Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations

Abstract

We derive residue formulas for the regularized integrals (introduced by Li and Zhou in Commun Math Phys 388:1403–1474, 2021) on configuration spaces of elliptic curves. Based on these formulas, we prove that the regularized integrals satisfy holomorphic anomaly equations, providing a mathematical formulation of the so-called contact term singularities. We also discuss residue formulas for the ordered A-cycle integrals and establish their relations with those for the regularized integrals.

A Some Combinatorial Properties of Residue Formulas

We discuss some combinatorial properties of residue formulas and their applications in this Appendix.

1.1 A.1 Further commutation relations of residue operators

We first give an extension of Lemma 3.3.

Lemma A.1

Introduce the notation $R^{(a)}_{0}=\int _{A} dz_{a}$. Then one has the following identities of operators acting on quasi-elliptic functions in z

• (i).
  $$\begin{aligned} R^{(a)}_{c}R^{(b)}_{c}-R^{(b)}_{c}R^{(a)}_{c}=R^{(b)}_{c}R^{(a)}_{b} =-R^{(a)}_{c}R^{(b)}_{a},\quad a,b\in [n],~ c\in [n] \cup \{0\}.\nonumber \\ \end{aligned}$$

  (A.1)
• (ii).
  $$\begin{aligned} R^{(e)}_{c}R^{(a)}_{b}=R^{(a)}_{b}R^{(e)}_{c}, \quad \textrm{if}\, \quad a,b,e\in [n], c\in [n]\cup \{0\},\quad \{a,b\}\cap \{c,e\}=\emptyset .\nonumber \\ \end{aligned}$$

  (A.2)

• (iii). When acting on almost-elliptic functions in z, one has

  $$\begin{aligned} R^{(b)}_{c}R^{(a)}_{b}+R^{(c)}_{a}R^{(b)}_{c}+R^{(a)}_{b}R^{(c)}_{a}=0, \quad \textrm{if}\, \quad a,b,c\in [n]\quad \text {are distinct}.\nonumber \\ \end{aligned}$$

  (A.3)

Here the various operators are understood to be composed in the natural operator ordering.

Proof

The first two identities with $c\ne 0$ are covered in Lemma 3.3. The first identity (A.1) with $c=0$ is proved by using the standard integral contour deformation argument on the complex plane. For the identity (A.2), by the global residue theorem and Lemma 3.3 one has

$$\begin{aligned} R^{(e)}_{0}R^{(a)}_{b}= & {} -(\sum _{\begin{array}{c} i:\, e\prec i\\ i\ne b \end{array}} R^{(e)}_{i} +R^{(e)}_{b}) R^{(a)}_{b}\nonumber \\= & {} -R^{(a)}_{b} \sum _{\begin{array}{c} i:\, e\prec i\\ i\ne b \end{array}} R^{(e)}_{i} -R^{(a)}_{b}R^{(e)}_{b}-R^{(a)}_{b}R^{(e)}_{a}+R^{(a)}_{b}R^{(e)}_{b}+R^{(a)}_{b}R^{(e)}_{a}-R^{(e)}_{b} R^{(a)}_{b}\nonumber \\= & {} R^{(a)}_{b} R^{(e)}_{0}. \end{aligned}$$

Here in the first equality we have used the fact that $R^{(a)}_{b}$ preserves meromorphicity, so that the global residue theorem can be applied to $R^{(e)}_{0}R^{(a)}_{b}$. The identity (A.3) is proved by a local calculation in a similar way as in Lemma 3.3. $\square$

The special case with $c=0$ in (A.2) reads that

$$\begin{aligned} R^{(a)}_{b}\int _{A_{i}}=\int _{A_{i}}R^{(a)}_{b} \,\quad \text {for distinct}~a,b,i,~\text {when acting on quasi-elliptic functions}. \end{aligned}$$

We also record the following interesting result that is parallel to the above commutation relation.

