Black Holes and Higher Depth Mock Modular Forms

Abstract

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi–Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4–D2–D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor ${\mathcal {D}}$, at the large volume attractor point. For ${\mathcal {D}}$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on ${\mathcal {D}}$ and is therefore known to be modular. Instead, when ${\mathcal {D}}$ is the sum of n irreducible divisors ${\mathcal {D}}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary n by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.

Appendices

A Proof of the Theorem

In this appendix we prove Theorem 1 presented in the Introduction. The crucial ingredient is provided by the following Lemma³²

Lemma 1

The number of ways of labelling the vertices of a rooted ordered tree T with increasing labels is given by

$$\begin{aligned} N_T=n_T! \prod _{v\in V_T}\frac{1}{n_v(T)}\, , \end{aligned}$$

(A.1)

where $n_T$ is the number of vertices of the tree T.

Proof

First, let us find a recursive relation between the numbers $N_T$. Namely, consider a vertex v and the set of its children $\mathrm{Ch}(v)$. Denote by $T_v$ the tree rooted at v and with leaves coinciding with leaves of T, for which one has $n_{T_v}=n_v$. Then it is clear that

$$\begin{aligned} N_{T_v}=\frac{\left( \sum _{v'\in \mathrm{Ch}(v)}n_{v'}\right) ! }{\prod _{v'\in \mathrm{Ch}(v)}n_{v'}!}\prod _{v'\in \mathrm{Ch}(v)}N_{T_{v'}}\, . \end{aligned}$$

(A.2)

From this relation, one easily shows by recursion that (A.1) follows. Indeed, assuming that it holds for the subtrees $T_{v'}$ and taking into account that $\sum _{v'\in \mathrm{Ch}(v)}n_{v'}=n_v-1$, the r.h.s. can be rewritten as

$$\begin{aligned} (n_v-1)!\prod _{v'\in \mathrm{Ch}(v)}\prod _{v''\in V_{T_{v'}}}\frac{1}{n_{v''}} =n_v!\prod _{v'\in V_{T_v}}\frac{1}{n_{v'}}, \end{aligned}$$

(A.3)

which coincides with (A.1) when v is the root of T. $\quad \square$

Given this Lemma, we can now rewrite the l.h.s. of (1.2) as

$$\begin{aligned} \sum _{T'\subset T}\frac{N_{T'}}{m!}\prod _{v\in V_{T'}}n_v(T) =\frac{n!}{N_{T}}\sum _{T'\subset T}\frac{N_{T'}}{m!}\prod _{T_r\subset T{\setminus } T'}\frac{N_{T_r}}{n_{T_r}!}, \end{aligned}$$

(A.4)

where the last product goes over the subtrees which complement $T'$ to the full tree T. On the other hand, it is easy to check that the relation (A.2) is the special case $m=1$ of a more general relation

$$\begin{aligned} N_{T}=\frac{\left( \sum _{T_r\subset T{\setminus } T'}n_{T_r}\right) !}{\prod _{T_r\subset T{\setminus } T'}n_{T_r}!} \sum _{T'\subset T}N_{T'}\prod _{T_r\subset T{\setminus } T'}N_{T_r}. \end{aligned}$$

(A.5)

where the sum runs over subtrees $T'$ with m vertices. Taking into account that $\sum _{T_r\subset T{\setminus } T'}n_{T_r}=n-m$ and substituting the resulting expression into (A.4), one recovers the binomial coefficient as stated in the Theorem.

B. D3-Instanton Contribution to the Contact Potential

To evaluate $(e^\phi )_{\mathrm{D3}}$, we first replace in (3.20) the full prepotential F by its classical, cubic part $F^{\mathrm{cl}}$ (2.22) and take into account that the sum over $\gamma \in -\Gamma _+$ is complex conjugate to the sum over $\gamma \in \Gamma _+$. This gives

$$\begin{aligned} e^{\phi } \approx \frac{\tau _2^2}{12}\,((\,\mathrm{Im}\,u)^3) +\frac{\mathrm {i}\tau _2}{8}\,\sum _{\gamma \in \Gamma _+}\,\mathrm{Re}\,\int _{\ell _\gamma } \frac{\text {d}t}{t} \left( t^{-1} Z_\gamma (u^a)-t{\bar{Z}}_\gamma ({\bar{u}}^a)\right) H_\gamma . \end{aligned}$$

(B.1)

Next, we substitute the quantum corrected mirror map (3.8), change the integration variable t to z, and take the combined limit $t^a\rightarrow \infty$, $z\rightarrow 0$. Keeping only the leading contributions, one obtains

$$\begin{aligned} (e^\phi )_{\mathrm{D3}}= & {} -\frac{\tau _2}{2}\sum _{\gamma \in \Gamma _+}\,\mathrm{Re}\,\int _{\ell _{\gamma }}\mathrm {d}z \left[ {\hat{q}}_0+{1\over 2}\, (q+b)^2 +2(pt^2)zz_\gamma -\frac{3}{2}\, z^2(pt^2)\right] H_{\gamma } \nonumber \\&-\frac{1}{4}\sum _{\gamma _1,\gamma _2\in \Gamma _+}(tp_1p_2) \left( \,\mathrm{Re}\,\int _{\ell _{\gamma _1}}\!\!\mathrm {d}z_1\, H_{\gamma _1}\right) \left( \,\mathrm{Re}\,\int _{\ell _{\gamma _2}}\!\!\mathrm {d}z_2\, H_{\gamma _2}\right) . \end{aligned}$$

(B.2)

To further simplify this expression, we note that

$$\begin{aligned} \begin{aligned} 0=&\, \frac{1}{4\pi }\sum _{\gamma \in \Gamma _+}\int _{\ell _{\gamma }}\mathrm {d}z\, \partial _z\left( z\,H_\gamma \right) \\ =&\,\sum _{\gamma \in \Gamma _+}\int _{\ell _{\gamma }}\mathrm {d}z \left[ \frac{1}{4\pi }+\tau _2(pt^2) (zz_\gamma - z^2) -\frac{\mathrm {i}}{4}\sum _{\gamma '\in \Gamma _+}\langle \gamma ,\gamma '\rangle \int _{\ell _{\gamma '}} \frac{\mathrm {d}z'}{z-z'}\, H_{\gamma '}\right] H_\gamma , \end{aligned} \nonumber \\ \end{aligned}$$

(B.3)

where we used the integral equation (3.15). Multiplying this identity by 3/4 and adding its real part to (B.2), one finds

$$\begin{aligned} (e^\phi )_{\mathrm{D3}}= & {} \frac{\tau _2}{2}\sum _{\gamma \in \Gamma _+}\,\mathrm{Re}\,\int _{\ell _{\gamma }}\mathrm {d}z\, a_{\gamma ,-\frac{3}{2}}(z)\,H_{\gamma } \nonumber \\&-\frac{1}{4}\sum _{\gamma _1,\gamma _2\in \Gamma _+}\left[ (tp_1p_2) \left( \,\mathrm{Re}\,\int _{\ell _{\gamma _1}}\!\!\mathrm {d}z_1\, H_{\gamma _1}\right) \left( \,\mathrm{Re}\,\int _{\ell _{\gamma _2}}\!\!\mathrm {d}z_2\, H_{\gamma _2}\right) \right. \nonumber \\&\left. +\frac{3}{4}\,\,\mathrm{Re}\,\left( \int _{\ell _{\gamma _1}}\!\!\mathrm {d}z_1\, H_{\gamma _1}\int _{\ell _{\gamma _2}}\!\!\mathrm {d}z_2\, H_{\gamma _2} \, \frac{\mathrm {i}\langle \gamma ,\gamma '\rangle }{z-z'}\right) \right] , \end{aligned}$$

(B.4)

where we introduced

$$\begin{aligned} a_{\gamma ,{\mathfrak {h}}}(z)=-\left( {\hat{q}}_0+{1\over 2}\,(q+b)^2 +\frac{1}{2}\, (pt^2)zz_\gamma +\frac{{\mathfrak {h}}}{4\pi \tau _2}\right) . \end{aligned}$$

(B.5)

The meaning of this function is actually very simple: it gives the action of the modular covariant derivative operator ${\mathcal {D}}_{{\mathfrak {h}}}$ (3.23) on the classical part of the Darboux coordinate (3.12),

$$\begin{aligned} {\mathcal {D}}_{{\mathfrak {h}}}{\mathcal {X}}^{\mathrm{cl}}_{\gamma } =a_{\gamma ,{\mathfrak {h}}}(z){\mathcal {X}}^{\mathrm{cl}}_{\gamma }. \end{aligned}$$

(B.6)

Combining this fact with equation (3.15), it is easy to check that the function (3.22) satisfies, for any value of the weight ${\mathfrak {h}}$,

$$\begin{aligned} {\mathcal {D}}_{{\mathfrak {h}}}{\mathcal {G}}= & {} \sum _{\gamma \in \Gamma _+}\int _{\ell _{\gamma }} \mathrm {d}z\,a_{\gamma ,{\mathfrak {h}}}(z)\, H_{\gamma } +\frac{1}{8\tau _2}\sum _{\gamma _1,\gamma _2\in \Gamma _+} \int _{\ell _{\gamma _1}}\mathrm {d}z_1\,\nonumber \\&\quad \int _{\ell _{\gamma _2}} \mathrm {d}z_2 \left[ (tp_1p_2)+2{\mathfrak {h}}\left( (tp_1p_2)+\frac{\mathrm {i}\langle \gamma _1,\gamma _2\rangle }{z_1-z_2} \right) \right] H_{\gamma _1}(z_1)H_{\gamma _2}(z_2), \end{aligned}$$

(B.7)

$$\begin{aligned} \partial _{{\tilde{c}}_a}{\mathcal {G}}= & {} 2\pi \mathrm {i}\sum _{\gamma \in \Gamma _+}p^a \int _{\ell _{\gamma }} \mathrm {d}z\, H_{\gamma }, \end{aligned}$$

(B.8)

which allows to rewrite (B.4) exactly as in (3.24).

C. Smoothness of the Instanton Generating Function

In this appendix we prove the smoothness of the function ${\mathcal {G}}$ across walls of marginal stability. The starting point is the representation (4.2) where the potential discontinuities are hidden in the kernel of the theta series (4.3). Given that $\Phi ^{\scriptscriptstyle \,\int }_{m}$ is represented as a sum over unrooted labelled trees (4.6), whereas $\Phi ^{\,g}_{n_k}$ appear as sums over flow trees (4.15), the kernel $\Phi ^{\mathrm{tot}}_{n}$ can be viewed as a sum over ‘blooming trees’ which are unrooted trees with a flow tree (the ‘flower’) growing from each vertex. Then the idea is that the discontinuities due to flow trees (i.e. due to DT invariants) of a blooming tree with m vertices are cancelled by the discontinuities due to exchange of integration contours in $\Phi ^{\scriptscriptstyle \,\int }_{m+1}$ corresponding to blooming trees with $m+1$ vertices.

Fig. 9

Combination of trees showing cancellation of discontinuities across the wall of marginal stability corresponding to the decay $\gamma _{\mathfrak {v}}\rightarrow \gamma _L+\gamma _R$. The parts corresponding to attractor flow trees are drown in blue

Following this idea, let us consider a tree which has an attractor flow tree $T_{\mathfrak {v}}$ growing from a vertex ${\mathfrak {v}}$ (see Fig. 9). We denote by ${\mathcal {T}}_i$, $i=1,\ldots ,n_{\mathfrak {v}}$, the blooming subtrees connected to this vertex and by $T_{L}$ and $T_{R}$ the two parts (which may be trivial) of $T_{\mathfrak {v}}$ with the total charges $\gamma _L$ and $\gamma _R$ so that $\gamma _{\mathfrak {v}}=\gamma _L+\gamma _R$. Together with the contribution of this tree, we consider the contributions of the trees obtained by splitting the vertex ${\mathfrak {v}}$ into two vertices ${\mathfrak {v}}_L$ and ${\mathfrak {v}}_R$ connected by an edge, carrying charges $\gamma _L$ and $\gamma _R$ and all possible allocations of the subtrees ${\mathcal {T}}_i$ to these two vertices. Different allocations are accounted for by the sum over permutations, whilethe weight $\frac{1}{\ell !(n_{\mathfrak {v}}-\ell )!}$ takes into account the fact that permutations between subtrees connected to one vertex are redundant. The attractor flow trees $T_{L}$ and $T_{R}$ are then connected to ${\mathfrak {v}}_L$ and ${\mathfrak {v}}_R$, respectively, as shown in Fig. 9.

The contribution corresponding to the first blooming tree has a discontinuity at the wall of marginal stability for the bound state $\gamma _L+\gamma _R$ and originating from the factor $\Delta ^{z}_{\gamma _L\gamma _R}$ assigned to the root vertex of $T_{\mathfrak {v}}$. The other contributions have discontinuities at the same wall due to the exchange of the contours $\ell _{\gamma _L}$ and $\ell _{\gamma _R}$ for the integrals assigned to ${\mathfrak {v}}_L$ and ${\mathfrak {v}}_R$, respectively. They are given by the residues at the pole of the integration kernel $K_{\gamma _L\gamma _R}$. It is clear that the structure of all jumps is very similar since different subtrees produce essentially the same weights. Let us analyze what differences may arise.

• First, the contributions of the flow trees $T_{L}$ and $T_{R}$ could differ in the two cases because they have different starting points for the attractor flows: for the trees on the right side of Fig. 9 this is $z^a\in {\mathcal {M}}_K$, whereas for the tree on the left this is the point on the wall for $\gamma _L+\gamma _R$ reached by the flow from $z^a$. But we are evaluating the discontinuity exactly on the wall where the two points coincide. Thus, the contributions are the same.

• Although the subtrees ${\mathcal {T}}_i$ give rise to the same contributions for all blooming trees shown in Fig. 9, the contributions of the edges connecting them to either ${\mathfrak {v}}$ or ${\mathfrak {v}}_L$, ${\mathfrak {v}}_R$ are not exactly the same. Each of them contributes the factor given by the kernel (3.16). After taking the residue, the z-dependence of these kernels for the trees shown on the left and the right sides of the picture is identical. However, their charge dependence is different: for the tree on the left they depend on $\gamma _{\mathfrak {v}}$, whereas for the trees on the right they depend on $\gamma _L$ or $\gamma _R$, depending on which vertex they are connected to. But it is easy to see that the sums over $\ell$ and permutations produce the standard binomial expansion of a single product of kernels which all depend on $\gamma _L+\gamma _R=\gamma _{\mathfrak {v}}$ and thus coinciding with the contribution of the tree on the left.

• Finally, one should take into account that the discontinuity of $\Delta ^{z}_{\gamma _L\gamma _R}$ in the first contribution gives the factor $\langle \gamma _L,\gamma _R\rangle$. But exactly the same factor arises as the residue of $K_{\gamma _L\gamma _R}$ corresponding to the additional edge.

