$\displaystyle v_{5}$ $\displaystyle=-\frac{v_{2}\sin\left(\frac{\pi
l}{L}\right)}{18n^{3}}\bigg{(}(11n^{3}+61n)\sin\left(\frac{2\pi
l}{L}\right)+i(61n^{2}+11)\cos\left(\frac{2\pi l}{L}\right)$
$\displaystyle\phantom{=}+i(29n^{2}-11)\bigg{)}\,.$ (A.20)

Note that for the dual fields we basically have to take the composition of the
uniformization map with the inversion, i.e. we have to repeat the latter
arguments for $w(1/z)$. Let us denote the local coordinate change by
$\theta_{0_{k}}(\rho)$. It is given by

$\theta_{0_{k}}(\rho)=\eta_{0_{k}}(\rho)|_{l\to-l}\,,$ (A.21)

so that the respective local action is given by
$\Gamma_{w(1/z)}=\Gamma_{w(z)}\Gamma_{1/z}\equiv\Gamma_{k,-l}$ .

###  A.2 Code to compute correlation functions of descendants

We report here the Mathematica codes with which we implemented the recursive
formula discussed in 3.1.2. A working Mathematica notebook with the present
codes is also available in the ancillary files of the arXiv version of
[BB21f]. We use the Mathematica implementation of the Virasoro algebra by M.
Headrick that can be downloaded from
http://people.brandeis.edu/~headrick/Mathematica/.

####  A.2.1 Any $N$-point function of vacuum descendants

The Mathematica code to compute any correlator of descendants of the vacuum is

    VacNptFct[stat_] := Module[{states, ntrivial, TMP, tmp0, tmp1, tmp2},

      (*reorders the states s.t. the descendants with more Virasoro
      generators are mostleft. This makes the recursion faster:*)
      states = Sort[stat, Length[
        GetSequence[#1[[1]]]]<Length[GetSequence[#2[[1]]]] &];

      (*Checks at which position there are non-trivial descendants:*)
      ntrivial = Position[Sign[level[states[[All, 1]]]], 1] // Flatten;

      (*When there are no non-trivial descendants the function returns 1.
      For only one descendant it returns 0 due to translation invariance.
      In any other case it uses the recursion:*)
      Which[Length[ntrivial] == 0, 1, Length[ntrivial] == 1, 0, True,
       TMP = Table[
         RecStep[states[[ntrivial[[1]], 1]], states[[ntrivial[[i]], 1]],
          states[[ntrivial[[1]], 2]], states[[ntrivial[[i]], 2]]], {i, 2,
          Length[ntrivial]}];
       Sum[Sum[
         TMP[[j - 1, nn, 1]] VacNptFct[
           Table[Which[
             i == ntrivial[[1]], {CutFirst[states[[i, 1]]],
              states[[i, 2]]},
             i == ntrivial[[j]], {TMP[[j - 1, nn, 2]], states[[i, 2]]},
             True, states[[i]]], {i, Length[states]}]], {nn,
          Length[TMP[[j - 1]]]}], {j, 2, Length[ntrivial]}]
       ]
      ]

where we define the functions

    RecStep[ L[n1__], L[n2__], z1_, z2_] :=
     Module[{tmp1, test},
      tmp1 = List @@
         Expand[Sum[- Coeff[-{n1}[[1]], n, z1, z2] L[n - 1] ** L[n2] **
             vac, {n, 0, -Plus[n2] + 1}]] ** vac;
      tmp1 = ReArr[tmp1];
      test = 1;
      While[test == 1, test = 0;
       For[nnn = 1, nnn < Length[tmp1], nnn++,
        If[GetSequence[tmp1[[nnn, 2]]]==GetSequence[tmp1[[nnn + 1, 2]]],
          test = 1; tmp1[[nnn, 1]] = tmp1[[nnn, 1]] + tmp1[[nnn + 1, 1]];
         tmp1 = Delete[tmp1, nnn + 1]; Break;]]];
      tmp1 = Simplify[tmp1]
      ]

    ReArr[a_ L[m__]] := {a, L[m]}
    ReArr[ L[m__]] := {1, L[m]}
    ReArr[a_ ] := {a, 1}
    SetAttributes[ReArr, Listable];

    GetSequence[L[m__]] := {m}
    GetSequence[a] := {}
    SetAttributes[GetSequence, Listable];

    Coeff[m_, n_, zi_, zj_] :=
        (-1)^n Binomial[n + m - 2, n] (zj - zi)^(1 - m - n)

    CutFirst[ L[n__]] :=  L[{n}[[2 ;; Length[{n}]]] /. List -> Sequence]

The function `VacNptFct` takes as arguments a list of descendants together
with their coordinates. The descendants are given in the form
$\verb+L+[-n_{1},...,-n_{k}]$, where $n_{i}\geq n_{i+1}$,
$n_{i}\in\mathbb{N}$. The coordinates can either be variables or specific
values. For example

        VacNptFct[{{L[-2],z},{L[-2],w}}]

gives the result for the two-point function of the energy momentum tensor,
$\frac{c/2}{(z-w)^{4}}$.

####  A.2.2 Any $N$-point function of non-vacuum descendants

Given a correlator of descendants of non-vacuum primaries, we compute the
differential operator acting on the correlator of primaries with the function
`NPtFct`:

    NPtFct[stat_] :=
     Which[

      (* checks the input is given in the correct form :*)
      And @@ Table[Length[stat[[i]]] != 2, {i, Length[stat]}],
      "The number of fields and coordinates do not match!",

      (* If there is only one descendant then it returns 0 due to
         translational invariance *)
      Length[stat] == 1, 0,
      True, Module[{states, virpos, derivative, tmp, rec, noone, pr},

       (*reorders the states s.t.
         the descendants with more Virasoro generators are most left.
         This makes the recursion faster :*)
       states =
        Sort[stat,
         Length[GetSequence[#1[[1]]]] <= Length[GetSequence[#2[[1]]]] &];
       pr = FindPermutation[stat, states];
       virpos = Position[Sign[level[states[[All, 1]]]], 1] // Flatten;
       derivative =
        Table[Length[GetSequence[states[[virpos[[i]], 1]]]] ==
          level[states[[virpos[[i]], 1]]], {i, 1, Length[virpos]}];

       (*When there are no non-
         trivial descendants the function returns corrp[...].
         If the descendants are only level 1 descendants,
         it returns the appropriate derivatives acting on corrp[...].
         In any other case it uses the recursion :*)
       Which[
        Length[virpos] == 0, corrp[stat[[All, 2]] /. List -> Sequence],
        And @@ derivative,
        Derivative[level[stat[[All, 1]]] /. List -> Sequence][corrp][
         stat[[All, 2]] /. List -> Sequence],
        True, noone = Position[derivative, False] // Flatten;
        rec =
         Drop[Table[i, {i, 1, Length[states]}], {virpos[[noone[[1]]]]}];
          tmp =
         Table[RecStep[states[[virpos[[noone[[1]]]], 1]], states[[i, 1]],
           states[[virpos[[noone[[1]]]], 2]], states[[i, 2]]], {i, rec}];
        Sum[tmp[[i, j, 1]] NPtFct[
           Permute[
            ReplacePart[
             states, {{virpos[[noone[[1]]]], 1} ->
               CutFirst[states[[virpos[[noone[[1]]]], 1]]], {rec[[i]],
                1} -> tmp[[i, j, 2]]}], Ordering[PermutationList[pr]]]
           ], {i, 1, Length[tmp]}, {j, 1, Length[tmp[[i]]]}]]]]

where we define the functions:

    RecStep[ L[n1__] ** prim[p1_], L[n2__] ** prim[p2_], z1_, z2_] :=
     Module[{tmp1, test},
      tmp1 = List @@
        Expand[Sum[-Coeff[-{n1}[[1]],n,z1,z2] L[n-1]**L[n2]**prim[p2],
                {n, 0, -Plus[n2] + 1}]];
      tmp1 = ReArr[tmp1];
      test = 1;
      While[test == 1, test = 0;
       For[nnn = 1, nnn < Length[tmp1], nnn++,
        If[GetSequence[tmp1[[nnn, 2]]]==GetSequence[tmp1[[nnn + 1, 2]]],
          test = 1; tmp1[[nnn, 1]] = tmp1[[nnn, 1]] + tmp1[[nnn + 1, 1]];
         tmp1 = Delete[tmp1, nnn + 1]; Break;]]];
      tmp1 = Simplify[tmp1]
      ]

    RecStep[ L[n1__] ** prim[p1_], prim[p2_], z1_, z2_] :=
     Module[{tmp1, test},
      tmp1 = List @@
        Expand[-Sum[
           Coeff[-{n1}[[1]], n, z1, z2] L[n - 1]**prim[p2], {n, 0, 1}]];
      tmp1 = ReArr[tmp1];
      test = 1;
      While[test == 1, test = 0;
       For[nnn = 1, nnn < Length[tmp1], nnn++,
        If[GetSequence[tmp1[[nnn, 2]]]==GetSequence[tmp1[[nnn + 1, 2]]],
          test = 1; tmp1[[nnn, 1]] = tmp1[[nnn, 1]] + tmp1[[nnn + 1, 1]];
         tmp1 = Delete[tmp1, nnn + 1]; Break;]]];
      tmp1 = Simplify[tmp1]
      ]

    ReArr[a_ L[m__] ** prim[p_]] := {a, L[m] ** prim[p]}
    ReArr[L[m__] ** prim[p_]] := {1, L[m] ** prim[p]}
    ReArr[a_ prim[p_]] := {a, prim[p]}
    ReArr[prim[p_]] := {1, prim[p]}
    SetAttributes[ReArr, Listable];

    CutFirst[ L[n__] ** prim[p_]] :=
     L[{n}[[2 ;; Length[{n}]]] /. List -> Sequence] ** prim[p]

    GetSequence[L[m__] ** prim[p_]] := {m}
    GetSequence[prim[p_]] := {}
    SetAttributes[GetSequence, Listable];

The function `NPtFct` takes as arguments a list of $N$ lists, where in the
innermost lists the first entry is the descendant and the second entry is the
coordinate. The descendants are given as
$\verb|L[|-n_{1},...,-n_{k}\verb|]**prim[p]|$, where again $n_{i}\geq
n_{i+1}$, $n_{i}\in\mathbb{N}$ and `prim[p]` denotes the primary state. For
instance,

        tp = NPtFct[{ {L[-2] ** prim[p], z}, {L[-1, -1] ** prim[p], w} }]

produces the output

$\displaystyle\frac{6\,\text{corrp}[z,w]h[p]}{(w-z)^{4}}-\frac{2(1+2h[p])\text{corrp}^{(0,1)}[z,w]}{(w-z)^{3}}$
$\displaystyle+\frac{(2+h[p])\text{corrp}^{(0,2)}[z,w]}{(w-z)^{2}}+\frac{\text{corrp}^{(0,3)}[z,w]}{-w+z}$

where `h[p]` is the conformal dimension of `prim[p]` and the function `corrp`,
which is a function of the insertion points, denotes the correlator of
primaries. The derivatives acting on it are displayed in the Mathematica
language. If we know the explicit expression of `corrp`, we can further
simplify the output; in our example we can for instance write:

        corrp[z1_, z2_] := 1/(z1 - z2)^(2 h[p])
        tp // Simplify
        Clear[corrp]

to get the explicit result

$6(-w+z)^{-2(2+h[p])}h[p]\left(3+5h[p]+2h[p]^{2}\right)$

###  A.3 Explicit results

We collect here some more results in addition to the ones already presented in
chap. 3 for the Rényi entanglement entropy, sandwiched Rényi divergence and
trace square distance of (non-)vacuum descendants. Henceforth we write
$x=l/L$.

####  A.3.1 Rényi entanglement entropy

##### Vacuum module

The second Rényi entanglement entropy for $L_{-3}|{0}\rangle$,
$L_{-4}|{0}\rangle$, and $L_{-5}|{0}\rangle$ are:

$\displaystyle F^{(2)}_{L_{-3}|{0}\rangle}$
$\displaystyle=\frac{c^{2}\sin^{8}(\pi x)\cos^{4}(\pi x)}{64}$
$\displaystyle\phantom{=}+\frac{c\sin^{4}(\pi x)\cos^{2}(\pi x)(255\cos(2\pi
x)+90\cos(4\pi x)+17\cos(6\pi x)+1686)}{8192}$
$\displaystyle\phantom{=}+\frac{\sin^{4}(\pi x)(8391\cos(2\pi x)+1890\cos(4\pi
x)+361\cos(6\pi x)+7790)}{16384c}$
$\displaystyle\phantom{=}+\frac{1}{8388608}(3032808\cos(2\pi
x)+819919\cos(4\pi x)-27612\cos(6\pi x)$
$\displaystyle\phantom{=}+386\cos(8\pi x)+8436\cos(10\pi x)+289\cos(12\pi
x)+4554382)\,,$ (A.22)

$\displaystyle F^{(2)}_{L_{-4}|{0}\rangle}$
$\displaystyle=\frac{c^{2}\sin^{8}(\pi x)(3\cos(2\pi x)+2)^{4}}{6400}$
$\displaystyle\phantom{=}+\frac{c\sin^{4}(\pi x)(3\cos(2\pi
x)+2)^{2}}{6553600}(12760\cos(2\pi x)+7396\cos(4\pi x)$
$\displaystyle\phantom{=}+2152\cos(6\pi x))+1263\cos(8\pi x)+140269)$
$\displaystyle\phantom{=}+\frac{\sin^{4}(\pi x)}{6553600c}(5444642\cos(2\pi
x)+2684168\cos(4\pi x)+913973\cos(6\pi x)$
$\displaystyle\phantom{=}+286934\cos(8\pi x)+59049\cos(10\pi x))$
$\displaystyle\phantom{=}+\frac{1}{53687091200}(16641312784\cos(2\pi
x)+4954285000\cos(4\pi x)$ $\displaystyle\phantom{=}+1976400688\cos(6\pi
x)-121298020\cos(8\pi x)-5870960\cos(10\pi x)$
$\displaystyle\phantom{=}+13794296\cos(12\pi x)+18977328\cos(14\pi
x)+1595169\cos(16\pi x)$ $\displaystyle\phantom{=}+30207894915)\,,$ (A.23)

$\displaystyle F^{(2)}_{L_{-5}|{0}\rangle}$
$\displaystyle=\frac{1}{3435973836800}\big{[}c^{2}(-16777216\cos(2\pi
x)-51380224\cos(4\pi x)$ $\displaystyle\phantom{=}-79691776\cos(6\pi
x))-98566144\cos(8\pi x)+121634816\cos(10\pi x)$
$\displaystyle\phantom{=}+17825792\cos(12\pi x)+8388608\cos(14\pi
x)-8388608\cos(16\pi x)$ $\displaystyle\phantom{=}-33554432\cos(18\pi
x)+16777216\cos(20\pi x)+123731968)$
$\displaystyle\phantom{=}+c(-4076806144\cos(2\pi x)-9140649984\cos(4\pi x)$
$\displaystyle\phantom{=}-14113284096\cos(6\pi x)-18862759936\cos(8\pi x)$
$\displaystyle\phantom{=}+18117304320\cos(10\pi x)-27205632\cos(12\pi
x)+101367808\cos(14\pi x)$ $\displaystyle\phantom{=}+114972672\cos(16\pi
x)-28581888\cos(18\pi x)+74997760\cos(20\pi x)$
$\displaystyle\phantom{=}+27840645120)+c^{-1}(-19727178304\cos(2\pi
x)-27011932672\cos(4\pi x)$ $\displaystyle\phantom{=}-22303010688\cos(6\pi
x)-10523886336\cos(8\pi x)$ $\displaystyle\phantom{=}+8735760000\cos(10\pi
x)+2720936448\cos(12\pi x)$ $\displaystyle\phantom{=}+1016710944\cos(14\pi
x)+879348416\cos(16\pi x)+919004192\cos(18\pi x)$
$\displaystyle\phantom{=}+65294248000)+967144492584\cos(2\pi
x)+295129895330\cos(4\pi x)$ $\displaystyle\phantom{=}+135116995760\cos(6\pi
x)+61785824936\cos(8\pi x)$ $\displaystyle\phantom{=}-5774059280\cos(10\pi
x)-260030763\cos(12\pi x)+179443820\cos(14\pi x)$
$\displaystyle\phantom{=}+597122990\cos(16\pi x)+506168268\cos(18\pi
x)+83814025\cos(20\pi x)$ $\displaystyle\phantom{=}+1981464169130\big{]}\,.$
(A.24)