Corollary A.2

One has the following identity of operators

Proof

One proof is already implicitly given when proving (3.4) in Theorem 3.1 using the residue formula for regularized integrals. An alternative proof using the relation between regularized integrals and ordered A-cycle integrals is given as follows.⁴ For any almost-elliptic function $\Psi$, we write

$$\begin{aligned} \Psi =\sum _{k} \Psi _{k} ({{\overline{z}}_i-{z}_i\over {\overline{\tau }}-\tau })^k, \end{aligned}$$

where $\Psi _{k}$ is a quasi-elliptic function in $z_{i}$. Then using the expression of $\textrm{vol}$ given in Remark 2.7 and the Stokes theorem (see [LZ21, Lemma 3.26]), we have

(A.4)

By (A.4), linearity and Lemma A.1 with $c=0$, it suffices to prove the following identity

$$\begin{aligned} R^{(a)}_{b}\sum _{r:\,r\ne i}R^{(i)}_{r} \left( ({\overline{z}}_{i}-z_{i})^{k+1} \Psi _k\right) =\sum _{r:\,r\ne i,a}R^{(i)}_{r} \left( ({\overline{z}}_{i}-z_{i})^{k+1} R^{(a)}_{b} \Psi _k\right) , \end{aligned}$$

where $\Psi _k$ is quasi-elliptic in $z_{i}$. This follows from Lemma 3.3 that is derived by local computations and is valid when acting on $({\overline{z}}_{i}-z_{i})^{k+1} \Psi _k$. $\square$

1.2 A.2 Summation over planar rooted labeled forests

The sequence r in Propositions  2.11, 4.4 corresponds to rooted, labeled forest, as has been discussed in [LZ21]. We now elaborate on this. For each sequence r we associate a map

$$\begin{aligned} \varvec{r}:[n-1]\rightarrow [n],\quad \quad \varvec{r}(j)=r_{j}, ~j\in [n-1]. \end{aligned}$$

(A.5)

satisfying

$$\begin{aligned} j<\varvec{r}(j),\quad j\in [n-1]. \end{aligned}$$

(A.6)

The data (A.5) gives rise to a rooted, labeled forest, with the vertices labeled by $\varvec{r}^{-1}(n)$ designated as the roots. Define an equivalence relation $\sim$ on $[n-1]$ by declaring

$$\begin{aligned} j\sim \varvec{r}(j),\quad j\in [n-1]. \end{aligned}$$

Then $[n-1]$ is decomposed into several disjoint subsets

$$\begin{aligned}{}[n-1]=\bigsqcup _{k\in \varvec{r}^{-1}(n)} \Gamma _{k}. \end{aligned}$$

(A.7)

We call $\varvec{r}|_{\Gamma _{k}}$ a tree in the forest $\varvec{r}$.

Consider the blackboard orientation in which vertices that have valency zero or one are ordered from right to left, vertices within a path in a tree are ordered from bottom to the root in the top. Due to the constraint (A.6), each sequence $\varvec{r}$ above gives a planar/embdedded, rooted, labelled forest (called forest for short), whose labelings are compatible with the blackboard orientation in the sense that with respect to <

• labelings for the roots $\varvec{r}^{-1}(n)$ increases from right to left;

• within a rooted tree, labelings increases from from bottom to top (towards the root);

• within a rooted tree, labelings of children of the same vertex increase from right to left.

By Lemma 3.3, when acting on almost-elliptic functions the operator $R^{[n-1]}_{\varvec{r}}= R^{(n-1)}_{\varvec{r}(n-1)} \cdots R^{(2)}_{\varvec{r}(2)} R^{(1)}_{\varvec{r}(1)}$ can be factored into a product of residue operators associated to the trees

$$\begin{aligned} R^{[n-1]}_{\varvec{r}}=\prod _{k}R_{\Gamma _{k}}. \end{aligned}$$

(A.8)

Similarly, the integration operator (2.10) factor into a product of commuting integration operators. The condition (A.6) or equivalently the blackboard orientation records the operator orderings in the residue and integration operators within each $\Gamma _{k}$ (permutations of different trees and of the leaves within the trees are not allowed by the embedding data). In particular, when acting on an elliptic function $\Phi$ one has

(A.9)

where the summation $\sum ^{<}$ on the right hand side is over forests whose labelings are compatible with the blackboard orientation, with respect to <.