Thus, it remains only to check that all numerical factors work out correctly. Leaving aside the factors which are common to both contributions, we have

$$\begin{aligned} -\frac{(-1)^{\langle \gamma _L,\gamma _R\rangle }}{2}\,\frac{\sigma _{\gamma _{\mathfrak {v}}}}{(2\pi )^2}\, \frac{2m}{ m!} +(2\pi \mathrm {i})(-2\pi \mathrm {i})\,\frac{\sigma _{\gamma _L}\sigma _{\gamma _R}}{(2\pi )^4}\,\frac{m(m+1)}{(m+1)!}. \end{aligned}$$

(C.1)

Here $-\frac{(-1)^{\langle \gamma _L,\gamma _R\rangle }}{2}$ comes from the factor $-\Delta ^{z}_{\gamma _L\gamma _R}$ in $T_{\mathfrak {v}}$, the factors with quadratic refinement are due to functions $H_\gamma$ (3.9) assigned to ${\mathfrak {v}}$ or ${\mathfrak {v}}_L$, ${\mathfrak {v}}_R$, $(2\pi \mathrm {i})$ is the standard weight of the residue, $(-2\pi \mathrm {i})$ is the residue of $K_{\gamma _L\gamma _R}$ (3.16), and factorials are the weights of the trees in the expansion (3.26). Finally, the factors 2m and $m(m+1)$ arise due to the freedom to relabel charges assigned to the marked vertices: on the left these are vertex ${\mathfrak {v}}$ and the two children of the root in $T_{\mathfrak {v}}$, whereas on the right these are ${\mathfrak {v}}_L$ and ${\mathfrak {v}}_R$. It is immediate to check that all these numerical weights cancel, which ensures that the function ${\mathcal {G}}$ is continuous across walls of marginal stability. Moreover, in this cancellation the condition that we sit on the wall was used only in locally constant factors. Therefore, this reasoning proves not only that ${\mathcal {G}}$ is continuous, but that it is actually smooth around these loci.

D. Indefinite Theta Series and Generalized Error Functions

1.1 Vignéras’ theorem

Let ${\varvec{\Lambda }}$ be a d-dimensional lattice equipped with a bilinear form $({{\varvec{x}}},{{\varvec{y}}})\equiv {{\varvec{x}}}\cdot {{\varvec{y}}}$, where ${{\varvec{x}}},{{\varvec{y}}}\in {\varvec{\Lambda }}\otimes \mathbb {R}$, such that its associated quadratic form has signature $(n,d-n)$ and is integer valued, i.e. ${{\varvec{k}}}^2\equiv {{\varvec{k}}}\cdot {{\varvec{k}}}\in \mathbb {Z}$ for ${{\varvec{k}}}\in {\varvec{\Lambda }}$. Furthermore, let ${{\varvec{p}}}\in {\varvec{\Lambda }}$ be a characteristic vector (such that ${{\varvec{k}}}\cdot ({{\varvec{k}}}+ {{\varvec{p}}})\in 2\mathbb {Z}$, $\forall \,{{\varvec{k}}}\in {\varvec{\Lambda }}$), ${\varvec{\mu }}\in {\varvec{\Lambda }}^*/{\varvec{\Lambda }}$ a glue vector, and $\lambda$ an arbitrary integer. We consider the following family of theta series

$$\begin{aligned}&\vartheta _{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau , {{\varvec{b}}}, {{\varvec{c}}})\nonumber \\&\quad =\tau _2^{-\lambda /2} \!\!\!\! \sum _{{{{\varvec{k}}}}\in {\varvec{\Lambda }}+{\varvec{\mu }}+{1\over 2}{{\varvec{p}}}}\!\! (-1)^{{{\varvec{k}}}\cdot {{\varvec{p}}}}\,\Phi (\sqrt{2\tau _2}({{\varvec{k}}}+{{\varvec{b}}}))\, \mathbf{e}\left( - \tfrac{\tau }{2}\,({{\varvec{k}}}+{{\varvec{b}}})^2+{{\varvec{c}}}\cdot ({{\varvec{k}}}+\textstyle {1\over 2}{{\varvec{b}}})\right) \qquad \end{aligned}$$

(D.1)

defined by a kernel $\Phi ({{\varvec{x}}})$ such that the function $f({{\varvec{x}}})\equiv \Phi ({{\varvec{x}}})\, e^{\tfrac{\pi }{2}\,{{\varvec{x}}}^2} \in L^1({\varvec{\Lambda }}\otimes \mathbb {R})$ so that the sum is absolutely convergent. Irrespective of the choice of this kernel and of the parameter $\lambda$, any such theta series satisfies the following elliptic properties

$$\begin{aligned} \begin{aligned} \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\left( {\Phi } ,\lambda ; \tau , {{\varvec{b}}}+{{\varvec{k}}},{{\varvec{c}}}\right)&=(-1)^{{{\varvec{k}}}\cdot {{\varvec{p}}}}\, \mathbf{e}\left( -\textstyle {1\over 2}\, {{\varvec{c}}}\cdot {{\varvec{k}}}\right) \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\left( {\Phi } ,\lambda ; \tau , {{\varvec{b}}},{{\varvec{c}}}\right) , \\ \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\left( {\Phi }, \lambda ; \tau , {{\varvec{b}}},{{\varvec{c}}}+{{\varvec{k}}}\right)&=(-1)^{{{\varvec{k}}}\cdot {{\varvec{p}}}}\, \mathbf{e}\left( \textstyle {1\over 2}\, {{\varvec{b}}}\cdot {{\varvec{k}}}\right) \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\left( {\Phi } ,\lambda ; \tau , {{\varvec{b}}},{{\varvec{c}}}\right) . \end{aligned} \end{aligned}$$

(D.2)

Now let us require that in addition the kernel satisfies the following two conditions:

• 1. Let $D({{\varvec{x}}})$ be any differential operator of order $\le 2$, and $R({{\varvec{x}}})$ any polynomial of degree $\le 2$. Then $f({{\varvec{x}}})$ defined above must be such that $f({{\varvec{x}}})$, $D({{\varvec{x}}})f({{\varvec{x}}})$ and $R({{\varvec{x}}})f({{\varvec{x}}})\in L^2({\varvec{\Lambda }}\otimes \mathbb {R})\bigcap L^1({\varvec{\Lambda }}\otimes \mathbb {R})$.

• 2. $\Phi ({{\varvec{x}}})$ must satisfy

  $$\begin{aligned} V_\lambda \cdot \Phi ({{\varvec{x}}})=0, \qquad V_\lambda = \partial _{{{\varvec{x}}}}^2 + 2\pi \left( {{\varvec{x}}}\cdot \partial _{{{\varvec{x}}}} - \lambda \right) . \end{aligned}$$

  (D.3)

Then in [43] it was proven that the theta series (D.1) transforms as a vector-valued modular form of weight $(\lambda +d/2,0)$ (see Theorem 2.1 in [28] for the detailed transformation under $\tau \rightarrow -1/\tau$). We refer to $V_\lambda$ as Vignéras’ operator. The simplest example is the Siegel theta series for which the kernel is $\Phi ({{\varvec{x}}})=e^{-\pi {{\varvec{x}}}_+^2}$ where ${{\varvec{x}}}_+$ is the projection of ${{\varvec{x}}}$ on a fixed positive plane of dimension n. This kernel is annihilated by $V_{-n}$.

In this paper we apply the Vignéras’ theorem to the case of ${\varvec{\Lambda }}=\oplus _{i=1}^n \Lambda _i$. Thus, the charges appearing in the description of the theta series (D.1) are of the type ${{\varvec{k}}}=(k_1^a,\ldots ,k_n^a)$, whereas the vectors ${{\varvec{b}}}$ and ${{\varvec{c}}}$ are taken with i-independent components, namely, ${{\varvec{b}}}_i^a=b^a$, ${{\varvec{c}}}_i^a=c^a$ for $i=1,\ldots , n$. The lattices $\Lambda _i$ carry the bilinear forms $\kappa _{i,ab}=\kappa _{abc}p_i^c$ which are all of signature $(1,b_2-1)$. This induces a natural bilinear form on ${\varvec{\Lambda }}$:

$$\begin{aligned} {{\varvec{x}}}\cdot {{\varvec{y}}}=\sum _{i=1}^n (p_ix_iy_i). \end{aligned}$$

(D.4)

Note also that the sign factor $(-1)^{{{\varvec{k}}}\cdot {{\varvec{p}}}}$ in (D.1) can be identified with the quadratic refinement provided we choose the latter as

$$\begin{aligned} \sigma _\gamma =\sigma _{p,q}\equiv \mathbf{e}\left( {1\over 2}\, p^a q_a\right) \sigma _p, \qquad \sigma _p=\mathbf{e}\left( {1\over 2}\, A_{ab}p^ap^b\right) . \end{aligned}$$

(D.5)

The matrix $A_{ab}$, satisfying

$$\begin{aligned} A_{ab} p^p - \frac{1}{2}\, \kappa _{abc} p^b p^c\in \mathbb {Z}\quad \text {for}\ \forall p^a\in \mathbb {Z}\, , \end{aligned}$$

(D.6)

appears due to the non-trivial quantization of charges on the type IIB side (2.19) and can be used to perform a symplectic rotation to identify them with mirror dual integer charges on the type IIA side [56]. It is easy to check that the quadratic refinement (D.5) satisfies (3.4).

1.2 Generalized error functions

An important class of solutions of Vignéras’ equation is given by the error function and its generalizations constructed in [28] and further elaborated in [29] (see [30, 31] for a more conceptual explanation of the origin of these functions). Let us take

$$\begin{aligned} M_1(u)= & {} -\text{ sgn }(u)\, \text {Erfc}(|u|\sqrt{\pi }) = \frac{\mathrm {i}}{\pi } \int _{\ell }\frac{\mathrm {d}z}{z}\, e^{-\pi z^2 -2\pi \mathrm {i}z u}, \end{aligned}$$

(D.7)

$$\begin{aligned} E_1(u)= & {} \text{ sgn }(u)+M_1(u) \nonumber \\= & {} {{\,\mathrm{Erf}\,}}(u\sqrt{\pi })= \int _{\mathbb {R}} \mathrm {d}u' \, e^{-\pi (u-u')^2} \mathrm{sgn}(u'), \end{aligned}$$

(D.8)

where the contour $\ell =\mathbb {R}-\mathrm {i}u$ runs parallel to the real axis through the saddle point at $z=-\mathrm {i}u$. Then, given a vector with a positive norm ${{\varvec{v}}}^2>0$ so that $|{{\varvec{v}}}|=\sqrt{{{\varvec{v}}}^2}$, we define

$$\begin{aligned} \Phi _1^E({{\varvec{v}}};{{\varvec{x}}})=E_1\left( \frac{{{\varvec{v}}}\cdot {{\varvec{x}}}}{|{{\varvec{v}}}|}\right) , \qquad \Phi _1^M({{\varvec{v}}};{{\varvec{x}}})=M_1\left( \frac{{{\varvec{v}}}\cdot {{\varvec{x}}}}{|{{\varvec{v}}}|}\right) . \end{aligned}$$

(D.9)

It is easy to check that the first function is a smooth solution of (D.3) with $\lambda =0$, whereas the second is exponentially suppressed at large ${{\varvec{x}}}$ and also solves the same equation, but only away from the locus ${{\varvec{v}}}\cdot {{\varvec{x}}}=0$ where it has a discontinuity.

Generalizing the integral representations (D.7) and (D.8), we define the generalized (complementary) error functions

$$\begin{aligned} M_n({\mathcal {M}};\mathbb {u})= & {} \left( \frac{\mathrm {i}}{\pi }\right) ^n |\,\mathrm{det}\, {\mathcal {M}}|^{-1} \int _{\mathbb {R}^n-\mathrm {i}\mathbb {u}}\mathrm {d}^n z\, \frac{e^{-\pi \mathbb {z}^{\mathrm{tr}} \mathbb {z}-2\pi \mathrm {i}\mathbb {z}^{\mathrm{tr}} \mathbb {u}}}{\prod ({\mathcal {M}}^{-1}\mathbb {z})}, \end{aligned}$$

(D.10)

$$\begin{aligned} E_n({\mathcal {M}};\mathbb {u})= & {} \int _{\mathbb {R}^n} \mathrm {d}\mathbb {u}' \, e^{-\pi (\mathbb {u}-\mathbb {u}')^\mathrm{tr}(\mathbb {u}-\mathbb {u}')} \mathrm{sgn}({\mathcal {M}}^{\mathrm{tr}} \mathbb {u}'), \end{aligned}$$

(D.11)

where $\mathbb {z}=(z_1,\ldots ,z_n)$ and $\mathbb {u}=(u_1,\ldots ,u_n)$ are n-dimensional vectors, ${\mathcal {M}}$ is $n\times n$ matrix of parameters, and we used the shorthand notations $\prod \mathbb {z}=\prod _{i=1}^n z_i$ and $\mathrm{sgn}(\mathbb {u})=\prod _{i=1}^n \mathrm{sgn}(u_i)$. The detailed properties of these functions can be found in [29]. Here we mention only a few:

• $M_n$ are exponentially suppressed for large $\mathbb {u}$ as $M_n\sim \frac{(-1)^n}{\pi ^n}|\,\mathrm{det}\, {\mathcal {M}}|^{-1}\, \frac{e^{-\pi \mathbb {u}^{\mathrm{tr}} \mathbb {u}}}{\prod ({\mathcal {M}}^{-1}\mathbb {u})}$, whereas $E_n$ are locally constant for large $\mathbb {u}$ as $E_n\sim \mathrm{sgn}({\mathcal {M}}^{\mathrm{tr}} \mathbb {u})$.

• More generally, $E_n$ can be expressed as a linear combination of $M_k$, $k=0,\ldots ,n$, multiplied by $n-k$ sign functions, generalizing the first relation in (D.8) (see Eq. (D.15) below for a precise statement).

• From (D.11) it follows that every identity between products of sign functions implies an identity between generalized error functions $E_n$. Moreover, expanding the $E_n$ functions in terms of $M_k$’s and sign functions, one obtains similar identities for functions $M_n$. For instance, the identity

  $$\begin{aligned} (\text{ sgn }(x_1)+\text{ sgn }(x_2))\,\text{ sgn }(x_1+x_2)=1+\text{ sgn }(x_1)\,\text{ sgn }(x_2) \end{aligned}$$

  (D.12)

  implies

  $$\begin{aligned} \begin{aligned} E_2((\mathbb {v}_1,\mathbb {v}_1+\mathbb {v}_2);\mathbb {u})+E_2((\mathbb {v}_2,\mathbb {v}_1+\mathbb {v}_2);\mathbb {u})=&\, 1+E_2((\mathbb {v}_1,\mathbb {v}_2);\mathbb {u}), \\ M_2((\mathbb {v}_1,\mathbb {v}_1+\mathbb {v}_2);\mathbb {u})+M_2((\mathbb {v}_2,\mathbb {v}_1+\mathbb {v}_2);\mathbb {u})=&\, M_2((\mathbb {v}_1,\mathbb {v}_2);\mathbb {u}), \end{aligned}\nonumber \\ \end{aligned}$$

  (D.13)

  where $\mathbb {v}_1,\mathbb {v}_2$ are two-dimensional vectors used to encode the $2\times 2$ matrix of parameters.