##### Ising model

Up to level 3 descendants of the energy density operator we find the following
results for the $n=2$ Rényi entanglement entropy:

$\displaystyle F^{(2)}_{L_{-1}|{\varepsilon}\rangle}$
$\displaystyle=\frac{(\cos(2\pi x)+7)(1558+439\cos(2\pi x)+26\cos(4\pi
x)+25\cos(6\pi x))}{16384}\,,$ (A.25) $\displaystyle
F^{(2)}_{L_{-2}|{\varepsilon}\rangle}$ $\displaystyle=\frac{\cos(2\pi
x)+7}{67108864}(6085442+1693410\cos(2\pi x)+514952\cos(4\pi x)$
$\displaystyle\phantom{=}+49813\cos(6\pi x)+9270\cos(8\pi x)+35721\cos(10\pi
x))$ (A.26) $\displaystyle=F^{(2)}_{L_{-1}^{2}|{\varepsilon}\rangle}\,,$
$\displaystyle F^{(2)}_{L_{-3}|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(2\pi
x)+7}{17179869184}(1523423468+432147835\cos(2\pi x)+111740030\cos(4\pi x)$
$\displaystyle\phantom{=}+65921129\cos(6\pi x)+7438836\cos(8\pi
x)+1584475\cos(10\pi x)$ $\displaystyle\phantom{=}+626850\cos(12\pi
x)+4601025\cos(14\pi x))$ (A.27)
$\displaystyle=F^{(2)}_{L_{-1}^{3}|{\varepsilon}\rangle}=F^{(2)}_{L_{-2}L_{-1}|{\varepsilon}\rangle}\,,$

where the common prefactor is due to the factorization of the holomorfic and
antiholomorfic parts of the correlator. Even though
$\mathcal{D}^{F(2)}_{L_{-1}^{2}}\neq\mathcal{D}^{F(2)}_{L_{-2}}$, at level 2
we find the same entanglement entropy for the different descendants and the
same happens at level 3. This reflects the existence of only one physical
state at level 2 and 3.

For $\sigma$ descendants:

$\displaystyle F^{(2)}_{L_{-1}|{\sigma}\rangle}$
$\displaystyle=\frac{435+60\cos(2\pi x)+17\cos(4\pi x)}{512}\,,$ (A.28)
$\displaystyle F^{(2)}_{L_{-2}|{\sigma}\rangle}$
$\displaystyle=\frac{1}{2097152}(1560707+438088\cos(2\pi x)+75420\cos(4\pi x)$
$\displaystyle\phantom{=}+8312\cos(6\pi x)+14625\cos(8\pi x))$ (A.29)
$\displaystyle=F^{(2)}_{L_{-1}^{2}|{\sigma}\rangle}\,,$ $\displaystyle
F^{(2)}_{L_{-3}|{\sigma}\rangle}$
$\displaystyle=\frac{1}{63438848}(42511910+16535144\cos(2\pi
x)+2825131\cos(4\pi x)$ $\displaystyle\phantom{=}+1123684\cos(6\pi
x)+179114\cos(8\pi x)+141364\cos(10\pi x)$
$\displaystyle\phantom{=}+122501\cos(12\pi x))\,,$ (A.30) $\displaystyle
F^{(2)}_{L_{-1}^{3}|{\sigma}\rangle}$
$\displaystyle=\frac{1}{5585604968448}(3968670881070+1175831066472\cos(2\pi
x)$ $\displaystyle\phantom{=}+306581016863\cos(4\pi x)+102222772068\cos(6\pi
x)$ $\displaystyle\phantom{=}+11235770850\cos(8\pi x)+5235592500\cos(10\pi x)$
$\displaystyle\phantom{=}+15827868625\cos(12\pi x))\,,$ (A.31) $\displaystyle
F^{(2)}_{L_{-2}L_{-1}|{\sigma}\rangle}$
$\displaystyle=\frac{1}{173946175488}(120071187054+40660967528\cos(2\pi x)$
$\displaystyle\phantom{=}+9353937343\cos(4\pi x)+2427785700\cos(6\pi x)$
$\displaystyle\phantom{=}+860494498\cos(8\pi x)+152046004\cos(10\pi
x)+419757361\cos(12\pi x))\,.$ (A.32)

In this case we have one physical state at level 2, while two physical states
at level 3 and we thus find different expressions for the REEs for degenerate
states at level 3.

##### Three-state Potts model

For the first descendant of the energy density in the three-states Potts model
we find:

$\displaystyle F^{(2)}_{L_{-1}|{\varepsilon}\rangle}$
$\displaystyle=\tfrac{1}{64}\,F\\!\left(-\tfrac{8}{5},-\tfrac{1}{5};-\tfrac{2}{5};\eta\right){}^{2}(\cos(2\pi
x)+7)^{2}$
$\displaystyle\phantom{=}+\tfrac{1}{166400}\big{\\{}F\left(-\tfrac{8}{5},-\tfrac{1}{5};-\tfrac{2}{5};\eta\right)\sin^{4}(\pi
x)\big{[}9\sin(\pi
x)\left(49F\\!\left(\tfrac{12}{5},\tfrac{19}{5};\tfrac{18}{5};\eta\right)\sin^{3}(\pi
x)\right.$
$\displaystyle\phantom{=}\left.+260F\\!\left(\tfrac{7}{5},\tfrac{14}{5};\tfrac{13}{5};\eta\right)\sin(2\pi
x)\right)+5200F\\!\left(-\tfrac{3}{5},\tfrac{4}{5};\tfrac{3}{5};\eta\right)\cos(\pi
x)$
$\displaystyle\phantom{=}+520F\\!\left(\tfrac{2}{5},\tfrac{9}{5};\tfrac{8}{5};\eta\right)(31\cos(2\pi
x)+41)\big{]}\big{\\}}$ $\displaystyle\phantom{=}-\left(111411200\
2^{2/5}\Gamma\left(-\tfrac{8}{5}\right)\Gamma\left(\tfrac{17}{10}\right)\Gamma\left(\tfrac{12}{5}\right)\right)^{-1}$
$\displaystyle\phantom{=}\times\bigg{\\{}\Gamma\left(-\tfrac{2}{5}\right)\Gamma\left(\tfrac{3}{10}\right)\Gamma\left(\tfrac{13}{5}\right)F\\!\left(\tfrac{6}{5},\tfrac{13}{5};\tfrac{12}{5};\eta\right)\sin^{\tfrac{28}{5}}(\pi
x)$
$\displaystyle\phantom{=}\times\big{[}34F\\!\left(\tfrac{6}{5},\tfrac{13}{5};\tfrac{12}{5};\eta\right)(14196\cos(2\pi
x)+15129\cos(4\pi x)+7667)$ $\displaystyle\phantom{=}+13\sin^{2}(\pi
x)\big{(}9016F\\!\left(\tfrac{26}{5},\tfrac{33}{5};\tfrac{32}{5};\eta\right)\sin^{6}(\pi
x)$
$\displaystyle\phantom{=}+17F\\!\left(\tfrac{11}{5},\tfrac{18}{5};\tfrac{17}{5};\eta\right)(8977\cos(\pi
x)+6479\cos(3\pi x))$
$\displaystyle\phantom{=}+79488F\\!\left(\tfrac{21}{5},\tfrac{28}{5};\tfrac{27}{5};\eta\right)\sin^{4}(\pi
x)\cos(\pi x)$
$\displaystyle\phantom{=}+99F\\!\left(\tfrac{16}{5},\tfrac{23}{5};\tfrac{22}{5};\eta\right)\sin^{2}(\pi
x)(1373\cos(2\pi x)+1003)\big{)}\big{]}\bigg{\\}}\,,$ (A.33)

where $F\equiv\;_{2}F_{1}$ is the hypergeometric function and
$\eta=\sin^{2}\left(\frac{\pi x}{2}\right)$. For higher-level descendants the
expressions are more involved, and we limit ourselves to show this simplest
example.

####  A.3.2 Sandwiched Rényi divergence

##### Vacuum module

Some explicit expressions for the SRD between the vaccum and light states are:

$\displaystyle\mathcal{F}^{(2)}_{L_{-3}|{0}\rangle}$
$\displaystyle=\frac{e^{12i\pi x}}{2048\left(1+e^{2i\pi
x}\right)^{12}}\big{[}c^{-1}\big{(}-202752\cos(2\pi x)-3775424\cos(4\pi x)$
$\displaystyle\phantom{=}+356352\cos(6\pi x)+888064\cos(8\pi
x)-184320\cos(10\pi x)-137\cos(12\pi x)$
$\displaystyle\phantom{=}+27648\cos(14\pi x)+14272\cos(16\pi x)+3072\cos(18\pi
x)+288\cos(20\pi x)$
$\displaystyle\phantom{=}+2874176\big{)}-86237760\cos(2\pi
x)+49130280\cos(4\pi x)$ $\displaystyle\phantom{=}-14301120\cos(6\pi
x)+2900567\cos(8\pi x)-122592\cos(10\pi x)$
$\displaystyle\phantom{=}+12228\cos(12\pi x)-1568\cos(14\pi x)-2062\cos(16\pi
x)-416\cos(18\pi x)$ $\displaystyle\phantom{=}+276\cos(20\pi x)+160\cos(22\pi
x)+25\cos(24\pi x)+57010590\big{]}\,,$ (A.34)

$\displaystyle\mathcal{F}^{(2)}_{L_{-4}|{0}\rangle}$
$\displaystyle=\frac{e^{16i\pi x}}{163840\left(1+e^{2i\pi
x}\right)^{16}}\big{[}c^{-1}\big{(}-2183086080\cos(2\pi
x)-14065619072\cos(4\pi x)$ $\displaystyle\phantom{=}+3329505280\cos(6\pi
x)+3373021952\cos(8\pi x)-1144576000\cos(10\pi x)$
$\displaystyle\phantom{=}-7513472\cos(12\pi x)-26378240\cos(14\pi
x)+22635008\cos(16\pi x)$ $\displaystyle\phantom{=}+21217280\cos(18\pi
x)+10149760\cos(20\pi x)+3225600\cos(22\pi x)$
$\displaystyle\phantom{=}+693504\cos(24\pi x)+92160\cos(26\pi
x)+5760\cos(28\pi x)+10666626560\big{)}$
$\displaystyle\phantom{=}-1927558100400\cos(2\pi x)+1107347224880\cos(4\pi x)$
$\displaystyle\phantom{=}-420523178000\cos(6\pi x)+102577128040\cos(8\pi x)$
$\displaystyle\phantom{=}-14129186800\cos(10\pi x)+1083586960\cos(12\pi
x)-18763600\cos(14\pi x)$ $\displaystyle\phantom{=}+2105260\cos(16\pi
x)-2374000\cos(18\pi x)-2296400\cos(20\pi x)$
$\displaystyle\phantom{=}-722000\cos(22\pi x)+205400\cos(24\pi
x)+286800\cos(26\pi x)$ $\displaystyle\phantom{=}+120720\cos(28\pi
x)+25200\cos(30\pi x)+2205\cos(32\pi x)$
$\displaystyle\phantom{=}+1161961353975\big{]}\,.$ (A.35)

##### Ising model

We present here some of the correlation functions related to the SRD
computation for non-vacuum descendants. For simplicity we show only the
results for $\varepsilon$ descendants in the Ising model:

$\displaystyle\mathcal{F}^{(2)}_{L_{-1}|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(4\pi x)+7}{4096\cos^{8}(\pi x)}(954-776\cos(2\pi
x)+319\cos(4\pi x)+4\cos(6\pi x)$ $\displaystyle\phantom{=}+6\cos(8\pi
x)+4\cos(10\pi x)+\cos(12\pi x))\,,$ (A.36)
$\displaystyle\mathcal{F}^{(2)}_{L_{-2}|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(4\pi x)+7}{1048576\cos^{12}(\pi
x)}(1546754-2155216\cos(2\pi x)+864034\cos(4\pi x)$
$\displaystyle\phantom{=}-139296\cos(6\pi x)+13320\cos(8\pi x)+224\cos(10\pi
x)+469\cos(12\pi x)$ $\displaystyle\phantom{=}+456\cos(14\pi x)+246\cos(16\pi
x)+72\cos(18\pi x)+9\cos(20\pi x))$ (A.37)
$\displaystyle=\mathcal{F}^{(2)}_{L_{-1}^{2}|{\varepsilon}\rangle}\,,$
$\displaystyle\mathcal{F}^{(2)}_{L_{-3}|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(4\pi x)+7}{67108864\cos^{16}(\pi
x)}(938450676-1469899184\cos(2\pi x)$
$\displaystyle\phantom{=}+710758371\cos(4\pi x)-201143980\cos(6\pi
x)+32581122\cos(8\pi x)$ $\displaystyle\phantom{=}-2510220\cos(10\pi
x)+99537\cos(12\pi x)+5080\cos(14\pi x)$
$\displaystyle\phantom{=}+13484\cos(16\pi x)+15448\cos(18\pi
x)+11059\cos(20\pi x)+5260\cos(22\pi x)$
$\displaystyle\phantom{=}+1630\cos(24\pi x)+300\cos(26\pi x)+25\cos(28\pi x))$
(A.38)
$\displaystyle=\mathcal{F}^{(2)}_{L_{-1}^{3}|{\varepsilon}\rangle}=\mathcal{F}^{(2)}_{L_{-2}L_{-1}|{\varepsilon}\rangle}\,.$

####  A.3.3 Trace square distance

##### Vacuum module

Here are some explicit expressions for the TSD between light states:

$\displaystyle T^{(2)}_{L_{-3}|{0}\rangle,|{0}\rangle}$
$\displaystyle=\frac{1}{64}c^{2}\sin^{8}(\pi x)\cos^{4}(\pi x)$
$\displaystyle\phantom{=}-\frac{c\sin^{6}(\pi x)\cos^{2}(\pi x)(124\cos(2\pi
x)+17\cos(4\pi x)+243)}{2048}$ $\displaystyle\phantom{=}+\frac{\sin^{4}(\pi
x)(8391\cos(2\pi x)+1890\cos(4\pi x)+361\cos(6\pi x)+7790)}{16384c}$
$\displaystyle\phantom{=}+\frac{1}{8388608}(-7864320\cos(\pi
x)+1984232\cos(2\pi x)-524288\cos(3\pi x)$
$\displaystyle\phantom{=}+295631\cos(4\pi x)-27612\cos(6\pi x)+386\cos(8\pi
x)+8436\cos(10\pi x)$ $\displaystyle\phantom{=}+289\cos(12\pi x)+6127246)\,,$
(A.39) $\displaystyle T^{(2)}_{L_{-4}|{0}\rangle,|{0}\rangle}$
$\displaystyle=\frac{c^{2}\sin^{8}(\pi x)(3\cos(2\pi x)+2)^{4}}{6400}$
$\displaystyle\phantom{=}-\frac{c\sin^{6}(\pi x)(3\cos(2\pi
x)+2)^{2}}{1638400}(15489\cos(2\pi x)+4678\cos(4\pi x)$
$\displaystyle\phantom{=}+1263\cos(6\pi x)+19530)$
$\displaystyle\phantom{=}+\frac{\sin^{4}(\pi x)}{6553600c}(5444642\cos(2\pi
x)+2684168\cos(4\pi x)+913973\cos(6\pi x)$
$\displaystyle\phantom{=}+286934\cos(8\pi x)+59049\cos(10\pi x)+3718434)$
$\displaystyle\phantom{=}+\frac{1}{53687091200}(-48486154240\cos(\pi
x)+12614780944\cos(2\pi x)$ $\displaystyle\phantom{=}-4445962240\cos(3\pi
x)+2269930440\cos(4\pi x)-754974720\cos(5\pi x)$
$\displaystyle\phantom{=}+634223408\cos(6\pi x)-121298020\cos(8\pi
x)-5870960\cos(10\pi x)$ $\displaystyle\phantom{=}+13794296\cos(12\pi
x)+18977328\cos(14\pi x)+1595169\cos(16\pi x)$
$\displaystyle\phantom{=}+38260958595)\,.$ (A.40)