Remark A.3

A rooted labeled forest $\varvec{r}$ above gives rise to a rooted labeled tree obtained by joining all the roots in the rooted forest to the vertex n now designated as the new root. The joint tree corresponds to an element in the basis of the cohomology group $H^{n-1}(\textrm{Conf}_{n}(\mathbb {C}),\mathbb {C})$ whose rank is the signed first Stirling number $(-1)^{n-1}s(n,1)=(n-1)!$. The relation $R^{(j)}_{n}R^{(i)}_{j}=-R^{(i)}_{n}R^{(j)}_{i}$ and the Jacobi identity

$$\begin{aligned}{}[[R^{(a)}_{n},R^{(b)}_{n} ],R^{(c)}_{n}]+[[R^{(b)}_{n},R^{(c)}_{n} ],R^{(a)}_{n}]+[[R^{(c)}_{n},R^{(a)}_{n} ],R^{(b)}_{n}]=0. \end{aligned}$$

from Lemma A.1 then correspond to the anti-symmetry and Arnold relation (assembling the same form as Lemma A.1 (iii)) in the cohomology ring $H^{*}(\textrm{Conf}_{n}(\mathbb {C}),\mathbb {C})$. See [Tot96, Get99] for details. It seems that our decomposition here is related to the mixed Hodge structures on $H^{*}(\textrm{Conf}_{n}(E),\mathbb {C})$. We hope to discuss the algebraic formulation of regularized integrals using the language of mixed Hodge structures in a future investigation, following the lines in [Tot96, Get99].

There is in fact a bijection between the set of planar, rooted, labeled forests that are compatible with blackboard orientation and $\mathfrak {S}_{[n-1]}$ given as follows.

• For each $\phi \in \mathfrak {S}_{[n-1]}$, consider its one-line notation $\phi =\phi _1 \phi _2\cdots \phi _{n-1}$. Read from left to right, the maxima correspond to roots of the trees. From the remaining entries of $\phi$, $\phi _{a}$ is a child of $\phi _{b}$ if $\phi _{b}$ is the rightmost entry that precedes $\phi _{a}$ and is larger than $\phi _{a}$. This gives a rooted, labeled forest and thus a unique planar one that is compatible with backboard orientation.

• For each planar, rooted, labeled forest compatible with backboard orientation, one starts at the vertex n of the forest and then traverses the forest clockwise. Reading off each label as it is reached, this yields an element in $\mathfrak {S}_{[n-1]}$ in its one-line notation.

Using the above correspondence, (A.9) can be alternatively written as

(A.10)

where $\varvec{r}(\phi )$ and equivalently $\Gamma (\phi )$ is the forest compatible with blackboard orientation that is determined by $\phi$.

A different ordering $\prec$ is determined by $\sigma \in \mathfrak {S}_{[n-1]}$ via the integration region $A_{\sigma ([n-1])}$ or $E_{\sigma ([n-1])}$. The ordering $\prec$ is related to the magnitude ordering $<:1<2<\cdots < n-1$ by the action by $\sigma$

$$\begin{aligned} \sigma _{1}\prec \sigma _{2}\prec \cdots \prec \sigma _{n-1}. \end{aligned}$$

(A.11)

Correspondingly, the map $\varvec{r}$ is changed to $\varvec{r}':\sigma ([n-1])=[n-1]\rightarrow [n]$ with the convention $\sigma (n)=n$. It satisfies

$$\begin{aligned} \sigma _{i}\prec \varvec{r}'(\sigma _{i}) \end{aligned}$$

(A.12)

instead of (A.6). The set of such functions $\varvec{r}'$ is in one-to-one correspondence with the set of the functions $\sigma (\varvec{r}):=\sigma \circ \varvec{r}\circ \sigma ^{-1}$, with the convention $\sigma (n)=n$. Correspondingly, we have

(A.13)

with $\varvec{r}$ satisfying (A.5), (A.6).

