b_{a\\{a_{1}\cdots
a_{l}\\}:\\{b\\}c}+l^{-1}b_{\\{b\\}\\{a_{1}\cdots
a_{l}\\}:ca}-l^{-1}b_{c\\{a_{1}\cdots a_{l}\\}:\\{b\\}a})$
$\displaystyle=\nabla^{a}(b_{\\{aa_{1}\cdots
a_{l}\\}:\\{b\\}c}+b_{ba_{1}\cdots a_{l}:ca}~{}~{}\eqref{eq:id-c1}$
$\displaystyle\quad{}-l^{-1}(l+1)b_{[ba_{1}\cdots
a_{l}]:ca}+l^{-1}b_{ba_{1}\cdots a_{l}:ca}~{}~{}\eqref{eq:id-asym}$
$\displaystyle\quad{}-l^{-1}b_{\\{ca_{1}\cdots
a_{l}\\}:\\{b\\}a}+l^{-1}b_{ba_{1}\cdots a_{l}:ca})~{}~{}\eqref{eq:id-c1}$
$\displaystyle=\nabla^{a}(b_{aa_{1}\cdots a_{l}:bc}+2l^{-1}b_{[b|a_{1}\cdots
a_{l}:|c]a})=\delta_{L}[b]_{a_{1}\cdots a_{l}:bc},$
$\displaystyle\delta_{L}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{bc\\}}$
$\displaystyle=\nabla^{a}(b_{a\\{a_{1}\cdots
a_{l}\\}:\\{bc\\}}~{}~{}\eqref{eq:id-c2}$
$\displaystyle\quad{}+l^{-1}(b_{b^{\prime}\\{a_{1}\cdots
a_{l}\\}:c^{\prime}a}-b_{c^{\prime}\\{a_{1}\cdots
a_{l}\\}:b^{\prime}a})\delta_{\\{b}^{b^{\prime}}\delta_{c\\}}^{c^{\prime}})~{}~{}\eqref{eq:id-c2a}$
$\displaystyle=\nabla^{a}(b_{\\{aa_{1}\cdots
a_{l}\\}:\\{bc\\}}+b_{b\\{a_{1}\cdots
a_{l}\\}:\\{c\\}a}~{}~{}\eqref{eq:id-c1}$
$\displaystyle\quad{}-l^{-1}(l-1)(b_{ba_{1}\cdots a_{l}:ca}-b_{ca_{1}\cdots
a_{l}:ba}))$ $\displaystyle=\nabla^{a}(b_{aa_{1}\cdots
a_{l}:bc}+b_{\\{ba_{1}\cdots a_{l}\\}:\\{c\\}a}-b_{ca_{1}\cdots a_{l}:ba}$
$\displaystyle\quad{}-(1-l^{-1})(b_{ba_{1}\cdots a_{l}:ca}-b_{ca_{1}\cdots
a_{l}:ba}))$ $\displaystyle=\nabla^{a}(b_{aa_{1}\cdots
a_{l}:bc}+l^{-1}b_{ba_{1}\cdots a_{l}:ca}-l^{-1}b_{ca_{1}\cdots a_{l}:ba})$
$\displaystyle=\delta_{L}[b]_{a_{1}\cdots a_{l}:bc}$

For the _right exterior derivative_ (108), the following rewriting makes the
intracolumn identities obvious:

$\displaystyle\mathrm{d}_{R}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=\nabla_{b}b_{a_{1}\cdots a_{l}:c}-\nabla_{c}b_{a_{1}\cdots
a_{l}:b}$
$\displaystyle\quad{}+(l-1)^{-1}\left(\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{\\{c\\}}b_{\\{a_{1}\cdots
a_{l}\\}:b}\right)~{}~{}\eqref{eq:id-asym}$
$\displaystyle=\nabla_{b}b_{a_{1}\cdots a_{l}:c}-\nabla_{c}b_{a_{1}\cdots
a_{l}:b}$ $\displaystyle\quad{}-(l-1)^{-1}(l+1)(\nabla_{[b}b_{a_{1}\cdots
a_{l}]:c}-\nabla_{[c}b_{a_{1}\cdots a_{l}]:b})$
$\displaystyle\quad{}+(l-1)^{-1}(\nabla_{b}b_{a_{1}\cdots
a_{l}:c}-\nabla_{c}b_{a_{1}\cdots a_{l}:b}).$

There are also two intercolumn identities to check:

$\displaystyle\mathrm{d}_{R}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}c}$
$\displaystyle=\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{c}b_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}}$
$\displaystyle\quad{}+(l-1)^{-1}(\nabla_{\\{\\{b\\}\\}}b_{\\{\\{a_{1}\cdots
a_{l}\\}\\}:c}-\nabla_{\\{\\{c\\}}b_{\\{a_{1}\cdots
a_{l}\\}\\}^{1}_{l}:\\{b\\}})$
$\displaystyle=\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{c}b_{a_{1}\cdots a_{l}:b}$
$\displaystyle\quad{}+(l-1)^{-1}l\nabla_{b}b_{a_{1}\cdots
a_{l}:c}-\nabla_{\\{b\\}}b_{\\{a_{1}\cdots a_{l}\\}:c}~{}~{}\eqref{eq:id-c1i}$
$\displaystyle\quad{}-(l-1)^{-1}\nabla_{b}b_{\\{b_{1}\cdots
a_{l}\\}:\\{c\\}}~{}~{}\eqref{eq:id-c1j}$
$\displaystyle\quad{}-(l-1)^{-1}(\nabla_{c^{\prime}}b_{a_{1}\cdots
a_{l}:b^{\prime}}-\nabla_{b^{\prime}}b_{a_{1}\cdots
a_{l}:c^{\prime}})\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}~{}~{}\eqref{eq:id-c2m}$
$\displaystyle=\nabla_{b}b_{a_{1}\cdots a_{l}:c}-\nabla_{c}b_{a_{1}\cdots
a_{l}:b}$ $\displaystyle\quad{}+(l-1)^{-1}(\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{\\{c\\}}b_{\\{a_{1}\cdots a_{l}\\}:b})$
$\displaystyle=\mathrm{d}_{R}[b]_{a_{1}\cdots a_{l}:bc},$
$\displaystyle\mathrm{d}_{R}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{bc\\}}$
$\displaystyle=(\nabla_{b^{\prime}}b_{\\{a_{1}\cdots
a_{l}\\}:c^{\prime}}-\nabla_{c^{\prime}}b_{\\{a_{1}\cdots
a_{l}\\}:b^{\prime}})\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}~{}~{}\eqref{eq:id-c2m},\eqref{eq:id-c1i2}$
$\displaystyle\quad{}+(l-1)^{-1}(\nabla_{\\{\\{b^{\prime}\\}}b_{\\{a_{1}\cdots
a_{l}\\}\\}^{a}_{l}:c^{\prime}}-\nabla_{\\{\\{c^{\prime}\\}}b_{\\{a_{1}\cdots
a_{l}\\}\\}^{a}_{l}:b^{\prime}})\delta^{b^{\prime}}_{\\{b}\delta^{c^{\prime}}_{c\\}}$
$\displaystyle=\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{\\{c\\}}b_{\\{a_{1}\cdots a_{l}\\}:b}$
$\displaystyle\quad{}+(\nabla_{b}b_{a_{1}\cdots
a_{l}:c}-\nabla_{c}b_{a_{1}\cdots a_{l}:b})$
$\displaystyle\quad{}-(l-1)^{-1}(l-2)(\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{\\{c\\}}b_{\\{a_{1}\cdots a_{l}\\}:b})$
$\displaystyle=\nabla_{b}b_{a_{1}\cdots a_{l}:c}-\nabla_{c}b_{a_{1}\cdots
a_{l}:b}$ $\displaystyle\quad{}+(l-1)^{-1}(\nabla_{\\{b\\}}b_{\\{a_{1}\cdots
a_{l}\\}:c}-\nabla_{\\{c\\}}b_{\\{a_{1}\cdots a_{l}\\}:b})$
$\displaystyle=\mathrm{d}_{R}[b]_{a_{1}\cdots a_{l}:bc}.$

For the _right divergence_ (109), the intracolumn identities are obvious, so
there is only one intercolumn identity to check:

$\displaystyle\delta_{R}[b]_{\\{a_{1}\cdots a_{l}\\}:\\{b\\}}$
$\displaystyle=\nabla^{c}b_{\\{a_{1}\cdots
a_{l}\\}:\\{b\\}c}=\nabla^{c}b_{a_{1}\cdots
a_{l}:bc}=\delta_{R}[b]_{a_{1}\cdots a_{l}:b}.$

###  A.4 Composition identities

Below, we list identities between some possible compositions of the operators
(104)–(109). These will be instrumental in the following Sec. A.5, where they
will be used to explicitly define the operators involved in the Calabi complex
(2) and the necessary identities between them. We do not show the necessary
explicit calculations, as they are lengthy but straightforward. It suffices to
make use of the key identities (113)–(122), as explicitly illustrated in Sec.
A.3.