The main reason to introduce these functions is that, similarly to the usual error and complementary error functions, they can be used to produce solutions of Vignéras’ equation on $\mathbb {R}^{n,d-n}$. To write them down, let us consider $d\times n$ matrix ${\mathcal {V}}$ which can be viewed as a collection of n vectors, ${\mathcal {V}}=({{\varvec{v}}}_1,\ldots ,{{\varvec{v}}}_n)$. We assume that these vectors span a positive definite subspace, i.e. ${\mathcal {V}}^{\mathrm{tr}}\cdot {\mathcal {V}}$ is a positive definite matrix. Let ${\mathcal {B}}$ be $n\times d$ matrix whose rows define an orthonormal basis for this subspace. Then we define the boosted generalized error functions

$$\begin{aligned} \Phi _n^M({\mathcal {V}};{{\varvec{x}}})=M_n({\mathcal {B}}\cdot {\mathcal {V}};{\mathcal {B}}\cdot {{\varvec{x}}}), \qquad \Phi _n^E({\mathcal {V}};{{\varvec{x}}})=E_n({\mathcal {B}}\cdot {\mathcal {V}};{\mathcal {B}}\cdot {{\varvec{x}}}). \end{aligned}$$

(D.14)

It can be shown that both these functions satisfy Vignéras’ equation (for $\Phi _n^M$ one should stay away from its discontinuities, i.e. loci where $\mathrm{sgn}(({\mathcal {B}}\cdot {\mathcal {V}})^{-1}{\mathcal {B}}\cdot {{\varvec{x}}})=0$). Moreover, they are symmetric under permutation of the vectors ${{\varvec{v}}}_i$. Since at large ${{\varvec{x}}}$ one has $\Phi _n^E\sim \mathrm{sgn}({\mathcal {V}}^{\mathrm{tr}}\cdot {{\varvec{x}}})=\prod _{i=1}^n \mathrm{sgn}({{\varvec{v}}}_i\cdot {{\varvec{x}}})$, one can think about this function as providing the modular completion for (indefinite) theta series with kernel given by a product of signs.

The relation between functions $E_n$ and $M_n$ mentioned above implies a similar relation between the functions (D.14). For generic n, it takes the following form³³

$$\begin{aligned} \Phi _n^E({\mathcal {V}};{{\varvec{x}}})=\sum _{{\mathcal {I}}\subseteq {\mathscr {Z}}_n}\Phi _{|{\mathcal {I}}|}^M(\{{{\varvec{v}}}_i\}_{i\in {\mathcal {I}}};{{\varvec{x}}}) \prod _{j\in {\mathscr {Z}}_n{\setminus } {\mathcal {I}}}\mathrm{sgn}({{\varvec{v}}}_{j\perp {\mathcal {I}}},{{\varvec{x}}}), \end{aligned}$$

(D.15)

where the sum goes over all possible subsets (including the empty set) of the set ${\mathscr {Z}}_{n}=\{1,\ldots ,n\}$, $|{\mathcal {I}}|$ is the cardinality of ${\mathcal {I}}$, and ${{\varvec{v}}}_{j\perp {\mathcal {I}}}$ denotes the projection of ${{\varvec{v}}}_j$ orthogonal to the subspace spanned by $\{{{\varvec{v}}}_i\}_{i\in {\mathcal {I}}}$. The cardinality $|{\mathcal {I}}|$ can also be interpreted as the number of directions in $\mathbb {R}^d$ along which the corresponding contribution has an exponential fall off.

Finally, note that a solution of Vignéras’ equation can be uplifted to a solution of the same equation with $\lambda$ shifted to $\lambda +1$ by acting with the differential operator

$$\begin{aligned} {\mathcal {D}}({{\varvec{v}}})={{\varvec{v}}}\cdot \left( {{\varvec{x}}}+\frac{1}{2\pi }\,\partial _{{\varvec{x}}}\right) , \end{aligned}$$

(D.16)

which realizes the action of the covariant derivative raising the holomorphic weight by 1. In particular, we can construct solutions with $\lambda =m$ which behave for large ${{\varvec{x}}}$ as products of n sign functions. To this end, it is enough to act on $\Phi _n^E$ by this operator m times. Thus, we define the uplifted boosted error function

$$\begin{aligned} {\widetilde{\Phi }}_{n,m}^E({\mathcal {V}},{\tilde{{\mathcal {V}}}};{{\varvec{x}}})=\left[ \prod _{i=1}^m {\mathcal {D}}({\tilde{{{\varvec{v}}}}}_i)\right] \Phi _n^E({\mathcal {V}};{{\varvec{x}}}), \end{aligned}$$

(D.17)

where ${\tilde{{\mathcal {V}}}}=({\tilde{{{\varvec{v}}}}}_1,\ldots ,{\tilde{{{\varvec{v}}}}}_m)$ encodes the vectors contracted with the covariant derivatives. Since the operators ${\mathcal {D}}({\tilde{{{\varvec{v}}}}}_i)$ commute, (D.17) is invariant under independent permutations of the vectors ${{\varvec{v}}}_i$ and ${\tilde{{{\varvec{v}}}}}_i$. In the case where all ${\tilde{{{\varvec{v}}}}}_i$ are mutually orthogonal, one finds the following asymptotics at large ${{\varvec{x}}}$

$$\begin{aligned} \lim _{{{\varvec{x}}}\rightarrow \infty }{\widetilde{\Phi }}_{n,m}^E({\mathcal {V}},{\tilde{{\mathcal {V}}}};{{\varvec{x}}})= \prod _{i=1}^m ({\tilde{{{\varvec{v}}}}}_i,{{\varvec{x}}})\prod _{j=1}^n \mathrm{sgn}({{\varvec{v}}}_j,{{\varvec{x}}}). \end{aligned}$$

(D.18)

Note that the derivative $\partial _{{\varvec{x}}}$ in (D.17) does not act on sign functions since $\Phi _n^E$ is smooth and all discontinuities due to signs are guaranteed to cancel. Similarly to (D.17), we can also define ${\widetilde{\Phi }}_{n,m}^M({\mathcal {V}},{\tilde{{\mathcal {V}}}};{{\varvec{x}}})$ where the action of derivatives on the discontinuities of $\Phi _n^M$ is ignored as well. In the particular case $m=n$, we will omit the second label and simply write ${\widetilde{\Phi }}_{n}^E$ or ${\widetilde{\Phi }}_{n}^M$.

E. Twistorial Integrals and Generalized Error Functions

In this appendix we evaluate the kernels $\Phi ^{\scriptscriptstyle \,\int }_n$ (4.6) and show that they can be expressed through the generalized error functions introduced in appendix D.2. To this end, let us note the following identity

$$\begin{aligned} \mathrm {i}\,{\mathcal {D}}({{\varvec{v}}}_{ij})\, \frac{W_{p_i}(x_i,z_i)\,W_{p_j}(x_j,z_j)}{z_i-z_j} = {{\hat{K}}}_{ij}\, W_{p_i}(x_i,z_i)\,W_{p_j}(x_j,z_j). \end{aligned}$$

(E.1)

By virtue of this relation, the kernel can be represented as

$$\begin{aligned} \Phi ^{\scriptscriptstyle \,\int }_n({{\varvec{x}}})= \frac{1}{n!}\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell }\left[ \prod _{e\in E_{\mathcal {T}}} {\mathcal {D}}({{\varvec{v}}}_{s(e) t(e)}) \right] \Phi _{\mathcal {T}}({{\varvec{x}}}), \end{aligned}$$

(E.2)

where

$$\begin{aligned} \Phi _{\mathcal {T}}({{\varvec{x}}}) =\frac{\mathrm {i}^{n-1}}{(2\pi )^n}\left[ \prod _{i=1}^n\int _{\ell _{\gamma _i}}\mathrm {d}z_i \, W_{p_i}(x_i,z_i)\right] \frac{1}{\prod _{e\in E_{\mathcal {T}}}\left( z_{s(e)}-z_{t(e)}\right) }\, . \end{aligned}$$

(E.3)

One may think that the representation (E.2) misses contributions from the covariant derivatives acting on each other. However, such contributions are proportional to the scalar products of two vectors ${{\varvec{v}}}_{s(e) t(e)}$ and are non-vanishing provided the two edges have a common vertex. If this is the case, the two edges generate the following factor

$$\begin{aligned} \frac{(p_{{\mathfrak {v}}_1}p_{{\mathfrak {v}}_2}p_{{\mathfrak {v}}_{3}})}{(z_{{\mathfrak {v}}_1}-z_{{\mathfrak {v}}_3})(z_{{\mathfrak {v}}_2}-z_{{\mathfrak {v}}_{3}})}\, , \end{aligned}$$

(E.4)

where ${\mathfrak {v}}_1,{\mathfrak {v}}_2,{\mathfrak {v}}_3$ are the tree vertices joint by the edges and ${\mathfrak {v}}_{3}=e_1\cap e_2$. The crucial observation is that if we pick up 3 subtrees, each with a marked vertex, there are exactly 3 ways to form a labelled tree out of them by joining the marked vertices as shown in Fig. 7 on page 7. Each subtree contributes the same factor in all 3 cases, whereas the joining edges and the sum over trees give rise to the vanishing factor

$$\begin{aligned} \frac{(p_1p_2p_3)}{(z_1-z_3)(z_2-z_3)}+\frac{(p_1p_2p_3)}{(z_1-z_2)(z_1-z_3)}-\frac{(p_1p_2p_3)}{(z_2-z_3)(z_1-z_2)}=0. \end{aligned}$$

(E.5)

This ensures that no additional contributions arise and thereby proves (E.2). Note that for this proof it was crucial that inequivalent labelled trees enter the sum with the same weight.

Then let us do the change of variables

$$\begin{aligned} z_i=z'-\frac{\mathrm {i}(pxt)}{\sqrt{2\tau _2}(pt^2)}+\sum _{\alpha =1}^{n-1} e_i^\alpha z'_\alpha , \end{aligned}$$

(E.6)

where $x^a=\kappa ^{ab}\sum \kappa _{i,bc} x^c_i$ (cf. (4.8)), and $e_i^\alpha$ are such that

$$\begin{aligned} \sum _{i=1}^n {\mathfrak {p}}_i e_i^\alpha =0, \qquad \sum _{i=1}^n {\mathfrak {p}}_i e_i^\alpha e_i^\beta =0, \quad \alpha \ne \beta \end{aligned}$$

(E.7)

and we introduced the convenient notation ${\mathfrak {p}}_i=(p_it^2)$. Labeling the $n-1$ edges of the tree by the same index $\alpha$, one can rewrite the function (E.3) in the new variables as

$$\begin{aligned} \Phi _{\mathcal {T}}({{\varvec{x}}})= & {} \frac{\mathrm {i}^{n-1} \sqrt{\Delta }}{(2\pi )^n}\, e^{-\frac{\pi (pxt)^2}{(pt^2)}}\int \mathrm {d}z' \, e^{-2\pi \tau _2(pt^2)z'^2}\prod _{\alpha =1}^{n-1}\nonumber \\&\times \int \frac{\mathrm {d}z'_\alpha \,e^{-2\pi \tau _2\, \Delta _\alpha (z'_\alpha )^2 -2\pi \mathrm {i}\sqrt{2\tau _2}\, w^\alpha z'_\alpha }}{\sum _{\beta =1}^{n-1}\left( e^\beta _{s(\alpha )}-e^\beta _{t(\alpha )}\right) z'_\beta }\, ,\nonumber \\ \end{aligned}$$

(E.8)

where

$$\begin{aligned} w^\alpha =\sum _{i=1}^n (p_ix_i t) e_i^\alpha , \qquad \Delta _\alpha =\sum _{i=1}^n {\mathfrak {p}}_i (e_i^\alpha )^2, \qquad \Delta =\frac{{\mathfrak {p}}\prod _{\alpha =1}^{n-1} \Delta _\alpha }{\prod _{i=1}^{n}{\mathfrak {p}}_i}\, . \end{aligned}$$

(E.9)

The integral over $z'$ is Gaussian and is easily evaluated. In the remaining integrals, rescaling the integration variables by $\sqrt{2\tau _2\Delta _\alpha }$, one recognizes the generalized error functions (D.10). Thus, one obtains

$$\begin{aligned} \Phi _{\mathcal {T}}({{\varvec{x}}}) =\frac{1}{2^{n-1}}\,\frac{\sqrt{\Delta }|\,\mathrm{det}\, {\mathcal {M}}|}{\prod _{\alpha =1}^{n-1} \Delta _\alpha }\, \Phi ^{\scriptscriptstyle \,\int }_1(x)\, M_{n-1}\left( {\mathcal {M}};\left\{ \frac{w^\alpha }{\sqrt{\Delta _\alpha }}\right\} \right) \end{aligned}$$

(E.10)

where we used the function $\Phi ^{\scriptscriptstyle \,\int }_1$ (4.13) and introduced the matrix ${\mathcal {M}}$ such that

$$\begin{aligned} {\mathcal {M}}^{-1}_{\alpha \beta }=(\Delta _\alpha \Delta _\beta )^{-1/2}\left( e^\beta _{s(\alpha )}-e^\beta _{t(\alpha )}\right) . \end{aligned}$$

(E.11)

A simple solution to the conditions (E.7) may be constructed as follows. Let T be a rooted ordered binary tree with n leaves decorated by $\gamma _i$, $i=1\ldots n$. As usual for such trees, other vertices v carry charges given by the sum of charges of their children, i.e. $\gamma _v=\sum _{i\in {\mathcal {I}}_v}\gamma _i$ where ${\mathcal {I}}_v$ is the set of leaves which are descendants of v. There are $n-1$ such vertices which we label by index $\alpha$. Then we can choose

$$\begin{aligned} e^\alpha _i=\sum _{j\in {\mathcal {I}}_{L(v_\alpha )}}\sum _{k\in {\mathcal {I}}_{R(v_\alpha )}} \left( \delta _{ij}\,{\mathfrak {p}}_k-\delta _{ik}\,{\mathfrak {p}}_j\right) , \end{aligned}$$

(E.12)

which satisfy (E.7) as can be easily checked. For this choice (cf. (4.14))

$$\begin{aligned} \begin{aligned} w^\alpha =&\, ({\tilde{{{\varvec{u}}}}}_{\alpha },{{\varvec{x}}}), \quad \text{ where }\quad {\tilde{{{\varvec{u}}}}}_\alpha =\sum _{i\in {\mathcal {I}}_{L(v_\alpha )}}\sum _{j\in {\mathcal {I}}_{R(v_\alpha )}}{{\varvec{u}}}_{ij}, \\ \Delta _\alpha =&\,{\tilde{{{\varvec{u}}}}}_\alpha ^2={\mathfrak {p}}_{v_\alpha } {\mathfrak {p}}_{L(v_\alpha )}{\mathfrak {p}}_{R(v_\alpha )}. \end{aligned} \end{aligned}$$

(E.13)

Note that the vectors ${\tilde{{{\varvec{u}}}}}_\alpha$ are mutually orthogonal.