Similarly to (A.9), (A.13) is a sum over the collection of embedded, rooted, labeled forests in (A.9), with the new labelings obtained by applying $\sigma$ on the original ones. Now the new labelings are compatible with the blackboard orientation not with respect to < but with respect to $\prec$ in (A.11). See Fig. 1 below for an illustration.

Fig. 1

From left to right: picking representatives compatible with blackboard orientation. From top to bottom: applying permutation $\sigma$ to obtain labeled rooted forests in the residue formulas of regularized integrals, with the map $\varvec{r}'$ given by $\sigma \circ \varvec{r}\circ \sigma ^{-1}$

By the independence on the ordering of the regularized integral, we have

(A.14)

Now summing over the orderings $\prec$ in (A.11) is equivalent to summing over the relabelings of the embedded, labeled rooted forests in (A.9), with the operator orderings for the residue and integration operators given by the blackboard orientation. It follows that

(A.15)

where $\sigma \circ \Gamma _{k}$ is the tree whose labelings are obtained from the $\sigma$-action on those in $\Gamma _{k}$. Denote the set of labelings of $\Gamma _{k}$ by $\rho _{k}\subseteq [n-1]$, then the summation can be further expressed as a sum

$$\begin{aligned} \sum _{\Gamma =(\Gamma _{k})}^{<} \sum _{\sigma \in \mathfrak {S}_{[n-1]}} = \sum _{[\Gamma ]=([\Gamma _{k}])} \sum _{\rho =(\rho _{k})\in \Gamma _{[n-1]}}\sum _{\tau \in \prod _{k}\mathfrak {S}_{\rho _{k}}}, \end{aligned}$$

where $[\Gamma ]$ is the forest without the labelings and $\Gamma _{[n-1]}$ is the set of compositions of $[n-1]$. Therefore, we obtain

(A.16)

One can also exchange the order of the summation by summing over $\rho$ in the outmost one. Then the summation is related to summation over different types of forests with fixed number of vertices in each of the trees.

Example A.4

Consider the $n=3$ case. The two forests in (A.9) are given by the top left two in Fig. 2 below. In this case, $\mathfrak {S}_{n-1}$ consists of two elements $\sigma =(1)(2), (12)$. Hence the forests in (A.15) are obtained by including two more forests indicated in the top right two in Fig. 2. The resummation in (A.16) is illustrated in the data in the bottom.

Fig. 2

The left two forests correspond to summands in $\int _{E_3}\int _{E_2}\int _{E_1}$, the right two correspond to the summands in $\int _{E_3}\int _{E_1}\int _{E_2}$

Similar results hold for ordered A-cycle integrals, except now one does not have (A.14) for a general elliptic function $\Phi$.

The structures in (A.9) and (A.16) elucidate many properties of regularized integrals and ordered A-cycle integrals. For example, Corollary 2.13 follows directly from this and the choice $F_{I}={\widehat{Z}}_{i_{1}}^{m}/m!$ given in (2.12) for a non-recurring sequence $I=(i_{1},\cdots , i_{m}),i_{m+1}=n$.

1.3 A.3 Holomorphic limit of regularized integrals

In Proposition 4.5 (ii), firstly established by [LZ21, Theorem 3.4 (2)], we used Proposition 4.4 (iii) proved by Oberdieck-Pixton in [OP18, Proposition 10]. We now give a proof of Proposition 4.5 (ii) that is simpler than the one in [LZ21], following the residue operator approach. According to Remark 4.6, this also provides a different proof of Proposition 4.4 (iii).

Lemma A.5

[LZ21, Lemma 3.26, Lemma 3.37]. As operators acting on elliptic functions in ${\mathcal {J}}_{n}$, one has

(A.17)

where

• $\prec$ is the prescribed ordering < by magnitude as in $\int _{A_{[n]}}$.