Recall that $\nabla$ denotes the Levi-Civita connection on a pseudo-Riemannian
space of constant curvature with metric $g$ and dimension $n$. The Riemann
tensor on this space is defined by the convention
$2\nabla_{[a}\nabla_{b]}\omega_{c}=\bar{R}_{ab:c}{}^{d}\omega_{d}$ and is
explicitly equal to

$\bar{R}_{ab:cd}=\frac{\lambda}{2}(g\odot
g)_{ab:cd}=\lambda(g_{ac}g_{bd}-g_{ad}g_{bc}),\quad\text{with}~{}~{}\lambda=\frac{k}{n(n-1)},$
(123)

such that $k=g^{ac}\bar{R}_{ab:c}{}^{b}$ is the curvature constant.

The simplest composition identity is of two left exterior derivatives ($l\geq
4$):

$\mathrm{d}_{L}\circ\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc}=0.$ (124)

The principal symbols of these operators augment the left index column of the
argument and antisymmetrize over it, thus composing to zero, as in the case of
the de Rham differential. This means that, at worst, the result of the
composition of the operators is of order zero and proportional to the
background curvature $\bar{R}$ given in (123). Now, note that the background
curvature is $\mathrm{GL}(n)$-equivariantly composed only out of the metric
and the composition $\mathrm{d}_{L}\circ\mathrm{d}_{L}$ is also an equivariant
operator (taking into account the transformation properties of the covariant
derivative and the metric). Then the result of the composition (Young type
$(2,2,1^{l-2})$) must be equivariantly composed only out of the metric $g$
(Young type $(2)$) and the argument $b$ (Young type $(2,2,1^{l-4})$). However,
according the Littlewood-Richardson rules [33, 59], there is no non-trivial
combination of that kind. Therefore, the composition of these operator must
vanish.

Next, we show the relation between the compositions
$\delta_{L}\circ\mathrm{d}_{L}$ and $\mathrm{d}_{L}\circ\delta_{L}$, along
with some auxiliary identities involving the curvature. These formulas hold
when the length of the left index column of the output is $l>2$.

$\displaystyle 2\nabla_{[a}\nabla_{b]}b_{a_{1}\cdots a_{l}:cd}$
$\displaystyle=(\bar{R}\cdot b)_{ab~{}a_{1}\cdots a_{l}:cd},$ (125)
$\displaystyle(\bar{R}\cdot b)_{ab~{}a_{1}\cdots a_{l}:cd}$
$\displaystyle=\bar{R}_{ab:\\{e\\}}{}^{e}b_{\\{a_{1}\cdots
a_{l}\\}:cd}+\bar{R}_{ab:c}{}^{e}b_{a_{1}\cdots
a_{l}:ed}+\bar{R}_{ab:d}{}^{e}b_{a_{1}\cdots a_{l}:ce}$
$\displaystyle=\lambda(g_{a\\{b\\}}-g_{b\\{a\\}})b_{\\{a_{1}\cdots
a_{l}\\}:cd}$ $\displaystyle\quad{}+\lambda(g_{ac}b_{a_{1}\cdots
a_{l}:bd}-g_{bc}b_{a_{1}\cdots a_{l}:ad})$
$\displaystyle\quad{}-\lambda(g_{ad}b_{a_{1}\cdots
a_{l}:bc}-g_{bd}b_{a_{1}\cdots a_{l}:ac});$ (126)

$\displaystyle(l+1)(\bar{R}\cdot b)_{a[b~{}a_{1}\cdots a_{l}]:cd}$
$\displaystyle=-l(l+1)\lambda g_{a[b}b_{a_{1}\cdots a_{l}]:cd}$
$\displaystyle\quad{}-(l+1)\lambda(g_{c[b}b_{a_{1}\cdots
a_{l}]:ad}-g_{d[b}b_{a_{1}\cdots a_{l}]:ac}),$
$\displaystyle(l+1)(\bar{R}\cdot b)^{a}{}_{[a~{}a_{1}\cdots a_{l}]:bc}$
$\displaystyle=-(l(n-l)+2)\lambda b_{a_{1}\cdots
a_{l}:bc}+(-)^{l}\lambda(g\odot\operatorname{tr}[b])_{a_{1}\cdots a_{l}:bc},$
$\displaystyle(l+1)(\bar{R}\cdot b)^{a}{}_{[b~{}a_{1}\cdots
a_{l}]:ca}-(l+1)(\bar{R}\cdot b)^{a}{}_{[c~{}a_{1}\cdots a_{l}]:ba}$
$\displaystyle=0;$

$\displaystyle\delta_{L}\circ\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=\square b_{a_{1}\cdots
a_{l}:bc}-\mathrm{d}_{L}\circ\delta_{L}[b]_{a_{1}\cdots
a_{l}:bc}+l^{-1}\mathrm{d}_{R}\circ\delta_{R}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle\quad{}-(l(n-l)+2)\lambda b_{a_{1}\cdots
a_{l}:bc}+(-)^{l}\lambda(g\odot\operatorname{tr}[b])_{a_{1}\cdots a_{l}:bc}.$
(127)

The main identity that will be useful in Section A.5 is the following:

$(\delta_{L}\circ\mathrm{d}_{L}+\mathrm{d}_{L}\circ\delta_{L})[b]_{a_{1}\cdots
a_{l}:bc}=\square b_{a_{1}\cdots
a_{l}:bc}+l^{-1}\mathrm{d}_{R}\circ\delta_{R}[b]_{a_{1}\cdots a_{l}:bc}\\\
-(l(n-l)+2)\lambda b_{a_{1}\cdots
a_{l}:bc}+(-)^{l}\lambda(g\odot\operatorname{tr}[b])_{a_{1}\cdots a_{l}:bc}.$
(128)

The composition $\delta_{L}\circ\mathrm{d}_{L}$ has a special form when the
length of the left index column of the output is $l=2$:

$\displaystyle\nabla_{\\{a\\}}r_{\\{a_{1}a_{2}\\}:bc}$
$\displaystyle=-(\nabla_{\\{c\\}}r_{\\{a_{1}a_{2}\\}:ba}+\nabla_{\\{b\\}}r_{\\{a_{1}a_{2}\\}:ca});$
(129)

$\displaystyle(\bar{R}\cdot r)^{a}{}_{b~{}a_{1}a_{2}:ca}$
$\displaystyle=-(n-1)\lambda
r_{a_{1}a_{2}:bc}+\lambda(g_{ba_{1}}\operatorname{tr}[r]_{a_{2}:c}-g_{ba_{2}}\operatorname{tr}[r]_{a_{1}:c}),$
$\displaystyle(\bar{R}\cdot r)^{a}{}_{b~{}a_{1}a_{2}:ca}-(\bar{R}\cdot
r)^{a}{}_{c~{}a_{1}a_{2}:ba}$ $\displaystyle=-2(n-1)\lambda
r_{a_{1}a_{2}:bc}+\lambda(g\odot\operatorname{tr}[r])_{a_{1}a_{2}:bc},$
$\displaystyle(\bar{R}\cdot r)^{a}{}_{[b~{}a_{1}a_{2}]:ca}-(\bar{R}\cdot
r)^{a}{}_{[c~{}a_{1}a_{2}]:ba}$ $\displaystyle=0;$

$\displaystyle\delta_{L}\circ\mathrm{d}_{L}[r]_{a_{1}a_{2}:bc}$
$\displaystyle=\square
r_{a_{1}a_{2}:bc}+\frac{1}{2}\mathrm{d}_{R}\circ\delta_{R}[r]_{a_{1}a_{2}:bc}$
$\displaystyle\quad{}-2(n-1)\lambda
r_{a_{1}a_{2}:bc}+\lambda(g\odot\operatorname{tr}[r])_{a_{1}a_{2}:bc}.$ (130)

Next, we show the relation between the compositions
$\operatorname{tr}{}\circ\mathrm{d}_{L}$ and
$\mathrm{d}_{L}\circ\operatorname{tr}$:

$\displaystyle\operatorname{tr}{}\circ\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:b}$
$\displaystyle=\mathrm{d}_{L}\circ\operatorname{tr}[b]_{a_{1}\cdots
a_{l}:b}+(-)^{l}\delta_{R}[b]_{a_{1}\cdots a_{l}:b}.$ (131)

Next, we show the relations between the compositions
$\mathrm{d}_{L}\circ\mathrm{d}_{R}$, $\mathrm{d}_{R}\circ\mathrm{d}_{L}$ and
the operator $(\nabla\nabla\odot-)$, along with some auxiliary identities
involving the curvature. Note that below we make use of the notation
$[\cdots]^{1}_{l}$ which denotes idempotent antisymmetrization of the indices
$a_{1}\cdots a_{l}$, as if they were given in that position and ignoring any
other indices appearing within the same brackets.