##### Ising model

Here are some results for $\varepsilon$ descendants in the Ising models:

$\displaystyle T^{(2)}_{L_{-1}|{\varepsilon}\rangle,|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(2\pi x)+7}{1024}\sin^{4}\left(\frac{\pi
x}{2}\right)(11+92\cos(\pi x)+148\cos(2\pi x)$
$\displaystyle\phantom{=}+100\cos(3\pi x)+25\cos(4\pi x))\,,$ (A.41)
$\displaystyle T^{(2)}_{L_{-2}|{\varepsilon}\rangle,|{\varepsilon}\rangle}$
$\displaystyle=\frac{\cos(2\pi x)+7}{4194304}\sin^{4}\left(\frac{\pi
x}{2}\right)(2347723+2430412\cos(\pi x)$
$\displaystyle\phantom{=}+1872304\cos(2\pi x)+1393796\cos(3\pi
x)+1144940\cos(4\pi x)$ $\displaystyle\phantom{=}+751500\cos(5\pi
x)+366480\cos(6\pi x)+142884\cos(7\pi x)$
$\displaystyle\phantom{=}+35721\cos(8\pi x))$ (A.42)
$\displaystyle=T^{(2)}_{L_{-1}^{2}|{\varepsilon}\rangle,|{\varepsilon}\rangle}\,,$
$\displaystyle T^{(2)}_{L_{-3}|{\varepsilon}\rangle,|{\varepsilon}\rangle}$
$\displaystyle=\frac{(\cos(2\pi x)+7)}{1073741824}\sin^{4}\left(\frac{\pi
x}{2}\right)(912429118+1218353112\cos(\pi x)$
$\displaystyle\phantom{=}+998414040\cos(2\pi x)+780711528\cos(3\pi
x)+629316847\cos(4\pi x)$ $\displaystyle\phantom{=}+497152212\cos(5\pi
x)+393829628\cos(6\pi x)+276532300\cos(7\pi x)$
$\displaystyle\phantom{=}+168888850\cos(8\pi x)+94527900\cos(9\pi
x)+46637100\cos(10\pi x)$ $\displaystyle\phantom{=}+18404100\cos(11\pi
x)+4601025\cos(12\pi x))$ (A.43)
$\displaystyle=T^{(2)}_{L_{-1}^{3}|{\varepsilon}\rangle,|{\varepsilon}\rangle}=T^{(2)}_{L_{-2}L_{-1}|{\varepsilon}\rangle,|{\varepsilon}\rangle}\,.$

For $\sigma$ descendants:

$\displaystyle T^{(2)}_{L_{-1}|{\sigma}\rangle,|{\sigma}\rangle}$
$\displaystyle=-\frac{\cos^{2}\left(\frac{\pi
x}{2}\right)\sqrt{\csc\left(\frac{\pi
x}{2}\right)}}{32768\sin^{\frac{7}{2}}\left(\frac{\pi x}{2}\right)(\cos(\pi
x)+1)^{5/4}}\Bigg{\\{}66752\ 2^{3/4}\sqrt{\sin^{7}\left(\frac{\pi
x}{2}\right)\sin(\pi x)}$ $\displaystyle\phantom{=}+\sqrt[4]{\cos(\pi
x)+1}\big{[}46488\cos(\pi x)-16137\cos(2\pi x)+3612\cos(3\pi x)$
$\displaystyle\phantom{=}-1006\cos(4\pi x)+76\cos(5\pi x)-7\cos(6\pi
x)\big{]}-33026\sqrt[4]{\cos(\pi x)+1}$
$\displaystyle\phantom{=}-16\sqrt[4]{2}\sin^{4}\left(\frac{\pi
x}{2}\right)\sqrt{\cos\left(\frac{\pi x}{2}\right)}\big{[}-3888\cos(\pi
x)-804\cos(2\pi x)$ $\displaystyle\phantom{=}+48\cos(3\pi x)+129\cos(4\pi
x)+7099\big{]}\Bigg{\\}}\,,$ (A.44) $\displaystyle
T^{(2)}_{L_{-2}|{\sigma}\rangle,|{\sigma}\rangle}$
$\displaystyle=\left(33554432\ 2^{3/4}\sin^{\frac{5}{4}}(\pi
x)\sqrt{\csc\left(\frac{\pi
x}{2}\right)}\right)^{-1}\Bigg{\\{}2\big{[}51995040\cos(\pi x)$
$\displaystyle\phantom{=}+6726368\cos(2\pi x)-50083264\cos(3\pi
x)-27705396\cos(4\pi x)$ $\displaystyle\phantom{=}-23115584\cos(5\pi
x)+3517116\cos(6\pi x)-1600624\cos(7\pi x)$
$\displaystyle\phantom{=}+688029\cos(8\pi x)-18480\cos(9\pi x)+36\cos(10\pi
x)+93860567\big{]}$ $\displaystyle\phantom{=}\times\sqrt[4]{\sin^{3}(\pi
x)(\cos(\pi x)+1)}+\big{[}-124036846\sin(\pi x)$
$\displaystyle\phantom{=}-131438432\sin(2\pi x)-31447124\sin(3\pi
x)+24870528\sin(4\pi x)$ $\displaystyle\phantom{=}+31182128\sin(5\pi
x)+21514960\sin(6\pi x)-2930095\sin(7\pi x)$
$\displaystyle\phantom{=}+1582144\sin(8\pi x)-453993\sin(9\pi
x)+18480\sin(10\pi x)$ $\displaystyle\phantom{=}-36\sin(11\pi
x)\big{]}\sqrt[4]{\cot\left(\frac{\pi x}{2}\right)}\Bigg{\\}}$ (A.45)
$\displaystyle=T^{(2)}_{L_{-1}^{2}|{\sigma}\rangle,|{\sigma}\rangle}\,,$
$\displaystyle T^{(2)}_{L_{-3}|{\sigma}\rangle,|{\sigma}\rangle}$
$\displaystyle=\left(64961380352\ 2^{3/4}\sin^{\frac{5}{4}}(\pi
x)\sqrt{\csc\left(\frac{\pi x}{2}\right)}\right)^{-1}$
$\displaystyle\phantom{=}\times\Bigg{\\{}\big{[}-804063279604\sin(\pi
x)-679262872576\sin(2\pi x)$ $\displaystyle\phantom{=}-676379752341\sin(3\pi
x)-151873435008\sin(4\pi x)$ $\displaystyle\phantom{=}+78179526785\sin(5\pi
x)+162043727616\sin(6\pi x)$ $\displaystyle\phantom{=}+203623756022\sin(7\pi
x)+116161263104\sin(8\pi x)$ $\displaystyle\phantom{=}+68995216314\sin(9\pi
x)+27584342272\sin(10\pi x)$ $\displaystyle\phantom{=}-1884177607\sin(11\pi
x)+2161088640\sin(12\pi x)$ $\displaystyle\phantom{=}-1097984829\sin(13\pi
x)+30000(1568\sin(14\pi x)$ $\displaystyle\phantom{=}+93\sin(15\pi
x))\big{]}\sqrt[4]{\cot\left(\frac{\pi
x}{2}\right)}-2\big{[}-463898496256\cos(\pi x)$
$\displaystyle\phantom{=}-339771582456\cos(2\pi x)+161014585088\cos(3\pi x)$
$\displaystyle\phantom{=}+347509883805\cos(4\pi x)+308827933824\cos(5\pi x)$
$\displaystyle\phantom{=}+269411694364\cos(6\pi x)+145953734016\cos(7\pi x)$
$\displaystyle\phantom{=}+65832431142\cos(8\pi x)+29792470912\cos(9\pi x)$
$\displaystyle\phantom{=}-3124129172\cos(10\pi x)+2208128640\cos(11\pi x)$
$\displaystyle\phantom{=}-1220635853\cos(12\pi x)+47040000\cos(13\pi
x)+2790000\cos(14\pi x)$
$\displaystyle\phantom{=}-619071617270\big{]}\sqrt[4]{\sin^{3}(\pi x)(\cos(\pi
x)+1)}\Bigg{\\}}\,.$ (A.46)

For degenerate states at level 3 the expressions are different, but we report
here only one for simplicity.

##### Three-states Potts model

We conclude this appendix with the TSD between the first-level descendant of
$\varepsilon$ and the primary state itself in the Potts model:

$\displaystyle T^{(2)}_{L_{-1}|{\varepsilon}\rangle,|{\varepsilon}\rangle}$
$\displaystyle=\frac{1}{8}F\\!\left(-\tfrac{8}{5},-\tfrac{1}{5};-\tfrac{2}{5};\eta\right){}^{2}\sin^{4}\\!\left(\frac{\pi
x}{2}\right)(4\cos(\pi x)+\cos(2\pi x)+19)$
$\displaystyle\phantom{=}+\frac{1}{5200}\Bigg{\\{}(F\\!\left(-\tfrac{8}{5},-\tfrac{1}{5};-\tfrac{2}{5};\eta\right)\sin^{10}\\!\left(\frac{\pi
x}{2}\right)\cos^{8}\\!\left(\frac{\pi x}{2}\right)\csc^{6}(\pi x)$
$\displaystyle\phantom{=}\times\bigg{[}(41600F\\!\left(-\tfrac{3}{5},\tfrac{4}{5};\tfrac{3}{5};\eta\right)(2\cos(\pi
x)+\cos(2\pi x)-15)+32\cos^{2}\\!\left(\frac{\pi x}{2}\right)$
$\displaystyle\phantom{=}\times\bigg{(}9\sin(\pi
x)\left(49F\\!\left(\tfrac{12}{5},\tfrac{19}{5};\tfrac{18}{5};\eta\right)\sin^{3}(\pi
x)+260F\\!\left(\tfrac{7}{5},\tfrac{14}{5};\tfrac{13}{5};\eta\right)\sin(2\pi
x)\right)$
$\displaystyle\phantom{=}+520F\\!\left(\tfrac{2}{5},\tfrac{9}{5};\tfrac{8}{5};\eta\right)(31\cos(2\pi
x)+9)\bigg{)}\bigg{]}\Bigg{\\}}$ $\displaystyle\phantom{=}-\left(1782579200\
2^{2/5}\Gamma\left(-\frac{8}{5}\right)\Gamma\left(\frac{17}{10}\right)\Gamma\left(\frac{12}{5}\right)\right)^{-1}$
$\displaystyle\phantom{=}\times\Bigg{\\{}\Gamma\left(-\frac{2}{5}\right)\Gamma\left(\frac{3}{10}\right)\Gamma\left(\frac{13}{5}\right)F\\!\left(\tfrac{6}{5},\tfrac{13}{5};\tfrac{12}{5};\eta\right)\sin^{\frac{28}{5}}(\pi
x)$
$\displaystyle\phantom{=}\times\bigg{[}544F\\!\left(\tfrac{6}{5},\tfrac{13}{5};\tfrac{12}{5};\eta\right)(9600\cos(\pi
x)-17164\cos(2\pi x)+15129\cos(4\pi x)$
$\displaystyle\phantom{=}-1293)+208\sin^{2}(\pi
x)\bigg{(}9016F\\!\left(\tfrac{26}{5},\tfrac{33}{5};\tfrac{32}{5};\eta\right)\sin^{6}(\pi
x)+17F\\!\left(\tfrac{11}{5},\tfrac{18}{5};\tfrac{17}{5};\eta\right)$
$\displaystyle\phantom{=}\times(3697\cos(\pi x)+6479\cos(3\pi
x)+800)+79488F\\!\left(\tfrac{21}{5},\tfrac{28}{5};\tfrac{27}{5};\eta\right)\sin^{4}(\pi
x)$ $\displaystyle\phantom{=}\times\cos(\pi
x)+99F\\!\left(\tfrac{16}{5},\tfrac{23}{5};\tfrac{22}{5};\eta\right)\sin^{2}(\pi
x)(1373\cos(2\pi x)+843)\bigg{)}\bigg{]}\Bigg{\\}}\,,$ (A.47)

where $F\equiv\;_{2}F_{1}$ is the hypergeometric function and
$\eta=\sin^{2}\left(\frac{\pi x}{2}\right)$. We computed also TSDs for higher-
level descendants, but the expressions are more complicated and we won’t show
them here.

##  Appendix B Conventions and further details on the fermionic models

###  B.1 Conventions on fermions and Dirac matrices

We collect here our conventions on Dirac matrices and Dirac and Weyl fermions.
They are employed in chap. 4 and were also used in [BB19l, BB19m, BB20f].

In a four-dimensional Minkowski spacetime we use a flat metric $\eta_{ab}$.
The Dirac matrices $\gamma^{a}$ satisfy

$\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}\,,$ (B.1)

and the conjugate Dirac spinor $\overline{\psi}$ is defined using
$\beta=i\gamma^{0}$ by

$\overline{\psi}=\psi^{\dagger}\beta\;.$ (B.2)

The hermitian chiral matrix $\gamma^{5}$ is given by

$\gamma^{5}=-i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\,,$ (B.3)

and used to define the chiral projectors

$P_{L}=\frac{\mathbb{I}+\gamma_{5}}{2}\,,\qquad
P_{R}=\frac{\mathbb{I}-\gamma_{5}}{2}\,,$ (B.4)

that split a Dirac spinor $\psi$ into its left and right Weyl components

$\psi=\lambda+\rho\;,\qquad\lambda=P_{L}\psi\;,\quad\rho=P_{R}\psi\;.$ (B.5)

The charge conjugation matrix $C$ satisfies

$C\gamma^{a}C^{-1}=-\gamma^{aT}\;,$ (B.6)

it is antisymmetric and used to define the charge conjugation of the spinor
$\psi$ by

$\psi_{c}=C^{-1}\overline{\psi}^{T}\,,$ (B.7)

for which the roles of particle and antiparticle get interchanged. Note that a
chiral spinor $\lambda$ has its charge conjugated field $\lambda_{c}$ of
opposite chirality. A Majorana spinor $\mu$ is a spinor that equals its
charged conjugated spinor

$\mu=\mu_{c}\;.$ (B.8)

This constraint is incompatible with the chiral constraint, and Majorana-Weyl
spinors do not exist in 4 dimensions.