• $J=(j_1,\cdots , j_\ell )$ is an non-recurring sequence: $j_1\prec j_2\prec \cdots \prec j_\ell$ with cardinality $|J|=\ell$.

• $\mathfrak {r}_{J}=(\mathfrak {r}_{j_1},\cdots , \mathfrak {r}_{j_\ell })$ satisfies $j_a\prec \mathfrak {r}_{j_{a}}, \mathfrak {r}_{j_{a}}\in [n]$ for $a=1,2,\cdots ,\ell$.

• $C_{J,\mathfrak {r}_{J}}=\int _{0}^{1}dx_{\mathfrak {r}_{j_{\ell }}}\int _{0}^{x_{\mathfrak {r}_{j_{\ell }}}}dx_{j_{\ell }}\cdots \int _{0}^{ x_{\mathfrak {r}_{j_{1}}}}dx_{j_{1}}$.

• $R^{J}_{\mathfrak {r}_{J}}=R^{(j_\ell )}_{\mathfrak {r}_{j_\ell }}\circ \cdots \circ R^{(j_1)}_{\mathfrak {r}_{j_1}}$ and the notation $\left( \int _{A_{[n]-J}} R^{J}_{\mathfrak {r}_{J}}\right) ^{\prec }$ stands for the operator arranged according the ordering $\prec$.

Proof

We only sketch the proof. For any almost-elliptic function $\Psi$, $\Psi =\sum _{k} \Psi _{k} ({{\overline{z}}_i-{z}_i\over {\overline{\tau }}-\tau })^k,$ continuing with (A.4), one has

$$\begin{aligned}= & {} \sum _{k} {1\over k+1}\int _{A} \Psi _{k} +2\pi i \sum _{k}{1\over k+1}\sum _{r} R^{(i)}_{r} \left( ({{\overline{z}}_i-{\overline{z}}_r+{\overline{z}}_r-z_{r}+z_{r}-z_{i}\over {\overline{\tau }}-\tau })^{k+1} \Psi _{k}\right) \nonumber \\= & {} \sum _{k} {1\over k+1}\int _{A} \Psi _{k}\\{} & {} +2\pi i \sum _{k}{1\over k+1}\sum _{\ell :\, 0\le \ell \le k+1}{k+1\atopwithdelims ()\ell } ({{\overline{z}}_r-z_{r}\over {\overline{\tau }}-\tau })^{k+1-\ell }({1\over {\overline{\tau }}-\tau })^{\ell } \sum _{r}R^{(i)}_{r} \left( (z_{r}-z_{i})^{\ell } \Psi _{k}\right) . \end{aligned}$$

Here in the last equality we have used (2.8). It follows that as operators on the almost-elliptic function $\Psi$ one has

One then expands iterated regularized integrals of almost-elliptic functions in terms of iterated A-cycle integrals and residues (see [LZ21, Lemma 3.26]). Applying the above reasoning to get rid of terms that do not contribute in the holomorphic limit, the desired claim then follows, similar to the proof of Proposition 2.11. Note that there is no term with $J=[n]$ by the above constraint on $J,\mathfrak {r}_{J}$, as it should be the case according to the behavior of the pole structure under residue operations. $\square$

Remark A.6

When performing the operations in Lemma A.5, we do not use $z_{n}$ as the reference as in Proposition 2.11, so that the $\mathfrak {S}_{n}$-symmetry among all the indices in [n] is retained. Of course the same relation holds if we do apply the referencing.