$\displaystyle 2\nabla_{[a}\nabla_{b]}b_{a_{1}\cdots a_{l}:c}$
$\displaystyle=(\bar{R}\cdot b)_{ab~{}a_{1}\cdots a_{l}:c},$ (132)
$\displaystyle(\bar{R}\cdot b)_{ab~{}a_{1}\cdots a_{l}:c}$
$\displaystyle=\bar{R}_{ab:\\{d\\}}{}^{d}b_{\\{a_{1}\cdots
a_{l}\\}:c}+\bar{R}_{ab:c}{}^{d}b_{a_{1}\cdots a_{l}:d}$
$\displaystyle=\lambda(g_{a\\{b\\}}-g_{b\\{a\\}})b_{\\{a_{1}\cdots
a_{l}\\}:c}$ $\displaystyle\quad{}+\lambda(g_{ac}b_{a_{1}\cdots
a_{l}:b}-g_{bc}b_{a_{1}\cdots a_{l}:a});$ (133)

$\displaystyle l(\bar{R}\cdot b)_{b[a_{1}~{}a_{2}\cdots a_{l}]:c}$
$\displaystyle=-l^{2}\lambda g_{b[a_{1}}b_{a_{2}\cdots
a_{l}]:c}+\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc},$ $\displaystyle
2l(\bar{R}\cdot b)_{[b|[a_{1}~{}a_{2}\cdots a_{l}]:|c]}$
$\displaystyle=-(l-2)\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc},$ $\displaystyle
l(\bar{R}\cdot b)_{\\{b\\}\\{[a_{1}~{}a_{2}\cdots a_{l}]\\}:c}$
$\displaystyle=-l^{2}\lambda g_{b[a_{1}}b_{a_{2}\cdots
a_{l}]:c}+\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc},$ $\displaystyle
2l(\bar{R}\cdot b)_{[\\{b\\}|\\{[a_{1}~{}a_{2}\cdots a_{l}]\\}:|c]}$
$\displaystyle=-(l-2)\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc},$ $\displaystyle
Q_{ba_{1}~{}a_{2}\cdots a_{l}:c}=(\bar{R}\cdot
b)_{a_{1}\\{b\\}~{}\\{a_{2}\cdots a_{l}\\}:c}$ $\displaystyle=-(\bar{R}\cdot
b)_{ba_{1}~{}a_{2}\cdots a_{l}:c}+l^{2}\lambda g_{a_{1}[b}b_{a_{2}\cdots
a_{l}]:c}-\lambda(g\odot b)_{ba_{2}\cdots a_{l}:a_{1}c},$ $\displaystyle
l(\bar{R}\cdot b)_{[a_{1}\\{b\\}~{}\\{a_{2}\cdots
a_{l}\\}]^{1}_{l}:c}-l(\bar{R}\cdot b)_{[a_{1}\\{c\\}~{}\\{a_{2}\cdots
a_{l}\\}]^{1}_{l}:b}$ $\displaystyle=lQ_{b[a_{1}~{}a_{2}\cdots
a_{l}]:c}-lQ_{c[a_{1}~{}a_{2}\cdots a_{l}]:b}$
$\displaystyle=2(l-2)\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc};$

$\displaystyle\mathrm{d}_{R}\circ\mathrm{d}_{L}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=(l-1)^{-1}l(\nabla\nabla\odot b)_{a_{1}\cdots a_{l}:bc}$
$\displaystyle\qquad{}-\frac{l(l-2)}{2(l-1)}\lambda(g\odot b)_{a_{1}\cdots
a_{l}:bc},$ (134)

$\displaystyle\mathrm{d}_{L}\circ\mathrm{d}_{R}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=(\nabla\nabla\odot b)_{a_{1}\cdots
a_{l}:bc}+\frac{l}{2}\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc}.$ (135)

The main identity that will be useful in Section A.5 is the following:

$\displaystyle
l^{-1}\mathrm{d}_{R}\circ\mathrm{d}_{L}-(l-1)^{-1}\mathrm{d}_{L}\circ\mathrm{d}_{R}$
$\displaystyle=-\lambda(g\odot b)_{a_{1}\cdots a_{l}:bc}.$ (136)

###  A.5 Calabi complex and its homotopy formulas

Below, we use the special differential operators introduced earlier in Section
A.2 to explicitly define the differential operators $B_{l}$, $E_{l}$ and
$P_{l}$ that make up the Calabi complex and its homotopy formulas, as
discussed in more detail in Section 2.2:

$\displaystyle B_{1}[v]_{a:b}$
$\displaystyle=\nabla_{a}v_{b}+\nabla_{b}v_{a},$ $\displaystyle
B_{2}[h]_{a_{1}a_{2}:bc}$ $\displaystyle=(\nabla\nabla\odot
h)_{a_{1}a_{2}:bc}+\lambda(g\odot h)_{a_{1}a_{2}:bc},$ $\displaystyle
B_{l}[b]_{a_{1}\cdots a_{l}:bc}$ $\displaystyle=\mathrm{d}_{L}[b]_{a_{1}\cdots
a_{l}:bc}\quad(l\geq 3),$ $\displaystyle E_{1}[h]_{a}$
$\displaystyle=\nabla^{b}h_{a:b}-\frac{1}{2}\nabla_{a}\operatorname{tr}[h],$
$\displaystyle E_{2}[b]_{a:b}$ $\displaystyle=\operatorname{tr}[b]_{a:b},$
$\displaystyle E_{l+1}[b]_{a_{1}\cdots a_{l}:bc}$
$\displaystyle=\left(\delta_{L}-(-)^{l}l^{-1}\mathrm{d}_{R}\circ\operatorname{tr}\right)[b]_{a_{1}\cdots
a_{l}:bc}\quad(l\geq 2).$

Explicit formulas for $B_{l}$ and $E_{l}$ with low $l$ have been given in
Section 2.2.

Further, we make use of the identities given in Section A.4 to show that these
operators satisfy the required identities, namely $B_{l+1}\circ B_{l}=0$. The
identities $B_{2}\circ B_{1}=0$ and $B_{3}\circ B_{2}=0$ have already been
shown to follow in Section 2.2 from the usual transformation properties of the
Riemann curvature tensor under diffeomorphisms and from its Bianchi
identities. The identities $B_{l+1}\circ B_{l}=0$ for $l>2$ then follow
directly from the composition identity $\mathrm{d}_{L}\circ\mathrm{d}_{L}=0$
in Equation (124).

Again, appealing to the identities of Section A.4, we give the homotopy
formulas $P_{l}=E_{l+1}\circ B_{l+1}+B_{l}\circ E_{l}$ for $l\leq 2$:

$\displaystyle E_{1}\circ B_{1}[v]_{a}$
$\displaystyle=\nabla^{b}(\nabla_{a}v_{b}+\nabla_{b}v_{a})-\frac{1}{2}\nabla_{a}(2\nabla^{b}v_{b})$
$\displaystyle P_{0}$ $\displaystyle=\square v_{a}+\lambda(n-1)v_{a},$

$\displaystyle(E_{2}\circ B_{2}+B_{1}\circ E_{1})[h]_{a:b}$
$\displaystyle=(\nabla\nabla\odot h)_{ac:b}{}^{c}+\lambda(g\odot
h)_{ac:b}{}^{c}$
$\displaystyle\quad{}+\nabla_{a}\left(\nabla^{c}h_{b:c}-\frac{1}{2}\nabla_{b}\operatorname{tr}[h]\right)+\nabla_{b}\left(\nabla^{c}h_{a:c}-\frac{1}{2}\nabla_{a}\operatorname{tr}[h]\right)$
$\displaystyle P_{1}$ $\displaystyle=\square h_{ab}-2\lambda h_{ab}+2\lambda
g_{ab}\operatorname{tr}[h],$