We find it convenient, as a check on our formulae, to use the chiral
representation of the gamma matrices. In terms of $2\times 2$ blocks they are
given by

$\gamma^{0}=-i\left(\begin{array}[]{cc}0&\mathbb{I}\\\
\mathbb{I}&0\end{array}\right)\,,\qquad\gamma^{i}=-i\left(\begin{array}[]{cc}0&\sigma^{i}\\\
-\sigma^{i}&0\end{array}\right)\,,$ (B.9)

where $\sigma^{i}$ are the Pauli matrices, so that

$\gamma^{5}=\left(\begin{array}[]{cc}\mathbb{I}&0\\\
0&-\mathbb{I}\end{array}\right)\;,\qquad\beta=i\gamma^{0}=\left(\begin{array}[]{cc}0&\mathbb{I}\\\
\mathbb{I}&0\end{array}\right)\;.$ (B.10)

The chiral representation makes evident that the Lorentz generators in the
spinor space
$M^{ab}=\frac{1}{4}[\gamma^{a},\gamma^{b}]=\frac{1}{2}\gamma^{ab}$ take a
block diagonal form

$M^{0i}=\frac{1}{2}\left(\begin{array}[]{cc}\sigma^{i}&0\\\
0&-\sigma^{i}\end{array}\right)\;,\qquad
M^{ij}=\frac{i}{2}\epsilon^{ijk}\left(\begin{array}[]{cc}\sigma^{k}&0\\\
0&\sigma^{k}\end{array}\right)\,,$ (B.11)

and do not mix the chiral components of a Dirac spinor (as $\gamma^{5}$ is
also block diagonal). The usual two-dimensional Weyl spinors appear inside a
four-dimensional Dirac spinor as follows

$\psi=\left(\begin{array}[]{c}l\\\
r\end{array}\right)\;,\qquad\lambda=\left(\begin{array}[]{c}l\\\
0\end{array}\right)\;,\qquad\rho=\left(\begin{array}[]{c}0\\\
r\end{array}\right)$ (B.12)

where $l$ and $r$ indicate two-dimensional independent spinors of opposite
chirality. In the chiral representation one may take the charge conjugation
matrix $C$ to be given by

$C=\gamma^{2}\beta=-i\left(\begin{array}[]{cc}\sigma^{2}&0\\\
0&-\sigma^{2}\end{array}\right)\,,$ (B.13)

and satisfies

$C=-C^{T}=-C^{-1}=-C^{\dagger}=C^{*}\,,$ (B.14)

and we remark that some of these relations are representation dependent. In
the chiral representation the Majorana constraint (B.8) takes the form

$\mu=\mu_{c}\qquad\to\qquad\left(\begin{array}[]{c}l\\\
r\end{array}\right)=\left(\begin{array}[]{c}i\sigma^{2}r^{*}\\\
-i\sigma^{2}l^{*}\end{array}\right)\,,$ (B.15)

which shows that the two-dimensional spinors $l$ and $r$ cannot be
independent. The Majorana condition can be solved in terms of the single two-
dimensional left-handed spinor $l$ as

$\mu=\left(\begin{array}[]{c}l\\\ -i\sigma^{2}l^{*}\end{array}\right)\,,$
(B.16)

which, evidently, contains the four-dimensional chiral spinors $\lambda$ and
$\lambda_{c}$ defined by

$\lambda=\left(\begin{array}[]{c}l\\\
0\end{array}\right)\;,\qquad\lambda_{c}=\left(\begin{array}[]{c}0\\\
-i\sigma^{2}l^{*}\end{array}\right)\;.$ (B.17)

In a four-dimensional spinors notation one can write

$\mu=\lambda+\lambda_{c}\;.$ (B.18)

Alternatively, the Majorana condition can be solved in terms of the two-
dimensional right-handed spinor $r$ as

$\mu=\left(\begin{array}[]{c}i\sigma^{2}r^{*}\\\ r\end{array}\right)\,,$
(B.19)

which contains the four-dimensional chiral spinors $\rho$ and $\rho_{c}$

$\rho=\left(\begin{array}[]{c}0\\\
r\end{array}\right)\;,\qquad\rho_{c}=\left(\begin{array}[]{c}i\sigma^{2}r^{*}\\\
0\end{array}\right)\,,$ (B.20)

and $\mu=\rho+\rho_{c}$. This solution is of course the same as the previous
one, as one may identify $\lambda=\rho_{c}$.

The explicit dictionary between Weyl and Majorana spinors shows that the field
theory of a Weyl spinor is equivalent to that of a Majorana spinor, as Lorentz
symmetry fixes uniquely their actions, which are bound to be identical.

Finally, we normalize our $\epsilon$ symbols by $\epsilon_{0123}=-1$ and
$\epsilon^{0123}=1$, so that

$\frac{1}{4}{\rm
tr}(\gamma^{5}\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d})=i\epsilon^{abcd}\;.$
(B.21)

###  B.2 Metric, vielbein and related connections

In a Lorentzian spacetime we use a metric $g_{\mu\nu}$ which is covariantly
constant

$\nabla_{\lambda}g_{\mu\nu}=\partial_{\lambda}g_{\mu\nu}-\Gamma^{\rho}_{\lambda\mu}g_{\nu\rho}-\Gamma^{\rho}_{\lambda\nu}g_{\mu\rho}=0\,,$
(B.22)

so that the Levi-Civita connection $\Gamma^{\lambda}_{\mu\nu}$ is given by

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\rho}\left(\partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\mu\rho}-\partial_{\rho}g_{\mu\nu}\right)\,.$
(B.23)

The covariant derivative on a vector field acts as

$\nabla_{\mu}V^{\lambda}=\partial_{\mu}V^{\lambda}+\Gamma_{\mu\nu}^{\lambda}V^{\nu}\,,$
(B.24)

and we remind our conventions for the curvature tensor

$[\nabla_{\mu},\nabla_{\nu}]V^{\lambda}=R_{\mu\nu}{}^{\lambda}{}_{\rho}V^{\rho}\,,\quad
R_{\mu\nu}=R_{\mu\lambda\nu}{}^{\lambda}\,,\quad R=R^{\mu}{}_{\mu}\,.$ (B.25)

Introducing the vielbein $e_{\mu}{}^{a}$ by setting

$g_{\mu\nu}=e_{\mu}{}^{a}\,e_{\nu}{}^{b}\,\eta_{ab}\,,$ (B.26)

we gain a new gauge symmetry, namely the local Lorentz transformations of the
tangent space; in this way we are able to couple a fermionic theory to
gravity. Latin letters are used for indices of the tangent space, and
$\eta_{ab}$ is a Minkowski metric. In four dimensions, we refer to
$e_{\mu}{}^{a}$ as the vierbein. The covariant derivative now needs a
corresponding connection, i.e. the spin connection, which in the absence of
torsion is defined by requiring the vielbein to be covariantly constant

$\nabla_{\mu}e_{\nu}{}^{a}=\partial_{\mu}e_{\nu}{}^{a}-\Gamma^{\lambda}_{\mu\nu}e_{\lambda}{}^{a}+\omega_{\mu}{}^{a}{}_{b}e_{\nu}{}^{b}=0\,.$
(B.27)

This condition can be solved by

$\omega_{\mu}{}^{ab}=e^{b\nu}\left(\Gamma^{\lambda}_{\mu\nu}e_{\lambda}{}^{a}-\partial_{\mu}e_{\nu}{}^{a}\right)\,,$
(B.28)

or equivalently

$\displaystyle\omega_{\mu}{}^{ab}$
$\displaystyle=\frac{1}{2}e^{a\lambda}e^{b\nu}e_{c\mu}\left(\partial_{\nu}e_{\lambda}{}^{c}-\partial_{\lambda}e_{\nu}{}^{c}\right)+\frac{1}{2}e^{a\nu}\left(\partial_{\mu}e_{\nu}{}^{b}-\partial_{\nu}e_{\mu}{}^{b}\right)-\frac{1}{2}e^{b\nu}\left(\partial_{\mu}e_{\nu}{}^{a}-\partial_{\nu}e_{\mu}{}^{a}\right)$
(B.29)

in which it is manifest that $\omega_{\mu}{}^{ab}+\omega_{\mu}{}^{ba}=0$.
Then, the covariant derivative acts on a spinor $\psi$ as

$\nabla_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{2}\omega_{\mu ab}M^{ab}\psi\,,$
(B.30)

where $M^{ab}$ are the generators of the Lorentz group, which in our
conventions are made explicit in eq. (B.11).

###  B.3 The stress tensor of the Bardeen model

In chap. 4 we have defined the stress tensor of the Bardeen model as

$T^{\mu}{}_{a}=\frac{1}{e}\frac{\delta S}{\delta e_{\mu}{}^{a}}\,,$ (B.31)

where $S=\int\,d^{4}x{\cal L}$ is the action associated to the lagrangian
(4.9). We review here its classical properties, which in sec. 4.1 we only need
in the flat space limit. The model depends on the background fields
$e_{\mu}{}^{a},A_{\mu},B_{\mu}$, and statisfies various background gauge
symmetries, responsible for the properties of the gauge currents and stress
tensor. Let us discuss the latter.

The infinitesimal background symmetries associated to general coordinate
invariance (with infinitesimal local parameters $\xi^{\mu}$), local Lorentz
invariance (with infinitesimal local parameters $\omega_{ab}$), and Weyl
invarance (with infinitesimal local parameter $\sigma$), take the form

$\displaystyle\delta e_{\mu}{}^{a}$
$\displaystyle=\xi^{\nu}\partial_{\nu}e_{\mu}{}^{a}+(\partial_{\mu}\xi^{\nu})e_{\nu}{}^{a}+\omega^{a}{}_{b}e_{\mu}{}^{b}+\sigma
e_{\mu}{}^{a}\,,$ (B.32) $\displaystyle\delta A_{\mu}$
$\displaystyle=\xi^{\nu}\partial_{\nu}A_{\mu}+(\partial_{\mu}\xi^{\nu})A_{\nu}\,,$
$\displaystyle\delta B_{\mu}$
$\displaystyle=\xi^{\nu}\partial_{\nu}B_{\mu}+(\partial_{\mu}\xi^{\nu})B_{\nu}\,,$
$\displaystyle\delta\psi$
$\displaystyle=\epsilon^{\mu}\partial_{\mu}\psi+\frac{1}{4}\omega_{ab}\gamma^{ab}\psi-\frac{3}{2}\sigma\psi\,.$

Under the Weyl symmetry $\delta_{\sigma}$ with local parameter $\sigma$, the
gauge fields do not transform, and the invariance of the action implies

$\displaystyle\delta_{\sigma}S$ $\displaystyle=\int d^{4}x\left(\frac{\delta
S}{\delta
e_{\mu}{}^{a}}\delta_{\sigma}e_{\mu}{}^{a}+\frac{\delta_{R}S}{\delta\psi}\delta_{\sigma}\psi+\delta_{\sigma}\overline{\psi}\frac{\delta_{L}S}{\delta\overline{\psi}}\right)$
(B.33) $\displaystyle=\int
d^{4}xe\,T^{\mu}{}_{a}\delta_{\sigma}e_{\mu}{}^{a}=\int
d^{4}xe\,T^{a}{}_{a}\sigma=0\,,$

where we have used left and right derivatives for the Grassmann valued fields
and in the second line we have implemented the equations of motion of the
spinor fields. Thus local Weyl invariance implies tracelessness of the stress
tensor, with the trace computed with the vierbein as
$T^{a}{}_{a}=T^{\mu}{}_{a}e_{\mu}{}^{a}$. Thus, the stress tensor is traceless
at the classical level.

Similarly, the Lorentz symmetry $\delta_{\omega}$ with local parameters
$\omega_{ab}$ implies

$\displaystyle\delta_{\omega}S$ $\displaystyle=\int d^{4}x\left(\frac{\delta
S}{\delta
e_{\mu}{}^{a}}\delta_{\omega}e_{\mu}{}^{a}+\frac{\delta_{R}S}{\delta\psi}\delta_{\omega}\psi+\delta_{\omega}\overline{\psi}\frac{\delta_{L}S}{\delta\overline{\psi}}\right)$
(B.34) $\displaystyle=\int
d^{4}xe\,T^{\mu}{}_{a}\delta_{\omega}e_{\mu}{}^{a}=\int
d^{4}xe\,T^{\mu}{}_{a}\omega^{a}{}_{b}e_{\mu}{}^{b}=\int
d^{4}xe\,T^{ba}\omega_{ab}=0\,,$

and constrains the antisymmetric part of the stress tensor to vanish on-shell.
Again, we have used the fermionic equations of motion and the fact that the
gauge fields $A_{\mu}$ and $B_{\mu}$ do not transform under local Lorentz
transformations. Considering the arbitrariness of the local parameters and
their antisymmetry, $\omega_{ab}=-\omega_{ba}$, we recognize that the stress
tensor with flat indices is symmetric, $T^{ab}=T^{ba}$. At the quantum level,
our PV regularization preserves this symmetry, and no anomalies can arise in
the local Lorentz sector.

Finally, a suitable conservation law of the stress tensor arises as a
consequence of the infinitesimal diffeomorphism invariance $\delta_{\xi}$. It
is actually useful to combine it with additional local Lorentz and gauge
symmetries (with composite parameters), so to obtain a conservation in the
following way

$\displaystyle\delta_{\xi}S$ $\displaystyle=\int d^{4}x\biggl{(}\frac{\delta
S}{\delta
e_{\mu}{}^{a}}\delta_{\xi}e_{\mu}{}^{a}+\frac{\delta_{R}S}{\delta\psi}\delta_{\xi}\psi+\delta_{\xi}\overline{\psi}\frac{\delta_{L}S}{\delta\overline{\psi}}+\frac{\delta
S}{\delta A_{\mu}^{\alpha}}\delta_{\xi}A_{\mu}^{\alpha}+\frac{\delta S}{\delta
B_{\mu}^{\alpha}}\delta_{\xi}B_{\mu}^{\alpha}\biggr{)}$ (B.35)
$\displaystyle=\int d^{4}xe\,\biggl{(}T^{\mu}{}_{a}{\cal
L}_{\xi}e_{\mu}{}^{a}+J^{\mu\alpha}{\cal
L}_{\xi}A_{\mu}^{\alpha}+J_{5}^{\mu\alpha}{\cal
L}_{\xi}B_{\mu}^{\alpha}\biggr{)}$ $\displaystyle=\int
d^{4}xe\,\biggl{(}T^{\mu}{}_{a}\nabla_{\mu}\xi^{a}+J^{\mu\alpha}\xi^{\nu}\hat{F}_{\nu\mu}^{\alpha}+J_{5}^{\mu\alpha}\xi^{\nu}\hat{G}_{\nu\mu}^{\alpha}\biggr{)}$
$\displaystyle=-\int
d^{4}xe\,\xi^{a}\biggl{(}\nabla_{\mu}T^{\mu}{}_{a}-J^{b\alpha}\hat{F}_{ab}^{\alpha}-J_{5}^{b\alpha}\hat{G}_{ab}^{\alpha}\biggr{)}=0\,,$

where in the second line we have used the fermion equations of motion (and
denoted with Lie derivatives ${\cal L}_{\xi}$ the transformation rules under
diffeomorphisms), and in the third line added for free to the Lie derivative
of the vierbein a spin connection term (it amounts to a local Lorentz
transformation with composite parameter, and it drops out once the stress
tensor is symmetric), and to the Lie derivatives of the gauge fields suitable
gauge transformations (which also drop out after partial integration as the
corresponding currents are covariantly conserved, recall eqs. (4.7) and (4.8)
which we use here in their curved space version), and then integrated by
parts. The arbitrariness of the local parameters $\xi^{a}(x)$ allows to derive
a covariant conservation law, which contains a contribution from the gauge
fields (that would vanish once the gauge fields are made dynamical, which
allows to use their equations of motion). It reads

$\nabla_{\mu}T^{\mu}{}_{a}=J^{b\alpha}\hat{F}_{ab}^{\alpha}+J_{5}^{b\alpha}\hat{G}_{ab}^{\alpha}\;.$
(B.36)

Our PV regularization preserves diffeomorphism invariance, and thus no anomaly
may appear in the Ward identities arising from this symmetry.

We end this appendix by presenting the explicit expression of the stress
tensor for the Bardeen model in flat spacetime:

$T_{ab}=\frac{1}{4}\overline{\psi}\Big{(}\gamma_{a}\overset{\leftrightarrow}{D}_{b}(A,B)+\gamma_{b}\overset{\leftrightarrow}{D}_{a}(A,B)\Big{)}\psi\,,$
(B.37)

where
$\overset{\leftrightarrow}{D}_{a}(A,B)=D_{a}(A,B)-\overset{\leftarrow}{D}_{a}(A,B)$,
with the second derivative meaning here
$\overset{\leftarrow}{D}_{a}(A,B)=\overset{\leftarrow}{\partial}_{a}-A_{a}-B_{a}\gamma^{5}$.
One can verify explicitly all the statements about the classical background
symmetry derived above, namely

$\partial_{a}T^{ab}=\overline{\psi}\gamma_{a}(\hat{F}^{ab}+\hat{G}^{ab}\gamma^{5})\psi\;,\qquad
T^{ab}=T^{ba}\;,\qquad T^{a}{}_{a}=0\;.$ (B.38)

The PV regularization used in the main text preserves the corresponding
quantum Ward identities except the last one, as the mass term in curved space
is not Weyl invariant, and a trace anomaly develops.