Before proceeding to give another proof of Proposition 4.5 (ii), we introduce some notations. Similar to the discussions in Appendix 4.2, for any pair $(J,\mathfrak {r}_{J})$ in Lemma A.5, we define a map

$$\begin{aligned} \varvec{r}:[n]\rightarrow [n]\cup \{0\},\quad \quad \varvec{r}(j)=\mathfrak {r}_{j}, ~j\in J\subseteq [n],\quad \varvec{r}(k)=0,~ k\in [n]-J. \end{aligned}$$

(A.18)

It is clear that the data $(J,\mathfrak {r}_{J})$ is equivalent to a map $\varvec{r}$ in (A.18) satisfying $j\prec \varvec{r}(j)$ for any $j\in \varvec{r}^{-1}([n])$. In what follows we shall not distinguish them. Denote

$$\begin{aligned} \mathcal {R}_{\varvec{r}}:= \left( \int _{A_{[n]-J}} R^{J}_{\mathfrak {r}_{J}}\right) ^{\prec }= R^{(n)}_{\varvec{r}(n)} \cdots R^{(2)}_{\varvec{r}(2)} R^{(1)}_{\varvec{r}(1)}. \end{aligned}$$

(A.19)

Again we call the data $\varvec{r}$ in (A.18) associated to $(J,\mathfrak {r}_{J})$ in Lemma A.5 a forest, and any element in [n] a vertex. By defining an equivalence relation $\sim$ on [n] by declaring

$$\begin{aligned} j\sim \varvec{r}(j),\quad j\in J, \end{aligned}$$

the set [n] is decomposed into several disjoint subsets called trees⁵

$$\begin{aligned}{}[n]=\bigsqcup _{k\in \varvec{r}^{-1}(0)} \Gamma _{k}. \end{aligned}$$

(A.20)

It is clear that the index set $\varvec{r}^{-1}(0)$ satisfies $|\varvec{r}^{-1}(0)|=1$ if and only if $|\mathfrak {r}_{J}-J|=1$, in which case the forest itself is a tree.

The following result Lemma A.7 relates the residue operators $\mathcal {R}_{\varvec{r}}$ associated to trees and forests to elements in the free algebra generated by the operator variables $R^{(i)}_{0},i\in [n]$.

Lemma A.7

Let the notations be as above.

• (i). (Residue operator associated to a tree) Assume $2\le m\le n$. Let $\mathfrak {r}=(\mathfrak {r}_{1},\cdots , \mathfrak {r}_{m-1})$ be a sequence satisfying

  $$\begin{aligned} \mathfrak {r}_{a}\in [m],~ \mathfrak {r}_{m-1}=m,\quad \quad a<\mathfrak {r}_{a},~\text {for}~a=1,2,\cdots , m-1. \end{aligned}$$

  Then for any $c\in [n]\cup \{0\}-[m]$, the operator on quasi-elliptic functions

  $$\begin{aligned} R^{(m)}_{c}\circ R^{[m-1]}_{\mathfrak {r}} :=R^{(m)}_{c}\circ R^{(m-1)}_{m}\circ \cdots R^{(2)}_{\mathfrak {r}_{2}}\circ R^{(1)}_{\mathfrak {r}_{1}} \end{aligned}$$

  is sum of nested commutators of m residue operators of the form $R^{(a)}_{c}, a\in [m]$.

• (ii). (Residue operator associated to a forest) The operator $\mathcal {R}_{\varvec{r}}$ in (A.19) is a product $L_{v}\cdots L_{1}$ of $v=|\varvec{r}^{-1}(0)|$ factors $L_{k},k=1,2,\cdots , v$, with the factors being the operators associated to the trees in the decomposition (A.7). In particular, each $L_{k}$ is either of the form $R^{(i)}_{0},i\in [n]$ or a sum of nested commutators of $m_k$ such operators, where $m_k$ is the number of vertices of the tree corresponding to $L_{k}$.⁶

The constructive proof of Lemma A.7 is essentially contained in [LZ21]. To prove Lemma A.7 (i), we first show the statement for operators of the form $R^{(b)}_{c}\prod _{i}R^{(i)}_{b}$ using Lemma A.1, then use induction on the height (i.e., the smallest number h such that $\varvec{r}^{h}(j)=0$) of the vertices $j\in [n]$. After that, one applies Lemma A.1 to move the residue operators $R^{(a)}_{b},b\ne 0$ across the $R^{(e)}_{0}$ operators and reduce the discussions on forests in Lemma A.7 (ii) to those on trees in Lemma A.7 (i). The details are elementary but tedious, and are therefore omitted.