$\displaystyle(E_{3}\circ B_{3}+B_{2}\circ E_{2})[r]_{a_{1}a_{2}:bc}$
$\displaystyle=\delta_{L}\circ\mathrm{d}_{L}[r]_{a_{1}a_{2}:bc}-\frac{1}{2}\mathrm{d}_{R}\circ\operatorname{tr}{}\circ\mathrm{d}_{L}[r]_{a_{1}a_{2}:bc}$
$\displaystyle\quad{}+(\nabla\nabla\odot\operatorname{tr}[r])_{a_{1}a_{2}:bc}+\lambda(g\odot\operatorname{tr}[r])_{a_{1}a_{2}:bc}$
$\displaystyle P_{2}$ $\displaystyle=\square r_{a_{1}a_{2}:bc}-2(n-1)\lambda
r_{a_{1}a_{2}:bc}+2\lambda(g\odot\operatorname{tr}[r])_{a_{1}a_{2}:bc}.$

Finally, the same set of identities also implies the following formulas for
$P_{l}$ with $l>2$:

$\displaystyle(E_{l+1}\circ B_{l+1}+B_{l}\circ E_{l})[b]_{a_{1}\cdots
a_{l}:bc}$
$\displaystyle=(\delta_{L}\circ\mathrm{d}_{L}+\mathrm{d}_{L}\circ\delta_{L})[b]_{a_{1}\cdots
a_{1}:bc}$
$\displaystyle\quad{}-(-)^{l}\left(l^{-1}\mathrm{d}_{R}\circ\operatorname{tr}{}\circ\mathrm{d}_{L}-(l-1)^{-1}\mathrm{d}_{L}\circ\mathrm{d}_{R}\circ\operatorname{tr}{}\right)[b]_{a_{1}\cdots
a_{l}:bc}$ $\displaystyle P_{l}$ $\displaystyle=\square b_{a_{1}\cdots
a_{l}:bc}-(l(n-l)+2)\lambda b_{a_{1}\cdots
a_{l}:bc}+(-)^{l}2\lambda(g\odot\operatorname{tr}[b])_{a_{1}\cdots a_{l}:bc}.$

Explicit formulas for $P_{l}$ with low $l$ have also been given in Section
2.2. Recall that, as in Equation (123), we have used the notation
$\lambda=\frac{k}{n(n-1)}$.

###  A.6 An adjoint operator

Here we derive Equation (30), which according to the general formula (22)
implies that $-n^{-1}\delta_{L}$ is the formal adjoint of $\mathrm{d}_{L}$
when acting on tensors of Young type $c_{a_{2}\cdots a_{n}:bc}$.

$\displaystyle\nabla_{a}(c^{aa_{2}\cdots a_{n}:bc}b_{a_{2}\cdots a_{n}:bc})$
$\displaystyle=c^{aa_{2}\cdots a_{n}:bc}\nabla_{a}b_{a_{2}\cdots
a_{n}:bc}+(\nabla_{a}c^{aa_{2}\cdots a_{n}:bc})b_{a_{2}\cdots a_{n}:bc}$
$\displaystyle=c^{aa_{2}\cdots a_{n}:bc}\nabla_{[a}b_{a_{2}\cdots
a_{n}]:bc}+\delta_{L}[c]^{a_{2}\cdots a_{n}:bc}b_{a_{2}\cdots a_{n}:bc}$
$\displaystyle\quad{}-\frac{1}{n-1}(\nabla_{a}c^{ba_{2}\cdots
a_{n}:ca}-\nabla_{a}c^{ca_{2}\cdots a_{n}:ba})b_{a_{2}\cdots a_{n}:bc}$
$\displaystyle=\frac{1}{n}c^{aa_{2}\cdots
a_{n}:bc}\mathrm{d}_{L}[b]_{aa_{2}\cdots a_{n}:bc}+\delta_{L}[c]^{a_{2}\cdots
a_{n}:bc}b_{a_{2}\cdots a_{n}:bc}$
$\displaystyle\quad{}-\frac{1}{n-1}(b_{[a_{2}\cdots
a_{n}:b]c}\nabla_{a}c^{ba_{2}\cdots a_{n}:ca}+b_{[a_{2}\cdots
a_{n}:c]b}\nabla_{a}c^{ca_{2}\cdots a_{n}:ba})~{}~{}\eqref{eq:id-asym}$
$\displaystyle=\frac{1}{n}c^{aa_{2}\cdots
a_{n}:bc}\mathrm{d}_{L}[b]_{aa_{2}\cdots a_{n}:bc}+\delta_{L}[c]^{a_{2}\cdots
a_{n}:bc}b_{a_{2}\cdots a_{n}:bc}.$ (137)

We have simply used the definitions of the $\mathrm{d}_{L}$ and $\delta_{L}$
differential operators as well as the fact that the contraction of two
tensors, one of which being totally anti-symmetric in a subset of indices,
allows the insertion of an anti-symmetrization over the corresponding indices
of the second tensor. Finally, some of the anti-symmetrizations annihilated
the corresponding tensors, due to their intercolumn identities and the
application of the identity (113).

##  Appendix B Homological algebra

Below we introduce some basic notions from homological algebra. A standard
text on the subject is [88], where more details can be found along with
complete proofs.

Let $A_{i}$, also denoted $A_{\bullet}$, be a sequence of vector spaces (real
vector spaces, for our purposes) with linear maps $A_{i}\to A_{i+1}$ between
them. If each successive pair of maps $A_{i-1}\to A_{i}\to A_{i+1}$ composes
to zero, this sequence is called a _complex (of vector spaces)_ or a _cochain
complex_ , with an element $a\in A_{i}$ being referred to as a _cochain (of
degree $i$)_, and the maps $A_{i}\to A_{i+1}$ referred to as _cochain
differentials_. Any complex gives rise to cohomologies

$H^{i}(A_{\bullet})=\ker(A_{i}\to A_{i+1})/\operatorname{im}(A_{i-1}\to
A_{i}).$ (138)

If all the cohomologies vanish, $H^{i}(A_{\bullet})=0$ or the image of each
map is equal to the kernel of the subsequent map, the complex is called
_exact_ or an _exact sequence_. Given two complexes $A_{\bullet}$ and
$B_{\bullet}$, the vertical maps in the diagram

${\cdots}$${A_{i}}$${A_{i+1}}$${\cdots}$${\cdots}$${B_{i}}$${B_{i+1}}$${\cdots}$
(139)

are called _cochain maps_ provided they make the diagram commute. Furthermore,
the diagonal maps in a diagram like

${\cdots}$${A_{i}}$${A_{i+1}}$${\cdots}$${\cdots}$${B_{i}}$${B_{i+1}}$${\cdots}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{h}$$\scriptstyle{\delta}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{h}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\mathrm{d}}$$\scriptstyle{\mathrm{d}}$
(140)

are called _cochain homotopies_. The homotopy maps induce vertical cochain
maps by the formula $h=\mathrm{d}\delta+\delta\mathrm{d}$. It is a basic fact
that cochain maps $A_{\bullet}\to B_{\bullet}$ naturally induce maps in
cohomology $H^{i}(A_{\bullet})\to H^{i}(B_{\bullet})$. Of course, identity
chain maps induce identity maps in cohomology and zero chain maps induces zero
maps in cohomology. Also, two cochain maps induce the same map in cohomology
when their difference is induced by a cochain homotopy.

A _short exact sequence_

${0}$${A_{\bullet}}$${B_{\bullet}}$${C_{\bullet}}$${0}$ (141)

between complexes $A_{\bullet}$, $B_{\bullet}$ and $C_{\bullet}$ consists of
cochain maps between them such that each instance of

${0}$${A_{i}}$${B_{i}}$${C_{i}}$${0}$ (142)

is an exact sequence of vector spaces. Another basic fact of homological
algebra is that a short exact sequence of complexes induces the following
_long exact sequence_ in cohomology

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.16151pt}{-24.9994pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{92.02266pt}{-24.9994pt}\pgfsys@lineto{115.22267pt}{-24.9994pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{115.42265pt}{-24.9994pt}\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (143)

where the maps $H^{i}(A_{\bullet})\to H^{i}(B_{\bullet})$ and
$H^{i}(B_{\bullet})\to H^{i}(C_{\bullet})$ are induced by the cochain maps
from the short exact sequence and the _connecting_ maps $H^{i}(C_{\bullet})\to
H^{i+1}(A_{\bullet})$ are induced by the cochain differential.

Finally, another standard result is the so-called _$5$ -lemma_ (or a simple
variant thereof). It states that the central vertical map in the commutative
diagram

${A_{-2}}$${A_{-1}}$${A_{0}}$${A_{1}}$${A_{2}}$${B_{-2}}$${B_{-1}}$${B_{0}}$${B_{1}}$${B_{2}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
(144)

is an isomorphism, provided that the top and bottom rows are exact sequences
and all the other vertical maps are isomorphisms themselves.