In principle, any rooted binary tree T is suitable for the above construction. However, given the unrooted tree ${\mathcal {T}}$, there is a simple (but non-unique) choice of T which simplifies the resulting matrix ${\mathcal {M}}$. To define it, let us construct a partially increasing family of subtrees of ${\mathcal {T}}$, such that two members in this family are either disjoint, or contained in one another. Moreover, we require that the largest subtree is ${\mathcal {T}}$ itself, while each subtree containing more than one vertex is obtained by joining two smaller subtrees along an edge of ${\mathcal {T}}$. Any such family contains $2n-1$ subtrees ${\mathcal {T}}_{{\hat{\alpha }}}$ labelled by ${\hat{\alpha }}=1,\ldots ,2n-1$. Among them, $n-1$ subtrees, which we label by $\alpha ={\hat{\alpha }}=1,\ldots ,n-1$, contain several vertices, while the remaining n subtrees with label ${\hat{\alpha }}=n,\ldots ,2n-1$ have only one vertex. For each subtree $T_\alpha$, we denote by $e_\alpha$ the edge of ${\mathcal {T}}$ which is used to reconstruct $T_\alpha$ from two smaller subtrees ${\mathcal {T}}_{\alpha _L}, {\mathcal {T}}_{\alpha _R}$. From this data, we construct a rooted binary tree T with $n-1$ vertices in one-to-one correspondence with the subtrees ${\mathcal {T}}_\alpha$ and n leaves in one-to-one correspondence with the one-vertex subtrees ${\mathcal {T}}_{{\hat{\alpha }}}$ with ${\hat{\alpha }}\ge n$. In this correspondence, the two children of a vertex associated to ${\mathcal {T}}_{\alpha }$ are the vertices associated to the two subtrees ${\mathcal {T}}_{\alpha _L}, {\mathcal {T}}_{\alpha _R}$. The ordering at each vertex is defined to be such that the subtree containing the source/target vertex of the corresponding edge $e_\alpha$ is on the left/right.³⁴ Of course, this construction is not unique since there are many ways to decompose ${\mathcal {T}}$ into such a set of subtrees (see Fig. 10).

Fig. 10

An example of an unrooted labelled tree with 4 vertices and two choices of decompositions into subtrees with the corresponding rooted binary trees. The edges are labelled $e_1,e_2,e_3$ from left to right along ${\mathcal {T}}$

Applying the above construction to this particular choice of rooted tree, one finds

$$\begin{aligned} \sqrt{\Delta _\alpha \Delta _\beta }\,{\mathcal {M}}^{-1}_{\alpha \beta }=\left\{ \begin{array}{l@{\quad }l} {\mathfrak {p}}_{v_\alpha }, \quad &{} \alpha =\beta , \\ \epsilon _{\alpha \beta } {\mathfrak {p}}_{L(v_\beta )}, \quad &{} e_\alpha \cap {\mathcal {T}}_{\beta _R} \ne \emptyset ,\ e_\alpha \nsubseteq {\mathcal {T}}_\beta , \\ \epsilon _{\alpha \beta } {\mathfrak {p}}_{R(v_\beta )}, \quad &{} e_\alpha \cap {\mathcal {T}}_{\beta _L} \ne \emptyset ,\ e_\alpha \nsubseteq {\mathcal {T}}_\beta , \\ 0, &{} e_\alpha \cap {\mathcal {T}}_\beta = \emptyset \ \text{ or } \ e_\alpha \subset {\mathcal {T}}_\beta , \end{array} \right. \end{aligned}$$

(E.14)

where $\epsilon _{\alpha \beta }=-1$ if the orientations of $e_\alpha$ and $e_\beta$ on the path joining them are the same and $+1$ otherwise. This result shows that the matrix ${\mathcal {M}}^{-1}$ turns out to be triangular which makes it much simpler to find its inverse. On the basis of (E.14), below we will prove the following

Lemma 2

One has ${\mathcal {B}}\cdot {\mathcal {V}}={\mathcal {M}}$ and ${\mathcal {B}}\cdot {{\varvec{x}}}=\left\{ \frac{w^\alpha }{\sqrt{\Delta _\alpha }}\right\}$ provided

$$\begin{aligned} \begin{aligned} {\mathcal {B}}=&\, \left( \frac{{\tilde{{{\varvec{u}}}}}_1}{\sqrt{\Delta _1}},\ldots , \frac{{\tilde{{{\varvec{u}}}}}_{n-1}}{\sqrt{\Delta _{n-1}}}\right) ^{\mathrm{tr}}, \\ {\mathcal {V}}=&\, {\mathfrak {p}}^{-1}\Bigl (\sqrt{\Delta _1}{{\varvec{u}}}_1,\ldots , \sqrt{\Delta _{n-1}}{{\varvec{u}}}_{n-1}\Bigr ), \end{aligned} \end{aligned}$$

(E.15)

where the vectors ${{\varvec{u}}}_\alpha$ are defined in (4.11). Moreover, the vectors ${\tilde{{{\varvec{u}}}}}_\alpha$ form an orthogonal basis in the subspace spanned by ${{\varvec{u}}}_\alpha$.

This lemma allows to reexpress the kernel $\Phi _{\mathcal {T}}$ (E.10) in terms of the boosted generalized error function $\Phi ^M_{n-1}$ (D.14). It is important that its argument ${\mathcal {V}}$ does not depend on the choice of the binary rooted tree T, but only on the unrooted tree ${\mathcal {T}}$. In addition, the function actually does not depend on the normalization of the vectors composing ${\mathcal {V}}$. Given also that the determinant of ${\mathcal {M}}$ is found to be

$$\begin{aligned} |\,\mathrm{det}\, {\mathcal {M}}|= \prod _{\alpha =1}^{n-1}\frac{ \Delta _\alpha }{{\mathfrak {p}}_{v_\alpha }} =\prod _{i=1}^{n}{\mathfrak {p}}_i\prod _{v\in V_{T}{\setminus }{\{v_0\}}}{\mathfrak {p}}_v =\sqrt{{\mathfrak {p}}^{-1}\,\prod _{i=1}^{n}{\mathfrak {p}}_i\,\prod _{\alpha =1}^{n-1} \Delta _\alpha }, \end{aligned}$$

(E.16)

so that the prefactor in (E.10) cancels, one arrives at

$$\begin{aligned} \Phi _{\mathcal {T}}({{\varvec{x}}}) =\frac{1}{2^{n-1}}\, \Phi ^{\scriptscriptstyle \,\int }_1(x)\, \Phi ^M_{n-1}(\{ {{\varvec{u}}}_e\};{{\varvec{x}}}). \end{aligned}$$

(E.17)

Finally, since the differential operator in (E.2) commutes with functions of x due to the orthogonality of ${{\varvec{v}}}_{ij}$ and ${{\varvec{t}}}$, one can write the kernel $\Phi ^{\scriptscriptstyle \,\int }_n$ as

$$\begin{aligned} \Phi ^{\scriptscriptstyle \,\int }_n({{\varvec{x}}})= \frac{\Phi ^{\scriptscriptstyle \,\int }_1(x)}{2^{n-1} n!}\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell }\left[ \prod _{e\in E_{\mathcal {T}}}{\mathcal {D}}( {{\varvec{v}}}_{s(e) t(e)})\right] \Phi ^M_{n-1}(\{ {{\varvec{u}}}_\alpha \};{{\varvec{x}}}). \end{aligned}$$

(E.18)

which is the same as (4.12).

1.1 Proof of Lemma 2

We start by proving that the vectors ${\tilde{{{\varvec{u}}}}}_\alpha$ form an orthogonal basis in the subspace spanned by ${{\varvec{u}}}_\alpha$. Since the orthogonality is ensured by construction based on a rooted binary tree, it remains to show that any vector ${{\varvec{u}}}_\alpha$ can be decomposed as a linear combination of ${\tilde{{{\varvec{u}}}}}_\alpha$. To this end, we show that the determinant of the Gram matrix constructed from the set of vectors $\{{\tilde{{{\varvec{u}}}}}_\alpha \}_{\alpha =1}^{n-1}\cup \{{{\varvec{u}}}_\beta \}$ vanishes. This requires to calculate the scalar product $({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\beta )$ which can be done using

$$\begin{aligned} ({{\varvec{u}}}_{ij},{{\varvec{u}}}_{kl})=\left\{ \begin{array}{l@{\quad }l} {\mathfrak {p}}_i{\mathfrak {p}}_j{\mathfrak {p}}_{i+j},\qquad &{} i=k,\ j=l, \\ {\mathfrak {p}}_i{\mathfrak {p}}_j{\mathfrak {p}}_{l},\qquad &{} i=k,\ j\ne l, \\ 0,\qquad &{} i,j\ne k,l. \end{array}\right. \end{aligned}$$

(E.19)

Summing i, j, k, l over appropriate subsets, it is immediate to see that $({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\beta )=0$ if ${\mathcal {T}}_{\alpha }\subset {\mathcal {T}}_\beta ^s$ or ${\mathcal {T}}_\beta ^t$, which are the two trees obtained by dividing the tree ${\mathcal {T}}$ into two parts by cutting the edge $e_\alpha$. In other words, it is non-vanishing only if $e_\beta \subseteq {\mathcal {T}}_\alpha$. Then there are two cases which give

$$\begin{aligned} ({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\beta )=\left\{ \begin{array}{l@{\quad }l} {\mathfrak {p}}_\alpha ^L\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}, \quad &{} \alpha =\beta , \\ {\mathfrak {p}}_{\alpha \beta }^{Ls}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}_\beta ^t-{\mathfrak {p}}_{\alpha \beta }^{Lt}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}_\beta ^s +{\mathfrak {p}}_{\alpha }^{L}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}_\beta ^s={\mathfrak {p}}_{\alpha \beta }^{Ls}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}},\quad &{} e_\beta \subset {\mathcal {T}}_\alpha , \end{array}\right. \end{aligned}$$

(E.20)

where we introduced

$$\begin{aligned} {\mathfrak {p}}_\alpha ^L=\sum _{i\in {\mathcal {I}}_{L(v_\alpha )}}{\mathfrak {p}}_i, \qquad {\mathfrak {p}}_{\beta }^{s}=\sum _{i\in V_{{\mathcal {T}}_\beta ^s}}{\mathfrak {p}}_i, \qquad {\mathfrak {p}}_{\alpha \beta }^{Ls}=\sum _{i\in {\mathcal {I}}_{L(v_\alpha )}\cap V_{{\mathcal {T}}_\beta ^s}}{\mathfrak {p}}_i, \qquad {\mathfrak {p}}_{\alpha \beta }^{st}=\sum _{i\in V_{{\mathcal {T}}_\alpha ^s}\cap V_{{\mathcal {T}}_\beta ^t}}{\mathfrak {p}}_i,\nonumber \\ \end{aligned}$$

(E.21)

and similarly for variables with labels R and t. In (E.20) in the second case we assumed that the orientation of edges is such that $e_\beta \subset {\mathcal {T}}_\alpha ^s$ and $e_\alpha \subset {\mathcal {T}}_\beta ^t$. If this is not the case, one should replace s by t, L by R and flip the sign for each change of orientation. Below we use the same assumption, but the computation can easily be generalized to a more general situation.

Given the result (E.20), $({\tilde{{{\varvec{u}}}}}_\alpha , {\tilde{{{\varvec{u}}}}}_\beta )=\Delta _\alpha \delta _{\alpha \beta }$ and $({{\varvec{u}}}_\beta , {{\varvec{u}}}_\beta )={\mathfrak {p}}_\beta ^s\,{\mathfrak {p}}_\beta ^t\,{\mathfrak {p}}$, the determinant of the Gram matrix is easily found to be

$$\begin{aligned} \,\mathrm{det}\, \mathrm{Gram}({\tilde{{{\varvec{u}}}}}_1,\ldots , {\tilde{{{\varvec{u}}}}}_{n-1},{{\varvec{u}}}_\beta )= {\mathfrak {p}}\prod _{\alpha =1}^{n-1}\Delta _\alpha \left[ {\mathfrak {p}}_\beta ^s\,{\mathfrak {p}}_\beta ^t-{\mathfrak {p}}\sum _{T_\alpha \supseteq e_\beta }\frac{({\mathfrak {p}}_{\alpha \beta }^{Ls})^2\,{\mathfrak {p}}_\alpha ^R}{{\mathfrak {p}}_{v_\alpha }{\mathfrak {p}}_\alpha ^L } \right] . \qquad \end{aligned}$$

(E.22)

Note that the subtrees ${\mathcal {T}}_\alpha$ containing the edge $e_\beta$ form an ordered set so that the sum in the square brackets goes over $\alpha _\ell$, $\ell =1,\ldots , m$, such that ${\mathcal {T}}_{\alpha _\ell }\subset {\mathcal {T}}_{\alpha _{\ell +1}}$. The first element of this set $\alpha _1=\beta$, whereas the last corresponds to the total tree, ${\mathcal {T}}_{\alpha _m}={\mathcal {T}}$. Due to ${\mathfrak {p}}_{\alpha _m}^L={\mathfrak {p}}_{\alpha _m}^s$, ${\mathfrak {p}}_{\alpha _m}^R={\mathfrak {p}}_{\alpha _m}^t$ and ${\mathfrak {p}}_{\alpha _m\beta }^{Ls}={\mathfrak {p}}_\beta ^s$, the first term in the square brackets together with the term in the sum corresponding to $\alpha _m$ gives

$$\begin{aligned} {\mathfrak {p}}_\beta ^s\left( {\mathfrak {p}}_\beta ^t -\frac{{\mathfrak {p}}_\beta ^s\,{\mathfrak {p}}_{\alpha _m}^t}{{\mathfrak {p}}_{\alpha _m}^s}\right) =\frac{{\mathfrak {p}}\,{\mathfrak {p}}_\beta ^s\,{\mathfrak {p}}_{\alpha _m\beta }^{st}}{{\mathfrak {p}}_{\alpha _m}^s}, \end{aligned}$$

(E.23)

where we have used that ${\mathfrak {p}}_\beta ^t={\mathfrak {p}}_{\alpha _m\beta }^{st}+{\mathfrak {p}}_{\alpha _m}^t$ and ${\mathfrak {p}}_{\alpha _m}^s={\mathfrak {p}}_{\alpha _m\beta }^{st}+{\mathfrak {p}}_\beta ^s$. Thus, the expression in the square brackets in (E.22) becomes

$$\begin{aligned} \frac{{\mathfrak {p}}}{{\mathfrak {p}}_{\alpha _m}^s}\left[ {\mathfrak {p}}_\beta ^s\,{\mathfrak {p}}_{\alpha _m\beta }^{st} -{\mathfrak {p}}_{\alpha _m}^s\sum _{\ell =1}^{m-1}\frac{({\mathfrak {p}}_{\alpha _\ell \beta }^{Ls})^2\,{\mathfrak {p}}_{\alpha _\ell }^R}{{\mathfrak {p}}_{v_{\alpha _\ell }}{\mathfrak {p}}_{\alpha _\ell }^L }\right] . \end{aligned}$$

(E.24)

The new expression in the square brackets is exactly the same as in (E.22) where tree ${\mathcal {T}}$ was replaced by subtree ${\mathcal {T}}_{\alpha _m}^s={\mathcal {T}}_{\alpha _{m-1}}$. Thus, one can repeat the above manipulation until one exhausts all terms in the sum. As a result, the determinant of the Gram matrix turns out to be proportional to ${\mathfrak {p}}_{\alpha _1\beta }^{st}$. But since $\alpha _1=\beta$, this quantity, and hence the whole determinant, trivially vanish.