###  B.4 Heat kernel traces

We outline here how to compute the anomalies in the Bardeen model and present
some intermediate results. Similarly, one also proceeds in the case of the MAT
background.

In Bardeen model, the role of the operator $H$ introduced in sec. 2.3.3 is
played by the regulators $R_{\psi}$ and $R_{\psi_{c}}$. Focusing on the
regulator ${R}_{\psi}=-D\\!\\!\\!\\!/\,^{2}(A,B)$, we expand it as

$\displaystyle{R}_{\psi}$
$\displaystyle=-D^{a}(A)D_{a}(A)+B^{2}-\gamma^{5}\left(D^{a}(A)B_{a}\right)$
$\displaystyle\phantom{=}-\frac{1}{2}\gamma^{ab}\left[\hat{F}_{ab}-4B_{a}B_{b}+\gamma^{5}\left(\hat{G}_{ab}-4B_{a}D_{b}(A)\right)\right]\,,$
(B.39)

where $\hat{F}_{ab}$ and $\hat{G}_{ab}$ are the Bardeen curvatures given in
(4.5), the covariant derivative is $D_{a}(A)=\partial_{a}+A_{a}$, the
covariant divergence of $B_{a}$ is
$D^{a}(A)B_{a}=(\partial^{a}B_{a})+[A^{a},B_{a}]$ and $B^{2}=B^{a}B_{a}$. We
compare this operator to the generic operator $H$ in (2.95), where in this
case we consider it in flat space and with only a gauge covariant derivative
with connection $W_{a}$, i.e. $H$ takes the form

$H=-\Box+V=-\partial^{a}\partial_{a}-2W^{a}\partial_{a}-(\partial_{a}W^{a})-W^{2}+V\,.$
(B.40)

Comparing with (B.4) we identify

$\displaystyle W_{a}$ $\displaystyle=A_{a}+\gamma_{ab}\gamma^{5}B^{b}\,,$
(B.41) $\displaystyle V$
$\displaystyle=-2B^{2}-\gamma^{5}\left(D^{a}(A)B_{a}\right)-\frac{1}{2}\gamma^{ab}\hat{F}_{ab}\,,$
(B.42) $\displaystyle{\cal F}_{ab}$
$\displaystyle=F_{ab}(A)+\left(\gamma_{ac}\gamma_{bd}-\gamma_{bc}\gamma_{ad}\right)B^{c}B^{d}+\gamma^{5}\left(\gamma_{ca}D_{b}(A)B^{c}-\gamma_{cb}D_{a}(A)B^{c}\right)\,,$
(B.43)

with

${\cal F}_{ab}=\partial_{a}W_{b}-\partial_{b}W_{a}+[W_{a},W_{b}]\,.$ (B.44)

Now the coefficient $a_{2}(R_{\psi})$, which enters in the anomaly
calculations in four dimensions, can be made explicit using (2.101) in flat
space. In this limit, which is the one of interest for the Bardeen model, it
reduces to

$a_{2}(x,H)=\frac{1}{2}V^{2}-\frac{1}{6}\Box V+\frac{1}{12}{\cal
F}_{ab}^{2}\,,$ (B.45)

and the Dirac traces can be directly computed. We list here some intermediate
results, grouping together the contributions from the same terms in (B.45). We
find (with $D_{a}=D_{a}(A)$, $DB=D_{a}B^{a}$, $B^{2}=B^{a}B_{a}$, and so on):

$i)$ from $a_{2}=\frac{1}{2}V^{2}$

$\displaystyle{\rm
tr}\left[\gamma^{5}T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\bigg{[}\frac{i}{2}\epsilon^{abcd}\hat{F}_{ab}\hat{F}_{cd}+4\\{B^{2},DB\\}\bigg{]}\,,$
(B.46) $\displaystyle{\rm
tr}\left[T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$ $\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\left[-\hat{F}^{ab}\hat{F}_{ab}+8B^{4}+2DBDB\right]\,,$
(B.47) $\displaystyle{\rm tr}\left[a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}\left[-\hat{F}^{ab}\hat{F}_{ab}+8B^{4}+2DBDB\right]\,.$
(B.48)

$ii)$ from $a_{2}=-\frac{1}{6}\nabla^{2}V$

$\displaystyle{\rm
tr}\left[\gamma^{5}T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\bigg{[}-\frac{2}{3}i\epsilon^{abcd}\left(B_{a}B_{b}\hat{F}_{cd}+2B_{a}\hat{F}_{bc}B_{d}+\hat{F}_{ab}B_{c}B_{d}\right)$
$\displaystyle\phantom{=}+\frac{2}{3}D^{2}DB+\frac{1}{3}[\hat{F}^{ab},\hat{G}_{ab}]+\frac{4}{3}[D^{a}\hat{F}_{ab},B^{b}]-2\\{B^{2},DB\\}$
$\displaystyle\phantom{=}+4B_{a}DBB^{a}\bigg{]}\,,$ (B.49) $\displaystyle{\rm
tr}\left[T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$ $\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\bigg{[}\frac{i}{6}\epsilon^{abcd}\left([\hat{G}_{ab},\hat{F}_{cd}]-4[B_{a},D_{b}\hat{F}_{cd}]\right)+8\left(B_{a}B^{2}B^{a}-B^{4}\right)$
$\displaystyle\phantom{=}+\frac{4}{3}D^{2}B^{2}+\frac{4}{3}\left(\hat{F}^{ab}B_{a}B_{b}+2B_{a}\hat{F}^{ab}B_{b}+B_{a}B_{b}\hat{F}^{ab}\right)\bigg{]}\,,$
(B.50) $\displaystyle{\rm tr}\left[a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}\bigg{[}\frac{4}{3}D^{2}B^{2}\bigg{]}\,.$ (B.51)

$iii)$ from $a_{2}=\frac{1}{12}{\cal F}_{ab}^{2}$

$\displaystyle{\rm
tr}\left[\gamma^{5}T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\bigg{[}i\epsilon^{abcd}\left(\frac{1}{6}\hat{G}_{ab}\hat{G}_{cd}-\frac{2}{3}\\{\hat{F}_{ab},B_{c}B_{d}\\}+\frac{16}{3}B_{a}B_{b}B_{c}B_{d}\right)$
$\displaystyle\phantom{=}-\frac{8}{3}\\{B^{2},DB\\}+\frac{4}{3}\\{\\{B^{a},B^{b}\\},D_{a}B_{b}\\}\bigg{]}\,,$
(B.52) $\displaystyle{\rm
tr}\left[T^{\alpha}a_{2}\left({R}_{\psi}\right)\right]$ $\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}T^{\alpha}\bigg{[}\frac{1}{3}\hat{F}^{ab}\hat{F}_{ab}-\frac{4}{3}\\{\hat{F}_{ab},B^{a}B^{b}\\}+\frac{8}{3}\left(B_{a}B_{b}B^{a}B^{b}-B^{4}\right)$
$\displaystyle\phantom{=}-8B_{a}B^{2}B^{a}-\frac{4}{3}D_{a}B_{b}D^{a}B^{b}-\frac{2}{3}DBDB\bigg{]}\,,$
(B.53) $\displaystyle{\rm tr}\left[a_{2}\left({R}_{\psi}\right)\right]$
$\displaystyle={\rm
tr}_{{}_{\\!\\!YM}}\bigg{[}\frac{1}{3}\hat{F}^{ab}\hat{F}_{ab}-\frac{8}{3}\hat{F}_{ab}B^{a}B^{b}+\frac{8}{3}B_{a}B_{b}B^{a}B^{b}-\frac{32}{3}B^{4}$
$\displaystyle\phantom{=}-\frac{4}{3}D_{a}B_{b}D^{a}B^{b}-\frac{2}{3}DBDB\bigg{]}\,.$
(B.54)

The analogous results for $a_{2}(R_{\psi_{c}})$ are obtained by replacing
$A\to-A^{T}$ and $B\to B^{T}$ (and also $T^{\alpha}\to T^{\alpha T}$ for the
explicit $T^{\alpha}$ appearing in the traces). Their effect is just to double
the contribution from $a_{2}(R_{\psi})$ in the chiral and trace anomalies.

###  B.5 Dirac mass in MAT background

We give here a brief description of a different regularization that could be
employed for a Dirac fermion coupled to the MAT backgound discussed in sec.
4.2. There, we used a Majorana mass for the PV fields, arguing that it was
most convenient as it breaks the least number of symmetries. We give support
here for that claim, by showing how a Dirac mass for the PV fields could be
coupled to the MAT background.

In a flat background the Dirac mass is given by

$-M\bar{\psi}\psi=\frac{1}{2}M\left(\psi_{c}^{T}C\psi+\psi^{T}C\psi_{c}\right)\;.$
(B.55)

This term breaks the $U(1)_{A}$ axial symmetry while maintaining the
$U(1)_{V}$ vector symmetry. This continues to be the case also when one tries
to MAT-covariantize it. There are various options to couple the Dirac mass to
the MAT geometry. We may choose to use only the metric $g_{\mu\nu}$ in the
mass term, without any axial extension, so that

$\Delta_{\scriptscriptstyle
D}\mathcal{L}=-\sqrt{g}M\bar{\psi}\psi=\frac{\sqrt{g}}{2}M(\psi^{T}_{c}C\psi+\psi^{T}C\psi_{c})$
(B.56)

has the virtue of preserving the vector-like diffeomorphisms and vector-like
local Lorentz transformations on top of the $U(1)_{V}$ symmetry, while
breaking all of their axial extensions. It also breaks both vector and axial
Weyl symmetries, which are then expected to be anomalous as well. Counterterms
should eventually be introduced to achieve the equivalence with the results in
sec. 4.2.4. Other choices are also possible, e.g.
$\sqrt{g}\to\tfrac{1}{2}(\sqrt{g_{+}}+\sqrt{g_{-}})$, which shares the same
property of preserving the vector-like diffeomorphisms and local Lorentz
transformations.

In the following we limit ourselves to derive the regulators to be used for
computing the anomalies in this scheme. We add to the lagrangian (4.37)
written in a symmetric form

$\mathcal{L}=\frac{1}{2}\psi_{c}^{T}C\sqrt{\bar{\hat{g}}}\hat{\nabla\\!\\!\\!\\!/\,}\psi+\frac{1}{2}\psi^{T}C\sqrt{\hat{g}}\bar{\hat{\nabla\\!\\!\\!\\!/\,}}\psi_{c}$
(B.57)

the mass term (B.56), and comparing it with (2.79) we find, choosing the field
basis $\phi=\left(\begin{array}[]{c}\psi\\\ \psi_{c}\end{array}\right)$,

$T\mathcal{O}=\left(\begin{array}[]{cc}0&C\sqrt{\hat{g}}\bar{\hat{\nabla\\!\\!\\!\\!/\,}}\\\
C\sqrt{\bar{\hat{g}}}{\hat{\nabla\\!\\!\\!\\!/\,}}&0\end{array}\right)\,,\quad\quad
T=\left(\begin{array}[]{cc}0&\sqrt{g}C\\\ \sqrt{g}C&0\end{array}\right)\,,$
(B.58)

and thus

$\mathcal{O}=\left(\begin{array}[]{cc}\sqrt{\frac{\bar{\hat{g}}}{g}}\,{\hat{\nabla\\!\\!\\!\\!/\,}}&0\\\
0&\sqrt{\frac{\hat{g}}{g}}\,\bar{\hat{\nabla\\!\\!\\!\\!/\,}}\end{array}\right)\,,\quad\quad\mathcal{O}^{2}=\left(\begin{array}[]{cc}\sqrt{\frac{\bar{\hat{g}}}{g}}\,{\hat{\nabla\\!\\!\\!\\!/\,}}\sqrt{\frac{\bar{\hat{g}}}{g}}\,{\hat{\nabla\\!\\!\\!\\!/\,}}&0\\\
0&\sqrt{\frac{\hat{g}}{g}}\,\bar{\hat{\nabla\\!\\!\\!\\!/\,}}\sqrt{\frac{\hat{g}}{g}}\,\bar{\hat{\nabla\\!\\!\\!\\!/\,}}\end{array}\right)\,,$
(B.59)

with the differential operators acting on everything placed on their right
hand side. This regulator $\mathcal{R}=-\mathcal{O}^{2}$ is difficult to work
with, but it has the virtue of being covariant under vector diffeomorphisms.
Its structure is again more transparent when splitting the Dirac fermion into
its chiral parts, so that on the basis

$\phi=\left(\begin{array}[]{c}\lambda\\\ \rho\\\ \rho_{c}\\\
\lambda_{c}\end{array}\right)\,,$ (B.60)

we find a block diagonal regulator

$\mathcal{R}=\left(\begin{array}[]{cccc}R_{\lambda}&0&0&0\\\ 0&R_{\rho}&0&0\\\
0&0&R_{\rho_{c}}&0\\\ 0&0&0&R_{\lambda_{c}}\end{array}\right)\,,$ (B.61)

with entries

$\displaystyle R_{\lambda}$
$\displaystyle=-\sqrt{\frac{g_{-}}{g}}\nabla\\!\\!\\!\\!/\,_{-}\sqrt{\frac{g_{+}}{g}}\nabla\\!\\!\\!\\!/\,_{+}P_{+}\;,\qquad
R_{\rho}=-\sqrt{\frac{g_{+}}{g}}\nabla\\!\\!\\!\\!/\,_{+}\sqrt{\frac{g_{-}}{g}}\nabla\\!\\!\\!\\!/\,_{-}P_{-}$
(B.62) $\displaystyle R_{\lambda_{c}}$
$\displaystyle=-\sqrt{\frac{g_{-}}{g}}\nabla\\!\\!\\!\\!/\,_{-}\sqrt{\frac{g_{+}}{g}}\nabla\\!\\!\\!\\!/\,_{+}P_{-}\;,\qquad
R_{\rho_{c}}=-\sqrt{\frac{g_{+}}{g}}\nabla\\!\\!\\!\\!/\,_{+}\sqrt{\frac{g_{-}}{g}}\nabla\\!\\!\\!\\!/\,_{-}P_{+}\;.$

The functions $\sqrt{\frac{g_{\mp}}{g}}$ are scalar functions under the
vector-like diffeomorphism, so that these regulators are covariant under that
symmetry. The projectors $P_{\pm}$ take just the unit value on the
corresponding chiral spinor space, but we have kept them to remind on which
space the different regulators act. A systematic analysis of all the
anomalies, including the axial gravitational anomaly, may be feasible in this
scheme, at least when treating $f_{\mu\nu}$ as a perturbation. The final
results have to agree with the anomalies presented in sec. 4.2.4 by adding
suitable counterterms to the effective action.

##  Appendix C Further details on the holographic anomalies with scalar fields

###  C.1 Ansätze for the backreaction

In this appendix we collect the Ansätze that we make to solve eqs. (5.21),
(5.22) for $n=1,2$. The explicit results can be found in the Mathematica
notebook provided in the supplementary material of [Bro22f].111The results
were obtained with the help of the xAct collection of Mathematica packages
[MPM07f, MYP08f, Mar08f, BMM09f, Nut14f]. In the following, we assume that
$g_{ij}\equiv g^{(0)}_{ij}$ and curvatures and covariant derivatives are
w.r.t. $g^{(0)}_{ij}$, and $\phi\equiv\phi_{(0)}$. We also choose $h_{0}=0$,
as explained in the text.