##  Appendix C Jets and jet bundles

In this appendix, we briefly introduce jet bundles, fix the relevant notation
and discuss differential operators in the context of jets. For simplicity, we
restrict ourselves to fields taking values in vector bundles. However, the
discussion could be straightforwardly generalized to general smooth bundles.
More details, as well as a coordinate independent definition, can be found in
the standard literature [61, 53].

Given a vector bundle $F\to M$ over a connected $n$-dimensional smooth
manifold $M$, the _$k$ -jet bundle_ $J^{k}F\to M$ is a vector bundle whose
defining characteristic is that for any (possibly non-linear) differential
operator $f\colon\Gamma(F)\to\Gamma(F^{\prime})$ of order $k$, there exists a
canonical factorization $f[u]=f\circ j^{k}u$ for any section $u\colon M\to F$,
where the _$k$ -jet prolongation_ $j^{k}\colon\Gamma(F)\to\Gamma(J^{k}F)$ is
composed with a smooth bundle map $f\colon J^{k}F\to F^{\prime}$, which by a
slight abuse of notation we denote using the same symbol as the original
differential operator. Composing the differential operator $f$ with an $l$-jet
prolongation canonically defines a new differential operator $p_{l}f\colon
J^{l+k}F\to J^{l}F^{\prime}$ called its _$l$ -prolongation_,
$j^{l}f[u]=p_{l}f\circ j^{k}u$. Given a trivializable restriction $F_{U}\to U$
of $F$ to a chart $U\subset M$ with local coordinates $(x^{i})$ and fiber-
adapted local coordinates $(x^{i},u^{a})$, there is a corresponding adapted
chart $J^{k}F_{U}\subset J^{k}F$ with adapted local coordinates
$(x^{i},u^{a}_{I})$, where $I=i_{1}\cdots i_{l}$ runs through multi-indices of
orders $|I|=l=0,\ldots,k$. In these coordinates, the $k$-jet prolongation is
given by $j^{k}u(x)=(x^{i},\partial_{I}u^{a}(x))$, while the $l$-prolongation
is given by $p_{l}f[u](x)=(x^{i},\partial_{I}f^{b}[u](x))$, where
$f[u](x)=(x^{i},f^{b}[u](x))$ in fiber-adapted local coordinates
$(x^{i},v^{b})$ on $F^{\prime}$. For any $l>k$, discarding the information
about all derivatives of order $>k$ defines a natural projection $J^{l}F\to
J^{k}F$. The projective limit $J^{\infty}F:=\varprojlim_{k\to\infty}J^{k}F$
defines the _$\infty$ -jet bundle_. The $\infty$-jet prolongation $j^{\infty}$
and $\infty$-prolongation $p_{\infty}$ are defined in the obvious way. By
composing with the natural projection $J^{\infty}F\to J^{k}F$, the
differential operator $f$ also canonically defines the smooth bundle map
$f\colon J^{\infty}F\to J^{k}F\stackrel{{\scriptstyle f}}{{\to}}F^{\prime}$,
which is again denoted by the same symbol $f$. Conversely, due to the
projective limit construction, any smooth bundle map $f\colon J^{\infty}F\to
F^{\prime}$ can only depend on finitely many coordinates of its domain, which
means that there exists a $k\geq 0$ such that this bundle map canonically
factors as $f\colon J^{\infty}F\to J^{k}F\stackrel{{\scriptstyle
f}}{{\to}}F^{\prime}$, with the smallest such $k$ being the _order_ of $f$.

Given vector bundles $F\to M$, $E\to M$ and a differential operator
$e\colon\Gamma(F)\to\Gamma(E)$, we write down the _partial differential
equation_ (PDE) $e[\psi]=0$, with $\psi\in\Gamma(F)$. Sometimes it is
convenient to refer to $F\to M$ as the _field bundle_ and to $E\to M$ as the
_equation bundle_. We will only consider linear PDEs below, where the
differential operator $e$ is linear. We denote the local spaces of solutions
by $\mathcal{S}_{e}(U)$, where $U\subseteq M$ is open and
$\psi\in\Gamma(F|_{U})$ belongs to $\mathcal{S}_{e}(U)$ iff $e[\psi]=0$ on
$U$. The PDE is said to be of order $k$ if it can be written as
$e[\psi]=e(j^{k}\psi)$, where on the right-hand side we have a (linear) bundle
map $e\colon J^{k}F\to E$.

In adapted coordinates $(x^{i},u^{a})$ on $F$, the PDE $e[\psi]=0$ has the
form $e^{I}(x)\partial_{I}\psi(x)=0$. When the PDE is of order $k$, the
coefficients $e^{I}(x)$ vanish for multi-indices with $|I|>k$. The
coefficients of the highest order derivatives, $e^{I}(x)$ with $|I|=k$, in
fact transform as a tensor under coordinate changes and define a linear bundle
map $\sigma e\colon F\otimes S^{k}T^{*}M\to E$ called the _principal symbol_
of $e$. If we fix $(x,p)\in T^{*}M$, then the corresponding linear map
$\sigma_{x,p}e=\sigma e(x)\cdot p^{\otimes k}\colon F_{x}M\to E_{x}M$ can also
be referred to as the value of the principal symbol of $e$ at $(x,p)$.

The PDE $e[\psi]=0$ is _equivalent_ to the PDE $e^{\prime}[\psi^{\prime}]=0$,
with $e^{\prime}\colon J^{k^{\prime}}F^{\prime}\to E^{\prime}$, if they have
isomorphic solution spaces. That is $e[\psi]=0$ implies that
$e^{\prime}[f[\psi]]=0$ and $e^{\prime}[\psi^{\prime}]=0$ implies that
$e[f^{\prime}[\psi^{\prime}]]=0$, for some differential operators $f$ and
$f^{\prime}$. In fact, it can be shown that the two PDEs are equivalent
precisely when they fit into the following diagram, where arrows are
differential operators and the bundle labels stand in for the corresponding
spaces of sections,

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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where differential operators satisfy the following identities:

$\displaystyle e^{\prime}\circ f$ $\displaystyle=g\circ e,$ $\displaystyle
f^{\prime}\circ f$ $\displaystyle=\mathrm{id}+q\circ e,$ (146) $\displaystyle
e\circ f^{\prime}$ $\displaystyle=g^{\prime}\circ e^{\prime},$ $\displaystyle
f\circ f^{\prime}$ $\displaystyle=\mathrm{id}+q^{\prime}\circ e^{\prime}.$
(147)

The reason we can express equivalence in this way, at least when all the
differential operators are linear, follows from linear algebra on jets. If we
replace the operators $e$, $e^{\prime}$, $f$ and $f^{\prime}$ by the
corresponding jet bundle maps, prolonged to the appropriate order, it follows
from basic linear algebra that there exist jet bundle maps that ostensibly
correspond to the operators $g$, $g^{\prime}$, $q$ and $q^{\prime}$. It then
follows from a deeper analysis of the properties of linear PDEs [38, 18, 73]
that once these differential operators are defined using prolongations of
sufficiently high order, the appropriate identities hold at all higher orders.
As a simple example, note that the equation $e[\psi]=0$ and its prolongation
$p_{k}e[\psi]=0$ are equivalent, with $f=f^{\prime}=\mathrm{id}$.

Consider vector bundles $E,F,G\to M$ and linear differential operators

$f\colon\Gamma(G)\to\Gamma(F)\quad\text{and}\quad
e\colon\Gamma(F)\to\Gamma(E),$ (148)

of respective orders $k$ and $l$, such that $e\circ f=0$. We say that the
composition of $e$ and $f$ is _formally exact_ if the composition
$p^{k+m}e\circ p^{m}f$ of jet bundle maps is exact in the usual linear algebra
sense (the image of $p^{m}f$ is equal to the kernel of $p^{k+m}e$). Formal
exactness is a powerful hypothesis. For instance, it implies that certain
differential operators factorise through either $e$ or $f$ [38, 63]. Namely,
if $g$ is any differential operator such that $g\circ f=0$, then there must
exist another differential operator $g^{\prime}$ such that $g=g^{\prime}\circ
e$. Similarly, if $g$ is any differential operator such that $e\circ g=0$,
then there must exist another differential operator $g^{\prime}$ such that
$g=f\circ g^{\prime}$.