Next, we prove that (E.15) is consistent with ${\mathcal {B}}\cdot {{\varvec{x}}}=\left\{ \frac{w^\alpha }{\sqrt{\Delta _\alpha }}\right\}$ and ${\mathcal {B}}\cdot {\mathcal {V}}={\mathcal {M}}$. The first relation is a direct consequence of (E.13). The second relation requires to show that $({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\beta )={\mathfrak {p}}\,\Delta _\alpha ^{1/2}\Delta _\beta ^{-1/2}{\mathcal {M}}_{\alpha \beta }$ or equivalently $\sum _{\gamma }({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\gamma ) \sqrt{\Delta _\gamma \Delta _\beta } {\mathcal {M}}_{\gamma \beta }^{-1}={\mathfrak {p}}\,\Delta _\alpha \delta _{\alpha \beta }$. Using the result (E.14) for the matrix ${\mathcal {M}}_{\gamma \beta }^{-1}$, this relation can be written explicitly as

$$\begin{aligned} {\mathfrak {p}}_{v_\beta }({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\beta ) +\left( \sum _{e_\gamma \in E_\beta ^R} \epsilon _{\gamma \beta } {\mathfrak {p}}_{L(v_\beta )} +\sum _{e_\gamma \in E_\beta ^L} \epsilon _{\gamma \beta } {\mathfrak {p}}_{R(v_\beta )}\right) ({\tilde{{{\varvec{u}}}}}_\alpha , {{\varvec{u}}}_\gamma ) ={\mathfrak {p}}\,\Delta _\alpha \delta _{\alpha \beta }, \qquad \end{aligned}$$

(E.25)

where $E_{\beta }^{L}=\{e\in E_{\mathcal {T}}:\ e\cap {\mathcal {T}}_{\beta _L}\ne \emptyset , \ e\nsubseteq {\mathcal {T}}_\beta \}$ and similarly for $E_{\beta }^{R}$.

Consider first the case $\alpha =\beta$. From (E.20), it immediately follows that the first term gives ${\mathfrak {p}}\,\Delta _\alpha$. On the other hand, the second contribution sums over edges for which ${\mathcal {T}}_\alpha \subseteq {\mathcal {T}}_\gamma ^s$ or ${\mathcal {T}}_\gamma ^t$, and as noted above (E.20) this leads to vanishing of the scalar product. Thus, in this case the relation (E.25) indeed holds.

Let us now show that it holds as well for $\alpha \ne \beta$. To this end, one should consider several options. If ${\mathcal {T}}_\alpha \cap {\mathcal {T}}_\beta =\emptyset$, then ${\mathcal {T}}_\alpha \subset {\mathcal {T}}_\beta ^s$ or ${\mathcal {T}}_\beta ^t$ which implies vanishing of the first term. But the second term vanishes as well because the conditions $e_\gamma \subseteq {\mathcal {T}}_\alpha$ and $e_\gamma \cap {\mathcal {T}}_\beta \ne \emptyset$ are inconsistent with ${\mathcal {T}}_\alpha \cap {\mathcal {T}}_\beta =\emptyset$.

Similarly, if ${\mathcal {T}}_\alpha \subset {\mathcal {T}}_\beta$, one has ${\mathcal {T}}_\alpha \subset {\mathcal {T}}_\beta ^s$ or $T_\beta ^t$ which again leads to the vanishing of the first term, whereas the vanishing of the second is a consequence of that $e_\gamma \subseteq {\mathcal {T}}_\alpha$ implies $e_\gamma \subset {\mathcal {T}}_\beta$ so that the sum over $e_\gamma$ is empty.

It remains to consider the case ${\mathcal {T}}_\beta \subset {\mathcal {T}}_\alpha$. It is clear that ${\mathcal {T}}_\beta \subset {\mathcal {T}}_\alpha ^s$ or ${\mathcal {T}}_\alpha ^t$. Without loss of generality, let us assume that ${\mathcal {T}}_\beta \subset {\mathcal {T}}_\alpha ^s$ and $e_\alpha \subset {\mathcal {T}}_\beta ^t$. Then according to (E.20), the first term gives ${\mathfrak {p}}_{v_\beta }\,{\mathfrak {p}}_{\alpha \beta }^{Ls}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}$. If instead we have chosen the orientation such that $e_\alpha \subset {\mathcal {T}}_\beta ^s$, then we would find $-{\mathfrak {p}}_{v_\beta }\,{\mathfrak {p}}_{\alpha \beta }^{Lt}\,{\mathfrak {p}}_\alpha ^R\,{\mathfrak {p}}$. Similar results are obtained for each term in the sum of the second contribution. Again without loss of generality we assume that for all relevant edges $e_\gamma$ one has $e_\alpha \subset {\mathcal {T}}_\gamma ^t$, otherwise one flips their orientation. Then the l.h.s. of (E.25) is proportional to

$$\begin{aligned} {\mathfrak {p}}_{v_\beta }\,{\mathfrak {p}}_{\alpha \beta }^{Ls} +\sum _{e_\gamma \in E_{\beta }^{R}} \epsilon _{\gamma \beta } {\mathfrak {p}}_\beta ^L\,{\mathfrak {p}}_{\alpha \gamma }^{Ls} +\sum _{e_\gamma \in E_{\beta }^{L}} \epsilon _{\gamma \beta } {\mathfrak {p}}_\beta ^R \,{\mathfrak {p}}_{\alpha \gamma }^{Ls}. \end{aligned}$$

(E.26)

Let $e_\alpha ^\star \in E_{\beta }^{R}$ is the edge belonging to the path from ${\mathcal {T}}_\beta$ to $e_\alpha$ (which may coincide with $e_\alpha$). Then our choice of orientation implies that $\epsilon _{\gamma \beta }=-1$ for $e_\gamma \in E_\beta ^L\cup \{e_\alpha ^\star \}$ and $\epsilon _{\gamma \beta }=1$ for $e_\gamma \in E_\beta ^R{\setminus }\{e_\alpha ^\star \}$. Furthermore, one has

$$\begin{aligned} {\mathfrak {p}}_{\alpha \gamma _\alpha ^\star }^{Ls}={\mathfrak {p}}_{v_\beta }+\sum _{e_\gamma \in E_\beta ^L\cup E_\beta ^R{\setminus }\{e_\alpha ^\star \}}{\mathfrak {p}}_{\alpha \gamma }^{Ls}, \qquad {\mathfrak {p}}_{\alpha \beta }^{Ls}={\mathfrak {p}}_\beta ^L+\sum _{e_\gamma \in E_\beta ^L}{\mathfrak {p}}_{\alpha \gamma }^{Ls}. \end{aligned}$$

(E.27)

As a result, one finds

$$\begin{aligned} ({\mathfrak {p}}_\beta ^L+{\mathfrak {p}}_\beta ^R)\left( {\mathfrak {p}}_\beta ^L+\sum _{e_\gamma \in E_\beta ^L}{\mathfrak {p}}_{\alpha \gamma }^{Ls}\right) -{\mathfrak {p}}_\beta ^L \,{\mathfrak {p}}_{\alpha \gamma _\alpha ^\star }^{Ls} +\sum _{e_\gamma \in E_{\beta }^{R}{\setminus }\{e_\alpha ^\star \}} {\mathfrak {p}}_\beta ^L\,{\mathfrak {p}}_{\alpha \gamma }^{Ls} -\sum _{e_\gamma \in E_{\beta }^{L}} {\mathfrak {p}}_\beta ^R \,{\mathfrak {p}}_{\alpha \gamma }^{Ls}=0.\nonumber \\ \end{aligned}$$

(E.28)

This completes the proof of the required statement (E.25).

F. Proofs of Propositions

In this appendix we prove several propositions which we stated in the main text.

1.1 Proposition 1

To prove the recursive equation (5.3), let us note that the tree index $g_{\mathrm{tr},n}$ satisfies a very similar equation (cf. (2.10) or (2.16)) which can be seen as the origin of its representation (2.3) in terms of attractor flow trees. The only difference is the absence of the last term in (5.3). Therefore, it is easy to see that this equation implies a simple relation between ${\widehat{g}}_n$ and $g_{\mathrm{tr},n}$

$$\begin{aligned} {\widehat{g}}_n(\{{{\check{\gamma }}}_i\},z^a)=\sum _{n_1+\cdots +n_m= n\atop n_k\ge 1} g_{\mathrm{tr},m}(\{{{\check{\gamma }}}'_k\},z^a) \prod _{k=1}^m W_{n_k}({{\check{\gamma }}}_{j_{k-1}+1},\ldots ,{{\check{\gamma }}}_{j_{k}}), \end{aligned}$$

(F.1)

where as usual we use notations from (2.17).

Next, we substitute this relation into the expansion (5.2). The result can be represented in the following form

$$\begin{aligned} h^{\mathrm{DT}}_{p,q}=\sum _{\sum _{i=1}^n {{\check{\gamma }}}_i={{\check{\gamma }}}} g_{\mathrm{tr},n}(\{{{\check{\gamma }}}_i\},z^a)\,e^{\pi \mathrm {i}\tau Q_n(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^n h^R_{p_i,\mu _i}(\tau ), \end{aligned}$$

(F.2)

where we introduced

$$\begin{aligned} \begin{aligned} h^{R}_{p,\mu }=&\, \sum _{\sum _{i=1}^n {{\check{\gamma }}}_i={{\check{\gamma }}}}W_n(\{{{\check{\gamma }}}_i\})\,e^{\pi \mathrm {i}\tau Q_n(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^n {\widehat{h}}_{p_i,\mu _i}(\tau ) \\ =&\, \sum _{\sum _{i=1}^n {{\check{\gamma }}}_i={{\check{\gamma }}}}\left[ \sum _{T\in \mathbb {T}_n^\mathrm{S}}(-1)^{n_T}\prod _{v\in V_T} R_{v}\right] \,e^{\pi \mathrm {i}\tau Q_n(\{{{\check{\gamma }}}_i\})} \\&\, \times \prod _{i=1}^n \left( h_{p_i,\mu _i} +\sum _{n_i=2}^\infty \sum _{\sum _{j=1}^{n_i} {{\check{\gamma }}}'_j={{\check{\gamma }}}} R_{n_i}(\{{{\check{\gamma }}}'_j\}) \, e^{\pi \mathrm {i}\tau Q_{n_i}(\{{{\check{\gamma }}}'_j\})} \prod _{j_i=1}^{n_i} h_{p'_{j_i},\mu '_{j_i}}\right) \end{aligned} \end{aligned}$$

(F.3)

and in the last relation we used the definition of $W_n$ (5.4) and the expansion of ${\widehat{h}}_{p,\mu }$ (5.1). The crucial observation is that if one picks up a factor $R_{n_i}$ from the second line of (F.3), appearing due to the expansion of ${\widehat{h}}_{p_i,\mu _i}$, and combines it with the contribution of a tree T from the first line, one finds the opposite of the contribution of another tree obtained from T by adding $n_i$ children to its ith leaf. As a result, all such contributions cancel and the function (F.3) reduces to the trivial term

$$\begin{aligned} h^{R}_{p,\mu }=h_{p,\mu }. \end{aligned}$$

(F.4)

Substituting this into (F.2), it gives back the original expansion (2.31) of the generating function of DT invariants, which proves the recursive equation (5.3).

Finally, let us evaluate (5.2) at the attractor point $z^a_\infty (\gamma )$. At this point the DT invariants coincide with MSW invariants so that the l.h.s. becomes the generating function $h_{p,\mu }$. Meanwhile, all factors $\Delta _{\gamma _L\gamma _R}^z$ vanish at the attractor point and ${\widehat{g}}_n$ reduces to $W_n$. As a result, the relation (5.2) reproduces (5.5), which completes the proof of the proposition.

1.2 Proposition 2

We prove Proposition 2 by induction. For $n=2$ the recursive relation (5.10) reads

$$\begin{aligned} g^{(0)}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2;c_1)-g^{(0)}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2;\beta _{21})=-\frac{1}{4}\, \bigl (\text{ sgn }(c_1)-\text{ sgn }(\beta _{21}) \bigr )\, \kappa (\gamma _{12}), \end{aligned}$$

(F.5)

where we took into account that $S_1=c_1$ and $\Gamma _{21}=\beta _{21}=-\gamma _{12}$. Since $g^{(0)}_2$ is supposed to have discontinuities only at walls of marginal stability, it must not involve signs of DSZ products. Therefore, we are led to take

$$\begin{aligned} g^{(0)}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2;c_1)=-\frac{1}{4}\, \text{ sgn }(c_1) \kappa (\gamma _{12}). \end{aligned}$$

(F.6)

Then (5.11) and (5.12) imply

$$\begin{aligned} {\mathcal {E}}_2=\frac{1}{4}\,\text{ sgn }(\gamma _{12})\,\kappa (\gamma _{12})+R_2, \end{aligned}$$

(F.7)

so that the ansatz (5.9) reads

$$\begin{aligned} {\widehat{g}}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2;c_1)= & {} g^{(0)}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2;c_1)-{\mathcal {E}}_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2)\nonumber \\= & {} -\frac{1}{4}\,\Bigl [\text{ sgn }(c_1)+\text{ sgn }(\gamma _{12})\Bigr ]\,\kappa (\gamma _{12})-R_2({{\check{\gamma }}}_1,{{\check{\gamma }}}_2), \end{aligned}$$

(F.8)

which reproduces the recursive equation (5.3). Furthermore, in appendix D.2 it is shown that there is a smooth solution of Vignéras’ equation which asymptotes the function $({{\varvec{v}}},{{\varvec{x}}})\mathrm{sgn}({{\varvec{v}}},{{\varvec{x}}})$, coinciding with the (rescaled) first term in ${\mathcal {E}}_2$ for ${{\varvec{v}}}={{\varvec{v}}}_{12}$ (4.10). It is given by

$$\begin{aligned} {\widetilde{\Phi }}_1^E({{\varvec{v}}},{{\varvec{v}}};{{\varvec{x}}})={{\varvec{v}}}\cdot \left( {{\varvec{x}}}+\frac{1}{2\pi }\,\partial _{{\varvec{x}}}\right) {{\,\mathrm{Erf}\,}}\left( \frac{\sqrt{\pi }{{\varvec{v}}}\cdot {{\varvec{x}}}}{|{{\varvec{v}}}|}\right) , \end{aligned}$$

(F.9)

which corresponds to the following choice of $R_2$ [14]

$$\begin{aligned} R_2=\frac{(-1)^{\gamma _{12}}}{8\pi }\, |\gamma _{12}| \,\beta _{\frac{3}{2}}\!\left( {\frac{2\tau _2\gamma _{12}^2 }{(pp_1p_2)}}\right) , \end{aligned}$$

(F.10)

where $\beta _{\frac{3}{2}}(x^2)=2|x|^{-1}e^{-\pi x^2}-2\pi \text {Erfc}(\sqrt{\pi } |x|)$. Note that the resulting ${\mathcal {E}}_2$ depends on the electric charges only through the DSZ product $\gamma _{12}$. Finally, it is immediate to see that the kernel $\Phi ^{\,{\widehat{g}}}_2$ (5.6) satisfies (5.7).