The solution of the PBH equation for $g^{(1)}_{ij}$ is given in (2.143) and it
is not modified by the presence of the backreaction. The most generic Ansatz
for $h^{(2)}_{ij}$ has thirty-five terms and six coefficients will not be
fixed by the modified PBH equation. We write the Ansatz as

$\displaystyle h^{(2)}_{ij}=$
$\displaystyle\;+h_{21}g_{ij}R_{kl}R^{kl}\phi^{2}+h_{22}R_{ij}R\phi^{2}+h_{23}R^{kl}R_{ikjl}\phi^{2}+h_{24}\phi^{2}\nabla_{j}\nabla_{i}R+h_{25}R_{ij}\phi\Box\phi$
$\displaystyle\phantom{=}+h_{26}g_{ij}R\phi\Box\phi+\beta_{1}R_{i}{}^{k}R_{jk}\phi^{2}+\beta_{2}g_{ij}R^{2}\phi^{2}+\beta_{3}R_{i}{}^{klm}R_{jklm}\phi^{2}$
$\displaystyle\phantom{=}+\beta_{4}g_{ij}R_{klmn}R^{klmn}\phi^{2}+\beta_{6}R\nabla_{i}\phi\nabla_{j}\phi+\beta_{5}(\phi\nabla_{i}\phi\nabla_{j}R+\phi\nabla_{i}R\nabla_{j}\phi)$
$\displaystyle\phantom{=}+\beta_{7}R\phi\nabla_{j}\nabla_{i}\phi+\beta_{8}(R_{j}{}^{k}\phi\nabla_{k}\nabla_{i}\phi+R_{i}{}^{k}\phi\nabla_{k}\nabla_{j}\phi)+\beta_{9}\phi^{2}\Box
R_{ij}+\beta_{10}g_{ij}\phi^{2}\Box R$
$\displaystyle\phantom{=}+\beta_{11}\nabla_{j}\nabla_{i}\phi\Box\phi+\beta_{12}(\nabla_{j}\phi\Box\nabla_{i}\phi+\nabla_{i}\phi\Box\nabla_{j}\phi)+\beta_{13}\phi\Box\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}+\beta_{14}g_{ij}\phi\nabla_{k}\phi\nabla^{k}R+\beta_{15}(\phi\nabla_{i}R_{jk}\nabla^{k}\phi+\phi\nabla_{j}R_{ik}\nabla^{k}\phi)$
$\displaystyle\phantom{=}+\beta_{16}(R_{jk}\nabla_{i}\phi\nabla^{k}\phi+R_{ik}\nabla_{j}\phi\nabla^{k}\phi)+\beta_{17}\phi\nabla_{k}R_{ij}\nabla^{k}\phi+\beta_{18}R_{ij}\nabla_{k}\phi\nabla^{k}\phi$
$\displaystyle\phantom{=}+\beta_{19}g_{ij}R\nabla_{k}\phi\nabla^{k}\phi+\beta_{20}\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\phi+\beta_{21}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi$
$\displaystyle\phantom{=}+\beta_{22}g_{ij}R^{kl}\phi\nabla_{l}\nabla_{k}\phi+\beta_{23}g_{ij}\Box\phi\Box\phi+\beta_{24}g_{ij}\nabla^{k}\phi\Box\nabla_{k}\phi+\beta_{25}g_{ij}\phi\Box^{2}\phi$
$\displaystyle\phantom{=}+\beta_{26}g_{ij}R_{kl}\nabla^{k}\phi\nabla^{l}\phi+\beta_{27}R_{ikjl}\nabla^{k}\phi\nabla^{l}\phi+\beta_{28}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi$
$\displaystyle\phantom{=}+\beta_{29}g_{ij}\nabla_{l}\nabla_{k}\phi\nabla^{l}\nabla^{k}\phi\,,$
(C.1)

and leave the coefficients $h_{21},\ldots,h_{26}$ free. The full solution is
in the aforementioned Mathematica notebook.

A similar Ansatz is made for $\tilde{h}^{(2)}_{ij}$ and its coefficients are
fixed in terms of $h_{1}$ and $h_{2}$ by the modified PBH equation.

The solution for $g^{(2)}_{ij}$ is given in (2.144) and it is not modified by
the presence of the backreaction. Thus, here we focus on ${h}^{(3)}_{ij}$,
which appears in eq. (5.36) only as ${h}^{(3)i}{}_{i}$ and we then do not
compute ${h}^{(3)}_{ij}$ but rather its trace that satisfies the following PBH
equation

$\delta g^{(2)i}{}_{i}+\delta{h}^{(3)i}{}_{i}=-2\sigma
g^{(2)i}{}_{i}+g^{(0)ij}\delta
g^{(2)}{}_{ij}+2\sigma\beta^{ij}g^{(2)}_{ij}-2\sigma{h}^{(3)i}{}_{i}+g^{(0)ij}\delta{h}^{(3)}_{ij}\,.$
(C.2)

The most generic Ansatz for ${h}^{(3)i}{}_{i}$ has sixty-six terms and nine
coefficients will not be fixed by the modified PBH equation. We write the
Ansatz as

$\displaystyle{h}^{(3)i}{}_{i}=$
$\displaystyle\;h_{31}R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi+h_{32}R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R+h_{33}R_{ij}R^{ij}R\phi^{2}+h_{34}R^{ij}R\phi\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}+h_{35}R_{ij}\phi\nabla^{i}R\nabla^{j}\phi+h_{36}R^{ij}R^{kl}R_{ikjl}\phi^{2}+h_{37}R^{ij}\phi^{2}\Box
R_{ij}+h_{38}R_{ij}R^{ij}\phi\Box\phi$
$\displaystyle\phantom{=}+h_{39}R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi+\gamma_{1}R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+\gamma_{2}R^{3}\phi^{2}+\gamma_{3}RR_{ijkl}R^{ijkl}\phi^{2}$
$\displaystyle\phantom{=}+\gamma_{4}R^{ij}R_{i}{}^{klm}R_{jklm}\phi^{2}+\gamma_{5}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}+\gamma_{6}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}$
$\displaystyle\phantom{=}+\gamma_{7}R\phi^{2}\Box
R+\gamma_{8}R^{2}\phi\Box\phi+\gamma_{9}R_{jklm}R^{jklm}\phi\Box\phi+\gamma_{10}\phi^{2}\nabla_{i}R\nabla^{i}R$
$\displaystyle\phantom{=}+\gamma_{11}R\phi\nabla_{i}\phi\nabla^{i}R+\gamma_{12}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\gamma_{13}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}+\gamma_{14}R^{2}\nabla_{i}\phi\nabla^{i}\phi+\gamma_{15}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+\gamma_{16}\nabla_{i}\phi\nabla^{i}\phi\Box
R+\gamma_{17}\phi\Box R\Box\phi$
$\displaystyle\phantom{=}+\gamma_{18}R\Box\phi\Box\phi+\gamma_{19}\nabla_{i}\phi\nabla^{i}R\Box\phi+\gamma_{20}\phi\nabla^{i}\phi\Box\nabla_{i}R+\gamma_{21}\phi\nabla^{i}R\Box\nabla_{i}\phi$
$\displaystyle\phantom{=}+\gamma_{22}R\nabla^{i}\phi\Box\nabla_{i}\phi+\gamma_{23}\phi^{2}\Box^{2}R+\gamma_{24}R\phi\Box^{2}\phi+\gamma_{25}R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}+\gamma_{26}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+\gamma_{27}R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi+\gamma_{28}R_{i}{}^{klm}R_{jklm}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}+\gamma_{29}\nabla^{i}\phi\nabla_{j}\nabla_{i}R\nabla^{j}\phi+\gamma_{30}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi+\gamma_{31}\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R$
$\displaystyle\phantom{=}+\gamma_{32}R_{i}{}^{klm}R_{jklm}\phi\nabla^{j}\nabla^{i}\phi+\gamma_{33}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi+\gamma_{34}R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}$
$\displaystyle\phantom{=}+\gamma_{35}R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi+\gamma_{36}R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi+\gamma_{37}\nabla^{i}\phi\nabla^{j}\phi\Box
R_{ij}$ $\displaystyle\phantom{=}+\gamma_{38}\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}+\gamma_{39}R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi+\gamma_{40}\nabla^{j}\Box\phi\Box\nabla_{j}\phi$
$\displaystyle\phantom{=}+\gamma_{41}R^{ij}\phi\Box\nabla_{j}\nabla_{i}\phi+\gamma_{42}\nabla^{j}\nabla^{i}\phi\Box\nabla_{j}\nabla_{i}\phi+\gamma_{43}\Box\phi\Box^{2}\phi+\gamma_{44}\nabla^{i}\phi\Box^{2}\nabla_{i}\phi$
$\displaystyle\phantom{=}+\gamma_{45}\phi\Box^{3}\phi+\gamma_{46}\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}+\gamma_{47}\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}+\gamma_{48}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}+\gamma_{49}R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi+\gamma_{50}\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi$
$\displaystyle\phantom{=}+\gamma_{51}\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi+\gamma_{52}\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi+\gamma_{53}R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}$
$\displaystyle\phantom{=}+\gamma_{54}R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}+\gamma_{55}R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi+\gamma_{56}R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi$
$\displaystyle\phantom{=}+\gamma_{57}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\,,$
(C.3)

and leave the coefficients $h_{31},\ldots,h_{39}$ free. The full solution is
in the aforementioned Mathematica notebook.

Similarly, since $\tilde{h}^{(3)}$ appears in eq. (5.36) only as
$\tilde{h}^{(3)i}{}_{i}$, we do not compute $\tilde{h}^{(3)}_{ij}$ but rather
its trace that satisfies the following PBH equation

$\delta\tilde{h}^{(3)i}{}_{i}=-2\sigma\tilde{h}^{(3)i}{}_{i}+g^{(0)ij}\delta\tilde{h}^{(3)}_{ij}\,.$
(C.4)

A similar Ansatz to (C.3) is made for $\tilde{h}^{(3)i}{}_{i}$ and its
coefficients are fixed in terms of $h_{1},h_{2},c_{1},c_{2}$ by the modified
PBH equation.

###  C.2 Gravitational action with backreaction

From (2.151) in FG coordinates with backreaction (5.6) we identify:

$\displaystyle\mathcal{L}_{h}$
$\displaystyle=\tfrac{1}{2}d(1+4\gamma)h^{i}{}_{i}$
$\displaystyle+\tfrac{1}{2}z(1+4\gamma)\big{[}2(d-1)g^{\prime}_{ij}h^{ij}-(d-1)g^{ij}g^{\prime}_{ij}h^{k}{}_{k}$
$\displaystyle\phantom{=}-2(d-1)g^{ij}h^{\prime}_{ij}+h^{ij}R_{ij}-\tfrac{1}{2}h^{i}{}_{i}R-\nabla^{j}\nabla^{i}h_{ij}+\Box
h^{i}{}_{i}\big{]}$
$\displaystyle+z^{2}\big{[}-2(1+4\gamma)g^{\prime\prime}_{ij}h^{ij}+(3-4(d-5)\gamma)g^{im}g^{jn}g^{\prime}_{ik}g^{\prime}_{mn}h^{jk}$
$\displaystyle\phantom{=}-(1+8\gamma)g^{ij}g^{\prime}_{ij}g^{\prime}_{kl}h^{kl}+(1+4\gamma)g^{ij}g^{\prime\prime}_{ij}h^{k}{}_{k}-2\gamma
R_{ikjl}\nabla^{l}\nabla^{k}h^{ij}$
$\displaystyle\phantom{=}-(\tfrac{3}{4}-(d-5)\gamma)g^{im}g^{jn}g^{\prime}_{ij}g^{\prime}_{mn}h^{k}{}_{k}+(\tfrac{1}{4}+2\gamma)(g^{ij}g^{\prime}_{ij})^{2}h^{k}{}_{k}$
$\displaystyle\phantom{=}-(3-4(d-5)\gamma)g^{ik}g^{jl}g^{\prime}_{ij}h^{\prime}_{kl}+2(1+4\gamma)g^{ij}h^{\prime\prime}_{ij}+2\gamma
g^{\prime}_{ij}R^{ij}h^{k}{}_{k}$
$\displaystyle\phantom{=}+(1+8\gamma)g^{ij}g^{kl}g^{\prime}{ij}h^{\prime}_{kl}+4\gamma
h^{\prime}_{ij}R^{ij}-8\gamma g^{\prime}_{ij}h^{i}{}_{k}R^{jk}-\gamma
h^{ij}R_{i}{}^{klp}R_{jklp}$ $\displaystyle\phantom{=}+\tfrac{1}{4}\gamma
h^{i}{}_{i}R_{jklp}R^{jklp}-2\gamma
g^{\prime}_{ij}\nabla^{j}\nabla^{i}h^{k}{}_{k}+4\gamma
g^{\prime}_{ij}\nabla_{k}\nabla^{j}h^{ik}-2\gamma g^{\prime}_{ij}\Box
h^{ij}\big{]}$ $\displaystyle+\gamma
z^{3}\big{[}-12g^{im}g^{jn}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime}_{jl}h^{kl}+8g^{ij}g^{mn}g^{\prime}_{ij}g^{\prime}_{lm}g^{\prime}_{mk}h^{kl}$
$\displaystyle\phantom{=}+4g^{im}g^{jn}g^{\prime}_{ij}g^{\prime}_{mn}g^{\prime}_{kl}h^{kl}+2g^{im}g^{jn}g^{kl}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime}_{jl}h^{p}{}_{p}$
$\displaystyle\phantom{=}-2g^{ij}g^{km}g^{nl}g^{\prime}_{ij}g^{\prime}_{kl}g^{\prime}_{mn}h^{p}{}_{p}+12g^{im}g^{jn}g^{kl}g^{\prime}_{ik}g^{\prime}_{mn}h^{\prime}_{jl}$
$\displaystyle\phantom{=}-8g^{ij}g^{km}g^{ln}g^{\prime}_{ij}g^{\prime}_{kl}h^{\prime}_{mn}-4g^{ik}g^{jl}g^{mn}g^{\prime}_{ij}g^{\prime}_{kl}h^{\prime}_{mn}-g^{\prime}_{ij}g^{\prime}_{kl}h^{p}{}_{p}R^{ikjl}$
$\displaystyle\phantom{=}-4g^{\prime}_{ij}h^{\prime}_{kl}R^{ikjl}-6g^{\prime}_{ij}g^{\prime}_{kl}h^{i}{}_{p}R^{jklp}-4g^{km}g^{ln}h^{ij}\nabla_{i}g^{\prime}_{kl}\nabla_{j}g^{\prime}_{mn}$
$\displaystyle\phantom{=}-4g^{\prime}_{ij}\nabla^{i}g^{\prime}_{kl}\nabla^{j}h^{kl}-8g^{il}\nabla^{j}h^{\prime}_{ik}\nabla^{k}g^{\prime}_{jl}+8g^{ik}g^{jl}\nabla_{k}h^{\prime}_{ij}\nabla^{k}g^{\prime}_{kl}$
$\displaystyle\phantom{=}+8g^{\prime}_{ij}\nabla^{i}g^{\prime}_{kl}\nabla^{l}h^{jk}-2g^{\prime}_{ij}g^{\prime}_{kl}\nabla^{l}\nabla^{j}h^{ik}+2g^{\prime
ij}g^{\prime kl}\nabla^{l}\nabla^{k}h^{ij}$
$\displaystyle\phantom{=}+8g^{kn}h^{ij}\nabla_{j}g^{\prime}_{kl}\nabla^{l}g^{\prime}_{in}+4g^{in}g^{\prime}_{jn}\nabla^{j}h^{kl}\nabla_{l}g^{\prime}_{ik}+4h^{ij}\nabla^{k}g^{\prime}_{jl}\nabla^{l}g^{\prime}_{ik}$
$\displaystyle\phantom{=}-4g^{in}g^{\prime}_{ij}\nabla^{k}h^{jl}\nabla_{l}g^{\prime}_{kn}-8g^{kn}h^{ij}\nabla_{l}g^{\prime}_{jk}\nabla^{l}g^{\prime}_{in}-4g^{in}g^{\prime}_{ij}\nabla_{l}h^{jk}\nabla^{l}g^{\prime}_{kn}$
$\displaystyle\phantom{=}-2g^{jn}h^{i}{}_{i}\nabla^{k}g^{\prime}_{jl}\nabla^{l}g^{\prime}_{kn}+2g^{jm}g^{km}h^{i}{}_{i}\nabla_{l}g^{\prime}_{jk}\nabla^{l}g^{\prime}_{mn}\big{]}$
$\displaystyle+\gamma
z^{4}\big{[}8g^{im}g^{kn}g^{\prime}_{ij}g^{\prime}_{kl}g^{\prime\prime}_{mn}h^{jl}-16g^{il}g^{\prime\prime}_{ik}g^{\prime\prime}_{jl}h^{jk}+16g^{im}g^{jn}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime\prime}_{jl}h^{kl}$
$\displaystyle\phantom{=}+4g^{ik}g^{jl}g^{\prime\prime}_{ij}g^{\prime\prime}_{kl}h^{k}{}_{k}-4g^{kq}g^{ir}g^{js}g^{\prime}_{ik}g^{\prime}_{rs}g^{\prime}_{jl}g^{\prime}_{pq}h^{lp}-8g^{im}g^{jn}g^{kl}g^{\prime}_{ik}g^{\prime}_{mn}h^{\prime\prime}_{jl}$
$\displaystyle\phantom{=}-4g^{ik}g^{jl}g^{mr}g^{ns}g^{pq}g^{\prime}_{ij}g^{\prime}_{kl}g^{\prime}_{mn}g^{\prime}_{rq}h^{ps}-4g^{im}g^{jn}g^{kl}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime\prime}_{jl}h^{p}{}_{p}$
$\displaystyle\phantom{=}+\tfrac{1}{2}g^{im}g^{jn}g^{kr}g^{ls}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime}_{jl}g^{\prime}_{rs}h^{p}{}_{p}+\tfrac{1}{2}(g^{ik}g^{jl}g^{\prime}_{ij}g^{\prime}_{kl})^{2}h^{p}{}_{p}-16g^{im}g^{jn}g^{kl}g^{\prime}_{mn}g^{\prime\prime}_{ik}h^{\prime}_{jl}$
$\displaystyle\phantom{=}+4g^{im}g^{jn}g^{kr}g^{ls}g^{\prime}_{ik}g^{\prime}_{mn}g^{\prime}_{jl}h^{\prime}_{rs}+4g^{im}g^{jn}g^{kr}g^{ls}g^{\prime}_{ij}g^{\prime}_{mn}g^{\prime}_{kl}h^{\prime}_{rs}+16g^{ik}g^{jl}g^{\prime\prime}_{ij}h^{\prime\prime}_{kl}\big{]}\,,$
(C.5)