##  Appendix D Deformations of flat principal bundles

The material below requires some familiarity with the theory of $G$-principal
bundles [81, 50, 57, 6]. Its main point is to show how one can reduce the
computation of the degree-$1$ cohomology space of a certain locally constant
sheaf on a manifold $M$ to the computation of the degree-$1$ group cohomology
of the fundamental group $\pi=\pi_{1}(M)$ with coefficients in a certain
corresponding representation. This reformulation is a significant
simplification because group cohomology calculations can often be reduced to
finite dimensional linear algebra and many explicit calculations of that sort
have already been performed and are available in the literature. The
connection between these sheaf and group cohomologies is established by
noticing that both of them describe equivalence classes of infinitesimal
deformations of flat principal bundles. Unfortunately, this argument is not
sufficient to establish an isomorphism between these sheaf and group
cohomologies in higher degrees, but degree-$1$ is already interesting because
it is the one relevant in the physical application we have in mind (Section
5).

We briefly recall some basic facts about principal $G$-bundles [81, 50, 57,
6]. The total space of the _principal bundle_ $P\to M$ has fibers that are
right principal homogeneous spaces of the group $G$. A _right principal
homogeneous space_ is defined by the possession a free, transitive action of
$G$. Thus, any principal homogeneous space is diffeomorphic to the manifold
underlying the Lie group $G$ and, if any particular point is identified with
the unit element of $G$, the action of $G$ coincides with right-
multiplication. The fiber-wise right action of $G$ on $P$ allows us to
construct so-called _associated bundles_. If $F$ is a left $G$-space, with
action $\rho\colon G\to\operatorname{Aut}(F)$, then we define the
corresponding associated bundle, denoted sometimes $F_{\rho}$ or $F_{P}$, as
$P\times_{\rho}F\cong(P\times F)/G$, where the quotient identifies the points
$(pg,f)=(p,gf)$, $p\in P$, $f\in F$, $g\in G$. In particular, we can define
the associated bundles $G_{P}=P\times_{\mathrm{Ad}}G$ and
$\mathfrak{g}_{P}=P\times_{\mathrm{Ad}}\mathfrak{g}$, where $\mathrm{Ad}$
denotes respectively the adjoint action of the Lie group on itself and its Lie
algebra, $\mathrm{Ad}(b)a=bab^{-1}$ and $\mathrm{Ad}(b)\alpha=b\alpha b^{-1}$,
with $a,b\in G$ and $\alpha\in\mathfrak{g}$. When convenient and for
simplicity of notation, we shall implicitly treat Lie group and Lie algebra
elements as if they were faithfully represented as matrices.

The principal $G$-bundle $P\to M$ is called _flat_ when it is endowed with a
flat connection or a notion of flat parallel transport, which are compatible
with the structure group action. The details of these notions are discussed in
the next subsections. The arguments presented therein roughly establish the
following

######  Proposition 17.

Let $P\to M$ be a flat principal $G$-bundle and $\pi=\pi(M)$ be the
fundamental group of $M$. We can define the following structures associated to
it: (a) the sheaf $\mathcal{F}_{\mathfrak{g}}$ of locally flat sections of the
associated bundle $\mathfrak{g}_{P}\to M$, (b) the twisted de Rham complex
$(\Lambda^{\bullet}M\otimes\mathfrak{g}_{P},D)$, and (c) the monodromy
representation $\rho\colon\pi\to G$. Then the following cohomology groups
(respectively the sheaf, twisted de Rham and group cohomologies) are all
isomorphic, by reason of each being isomorphic to the space of equivalence
classes of infinitesimal deformations of the flat principal $G$-bundle
structure of $P\to M$:

$H^{1}(M,\mathcal{F}_{\mathfrak{g}})\cong
H^{1}(\Lambda^{\bullet}M\otimes\mathfrak{g}_{P},D)\cong
H^{1}(\pi,\mathrm{Ad}_{\rho}).$ (149)

We defer to the standard references [50, 57, 6] for detailed proofs.

###  D.1 Flat principle bundle cocycle

There are multiple ways to construct a principal $G$-bundle over a manifold
$M$. The one that will be important for us here defines also a bit more
structure than principal bundle itself, it also defines a flat connection
thereon. We shall refer to these structures as _flat principal $G$-bundles_.
It is well known that this data can be specified as follows. Let
$\mathcal{U}=(U_{i})$ be an open cover of $M$ and $(U,V)\mapsto t_{U,V}\in G$
an assignment of a structure group element to every ordered pair of opens
$U,V\in\mathcal{U}$. Each $t_{U,V}$ is called a _transition map_. The
transition maps define a principle $G$-bundle with a flat connection if they
satisfy the following _cocycle identities_ ,

$\displaystyle t_{U,V}t_{V,U}=\mathrm{id},$ (150) $\displaystyle
t_{U,V}t_{V,W}t_{W,U}=\mathrm{id}.$ (151)

A change of trivialization is an assignment $U\mapsto a_{U}\in G$ for every
open $U\in\mathcal{U}$. The modified transition functions
$t^{\prime}_{U,V}=a_{U}t_{U,V}a_{V}^{-1}$ define an equivalent flat principal
$G$-bundle.

Next, we describe infinitesimal deformations of a flat bundle cocycle
$t_{U,V}$. Namely, suppose that $t_{U,V}(s)$ is a smooth $1$-parameter family
of flat bundle cocycles, with $t_{U,V}(0)=t_{U,V}$. Let us denote the
derivative at $s=0$ as $\dot{t}_{U,V}=\tau_{U,V}t_{U,V}$, with
$\tau_{U,V}\in\mathfrak{g}$. Then, the defining relations (150) and (151)
impose the following constraints on the infinitesimal deformation
$\tau_{U,V}$:

$\displaystyle\tau_{U,V}=-t_{V,U}^{-1}\tau_{V,U}t_{V,U},$ (152)
$\displaystyle\tau_{U,V}+t_{U,V}\tau_{V,W}t_{U,V}^{-1}-\tau_{W,U}=0.$ (153)

On the other hand, suppose that $a_{U}(s)$ is a smooth $1$-parameter family of
trivialization changes, with $a_{U}(0)=\mathrm{id}$. Let us write the
derivative at $s=0$ as $\dot{a}_{U}=-\sigma_{U}$. The induced infinitesimal
deformation in the transition functions $t_{U,V}(s)=a_{U}(s)t_{U,V}a_{V}^{-1}$
is

$\tau_{U,V}=-\sigma_{U}+t_{U,V}\sigma_{V}t_{U,V}^{-1}.$ (154)

The point of the above calculations is to show that infinitesimal deformations
of the flat principal bundle cocycle, up to infinitesimal trivialization
changes, correspond precisely to the cohomology classes of a certain sheaf. To
complete the argument, we need only introduce the basic definitions of Čech
cohomology, which is known to compute the cohomology vector spaces of a
corresponding sheaf [16, 45]. We will take the sheaf to be
$\mathcal{F}_{\mathfrak{g}}$, where $\mathcal{F}_{\mathfrak{g}}(U)$ consists
of the locally flat sections of the bundle $\mathfrak{g}_{P}\to M$, associated
to the flat principal $G$-bundle $P\to M$. Let us now fix an open cover
$\mathcal{U}=(U_{i})$ of $M$ such that each $U_{i}$ is contractible and any
multiple intersection of the $U_{i}$ is also contractible. On a manifold, any
open cover can be refined to such a _good cover_ [15, Thm.5.1]. In particular,
the flat principal bundle cocycle can always be refined to a good cover. The
good cover hypothesis ensures that the Čech cohomology spaces are in fact
isomorphic to the actual sheaf cohomologies.

We define a Čech $q$-cochain $\sigma$ as an assignment
$(U_{i_{1}},\ldots,U_{i_{q+1}})\mapsto\sigma_{i_{1}\cdots
i_{q+1}}\in\mathcal{F}_{\mathfrak{g}}(U_{i}\cap\cdots\cap U_{i_{q+1}})$ to
every ordered $(q+1)$-tuple of opens from $\mathcal{U}$. By local flatness,
for any $U\in\mathcal{U}$,
$\mathcal{F}_{\mathfrak{g}}(U)\cong\bar{F}_{\mathfrak{g}}\cong\mathfrak{g}$
(cf. Section 3.1). It is convenient to think of a Čech cocycle
$\sigma_{i_{1}\cdots i_{q+1}}$ as taking values in
$\mathfrak{g}\cong\mathcal{F}_{\mathfrak{g}}(U_{i})$. This means that
$\sigma_{i\cdots}$ and $\sigma_{j\cdots}$ restrict to the same element of
$\mathcal{F}_{\mathfrak{g}}(U_{i}\cap U_{j}\cap\cdots)$ only if
$\sigma_{i\cdots}=\mathrm{Ad}(t_{U_{i},V_{i}})\sigma_{j\cdots}=(t_{U_{i},V_{j}})\sigma_{j\cdots}(t_{U_{i},V_{j}}^{-1})$.
We shall only need the Čech differential to be defined on $0$\- and
$1$-cochains:

$\displaystyle(\delta\sigma)_{ij}$ $\displaystyle=\sigma_{j}|_{U_{i}\cap
U_{j}}-\sigma_{i}|_{U_{i}\cap U_{j}}$
$\displaystyle=\mathrm{Ad}(t_{U_{i},U_{j}})\sigma_{j}-\sigma_{i}$
$\displaystyle=t_{U_{i},U_{j}}\sigma_{j}t_{U_{i},U_{j}}^{-1}-\sigma_{i},$
(155) $\displaystyle(\delta\tau)_{ijk}$ $\displaystyle=\tau_{jk}|_{U_{i}\cap
U_{j}\cap U_{k}}-\tau_{ik}|_{U_{i}\cap U_{j}\cap U_{k}}+\tau_{ij}|_{U_{i}\cap
U_{j}\cap U_{k}}$
$\displaystyle=\mathrm{Ad}(t_{U_{i},U_{j}})\tau_{jk}-\tau_{ik}+\tau_{ij}$
$\displaystyle=t_{U_{i},U_{j}}\tau_{jk}t_{U_{i},U_{j}}^{-1}-\tau_{ik}+\tau_{ij}.$
(156)

The space of closed Čech $q$-cocycles modulo the Čech coboundaries then is
isomorphic to the sheaf cohomology group in degree $q$, which in our case is
$H^{q}(\mathcal{F}_{\mathfrak{g}})$.

It should now be clear, from Equations (153) and (154), that the infinitesimal
deformation of the flat bundle cocycle defines a Čech $1$-cocycle
$\tau_{ij}=\tau_{U_{i},U_{j}}$ and an infinitesimal change in trivialization
defines a Čech coboundary $\tau_{ij}=(\delta\sigma)_{ij}$, with
$\sigma_{i}=\sigma_{U_{i}}$.

###  D.2 Flat connection on a principal bundle

Another, ultimately equivalent, way to specify a principal bundle with a flat
connection is as follows.

A _principal $G$-connection_ on a principal $G$-bundle $P$ is a
$\mathfrak{g}$-valued $1$-form $\omega$ on the total space $P$ (an element of
$\Omega^{1}(P)\otimes\mathfrak{g}$) such that (i) $\omega$ is
$\mathrm{Ad}$-equivariant ($R_{a}^{*}\omega=\mathrm{Ad}(a^{-1})\omega$, where
$R_{a}\colon P\to P$ is the action of $a\in G$ on $P$ by right multiplication)
and (ii) $\omega(\beta)=\beta$ for any vertical vector $\beta\in TP$. Recall
that vertical vectors are those annihilated by the tangent map of the
projection $P\to M$ and that the vertical subspace of $TP$ at any point of $P$
may be naturally identified with $\mathfrak{g}$, which we have used in the
preceding definition. The defining condition on the form $\omega$ is clearly
linear inhomogeneous. Thus, the space of all principal $G$-connections forms
an affine subspace of $\Omega^{1}(P)\otimes\mathfrak{g}$. So, the difference
$A=\omega^{\prime}-\omega$ between any two principal connections belongs to
the subspace of $\Omega^{1}(P)\otimes\mathfrak{g}$ that is
$\mathrm{Ad}$-equivariant and horizontal (annihilates vertical vectors). This
subspace is in fact isomorphic, by pullback along the projection $P\to
P/G\cong M$, to the space of sections
$\Gamma(\Lambda^{1}M\otimes\mathfrak{g}_{P})$ of the associated bundle
$\Lambda^{1}M\otimes\mathfrak{g}_{P}\to M$. In fact, we can identify the
spaces of sections $\Gamma(\Lambda^{p}M\otimes\mathfrak{g}_{P})$ with the
$\mathrm{Ad}$-equivariant, horizontal subspaces of
$\Omega^{p}(P)\otimes\mathfrak{g}$. The first order differential operator
$DA=\mathrm{d}A+[\omega\wedge A]$ (see below for notation) preserves these
subspaces and hence can be projected down to a first order differential
operator
$D\colon\Gamma(\Lambda^{p}M\otimes\mathfrak{g}_{P})\to\Gamma(\Lambda^{p+1}M\otimes\mathfrak{g}_{P})$,
which we shall refer to as the _twisted differential_ (cf. Section 2.4).

The curvature $\Omega$ of a principal $G$-connection $\omega$ is defined to be
the following $\mathfrak{g}$-valued $2$-form on $P$:

$\Omega=\mathrm{d}\omega+\frac{1}{2}[\omega\wedge\omega],$ (157)

where the bracketed wedge product is definite to satisfy
$[(\lambda\otimes\alpha)\wedge(\mu\otimes\beta)]=\lambda\wedge\mu\otimes[\alpha,\beta]$
for any $\lambda,\mu\in\Omega^{1}(P)$ and $\alpha,\beta\in\mathfrak{g}$. Since
$\Omega$ is $\mathrm{Ad}$-equivariant and horizontal, we can equally write
$\Omega\in\Gamma(\Lambda^{2}M\otimes\mathfrak{g}_{P})$. The twisted
differential $D$ is not nilpotent, $D^{2}\neq 0$. However, its square is
$C^{\infty}(M)$-linear and so is a differential operator of order $0$. In
fact, we can compute it to be

$D^{2}A=[\Omega\wedge A],$ (158)

for any $A\in\Gamma(\Lambda^{p}M\otimes\mathfrak{g}_{P})$. If $\Omega=0$, then
the connection is said to be _flat_. This is a sufficient condition for the
twisted differential to become nilpotent, $D^{2}=0$. A necessary and
sufficient condition would simply be that the curvature $\Omega$ takes values
in the center of $\mathfrak{g}$, upon local trivialization of
$\mathfrak{g}_{P}$.

Given any two flat connections $\omega$ and $\omega^{\prime}$, their
difference can be represented by a section
$\omega^{\prime}-\omega=A\in\Gamma(\Lambda^{1}M\otimes\mathfrak{g}_{P})$ (or
rather its pullback to $P$) that necessarily satisfies the following equation:

$\displaystyle 0$
$\displaystyle=\mathrm{d}\omega^{\prime}+\frac{1}{2}[\omega^{\prime}\wedge\omega^{\prime}]$
(159)
$\displaystyle=\mathrm{d}(\omega+A)+\frac{1}{2}[(\omega+A)\wedge(\omega+A)]$
(160) $\displaystyle=\mathrm{d}A+[\omega\wedge A]+\frac{1}{2}[A\wedge A]$
(161) $\displaystyle=DA+\frac{1}{2}[A\wedge A].$ (162)

Where the last expression can be interpreted as computed on $M$ rather than on
$P$. Equating this last expression to zero gives a differential equation on
sections $A\in\Gamma(\Lambda^{1}M\otimes\mathfrak{g}_{P})$ identifying those
that parametrize the space of flat principal $G$-connections (relative to
$\omega$, which defines the twisted differential $D$).

An automorphism of a principal $G$-bundle $P\to M$ is a bundle map $f\colon
P\to P$ that covers the identity on $M$ and is equivariant with respect to the
right action of $G$ [1, 6]. It is a standard fact that such maps can be
expressed as functions $a_{f}\colon P\to G$ that are $\mathrm{Ad}$-equivariant
(where $\mathrm{Ad}$ is the left action of $G$ on $G$ by conjugation) with
respect to the right action of $G$ on $P$. In turn, the set of such maps is in
bijection with the space of sections of the associated bundle
$G_{P}=P\times_{\mathrm{Ad}}G$. Since the map $f\colon P\to P$ is an
automorphism, the pullback connection $f^{*}\omega$ is considered equivalent
to the original one. Given its equivariance, the map $f$ corresponds to a
section $a_{f}\in\Gamma(G_{P})$. Equivalently, given a section
$a\in\Gamma(G_{P})$, we can define the corresponding automorphism map
$f_{a}\colon P\to P$. It is not hard to compute that

$f_{a}^{*}\omega=\mathrm{Ad}(a^{-1})\omega+a^{-1}\mathrm{d}a.$ (163)

Naturally, the pullback $f^{*}\omega$ of a flat connection $\omega$ is also a
flat connection.