Now we assume that (5.3) is consistent with the ansatz (5.9) for all orders up to $n-1$ and check it at order n. Denoting the second term in (5.9) by ${\widehat{g}}^{(+)}_n$ and substituting this ansatz into the r.h.s. of (5.3), one finds

$$\begin{aligned} \,\mathrm{Sym}\, \left\{ \sum _{\ell =1}^{n-1}g_2^\ell \, \Bigl [g^{(0)}_\ell \,g^{(0)}_{n-\ell }-{\widehat{g}}^{(+)}_\ell \,{\widehat{g}}_{n-\ell }-{\widehat{g}}_\ell \,{\widehat{g}}^{(+)}_{n-\ell }-{\widehat{g}}^{(+)}_\ell \,{\widehat{g}}^{(+)}_{n-\ell }\Bigr ]_{c_i\rightarrow c_i^{(\ell )}} \right\} +W_n, \qquad \end{aligned}$$

(F.11)

where

$$\begin{aligned} g_2^\ell =-{1\over 2}\,\Delta _{\gamma _L^\ell \gamma _R^\ell }^z\,\kappa (\gamma _{LR}^\ell ) =\frac{1}{4}\, \bigl (\text{ sgn }(S_\ell )-\text{ sgn }(\Gamma _{n\ell })\bigr )\,\kappa (\Gamma _{n\ell }). \end{aligned}$$

(F.12)

In the first term proportional to $g^{(0)}_\ell \,g^{(0)}_{n-\ell }$, one can apply the relation (5.10), which together with (5.11) gives $g^{(0)}_n-{\mathcal {E}}^{(0)}_n$. The other terms in the sum over $\ell$ can be reorganized as follows

$$\begin{aligned} \begin{aligned}&-\,\mathrm{Sym}\, \left\{ \sum _{\ell =1}^{n-1} g_2^\ell \sum _{n_1+\cdots +n_m= n\atop n_k\ge 1, \ m<n, \ \ell \in \{j_k\}} {\widehat{g}}_{k_0}(c_i^{(\ell )})\, {\widehat{g}}_{m-k_0}(c_i^{(\ell )})\prod _{k=1}^m {\mathcal {E}}_{n_k}\right\} \\&\quad = -\,\mathrm{Sym}\, \left\{ \sum _{n_1+\cdots +n_m= n\atop n_k\ge 1, \ 1<m<n} \left[ \,\sum _{k_0=1}^{m-1} g_2^{j_{k_0}}\, {\widehat{g}}_{k_0}(c_i^{(\ell )})\, {\widehat{g}}_{m-k_0}(c_i^{(\ell )})\,\right] \prod _{k=1}^m {\mathcal {E}}_{n_k}\right\} . \end{aligned} \end{aligned}$$

(F.13)

Here we first combined three contributions into one sum over splittings by adding the condition $\ell \in \{j_k\}$, with $k_0$ being the index for which $j_{k_0}=\ell$, and then interchanged the two sums which allows to drop the condition $\ell \in \{j_k\}$, but adds the requirement $m>1$ (following from $\ell \in \{j_k\}$ in the previous representation). In square brackets one recognizes the first contribution from the r.h.s. of (5.3) with n replaced by $m<n$. Hence, it is subject to the induction hypothesis which allows to replace this expression by ${\widehat{g}}_{m}(\{\gamma '_k\},z^a)-W_m(\{\gamma '_k\})$. Combining all contributions together, one concludes that (F.11) is equal to

$$\begin{aligned} g^{(0)}_n-{\mathcal {E}}^{(0)}_n -\,\mathrm{Sym}\, \left\{ \sum _{n_1+\cdots +n_m= n\atop n_k\ge 1, \ 1<m<n} \bigl ({\widehat{g}}_{m}-W_m\bigr )\prod _{k=1}^m {\mathcal {E}}_{n_k}\right\} +W_n. \end{aligned}$$

(F.14)

The contributions involving W’s can be combined into one sum by dropping the condition $m<n$. The resulting sum coincides with the r.h.s. of (5.12) so that these contributions can be replaced by $-{\mathcal {E}}^{(+)}_n$. Combined with $-{\mathcal {E}}^{(0)}_n$, this gives $-{\mathcal {E}}_n$ and allows to drop the condition $m>1$ in the remaining sum with ${\widehat{g}}_{m}$. As a result, (F.14) becomes equivalent to the r.h.s. of (5.9), which proves the consistency of this ansatz with the recursive equation.

Finally, let us show that the ansatz satisfies the modularity constraint (5.7). The crucial observation is that the vectors ${{\varvec{v}}}_{ij}$ and ${{\varvec{u}}}_{ij}$ (4.10) satisfy

$$\begin{aligned} ({{\varvec{v}}}_{i+j,k},{{\varvec{v}}}_{ij})=({{\varvec{u}}}_{i+j,k},{{\varvec{v}}}_{ij})=0, \end{aligned}$$

(F.15)

where we abused notation and denoted ${{\varvec{v}}}_{i+j,k}={{\varvec{v}}}_{ik}+{{\varvec{v}}}_{jk}$, etc. These orthogonality relations, together with the assumption that ${\mathcal {E}}_n$ depend on electric charges only through the DSZ products $\gamma _{ij}\sim ({{\varvec{v}}}_{ij},{{\varvec{x}}})$, imply factorization of the action of Vignéras’ operator on the kernel corresponding to the second term in (5.9). Indeed, all contributions of $\partial _{{\varvec{x}}}^2$ where two derivatives act on different factors vanish and the action reduces to the sum of terms where Vignéras’ operator acts on one of the factors. But since it is supposed to vanish on $\Phi ^{\,{\mathcal {E}}}_n$, one obtains the simple result

$$\begin{aligned} V_{n-1} \cdot \Phi ^{\,{\widehat{g}}}_{n}=V_{n-1} \cdot \Phi ^{\,g^{(0)}}_{n} -\,\mathrm{Sym}\, \left\{ \sum _{n_1+\cdots +n_m= n\atop n_k\ge 1, \ m<n} (V_{m-1}\cdot \Phi ^{\,{\widehat{g}}}_m) \prod _{k=1}^m \Phi ^{\,{\mathcal {E}}}_{n_k}\right\} . \end{aligned}$$

(F.16)

Since for $n=2$ the constraint was already shown to hold, one can proceed by induction. Then in the second term one can substitute the r.h.s. of (5.7), whereas the first term can be evaluated using the recursive relation (5.10). First of all, by the same reasoning as above, away from discontinuities, the action of Vignéras’ operator is factorized and actually vanishes. Furthermore, since $g^{(0)}_n$ have discontinuities only at walls of marginal stability, to obtain the complete action, it is enough to consider it only on $\text{ sgn }(S_\ell )$. Since at $S_\ell =0$ one has $c^{(\ell )}_i=c_i$ (see (2.15)), one finds that $\Phi ^{\,g^{(0)}}_n$ satisfy exactly the same constraint as (5.7). Thus, one can rewrite (F.16) as

$$\begin{aligned}&V_{n-1} \cdot \Phi ^{\,{\widehat{g}}}_{n}\nonumber \\&\quad = \,\mathrm{Sym}\, \sum _{\ell =1}^{n-1} \Bigl ({{\varvec{u}}}_\ell ^2\,\Delta _{n,\ell }^{g^{(0)}} \,\delta '({{\varvec{u}}}_\ell \cdot {{\varvec{x}}}) + 2{{\varvec{u}}}_\ell \cdot \partial _{{\varvec{x}}}\Delta _{n,\ell }^{g^{(0)}} \,\delta ({{\varvec{u}}}_\ell \cdot {{\varvec{x}}})\Bigr ) \nonumber \\&\qquad -\,\mathrm{Sym}\, \left\{ \sum _{n_1+\cdots +n_m= n\atop n_k\ge 1, \ 1<m<n}\left[ \sum _{\ell =1}^{n-1} \Bigl ({{\varvec{u}}}_\ell ^2\,\Delta _{m,\ell }^{{\widehat{g}}} \,\delta '({{\varvec{u}}}_\ell \cdot {{\varvec{x}}}) +2{{\varvec{u}}}_\ell \cdot \partial _{{\varvec{x}}}\Delta _{m,\ell }^{{\widehat{g}}} \,\delta ({{\varvec{u}}}_\ell \cdot {{\varvec{x}}})\Bigr )\right] \prod _{k=1}^m \Phi ^{\,{\mathcal {E}}}_{n_k}\right\} .\nonumber \\ \end{aligned}$$

(F.17)

Note that the orthogonality relation allows to include $\Phi ^{\,{\mathcal {E}}}_{n_k}$ under the derivative in the last term. Then one can perform the same manipulations with the sum over splittings as in (F.13) but in the inverse direction, which directly leads to the constraint (5.7).

1.3 Proposition 3

Our goal is to find an explicit expression for ${\widetilde{\Phi }}^{(0)}_n$. Using the fact that the limit of large ${{\varvec{x}}}$ of the generalized error function $\Phi _{n}^E$ is the product of n sign functions, one immediately obtains

$$\begin{aligned} {\widetilde{\Phi }}^{(0)}_n({{\varvec{x}}})= \frac{1}{2^{n-1} n!}\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell }\left[ \prod _{e\in E_{\mathcal {T}}}{\mathcal {D}}( {{\varvec{v}}}_{s(e) t(e)})\right] \left[ \prod _{e\in E_{{\mathcal {T}}}}\mathrm{sgn}({{\varvec{u}}}_e,{{\varvec{x}}})\right] , \end{aligned}$$

(F.18)

where ${\mathcal {D}}({{\varvec{v}}})$ are the covariant derivative operators (D.16). The action of the derivatives $\partial _{{\varvec{x}}}$ on the sign functions can be ignored (since the original function is smooth), however there are additional contributions due to the mutual action of the operators ${\mathcal {D}}$. Similar contributions were discussed in a similar context in appendix E, where they were shown to cancel, but here they turn out to leave a finite remainder. The contribution generated by the mutual action of two operators ${\mathcal {D}}( {{\varvec{v}}}_{s(e) t(e)})$ is non-vanishing only if the two edges $e_1,e_2$ have a common vertex. In this case it contributes the factor

$$\begin{aligned} \frac{(p_{{\mathfrak {v}}_1}p_{{\mathfrak {v}}_2}p_{{\mathfrak {v}}_{3}})}{2\pi }\, \mathrm{sgn}({{\varvec{u}}}_{e_1},{{\varvec{x}}})\, \mathrm{sgn}({{\varvec{u}}}_{e_2},{{\varvec{x}}}) , \end{aligned}$$

(F.19)

where ${\mathfrak {v}}_1,{\mathfrak {v}}_2,{\mathfrak {v}}_3$ are the tree vertices joint by the edges $e_1=({\mathfrak {v}}_2,{\mathfrak {v}}_3)$, $e_2=({\mathfrak {v}}_1,{\mathfrak {v}}_3)$. Again, considering the three trees shown in Fig. 7, one can note that the vectors ${{\varvec{u}}}_e$ defined by these trees satisfy the following relations: ${{\varvec{u}}}_{e_1}^{(1)}=-{{\varvec{u}}}_{e_3}^{(2)}$, ${{\varvec{u}}}_{e_2}^{(1)}={{\varvec{u}}}_{e_3}^{(3)}$, ${{\varvec{u}}}_{e_2}^{(2)}={{\varvec{u}}}_{e_1}^{(3)}$ and ${{\varvec{u}}}_{e_2}^{(2)}={{\varvec{u}}}_{e_1}^{(1)}+{{\varvec{u}}}_{e_2}^{(1)}$, where $e_3=({\mathfrak {v}}_1,{\mathfrak {v}}_2)$ and we indicated by upper index the tree with respect to which the vector is defined. Therefore, the contributions generated by these trees combine into

$$\begin{aligned}&\frac{(p_{{\mathfrak {v}}_1}p_{{\mathfrak {v}}_2}p_{{\mathfrak {v}}_{3}})}{2\pi }\Bigl [ \mathrm{sgn}({{\varvec{u}}}_{e_1}^{(1)},{{\varvec{x}}})\, \mathrm{sgn}({{\varvec{u}}}_{e_2}^{(1)},{{\varvec{x}}}) +\mathrm{sgn}({{\varvec{u}}}_{e_2}^{(2)},{{\varvec{x}}})\, \mathrm{sgn}({{\varvec{u}}}_{e_3}^{(2)},{{\varvec{x}}})\nonumber \\&\quad \quad -\mathrm{sgn}({{\varvec{u}}}_{e_1}^{(3)},{{\varvec{x}}})\, \mathrm{sgn}({{\varvec{u}}}_{e_3}^{(3)},{{\varvec{x}}})\Bigr ] \nonumber \\&\quad = -\frac{(p_{{\mathfrak {v}}_1}p_{{\mathfrak {v}}_2}p_{{\mathfrak {v}}_{3}})}{2\pi }, \end{aligned}$$

(F.20)

where we used the above relations between the vectors and the sign identity (D.12) for $x_s=({{\varvec{u}}}_{e_s}^{(1)},{{\varvec{x}}})$, $s=1,2$, Thus, unlike in (E.5), we now get a non-vanishing result, due to the mutual action of derivative operators.