Remark A.8

We can in fact show that the operator $R^{(m)}_{c}\circ R^{[m-1]}_{\mathfrak {r}}$ in Lemma A.7 (i) and each $L_{k}$ in Lemma A.7 (ii), provided that the total number $m_{k}$ of vertices in the corresponding tree is at least 2, can be expressed as a nested commutator instead of a sum of nested commutators. A graphic rule, which is essentially given by the blackboard orientation, can be found in [LZ21, Section 3.4].

Furthermore, using Lemma A.1 one can show that the compositions of residue operators $\mathcal {R}_{\varvec{r}}$ can be conveniently described using the gluing of labeled rooted forests in the indicated orderings. Lemma A.1 can then be interpreted as saying that the algebra generated by the residue operators $R^{(a)}_{b},a\in [n],b\in [n]\cup \{0\}$ is acted on by the underlying operad of labeled rooted forests.

Another proof of Proposition 4.5 (ii)

Consider the average of (A.17) over $\sigma \in \mathfrak {S}_{n}$. Observe that $C_{J,\mathfrak {r}_{J}}$ is independent of the ordering of elements in the underlying set of $J\cup \mathfrak {r}_{J}$, as can be seen from its expression. Thanks to this property, to prove the desired statement Proposition 4.5 (ii) it is enough to show that⁷

$$\begin{aligned} \sum _{\sigma \in \mathfrak {S}_{n}}\mathcal {R}_{\sigma (\varvec{r})}=0,\quad ~\text {if}\quad \varvec{r}^{-1}([n])\ne \emptyset . \end{aligned}$$

(A.21)

Here as in (A.13) and (A.15), we have $\mathcal {R}_{\sigma (\varvec{r})}= R^{(\sigma ([n-1]))}_{\sigma (\varvec{r})(\sigma ([n-1]))},\sigma (\varvec{r})=\sigma \circ \varvec{r}\circ \sigma ^{-1}$ with the convention $\sigma (0)=0$.

For any such $\varvec{r}$, applying Lemma A.7 (ii) we obtain a decomposition $\mathcal {R}_{\varvec{r}}=L_{v}\cdots L_1$. The constraint on $\varvec{r}$ in Lemma A.5 implies that the number $v=|\varvec{r}^{-1}(0)|$ of trees in the decomposition (A.20) satisfies $v\le n-1$. This tells that there is at least a factor above, say $L_{*}$, such that the number m of vertices of the corresponding tree T is at least 2. Therefore, $L_{*}$ is a sum of m-nested commutators. Denote the set of vertices of the tree T by V. The summation over $\mathfrak {S}_{n}$ can be organized into a multi-summation similar to (A.16): over the choices for the underlying set of V, over the orderings of elements in the underlying set of V, and over the orderings of the rest of the elements in $\{1,2,\cdots , n\}$. Now to prove (A.21) it suffices to show that

$$\begin{aligned} \sum _{\sigma \in \mathfrak {S}(V)}\sigma (L_{*})=0, \end{aligned}$$

(A.22)

where $\sigma (L_{*})$ is the operator obtained by permuting the indices of the m-nested residue operators in $L_{*}$ using $\sigma$ from the permutation group $\mathfrak {S}(V)$.

Let K be any m-nested commutator summand of $L_{*}$. Consider the upper-indices, say $a_1,a_2$, of an inner-most commutator of K. The summation $\sum _{\sigma \in \mathfrak {S}(V)}\sigma (K)$ above is a multi-summation, with one of them being the sum over the permutation group of $\{a_1,a_2\}$. The result is therefore zero by the anti-symmetry⁸ of the nested commutator K in $a_1,a_2$. By linearity, this then implies (A.22). The proof is now complete. $\square$