Next, we describe infinitesimal deformations of a flat principle
$G$-connection. Given a flat connection described by a $\mathfrak{g}$-valued
$1$-form $\omega$ on $P$, any smooth $1$-parameter family of principal
connection can be written as $\omega+A(s)$, with $A(0)=0$, where $A(s)$ can,
for fixed $s$, be considered as a section of the associated bundle
$\Lambda^{1}M\otimes\mathfrak{g}_{P}\to M$. This family consists of flat
connections if and only if the equation $DA(s)+\frac{1}{2}[A(s)\wedge A(s)]=0$
is satisfied, where $D$ is the twisted differential defined by $\omega$. If
$\dot{A}(0)=A$, then the preceding identity imposes the condition $DA=0$ on
this infinitesimal deformation. Also, if $a(s)\in\Gamma(G_{P})$, with
$a(0)=\mathrm{id}$ and $\dot{a}(0)=\alpha\in\Gamma(\mathfrak{g}_{P})$, defines
a smooth $1$-parameter family of automorphisms $f_{a(s)}\colon P\to P$, then
the corresponding infinitesimal deformation of the original flat connection is
is equal to $A=\mathrm{d}\alpha-[\alpha,\omega]=D\alpha$.

It should now be clear that infinitesimal deformations of a given flat
principal $G$-connection, up to infinitesimal automorphisms of the underlying
principal $G$-bundle, are in bijections with the cohomology vector space
$H^{1}(\Lambda^{\bullet}\otimes\mathfrak{g}_{P},D)$ of the twisted de Rham
complex defined by the original flat connection.

###  D.3 Monodromy representation

A connection, in the sense of Ehresemann, can be defined as a splitting of the
tangent space of $P$ into $T_{x,a}P\cong T_{x}M\oplus\mathfrak{g}$, for
$(x,a)\in P$, with the $\mathfrak{g}$ summand canonically identified with the
subspace of vertical vectors, such that the splitting is smooth in $x$ and
equivariant in $a$. The $T_{x}M$ summand is called the _horizontal subspace_
of $T_{x,a}P$. This formulation leads naturally to the idea of parallel
transport. Given a point $(x,a)\in P$ and a smooth curve $\gamma\colon[0,1]\to
M$ such that $\gamma(0)=x$ and $\gamma(1)=1$, there exists a unique lift
$\tilde{\gamma}$ of $\gamma$ to $P$ such that $\tilde{\gamma}(0)=(x,a)$ and
the tangent vector $\dot{\tilde{\gamma}}$ is always horizontal. With $x$ and
$y$ fixed, the endpoint $(y,b)=\tilde{\gamma}(1)$ defines $b$ as the image of
$a$ parallel transported along $\gamma$. Since the splitting of $TP$ is
equivariant with respect to the right action of $G$ on $P$, so is parallel
transport. It is easy to see that parallel transport does not depend on the
parametrization of $\gamma$, is well defined also when $\gamma$ is piecewise
smooth, and respects concatenation, $a_{\gamma\eta}=a_{\gamma}a_{\eta}$ for
$\gamma(1)=\eta(0)$ and $\gamma\eta$ being the concatenated curve. In
particular, if $\gamma$ is a closed curve based at $x\in M$
($\gamma(0)=\gamma(1)=x$), then the effect of parallel transport is equivalent
to the right action on the fiber $P_{x}$ by some element $a_{\gamma}\in G$.

Let $\pi=\pi_{1}(M,x)$ be the fundamental group of $M$ based at some point
$x$. The connection splitting of $TP$ is called _flat_ when the parallel
transport along any closed contractible curve $\gamma$ is trivial,
$a_{\gamma}=\mathrm{id}$, and thus the group element $a_{\gamma}$ effecting
parallel transport along a closed curve $\gamma$ based at $x$ depends only on
its homotopy type $[\gamma]\in\pi$. In other words, parallel transport defines
a homomorphism $\rho\colon\pi\to G$, $\rho([\gamma])=a_{\gamma}$, which we
call the _monodromy representation_ of the fundamental group of $M$ in the
structure group of $P\to M$ (cf. the introduction to Section 4).

Thus, any flat principal $G$-bundle gives rise to a representation
$\rho\colon\pi\to G$. Two isomorphic flat principal bundles give rise to
equivalent monodromy representations, where two representations
$\rho^{\prime}$ and $\rho$ are equivalent if there exists an element $a\in G$
such that $\rho^{\prime}([\gamma])=a\rho([\gamma])a^{-1}$. Conversely, any
homomorphism $\rho\colon\pi\to G$ allows us to construct a flat principal
$G$-bundle with a monodromy representation equivalent to $\rho$. Namely,
consider the universal cover $\tilde{M}\to M$ as a principal $\pi$-bundle and
define the total space of the corresponding principal $G$-bundle as
$P=\tilde{M}\times_{\rho}G$. A flat connection can be defined on the trivial
principal $G$-bundle $\tilde{M}\times G$ using the construction of Section D.1
applied to an cover by contractible open sets and transition maps defined by
$\rho$. This flat connection then projects down to $P$.

Next, we describe infinitesimal deformations of a fixed monodromy
representation $\rho$. Let $\rho_{s}\colon\pi\to G$ be a smooth $1$-parameter
family of monodromy representations, with $\rho(s)=\rho$ and
$\dot{\rho}_{s}(a)=\tau(s)\rho(a)$ for some $\tau\colon\pi\to\mathfrak{g}$.
The representation property
$\rho_{s}([\gamma][\eta])=\rho_{s}([\gamma])\rho_{s}([\eta])$ imposes the
following constraint on the infinitesimal deformation:

$\tau([\gamma])+\rho([\gamma])\tau([\eta])\rho([\gamma])^{-1}-\tau([\gamma][\eta])=0.$
(164)

A family of trivial deformations is given by
$\rho_{s}([\gamma])=a_{s}\rho([\gamma])a_{s}^{-1}$ for a smooth $1$-parameter
family $a_{s}\in G$, with $a_{0}=\mathrm{id}$ and
$\dot{a}_{0}=-\sigma\in\mathfrak{g}$. The corresponding infinitesimal
deformation of the representation is given by

$\tau([\gamma])=-\sigma+\rho([\gamma])\sigma\rho([\gamma])^{-1}.$ (165)

The point of the above calculations is to show that these infinitesimal
deformations can be identified with certain _group cohomology_ classes. To see
that, we need to introduce some basic definitions [88, Ch.6], [91]. Group
cohomology is defined given a group and a representation thereof. We will give
the definitions by directly taking the group to be $\pi$ and the
representation to be the composite adjoint representation of $\pi$ on
$\mathfrak{g}$,
$\mathrm{Ad}_{\rho}=\mathrm{Ad}\circ\rho\colon\pi\to\mathrm{GL}(\mathfrak{g})$.
The vector space $C^{p}(\pi,\mathrm{Ad}_{\rho})$ of $p$-cochains consists of
functions $\sigma\colon\pi^{p}\to\mathfrak{g}$, where
$\pi^{p}=\pi\times\cdots\times\pi$ is the $p$-fold product. The cochain
differentials $\delta\colon C^{p}(\pi,\mathrm{Ad}_{\rho})\to
C^{p+1}(\pi,\mathrm{Ad}_{\rho})$ are defined by the formula

$\delta\sigma([\gamma_{1}],\ldots,[\gamma_{p+1}])=(-1)^{p+1}\sigma([\gamma_{1}],\ldots,[\gamma_{p}])\\\
{}+\mathrm{Ad}_{\rho}([\gamma_{1}])\sigma([\gamma_{2}],\ldots,[\gamma_{p+1}])\\\
+\sum_{q=1}^{p}(-1)^{q}\sigma(\ldots,[\gamma_{q}][\gamma_{q+1}],\ldots).$
(166)

For $0$\- and $1$-cochains, we have the following explicit formulas:

$\displaystyle\delta\sigma([\gamma])$
$\displaystyle=-\sigma+\mathrm{Ad}_{\rho}([\gamma])\sigma$
$\displaystyle=-\sigma+\rho([\gamma])\sigma\rho([\gamma])^{-1},$ (167)
$\displaystyle\delta\tau([\gamma],[\eta])$
$\displaystyle=\tau([\gamma])+\mathrm{Ad}_{\rho}([\gamma])\tau([\eta])-\tau([\gamma][\eta])$
$\displaystyle=\tau([\gamma])+\rho([\gamma])\tau([\eta])\rho([\gamma])^{-1}-\tau([\gamma][\eta]).$
(168)

It is worth noting that the degree-$0$ group cohomology is isomorphic to the
subspace of the representation on which the group acts trivially,
$H^{0}(\pi,\mathrm{Ad}_{\rho})\cong\mathfrak{g}^{\pi}$.

It should now be clear from Equations (164) and (165) that infinitesimal
deformations of a monodromy representations $\rho\colon\pi\to G$, up to
deformations by conjugation, are in bijection with the group cohomology
$H^{1}(\pi,\mathrm{Ad}_{\rho})$ of the group $\pi$ with coefficients in the
composite adjoint representation of $\pi$ on $\mathfrak{g}$.

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