Each pair of intersecting edges leads to a contribution which recombines the contributions of the edges from the three trees into a single factor (F.20). Furthermore, the sum over trees implies that we have to sum over all possible subtrees ${\mathcal {T}}_1,{\mathcal {T}}_2,{\mathcal {T}}_3$, in particular, over all possible allocations of different subtrees to the vertices ${\mathfrak {v}}_1,{\mathfrak {v}}_2,{\mathfrak {v}}_3$. The sign factors $\mathrm{sgn}({{\varvec{u}}}_e,{{\varvec{x}}})$ for edges of these subtrees do not depend on this allocation, but the factors ${\mathcal {D}}( {{\varvec{v}}}_{s(e) t(e)})$ do depend for the edges connecting to one of these vertices. It is easy to see that the sum over allocations effectively replaces the three vertices by a single one labelled by the total charge $p_{{\mathfrak {v}}_1}+p_{{\mathfrak {v}}_2}+p_{{\mathfrak {v}}_{3}}$. As a result, one obtains that the function (F.18) can be represented as a sum over marked trees where a mark corresponds to a collapse of a pair of intersecting edges and contributes the factor (F.20). More precisely, one has

$$\begin{aligned} {\widetilde{\Phi }}^{(0)}_n({{\varvec{x}}})= & {} \frac{1}{2^{n-1} n!}\sum _{m=0}^{[(n-1)/2]}\frac{(-1)^m}{(2\pi )^m} \sum _{{\mathcal {T}}\in \, \mathbb {T}_{n-2m,m}^\ell } \left[ \prod _{{\mathfrak {v}}\in V_{\mathcal {T}}}{\mathcal {P}}_{m_{\mathfrak {v}}}(\{p_{{\mathfrak {v}},s}\})\right] \nonumber \\&\times \prod _{e\in E_{{\mathcal {T}}}}({{\varvec{v}}}_{s(e) t(e)},{{\varvec{x}}})\,\mathrm{sgn}({{\varvec{u}}}_e,{{\varvec{x}}}), \end{aligned}$$

(F.21)

where

$$\begin{aligned} {\mathcal {P}}_{m}(\{p_{s}\})=\sum _{{\mathcal {I}}_1\cup \cdots \cup {\mathcal {I}}_{m}={\mathscr {Z}}_{2m+1}}\, \prod _{j=1}^{m}(p_{j_1}p_{j_2}p_{j_3}). \end{aligned}$$

(F.22)

This factor collects the weights (F.20) assigned to a vertex due to collapse of m pairs of edges. It is represented as a sum over all possible splittings of the set ${\mathscr {Z}}_{2m+1}=\{1,\ldots ,2m+1\}$ into union of triples ${\mathcal {I}}_j=\{j_1,j_2,j_3\}$ such that the labels in one triple ${\mathcal {I}}_j$ are all different, two different triples can have at most one common label, and there are no closed cycles in the sense that there are no subsets $\{{j_k}\}_{k=1}^r$ such that ${\mathcal {I}}_{j_k}\cap {\mathcal {I}}_{j_{k+1}}\ne \emptyset$ where $j_{r+1}\equiv j_1$. This sum simply counts all possible splittings of a collection of 2m joint edges into m intersecting pairs, suppressing for each pair the distinction between the three configurations of Fig. 7.

However, the representation (F.22) is not very convenient for our purposes. An alternative representation can be obtained by noting that, instead of collapsing all edges at once, one can collapse first one pair, sum over all configurations (i.e. allocations of subtrees), then collapse another pair, and so on. In this approach at each step the sum over different configurations ensures that all factors assigned to the elements of the surviving tree depend only on the sum of collapsing charges. As a result, one obtains a hierarchical structure described by a rooted ternary tree T, where the leaves correspond to the vertices of the original unrooted tree which have collapsed into one vertex with $m_{\mathfrak {v}}$ marks corresponding to the root of T. The other vertices of T are then in one-to-one correspondence with marked vertices appearing at intermediate stages of the above process.

This procedure gives rise to the representation (5.19) of the weight factor ${\mathcal {P}}_m$ in terms of a sum over rooted ternary trees. A non-trivial point which must be taken into account is that the procedure leading to this representation overcounts different configurations. As a result, each tree is weighted by the rational coefficient $N_{{{\hat{T}}}}/m!$ where ${{\hat{T}}}$ is the rooted tree obtained from T by dropping all leaves and $N_{{{\hat{T}}}}$ is the number of ways of labelling the vertices of ${{\hat{T}}}$ with increasing labels, which already appeared in appendix A. Here the numerator takes into account that the tree T is generated $N_{{{\hat{T}}}}$ times in the collapsing process, whereas the denominator removes the overcounting produced by specifying the order in which the m pairs of edges are collapsed. Finally, we apply Lemma 1 from appendix A. Since $n_{{{\hat{T}}}}=m$, the coefficients coincides with the inverse of the tree factorial (5.20). Taking into account that substitution of (F.21) into (5.15) gives exactly (5.17), this completes the proof of the proposition.

1.4 Proposition 4

Our aim is to show that the recursive formula (5.24) solves the equations (5.23). We will proceed by induction, starting with the case $n=3$ where there is a single unrooted tree and the system (5.23) contains a single equation corresponding to the trivial trees ${\mathcal {T}}_r$ consisting of one vertex. Thus, it is solved by $a_{\bullet \!\text{- }\!\bullet \!\text{- }\!\bullet }=\frac{1}{3}$ consistently with (5.24).

Let us now consider trees with n vertices assuming that for all trees with less number of vertices (5.24) holds. We start by noting that for every vertex ${\mathfrak {v}}\in V_{{\mathcal {T}}_r}$, among the subtrees ${\mathcal {T}}_{r,s}({\mathfrak {v}})\subset {\mathcal {T}}_r$ obtained by removing the vertex ${\mathfrak {v}}$, there is one which contains ${\mathfrak {v}}_r$, which we denote by ${\mathcal {T}}_{r,s_0}({\mathfrak {v}})$. (If ${\mathfrak {v}}={\mathfrak {v}}_r$, we take ${\mathcal {T}}_{r,s_0}({\mathfrak {v}})=\emptyset$.) Since every tree ${\hat{{\mathcal {T}}}}_r$ is a union of the three trees ${\mathcal {T}}_r$, substituting (5.24) into the l.h.s. of (5.23), one obtains that the sum over vertices of ${\hat{{\mathcal {T}}}}_r$ can be represented as

$$\begin{aligned} \frac{1}{n}\sum _{r=1}^3 \sum _{{\mathfrak {v}}\in V_{{\mathcal {T}}_r}} \epsilon _{\mathfrak {v}}\left( a_{{\hat{{\mathcal {T}}}}_{1,r}({\mathfrak {v}})} +a_{{\hat{{\mathcal {T}}}}_{2,r}({\mathfrak {v}})} -a_{{\hat{{\mathcal {T}}}}_{3,r}({\mathfrak {v}})}\right) \prod _{s=1\atop s\ne s_0}^{n_{\mathfrak {v}}} a_{{\mathcal {T}}_{r,s}({\mathfrak {v}})}, \end{aligned}$$

(F.23)

where ${\hat{{\mathcal {T}}}}_{r',r}({\mathfrak {v}})$ is obtained from ${\hat{{\mathcal {T}}}}_{r'}$ by replacing ${\mathcal {T}}_r$ by ${\mathcal {T}}_{r,s_0}({\mathfrak {v}})$. Applying the equations (5.23), this expression gives

$$\begin{aligned}&\frac{1}{n}\left[ a_{{\mathcal {T}}_2} a_{{\mathcal {T}}_3}\sum _{{\mathfrak {v}}\in V_{{\mathcal {T}}_1}}\epsilon _{\mathfrak {v}}\prod _{s=1}^{n_{\mathfrak {v}}} a_{{\mathcal {T}}_{1,s}({\mathfrak {v}})} +a_{{\mathcal {T}}_1} a_{{\mathcal {T}}_3}\sum _{{\mathfrak {v}}\in V_{{\mathcal {T}}_2}}\epsilon _{\mathfrak {v}}\prod _{s=1}^{n_{\mathfrak {v}}} a_{{\mathcal {T}}_{2,s}({\mathfrak {v}})}\nonumber \right. \\&\quad \left. +a_{{\mathcal {T}}_1} a_{{\mathcal {T}}_2}\sum _{{\mathfrak {v}}\in V_{{\mathcal {T}}_3}}\epsilon _{\mathfrak {v}}\prod _{s=1}^{n_{\mathfrak {v}}} a_{{\mathcal {T}}_{3,s}({\mathfrak {v}})} \right] . \end{aligned}$$

(F.24)

Since the trees ${\mathcal {T}}_{r}$ have less than n vertices, they are subject to the induction hypothesis which allows to replace the sums over vertices by $n_r a_{{\mathcal {T}}_r}$ where $n_r$ is the number of vertices in ${\mathcal {T}}_r$. Taking into account that $n_1+n_2+n_3=n$, the expression (F.24) reduces to $a_{{\mathcal {T}}_1} a_{{\mathcal {T}}_2}a_{{\mathcal {T}}_3}$, which proves that the equations (5.23) are indeed satisfied.

1.5 Proposition 5

The evaluation of the large ${{\varvec{x}}}$ limit of the function ${\widetilde{\Phi }}^{\,{\mathcal {E}}}_n({{\varvec{x}}})$ is very similar to the calculation done in the proof of Proposition 3. First, using the asymptotics of the generalized error functions, we can write

$$\begin{aligned} \lim _{{{\varvec{x}}}\rightarrow \infty }{\widetilde{\Phi }}^{\,{\mathcal {E}}}_n({{\varvec{x}}})= & {} \frac{1}{2^{n-1} n!} \!\! \sum _{m=0}^{[(n-1)/2]} \!\!\! \sum _{{\mathcal {T}}\in \, \mathbb {T}_{n-2m,m}^\ell } \left[ \prod _{{\mathfrak {v}}\in V_{\mathcal {T}}}{\mathcal {D}}_{m_{\mathfrak {v}}}(\{{{\check{\gamma }}}_{{\mathfrak {v}},s}\})\right] \nonumber \\&\times \left[ \prod _{e\in E_{\mathcal {T}}}{\mathcal {D}}( {{\varvec{v}}}_{s(e) t(e)})\right] \left[ \prod _{e\in E_{{\mathcal {T}}}}\mathrm{sgn}({{\varvec{u}}}_e,{{\varvec{x}}})\right] \! .\nonumber \\ \end{aligned}$$

(F.25)

We then observe that the two last factors depend only on sums of charges appearing in the operators ${\mathcal {D}}_{m_{\mathfrak {v}}}$ in the first factor. This implies that the vectors on which these factors depend are orthogonal to the vectors determining the operators in (5.22). Therefore, these operators effectively act on a constant and can be expanded as

$$\begin{aligned} {\mathcal {D}}_{m}\cdot 1 = \sum _{k=0}^m{\mathcal {V}}_{m,k}, \end{aligned}$$

(F.26)

where ${\mathcal {V}}_{m,k}$ are homogeneous polynomials in ${{\varvec{x}}}$ of degree $2(m-k)$. In particular, the highest degree term coincides with the function defined in (5.26), ${\mathcal {V}}_{m,0}(\{{{\check{\gamma }}}_{s}\})= {\mathcal {V}}_{m}(\{{{\check{\gamma }}}_{s}\})$.

Next, the mutual action of the derivative operators from the product over edges in the second factor generates contributions described by trees with pairs of collapsed edges replaced by marks. The difference here is that the original trees were also marked. This fact does not change the structure of the result, which is again given by a sum over marked trees, but it affects the weight associated with marks. Denoting this weight for a vertex with total m marks (old and new) by ${\mathcal {V}}^{\mathrm{tot}}_m$, we recover the equation (5.25) in the statement of the proposition, provided that ${\mathcal {V}}^{\mathrm{tot}}_m={\mathcal {V}}_m$. We now proceed to prove the latter identity.

The weight factor ${\mathcal {V}}^{\mathrm{tot}}_m$ coming from the above procedure is given by

$$\begin{aligned}&{\mathcal {V}}^{\mathrm{tot}}_m(\{{{\check{\gamma }}}_{s}\})\nonumber \\&\quad =\,\mathrm{Sym}\, \!\! \left\{ \sum _{m_0=0}^{m}\frac{(-1)^{m_0}}{(2\pi )^{m_0}} \!\!\!\! \sum _{\sum \limits _{r=1}^{2m_0+1}m_r=m-m_0} \!\!\!\!\!\! \!\!\!\!\!\! C(\{m_r\}) \,{\mathcal {P}}_{m_0}(\{p'_{r}\}) \!\! \prod _{r=1}^{2m_0+1}\sum _{k_r=0}^{m_r}{\mathcal {V}}_{m_r,k_r}({{\check{\gamma }}}_{j_{r-1}+1},\ldots ,{{\check{\gamma }}}_{j_{r}}) \right\} ,\nonumber \\ \end{aligned}$$

(F.27)

where the second sum goes over all ordered decompositions of $m-m_0$ into non-negative integers, $C(\{m_r\})=\frac{(2m+1)!}{\prod _r (2m_r+1)!}$, and we used notations similar to (2.17),

$$\begin{aligned} j_0=0, \qquad j_r=m_1+\cdots + m_r, \qquad \gamma '_r=\gamma _{j_{r-1}+1}+\cdots +\gamma _{j_{r}}. \end{aligned}$$

(F.28)

Here the first factor ${\mathcal {P}}_{m_0}$ arises due to collapse of $m_0$ pairs of edges in the marked trees one sums over in (F.25), whereas the factors given by the sums over $k_r$ are the ones corresponding to the “old" marks assigned to that trees.

Now let us use the equations (5.23) determining the coefficients $a_{\mathcal {T}}$. It is easy to see that they imply the following constraint on the next-to-highest degree term in the expansion (F.26)

$$\begin{aligned} {\mathcal {V}}_{m,1}(\{{{\check{\gamma }}}_{s}\})=\frac{1}{6\cdot 2\pi } \sum _{m_1+m_2+m_3=m-1\atop m_r\ge 0}C(\{m_r\}) \,\mathrm{Sym}\, \left\{ (p'_1 p'_2 p'_3)\prod _{r=1}^3 {\mathcal {V}}_{m_r}(\{{{\check{\gamma }}}_i\}_{i=j_{r-1}+1}^{j_r}) \right\} ,\nonumber \\ \end{aligned}$$

(F.29)

where $j_r=\sum _{s=1}^r (2m_s+1)$ and $p'_r=\sum _{i=1}^{2m_r+1}p_{j_{r-1}+i}$. Applying this constraint recursively, one can express all ${\mathcal {V}}_{m,k}$ for $k>0$ through ${\mathcal {V}}_{m}$. The idea is to replace the factors ${\mathcal {V}}_{m_r}$ by the operators ${\mathcal {D}}_{m_r}$. Then one can realize that, extracting from the resulting function the terms homogeneous in ${{\varvec{x}}}$ of order $2(m-2)$ (which have two factors of $(p^3)$), one obtains the result for $2{\mathcal {V}}_{m,2}$. Proceeding in this way, one arrives at the representation very similar to (F.27),

$$\begin{aligned} {\mathcal {V}}_{m,k}(\{{{\check{\gamma }}}_{s}\})=\frac{1}{(2\pi )^k}\,\mathrm{Sym}\, \left\{ \sum _{\sum \limits _{r=1}^{2k+1}m_r=m-k} \!\!\!\! C(\{m_r\})\, {\mathcal {P}}_{k}(\{p'_{r}\}) \prod _{r=1}^{2k+1}{\mathcal {V}}_{m_r}({{\check{\gamma }}}_{j_{r-1}+1},\ldots ,{{\check{\gamma }}}_{j_{r}}) \right\} .\nonumber \\ \end{aligned}$$

(F.30)