while $\mathcal{L}_{g}$ is still given by (2.152). Notice that also terms of
$\mathcal{O}(z^{4})$ contribute to the anomaly in $d=4$, since
$h^{\prime\prime}_{ij}=-z^{-2}\tilde{h}^{(1)}_{ij}+\mathcal{O}(z^{-1})$.

###  C.3 Coefficients renormalisation

Evaluating (2.154) and (5.36) on the PBH solutions, we find that
$\mathcal{L}_{g}^{(2)}+\mathcal{L}_{h}^{(2)}$ satisfies the WZ condition
(5.34), even though it is singular in $d=4$. In particular, expanding around
$d=4$ we find that

$\mathcal{L}_{g}^{(2)}+\mathcal{L}_{h}^{(2)}=\frac{1}{d-4}I^{(-1)}+I^{(0)}+(d-4)I^{(1)}+(d-4)^{2}I^{(2)}+\ldots\,,$
(C.6)

with ($g_{ij}\equiv g^{(0)}_{ij}$ and $\phi\equiv\phi_{(0)}$)

$\displaystyle I^{(-1)}$
$\displaystyle=c\,h_{1}\Bigg{(}\frac{7}{24}R_{ij}R^{ij}R\phi^{2}-\frac{1}{24}R^{3}\phi^{2}-\frac{1}{2}R^{ij}R^{kl}R_{ikjl}\phi^{2}+\frac{1}{24}R\phi^{2}\Box
R-\frac{1}{12}R^{2}\phi\Box\phi$
$\displaystyle\phantom{=}+\frac{1}{12}R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R+\frac{1}{3}R^{ij}R\phi\nabla_{j}\nabla_{i}\phi+\frac{1}{12}\phi\Box
R\Box\phi+\frac{1}{6}\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R$
$\displaystyle\phantom{=}-\frac{1}{4}R^{ij}\phi^{2}\Box
R_{ij}-\frac{1}{2}\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}+\frac{1}{4}R_{ij}R^{ij}\phi\Box\phi-R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi\Bigg{)}\,.$
(C.7)

Notice that $I^{(-1)}$ is entirely type B and $\delta I^{(-1)}=-4\sigma
I^{(-1)}$ under Weyl transformation up to $\mathcal{O}(\phi^{2})$. However,
$\mathcal{L}_{g}^{(2)}+\mathcal{L}_{h}^{(2)}$ contains also several free
coefficients, which are not fixed by the PBH equations, i.e.
$\\{c_{1},c_{2},h_{1},h_{21},\ldots,h_{26},h_{31},\ldots,h_{39}\\}$, and it is
possible to regularise (C.6) by renormalising some of these coefficients. In
particular, there is a unique shift which cancels $I^{(-1)}$ and it consists
of shifting some of the free coefficients which appear in $I^{(2)}$ as
follows:

$\displaystyle h_{32}\to h_{32}+\frac{c\,h_{1}}{18a(d-4)^{2}}\,,\quad$
$\displaystyle h_{33}\to h_{33}+\frac{7c\,h_{1}}{36a(d-4)^{2}}\,,\quad$
$\displaystyle h_{34}\to h_{34}+\frac{2c\,h_{1}}{9a(d-4)^{2}}\,,$
$\displaystyle h_{36}\to h_{36}-\frac{c\,h_{1}}{3a(d-4)^{2}}\,,\quad$
$\displaystyle h_{37}\to h_{37}-\frac{c\,h_{1}}{6a(d-4)^{2}}\,,\quad$
$\displaystyle h_{38}\to h_{38}+\frac{c\,h_{1}}{6a(d-4)^{2}}\,.$

Since the shifted coefficients only appear in $I^{(2)}$, this shift does not
introduce new singularities, and of course the regularised expression
satisfies the Wess-Zumino condition.

###  C.4 Complete expression of the anomaly

We provide here the complete expression of the holographic anomaly
$\mathcal{A}^{4d}$ as in (5.37), recalling that in the type A and B pure
gravity anomalies with the respective scalar field contributions we identify
the following terms

$\displaystyle\mathcal{A}^{4d}_{\text{B}}$
$\displaystyle=\mathcal{I}^{\text{pg}}_{\text{B}}+\mathcal{I}_{1}(h_{1})+\mathcal{I}_{2}(c_{1},h_{1},h_{2})+\mathcal{I}_{3}(c_{2},h_{1},h_{2})\,,$
(C.8) $\displaystyle\mathcal{A}^{4d}_{\text{A}}$
$\displaystyle=\mathcal{I}^{\text{pg}}_{\text{A}}+\mathcal{I}_{4}(h_{1})+\mathcal{I}_{5}(h_{2})+\mathcal{I}_{6}(h_{2})+\mathcal{I}_{7}(c_{1},h_{1},h_{2})+\mathcal{I}_{8}(c_{2},h_{1},h_{2})$
$\displaystyle\phantom{=}+\mathcal{I}_{9}(h_{21})+\mathcal{I}_{10}(h_{22})+\mathcal{I}_{11}(h_{23})+\mathcal{I}_{12}(h_{24})+\mathcal{I}_{13}(h_{25})+\mathcal{I}_{14}(h_{26})\,.$
(C.9)

We proceed by presenting and analysing them individually. Again, we assume
that $g_{ij}\equiv g^{(0)}_{ij}$ and curvatures and covariant derivatives are
w.r.t. $g^{(0)}_{ij}$, and $\phi\equiv\phi_{(0)}$. The interested reader can
also find the expressions of $\mathcal{A}^{4d}_{\text{B}}$ and
$\mathcal{A}^{4d}_{\text{A}}$ in the Mathematica notebook provided in the
supplementary material of [Bro22f].

As for $\mathcal{A}^{4d}_{\text{B}}$, we already presented
$\mathcal{I}^{\text{pg}}_{\text{B}}$ and $\mathcal{I}_{1}$ in (5.42). We move
then to $\mathcal{I}_{2}$, which is given by

$\mathcal{I}_{2}(c_{1},h_{1},h_{2})=4c\,c_{1}(h_{1}+4h_{2})\hat{R}\,C^{2}\,.$
(C.10)

This tensorial structure leads to a Weyl invariant anomaly integrand, as
already pointed out in [ST19f]. Thus, in the CFT it contributes to the anomaly
as an ‘ordinary’ anomaly, which satisfy the WZ condition without need of the
beta-function. Henceforth we will refer to this kind of structures as
‘ordinary anomaly’ for short. Notice that $\mathcal{I}_{2}$ vanishes in flat
space. A similar discussion follows for $\mathcal{I}_{3}$ since it is related
to $\mathcal{I}_{2}$ by

$\mathcal{I}_{3}(c_{2},h_{1},h_{2})=\mathcal{I}_{2}\left(\frac{c_{2}}{4},h_{1},h_{2}\right)\,.$
(C.11)

As for $\mathcal{A}^{4d}_{\text{A}}$, the tensorial structures
$\mathcal{I}^{\text{pg}}_{\text{A}}$ and $\mathcal{I}_{4}$ are given in (5.43)
and $\mathcal{I}_{5}$ is also already presented and discussed in the main
text. Then, we move to $\mathcal{I}_{6}$ which corresponds to ordinary
anomalies and thus satisfies the WZ condition without need of the metric beta-
function. Explicitly, it reads

$\displaystyle\mathcal{I}_{6}(h_{2})$
$\displaystyle=a\,h_{2}\Big{(}3\nabla^{j}\Box\phi\Box\nabla_{j}\phi-\tfrac{15}{2}\nabla^{j}\nabla^{i}\phi\Box\nabla_{j}\nabla_{i}\phi+\tfrac{15}{8}\Box\phi\Box^{2}\phi+\tfrac{9}{8}\phi\Box^{2}\phi$
$\displaystyle\phantom{=}-\tfrac{15}{2}\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi-\tfrac{3}{8}R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+\tfrac{5}{2}R_{ij}R^{ij}R\phi^{2}-\tfrac{23}{64}R^{3}\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{39}{8}R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{17}{64}RR_{ijkl}R^{ijkl}\phi^{2}+\tfrac{33}{4}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{9}{32}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}-\tfrac{3}{16}R\phi^{2}\Box
R-\tfrac{17}{8}R^{2}\phi\Box\phi-\tfrac{15}{8}R_{jklm}R^{jklm}\phi\Box\phi$
$\displaystyle\phantom{=}-\tfrac{1}{2}\phi^{2}\nabla_{i}R\nabla^{i}R-\tfrac{15}{4}R\phi\nabla_{i}\phi\nabla^{i}R+\tfrac{99}{4}R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{129}{8}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\tfrac{189}{8}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{23}{8}R^{2}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{51}{4}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{19}{8}R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R+6R^{ij}R\phi\nabla_{j}\nabla_{i}\phi+\tfrac{17}{8}\phi\Box
R\Box\phi$
$\displaystyle\phantom{=}+\tfrac{17}{8}R\Box\phi\Box\phi+\tfrac{1}{8}\nabla_{i}\phi\nabla^{i}R\Box\phi+\tfrac{3}{2}\phi\nabla^{i}\phi\Box\nabla_{i}R+\tfrac{21}{8}\phi\nabla^{i}R\Box\nabla_{i}\phi$
$\displaystyle\phantom{=}-\tfrac{15}{4}R\nabla^{i}\phi\Box\nabla_{i}\phi+\tfrac{3}{16}\phi^{2}\Box^{2}R+\tfrac{3}{2}R\phi\Box^{2}\phi-\tfrac{27}{4}R_{ij}\phi\nabla^{i}R\nabla^{j}\phi$
$\displaystyle\phantom{=}+\tfrac{27}{2}R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-\tfrac{37}{4}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+\tfrac{15}{2}R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}+\tfrac{7}{4}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi+\tfrac{23}{4}\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R-\tfrac{59}{8}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi$
$\displaystyle\phantom{=}+\tfrac{57}{4}R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}+\tfrac{15}{4}R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi+\tfrac{27}{2}R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}+\tfrac{9}{4}R^{ij}\phi^{2}\Box
R_{ij}-\tfrac{39}{4}\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}+\tfrac{21}{2}R_{ij}R^{ij}\phi\Box\phi-\tfrac{15}{4}R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi$
$\displaystyle\phantom{=}-9R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi+\tfrac{129}{16}\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}-\tfrac{9}{32}\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}+\tfrac{3}{2}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}+\tfrac{45}{4}R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi+3\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi$
$\displaystyle\phantom{=}-\tfrac{21}{2}\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi-18R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}-\tfrac{75}{4}R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-\tfrac{99}{4}R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi-\tfrac{3}{4}R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi-\tfrac{237}{64}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,.$
(C.12)

Notice that $\mathcal{I}_{6}$ vanishes in flat space after integration by
parts.

The tensorial structures in $\mathcal{I}_{7}$ correspond to Weyl invariant
ordinary anomalies and they are thus solutions of the WZ condition without
need of the metric beta-function. Explicitly, we have

$\displaystyle\mathcal{I}_{7}(c_{1},h_{1},h_{2})$
$\displaystyle=a\,c_{1}\Big{(}-48h_{1}R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+4(9h_{1}+8h_{2})R_{ij}R^{ij}R\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{16}{3}(h_{1}+h_{2})R^{3}\phi^{2}-4(h_{1}+4h_{2})RR_{ijkl}R^{ijkl}\phi^{2}-4h_{1}R\phi^{2}\Box
R$
$\displaystyle\phantom{=}-4(5h_{1}+8h_{2})R^{2}\phi\Box\phi-12(h_{1}+8h_{2})R_{jklm}R^{jklm}\phi\Box\phi$
$\displaystyle\phantom{=}-8h_{1}R\phi\nabla_{i}\phi\nabla^{i}R+48h_{1}R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi+24h_{1}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi$
$\displaystyle\phantom{=}-48(3h_{1}+8h_{2})R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi+8(3h_{1}+8h_{2})R^{2}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}+24(3h_{1}+8h_{2})R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+64h_{1}R^{ij}R\phi\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}-16h_{1}R\Box\phi\Box\phi+48h_{1}R_{ij}\phi\nabla^{i}R\nabla^{j}\phi+16h_{1}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi$
$\displaystyle\phantom{=}-96h_{1}R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}-96h_{1}R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi+24h_{1}R^{ij}\phi^{2}\Box
R_{ij}$
$\displaystyle\phantom{=}+24(3h_{1}+8h_{2})R_{ij}R^{ij}\phi\Box\phi+96h_{1}R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi$
$\displaystyle\phantom{=}-96h_{1}R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi+192h_{1}R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}$
$\displaystyle\phantom{=}-48h_{1}R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}-96h_{1}R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi$
$\displaystyle\phantom{=}-96h_{1}R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi\Big{)}\,.$
(C.13)

Notice that $\mathcal{I}_{7}$ vanishes in flat space. Similar properties hold
for $\mathcal{I}_{8}$ since it is related to $\mathcal{I}_{7}$ by

$\mathcal{I}_{8}(c_{2},h_{1},h_{2})=\mathcal{I}_{7}\left(\frac{c_{2}}{4},h_{1},h_{2}\right)\,.$
(C.14)

The remaining tensorial structures $\mathcal{I}_{9},\ldots,\mathcal{I}_{14}$
are parametrised by the free coefficients which appear in $h^{(2)}_{ij}$. Each
of them solves the WZ condition without need of the metric beta-function and
in the field theory they can be eliminated with counterterms. Thus, they
correspond to trivial solutions of the WZ condition. Explicitly, they read:

$\displaystyle\mathcal{I}_{9}(h_{21})$
$\displaystyle=a\,h_{21}\Big{(}-6R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+6R_{ij}R^{ij}R\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{3}{4}R^{3}\phi^{2}-6R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{3}{4}RR_{ijkl}R^{ijkl}\phi^{2}+6R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{3}{2}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}-\tfrac{1}{2}R\phi^{2}\Box
R-\tfrac{1}{2}R^{2}\phi\Box\phi-\tfrac{3}{2}R_{jklm}R^{jklm}\phi\Box\phi$
$\displaystyle\phantom{=}-\tfrac{1}{2}\phi^{2}\nabla_{i}R\nabla^{i}R-2R\phi\nabla_{i}\phi\nabla^{i}R+12R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi-6R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi$
$\displaystyle\phantom{=}+3R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{1}{2}R^{2}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{3}{2}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+3R^{ij}\phi^{2}\Box
R_{ij}$
$\displaystyle\phantom{=}+3R_{ij}R^{ij}\phi\Box\phi+3\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}-6R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-\tfrac{3}{2}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,,$
(C.15)

$\displaystyle\mathcal{I}_{10}(h_{22})$
$\displaystyle=a\,h_{22}\Big{(}3\nabla^{j}\Box\phi\Box\nabla_{j}\phi-3\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi+3R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{3}{2}R_{ij}R^{ij}R\phi^{2}+\tfrac{3}{16}R^{3}\phi^{2}+\tfrac{3}{16}RR_{ijkl}R^{ijkl}\phi^{2}-\tfrac{3}{2}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}$
$\displaystyle\phantom{=}-\tfrac{3}{8}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}+\tfrac{1}{2}R\phi^{2}\Box
R+\tfrac{3}{4}R^{2}\phi\Box\phi+\tfrac{3}{4}R_{jklm}R^{jklm}\phi\Box\phi$
$\displaystyle\phantom{=}+\tfrac{5}{8}\phi^{2}\nabla_{i}R\nabla^{i}R+\tfrac{3}{2}R\phi\nabla_{i}\phi\nabla^{i}R-9R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi+\tfrac{9}{4}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{3}{2}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{3}{8}R^{2}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{3}{8}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R$
$\displaystyle\phantom{=}-2R^{ij}R\phi\nabla_{j}\nabla_{i}\phi+\phi\Box
R\Box\phi+R\Box\phi\Box\phi+\tfrac{1}{2}\nabla_{i}\phi\nabla^{i}R\Box\phi+\tfrac{3}{2}\phi\nabla^{i}R\Box\nabla_{i}\phi$
$\displaystyle\phantom{=}-R_{ij}\phi\nabla^{i}R\nabla^{j}\phi+3R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-\tfrac{3}{2}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+3R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}-\tfrac{1}{2}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi+2\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R-R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi$
$\displaystyle\phantom{=}+3R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}+6R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi-3R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}-3R^{ij}\phi^{2}\Box
R_{ij}-6\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}-3R_{ij}R^{ij}\phi\Box\phi-6R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi$
$\displaystyle\phantom{=}+6R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi+3\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}-\tfrac{9}{2}\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-3\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}+3R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi-3\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi$
$\displaystyle\phantom{=}+3\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi-3R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}+9R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi$
$\displaystyle\phantom{=}+\tfrac{3}{8}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,,$
(C.16)

$\displaystyle\mathcal{I}_{11}(h_{23})$
$\displaystyle=a\,h_{23}\Big{(}\tfrac{3}{2}\nabla^{j}\Box\phi\Box\nabla_{j}\phi-\tfrac{3}{2}\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi-\tfrac{3}{2}R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{9}{4}R_{ij}R^{ij}R\phi^{2}-\tfrac{9}{32}R^{3}\phi^{2}-3R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{9}{32}RR_{ijkl}R^{ijkl}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{9}{4}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}+\tfrac{9}{16}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}+\tfrac{1}{8}R^{2}\phi\Box\phi$
$\displaystyle\phantom{=}-\tfrac{3}{8}R_{jklm}R^{jklm}\phi\Box\phi+\tfrac{1}{16}\phi^{2}\nabla_{i}R\nabla^{i}R-\tfrac{1}{4}R\phi\nabla_{i}\phi\nabla^{i}R+\tfrac{3}{2}R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{15}{8}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\tfrac{3}{4}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{1}{16}R^{2}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{9}{16}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{1}{2}R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R-R^{ij}R\phi\nabla_{j}\nabla_{i}\phi+\tfrac{1}{2}\phi\Box
R\Box\phi$
$\displaystyle\phantom{=}+\tfrac{1}{2}R\Box\phi\Box\phi+\tfrac{1}{4}\nabla_{i}\phi\nabla^{i}R\Box\phi+\tfrac{3}{4}\phi\nabla^{i}R\Box\nabla_{i}\phi-\tfrac{1}{2}R_{ij}\phi\nabla^{i}R\nabla^{j}\phi$
$\displaystyle\phantom{=}+\tfrac{3}{2}R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-\tfrac{3}{4}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+\tfrac{3}{2}R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}-\tfrac{1}{4}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi+\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R-\tfrac{1}{2}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi$
$\displaystyle\phantom{=}+\tfrac{3}{2}R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}+3R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi-\tfrac{3}{2}R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}-3\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}-3R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi+3R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi+\tfrac{3}{2}\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-\tfrac{3}{4}\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}-\tfrac{3}{2}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}+\tfrac{3}{2}R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi$
$\displaystyle\phantom{=}-\tfrac{3}{2}\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi+\tfrac{3}{2}\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi-\tfrac{3}{2}R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}$
$\displaystyle\phantom{=}-3R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}+\tfrac{9}{2}R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi-\tfrac{9}{16}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,,$
(C.17)

$\displaystyle\mathcal{I}_{12}(h_{24})$
$\displaystyle=a\,h_{24}\Big{(}3\nabla^{j}\Box\phi\Box\nabla_{j}\phi-3\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi-6R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{15}{2}R_{ij}R^{ij}R\phi^{2}-\tfrac{15}{16}R^{3}\phi^{2}-9R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{15}{16}RR_{ijkl}R^{ijkl}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{15}{2}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}+\tfrac{15}{8}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}-\tfrac{1}{4}R\phi^{2}\Box
R+\tfrac{1}{2}R^{2}\phi\Box\phi$
$\displaystyle\phantom{=}-\tfrac{3}{2}R_{jklm}R^{jklm}\phi\Box\phi-\tfrac{1}{8}\phi^{2}\nabla_{i}R\nabla^{i}R-\tfrac{3}{2}R\phi\nabla_{i}\phi\nabla^{i}R+9R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{27}{4}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\tfrac{3}{2}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{1}{8}R^{2}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{15}{8}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R-4R^{ij}R\phi\nabla_{j}\nabla_{i}\phi-\tfrac{1}{2}\nabla_{i}\phi\nabla^{i}\phi\Box
R$ $\displaystyle\phantom{=}+\tfrac{1}{2}\phi\Box
R\Box\phi+R\Box\phi\Box\phi+\tfrac{1}{2}\nabla_{i}\phi\nabla^{i}R\Box\phi+\tfrac{3}{2}\phi\nabla^{i}R\Box\nabla_{i}\phi-
R_{ij}\phi\nabla^{i}R\nabla^{j}\phi$
$\displaystyle\phantom{=}+3R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-\tfrac{7}{2}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+9R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}-\nabla^{i}\phi\nabla_{j}\nabla_{i}R\nabla^{j}\phi-\tfrac{1}{2}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi+\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R$
$\displaystyle\phantom{=}-R\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}\phi+3R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}+6R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi$
$\displaystyle\phantom{=}-3R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi+\tfrac{3}{2}R^{ij}\phi^{2}\Box
R_{ij}+3\nabla^{i}\phi\nabla^{j}\phi\Box
R_{ij}-3\phi\nabla^{j}\nabla^{i}\phi\Box R_{ij}$
$\displaystyle\phantom{=}-6R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi+6R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi+3\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-3\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}+3R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi-3\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi$
$\displaystyle\phantom{=}+3\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi-3R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}-9R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}+6R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi+9R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi-\tfrac{15}{8}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,,$
(C.18)

$\displaystyle\mathcal{I}_{13}(h_{25})$
$\displaystyle=a\,h_{25}\Big{(}\tfrac{1}{2}\nabla^{j}\Box\phi\Box\nabla_{j}\phi-4\nabla^{j}\nabla^{i}\phi\Box\nabla_{j}\nabla_{i}\phi+\Box\phi\Box^{2}\phi$
$\displaystyle\phantom{=}-\tfrac{7}{2}\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi-\tfrac{5}{2}R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+\tfrac{9}{4}R_{ij}R^{ij}R\phi^{2}-\tfrac{9}{32}R^{3}\phi^{2}$
$\displaystyle\phantom{=}-2R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{9}{32}RR_{ijkl}R^{ijkl}\phi^{2}+\tfrac{9}{4}R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{9}{16}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}-\tfrac{3}{4}R_{jklm}R^{jklm}\phi\Box\phi-\tfrac{1}{16}\phi^{2}\nabla_{i}R\nabla^{i}R$
$\displaystyle\phantom{=}+2R^{jk}\phi\nabla_{i}R_{jk}\nabla^{i}\phi-\tfrac{21}{8}R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\tfrac{3}{4}R_{jk}R^{jk}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{1}{16}R^{2}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{13}{16}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi-\tfrac{1}{4}R^{ij}\phi^{2}\nabla_{j}\nabla_{i}R+\tfrac{1}{4}\phi\Box
R\Box\phi$
$\displaystyle\phantom{=}+\tfrac{1}{2}R\Box\phi\Box\phi+\tfrac{3}{4}\nabla_{i}\phi\nabla^{i}R\Box\phi+\tfrac{3}{4}\phi\nabla^{i}R\Box\nabla_{i}\phi+R\nabla^{i}\phi\Box\nabla_{i}\phi+\tfrac{1}{2}R\phi\Box^{2}\phi$
$\displaystyle\phantom{=}-R_{ij}\phi\nabla^{i}R\nabla^{j}\phi+\tfrac{1}{2}R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-\tfrac{3}{4}R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+\tfrac{7}{2}R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}+\tfrac{1}{4}\nabla^{i}R\nabla_{j}\nabla_{i}\phi\nabla^{j}\phi-\tfrac{1}{2}\phi\nabla_{j}\nabla_{i}\phi\nabla^{j}\nabla^{i}R-\tfrac{1}{2}R^{jk}\phi\nabla^{i}\phi\nabla_{k}R_{ij}$
$\displaystyle\phantom{=}-R_{i}{}^{k}R^{ij}\phi\nabla_{k}\nabla_{j}\phi-\tfrac{7}{2}R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi+\tfrac{1}{2}R^{ij}\phi^{2}\Box
R_{ij}-\tfrac{1}{2}\phi\nabla^{j}\nabla^{i}\phi\Box R_{ij}$
$\displaystyle\phantom{=}+\tfrac{1}{2}R_{ij}R^{ij}\phi\Box\phi-2R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi-2R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi-2R^{ij}\phi\Box\nabla_{j}\nabla_{i}\phi$
$\displaystyle\phantom{=}-\phi^{2}\nabla_{j}R_{ik}\nabla^{k}R^{ij}+\tfrac{5}{4}\phi^{2}\nabla_{k}R_{ij}\nabla^{k}R^{ij}-\tfrac{7}{2}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}+\tfrac{1}{2}R^{ij}\nabla_{k}\nabla_{j}\phi\nabla^{k}\nabla_{i}\phi-\tfrac{5}{2}\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi-\tfrac{3}{2}\nabla^{i}\phi\nabla_{k}R_{ij}\nabla^{k}\nabla^{j}\phi$
$\displaystyle\phantom{=}-\tfrac{1}{2}R_{ijkl}\phi\nabla^{i}\phi\nabla^{l}R^{jk}-\tfrac{3}{2}R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}+R^{ij}R_{ikjl}\phi\nabla^{l}\nabla^{k}\phi$
$\displaystyle\phantom{=}-\tfrac{1}{2}R_{ikjl}\nabla^{j}\nabla^{i}\phi\nabla^{l}\nabla^{k}\phi-\tfrac{9}{16}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,,$
(C.19)

$\displaystyle\mathcal{I}_{14}(h_{26})$
$\displaystyle=a\,h_{26}\Big{(}3\nabla^{j}\Box\phi\Box\nabla_{j}\phi-12\nabla^{j}\nabla^{i}\phi\Box\nabla_{j}\nabla_{i}\phi+3\Box\phi\Box^{2}\phi$
$\displaystyle\phantom{=}-12\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}\nabla^{j}\nabla^{i}\phi-6R_{i}{}^{k}R^{ij}R_{jk}\phi^{2}+6R_{ij}R^{ij}R\phi^{2}-\tfrac{3}{4}R^{3}\phi^{2}$
$\displaystyle\phantom{=}-6R^{ij}R^{kl}R_{ikjl}\phi^{2}-\tfrac{3}{4}RR_{ijkl}R^{ijkl}\phi^{2}+6R_{i}{}^{m}{}_{k}{}^{n}R^{ijkl}R_{jmln}\phi^{2}$
$\displaystyle\phantom{=}+\tfrac{3}{2}R_{ij}{}^{mn}R^{ijkl}R_{klmn}\phi^{2}+\tfrac{1}{4}R\phi^{2}\Box
R+\tfrac{1}{4}R^{2}\phi\Box\phi-\tfrac{3}{2}R_{jklm}R^{jklm}\phi\Box\phi$
$\displaystyle\phantom{=}+\tfrac{1}{4}\phi^{2}\nabla_{i}R\nabla^{i}R+R\phi\nabla_{i}\phi\nabla^{i}R-6R^{jklm}\phi\nabla_{i}R_{jklm}\nabla^{i}\phi+\tfrac{1}{4}R^{2}\nabla_{i}\phi\nabla^{i}\phi$
$\displaystyle\phantom{=}-\tfrac{3}{2}R_{jklm}R^{jklm}\nabla_{i}\phi\nabla^{i}\phi+\tfrac{3}{2}\phi\Box
R\Box\phi+\tfrac{3}{2}R\Box\phi\Box\phi+3\nabla_{i}\phi\nabla^{i}R\Box\phi$
$\displaystyle\phantom{=}+3\phi\nabla^{i}R\Box\nabla_{i}\phi+3R\nabla^{i}\phi\Box\nabla_{i}\phi+\tfrac{3}{2}R\phi\Box^{2}\phi-3R_{ij}\phi\nabla^{i}R\nabla^{j}\phi$
$\displaystyle\phantom{=}+3R_{i}{}^{k}R_{jk}\nabla^{i}\phi\nabla^{j}\phi-3R_{ij}R\nabla^{i}\phi\nabla^{j}\phi+12R^{kl}R_{ikjl}\nabla^{i}\phi\nabla^{j}\phi$
$\displaystyle\phantom{=}-12R^{jk}\nabla^{i}\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi-6\phi\nabla^{j}\nabla^{i}\phi\Box
R_{ij}-6R^{ij}\nabla_{j}\nabla_{i}\phi\Box\phi$
$\displaystyle\phantom{=}-3R_{i}{}^{j}\nabla^{i}\phi\Box\nabla_{j}\phi-6R^{ij}\phi\Box\nabla_{j}\nabla_{i}\phi-12\phi\nabla_{k}\nabla_{j}\nabla_{i}\phi\nabla^{k}R^{ij}$
$\displaystyle\phantom{=}-12\nabla_{i}R_{jk}\nabla^{i}\phi\nabla^{k}\nabla^{j}\phi-6R_{ikjl}\phi^{2}\nabla^{l}\nabla^{k}R^{ij}-\tfrac{3}{2}\phi^{2}\nabla_{m}R_{ijkl}\nabla^{m}R^{ijkl}\Big{)}\,.$
(C.20)

Notice that $\mathcal{I}_{9}$ vanishes in flat space and similarly
$\mathcal{I}_{10},\ldots,\mathcal{I}_{14}$ after integration by parts.