Substituting it into (F.27) and using the expression (5.19) for the factors ${\mathcal {P}}_{m}$ through the sum over rooted ternary trees, one can recombine all these sums in the following way

$$\begin{aligned} \begin{aligned} {\mathcal {V}}^{\mathrm{tot}}_m&= \,\,\mathrm{Sym}\, \Biggl \{\sum _{k=0}^{m}\sum _{\sum \limits _{r=1}^{2k+1}m_r=m-k} \!\!\!\!\!\! C(\{m_r\}) \sum _{T\in \, \mathbb {T}_{2k+1}^{(3)}(\{{{\check{\gamma }}}'_r\})} \frac{1}{T!} \sum _{T'\subseteq T} (-1)^{n_{T'}} \prod _{v\in V_{T'}}\frac{n_v(T)}{n_v(T')} \\&\quad \,\times \prod _{v\in V_{T}}\frac{(p_{d_1(v)}p_{d_2(v)}p_{d_3(v)})}{2\pi } \prod _{r=1}^{2k+1}{\mathcal {V}}_{m_r}({{\check{\gamma }}}_{j_{r-1}+1},\ldots ,{{\check{\gamma }}}_{j_{r}}) \Biggr \}, \end{aligned} \end{aligned}$$

(F.31)

where the sum over $T'$ is the sum over subtrees of T having the same root. In terms of the variables appearing in (F.27), one can identify $k=m_0+\sum _r k_r$ and $n_{T'}=m_0$. Thus, $T'$ is the tree appearing in the decomposition of ${\mathcal {P}}_{m_0}$, whereas T is its union with $2m_0+1$ trees $T_r$ appearing in the decomposition of ${\mathcal {P}}_{k_r}$, i.e. $T=T'\cup \left( \cup _r T_r\right)$ where $T_r$ are the trees rooted at leaves of $T'$, Finally, we took into account that for such trees one has

$$\begin{aligned} T!=T'!\,\prod _r T_r!\,\prod _{v\in V_{T'}}\frac{n_v(T)}{n_v(T')}\,. \end{aligned}$$

(F.32)

Remarkably, the sum over subtrees in (F.31) factorizes and for a fixed number of vertices $n_{T'}=m_0$ is subject to Theorem 1 where the rôle of the trees is played by rooted ternary trees T and $T'$ after stripping out their leaves. As a result, one finds for $k>0$

$$\begin{aligned} \sum _{T'\subseteq T} (-1)^{n_{T'}} \prod _{v\in V_{T'}}\frac{n_v(T)}{n_v(T')} =\sum _{m_0=0}^k\frac{(-1)^{m_0}k!}{m_0!(k-m_0)!}=(1-1)^k = 0. \end{aligned}$$

(F.33)

Thus, the only non-vanishing contribution is the one with $k=0$ which coincides with ${\mathcal {V}}_m$. This is what we had to show and therefore completes the proof of the proposition.

1.6 Proposition 6

For simplicity, let us first show that the recursive equation (5.10) is satisfied by the contribution to $g^{(0)}_n(\{{{\check{\gamma }}}_i,c_i\})$ given by the trees without any marks, i.e. by the function

$$\begin{aligned} g^{\star }_n(\{{{\check{\gamma }}}_i,c_i\})=\frac{(-1)^{n-1+\sum _{i<j} \gamma _{ij} }}{2^{n-1} n!} \sum _{{\mathcal {T}}\in \, \mathbb {T}_{n}^\ell } \prod _{e\in E_{{\mathcal {T}}}}\gamma _{s(e) t(e)}\,\mathrm{sgn}(S_e). \end{aligned}$$

(F.34)

Then the inclusion of marks will be straightforward because the corresponding contributions can be dealt with essentially in the same way as the contribution (F.34).

To start with, we substitute $g^{\star }_n$ into the l.h.s. of the recursive equation and decompose $\Gamma _{n\ell }=-\sum _{i=1}^\ell \sum _{j=\ell +1}^n \gamma _{ij}$. Then the crucial observation is that this double sum, the sum over $\ell$ and the two sums over trees (over $\mathbb {T}_\ell ^\ell$ and $\mathbb {T}_{n-\ell }^\ell$) are all equivalent to a single sum over trees with n vertices, i.e. over $\mathbb {T}_n^\ell$, supplemented by the sum over edges. Namely, one can do the following replacement

$$\begin{aligned} {1\over 2}\,\mathrm{Sym}\, \sum _{\ell =1}^{n-1}\frac{1}{\ell !(n-\ell )!}\sum _{{\mathcal {T}}_L\in \, \mathbb {T}_\ell ^\ell }\sum _{{\mathcal {T}}_R\in \, \mathbb {T}_{n-\ell }^\ell }\sum _{i=1}^\ell \sum _{j=\ell +1}^n =\frac{1}{n!}\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell }\sum _{e\in E_{\mathcal {T}}}. \end{aligned}$$

(F.35)

The idea is that on the l.h.s. one sums over all possible splittings of unrooted labelled trees with n vertices into two trees with $\ell$ and $n-\ell$ vertices. Such splitting can be done by cutting an edge and then i, j correspond to the labels of the vertices joined by the cutting edge. The binomial coefficient $\frac{n!}{\ell !(n-\ell )!}$ takes into account that after splitting the vertices of the first tree can have arbitrary labels from the set $\{1,\ldots ,n\}$ and not necessarily $\{1,\ldots ,\ell \}$, whereas ${1\over 2}$ avoids doubling due to the symmetry between ${\mathcal {T}}_L$ and ${\mathcal {T}}_R$.

It is easy to check that all factors in (5.10) fit this interpretation and the l.h.s. takes the following form

$$\begin{aligned}&\frac{(-1)^{n-1+\sum _{i<j} \gamma _{ij} }}{2^{n-1} n!}\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell } \prod _{e\in E_{{\mathcal {T}}}}\gamma _{s(e) t(e)}\sum _{e\in E_{\mathcal {T}}} \bigl ( \mathrm{sgn}(S_e)-\mathrm{sgn}(\Gamma _{e})\bigr ) \nonumber \\&\quad \times \prod _{e'\in E_{{\mathcal {T}}}{\setminus }\{e\}}\mathrm{sgn}\left( S_{e'}-\frac{\Gamma _{e'}}{\Gamma _{e}}\, S_e\right) , \end{aligned}$$

(F.36)

where $\Gamma _e$ was defined in (5.30). Finally, we apply the following sign identity, established in [23, Eq.(A.7)],

$$\begin{aligned} \sum _{\beta =1}^m \left( \text{ sgn }(x_\beta )-1\right) \prod _{\alpha =1\atop \alpha \ne \beta }^m \text{ sgn }(x_\alpha -x_\beta )= \prod _{\alpha =1}^m \text{ sgn }(x_\alpha )-1, \end{aligned}$$

(F.37)

where one should take the label $\alpha$ to run over $m=n-1$ edges of a tree ${\mathcal {T}}$, identify $x_\alpha =S_\alpha /\Gamma _{\alpha }$, and multiply it by $\prod _{\alpha =1}^{n-1}\text{ sgn }(\Gamma _\alpha )$. Then the expression (F.36) reduces to $g^{\star }_n(\{{{\check{\gamma }}}_i,c_i\})-g^{\star }_n(\{{{\check{\gamma }}}_i,\beta _{ni}\})$, i.e. the r.h.s. of (5.10) evaluated for function (F.34). This proves that this function solves the recursive equation.

The generalization of this proof to the full ansatz (5.27) is elementary. Instead of the relation (F.35), one now has

$$\begin{aligned}&{1\over 2}\,\mathrm{Sym}\, \sum _{\ell =1}^{n-1}\frac{1}{\ell !(n-\ell )!}\sum _{k_L=0}^{[(\ell -1)/2]} \!\! \sum _{{\mathcal {T}}_L\in \, \mathbb {T}_{\ell -2k_L,k_L}^\ell } \!\!\!\!\! \sum _{k_R=0}^{[(n-\ell -1)/2]} \!\!\!\! \sum _{{\mathcal {T}}_R\in \, \mathbb {T}_{n-\ell -2k_R,k_R}^\ell } \!\! \sum _{i=1}^\ell \sum _{j=\ell +1}^n\nonumber \\&\quad =\frac{1}{n!}\sum _{k=0}^{[(n-1)/2]} \!\!\!\! \sum _{{\mathcal {T}}\in \, \mathbb {T}_{n-2k,k}^\ell }\sum _{e\in E_{\mathcal {T}}}, \end{aligned}$$

(F.38)

which again reflects the fact the sum over marked unrooted labelled trees can be represented as a sum over all possible splittings into two such trees by cutting them along an edge. If the cutting edge joins a marked vertex, it counts $1+2m_{\mathfrak {v}}$ times, producing the right factor $\gamma _{{\mathfrak {v}}_L{\mathfrak {v}}_R}$, which is reflected by the fact that the sums over i, j run over $\ell$ and $n-\ell$ values, respectively. Other numerical factors work in the same way as before. Thus, the l.h.s. of (5.10) can again be rewritten as in (F.36) with the only difference that $\sum _{{\mathcal {T}}\in \, \mathbb {T}_n^\ell }$ should now be replaced by $\sum _{k=0}^{[(n-1)/2]}\sum _{{\mathcal {T}}\in \, \mathbb {T}_{n-2k,k}^\ell }\prod _{{\mathfrak {v}}\in V_{\mathcal {T}}}{\tilde{{\mathcal {V}}}}_{\mathfrak {v}}$. Applying the same sign identity (F.37) for $m=n-1-2k$ and the same identification for $x_\alpha$, one recovers the r.h.s. of (5.10). This completes the proof of the proposition.

1.7 Proposition 7

This proposition trivially follows from (5.9).

1.8 Proposition 8

The easiest way to prove (5.34) is to substitute it into (5.4) and then check that the result is consistent with the constraint (5.12). The substitution generates a sum over trees which resemble the blooming trees of appendix C: these are trees with vertices from which other trees grows. But now the two types of trees, representing the ‘base’ and the ‘flowers’, are actually the same—both of them are Schröder trees. The only difference is that vertices of the ‘base’ carry weights ${\mathcal {E}}^{(+)}_v$, whereas the vertices of ‘flowers’ have weights ${\mathcal {E}}^{(0)}_v$. Since the leaves of a ‘flower’ are in one-to-one correspondence with the children of the vertex of the ‘base’ tree from which this flower grows, such blooming trees can be equivalently represented by the usual Schröder trees obtained by replacing the vertices of the ‘base’ by their ‘flowers’. In this way, we obtain

$$\begin{aligned} W_n= \,\mathrm{Sym}\, \left\{ \sum _{T\in \mathbb {T}_n^{\mathrm{S}}}(-1)^{n_T} \sum _{\cup \, T'=T} \prod _{T'}\left[ {\mathcal {E}}^{(+)}_{v_0(T')}\prod _{v\in V_{T'}{\setminus } \{v_0(T')\}}{\mathcal {E}}^{(0)}_{v}\right] \right\} , \end{aligned}$$

(F.39)

where the second sum goes over decompositions of T into subtrees such that the root $v_0(T')$ of a subtree $T'$ is a leaf of another subtree (except, of course, the root of the total tree).

Let us now consider a vertex v whose only children are leaves of T. Then all decompositions into subtrees $T=\cup _i\, T'_i$ can be split into pairs such that two decompositions differ only by whether v (together with its leaves) represents a separate subtree or it is a part of a bigger subtree. The contributions of two such decompositions into (F.39) differ only by the factors assigned to the vertex v and therefore they combine into the factor ${\mathcal {E}}_v$ assigned to this vertex. As a result, ${\mathcal {E}}_v$ appears as a common factor and the vertex v can be excluded from the following consideration. Proceeding in the same way with the tree obtained by removing this vertex, one finds that the sum over decompositions can be evaluated explicitly and gives

$$\begin{aligned} W_n= \,\mathrm{Sym}\, \left\{ \sum _{T\in \mathbb {T}_n^{\mathrm{S}}}(-1)^{n_T} {\mathcal {E}}^{(+)}_{v_0}\prod _{v\in V_T{\setminus }{\{v_0\}}}{\mathcal {E}}_{v}\right\} . \end{aligned}$$

(F.40)

Given this result, the proof of the constraint (5.12) is analogous to the proof of the relation (F.4): the contribution of each tree T (from the sum in (F.40)) and a splitting with $n_k>1$ (from the sum in (5.12)) is cancelled by the contribution of another tree obtained from T by adding $n_k$ children to its kth leaf and the same splitting but with $n_k$ replaced by $1+\cdots +1$ (repeated $n_k$ times). The only contribution which survives is the one generated by the tree with a single vertex and n leaves and the splitting with all $n_k=1$. It is given by ${\mathcal {E}}^{(+)}_n$, which verifies the constraint and proves the proposition.

1.9 Proposition 9

Our starting point to prove the proposition is the formula

$$\begin{aligned} \partial _{{\bar{\tau }}}{\widehat{h}}_{p,\mu }(\tau )= \frac{\mathrm {i}}{2}\sum _{n=2}^\infty \sum _{\sum _{i=1}^n {{\check{\gamma }}}_i={{\check{\gamma }}}} \partial _{\tau _2}R_n(\{{{\check{\gamma }}}_i\},\tau _2) \, e^{\pi \mathrm {i}\tau Q_n(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^n h_{p_i,\mu _i}(\tau ). \nonumber \\ \end{aligned}$$

(F.41)

Substituting $\partial _{\tau _2}R_n$ following from (5.34) and the inverse formula (5.5) expressing $h_{p,\mu }$ in terms of the completion, with the functions $W_n$ found in (F.40), one arrives at the result (5.35) where the functions ${\mathcal {J}}_n$ are given by

$$\begin{aligned} {\mathcal {J}}_n = \frac{\mathrm {i}}{2}\,\mathrm{Sym}\, \left\{ \sum _{T\in \mathbb {T}_n^{\mathrm{S}}}(-1)^{n_T-1} \partial _{\tau _2}{\mathcal {E}}_{v_0} \sum _{T'\subseteq T}\prod _{v\in V_{T'}{\setminus }{\{v_0\}}}{\mathcal {E}}^{(0)}_{v} \prod _{v\in L_{T'}}{\mathcal {E}}^{(+)}_{v}\prod _{v\in V_T{\setminus } (V_{T'}\cup L_{T'})}{\mathcal {E}}_{v} \right\} , \nonumber \\ \end{aligned}$$

(F.42)

where the second sum goes over all subtrees $T'$ of T containing its root and $L_{T'}$ is the set of their leaves. Here the subtree $T'$ corresponds to the tree in the formula (5.34) for $R_n$, whereas the subtrees starting from its leaves correspond to the trees in the expression (F.40) for $W_n$. The sum over subtrees can be evaluated in the same way as the sum over decompositions into subtrees in (F.39): the two contributions differing only by whether the vertex v belongs to $V_{T'}$ or $L_{T'}$ combine into the factor ${\mathcal {E}}_v$ assigned to this vertex. Performing this recombination for all vertices of the tree T, one obtains the expression (5.36), which proves the proposition.

1.10 Proposition 10

Our goal is to prove that in the expression

$$\begin{aligned} g_{\mathrm{tr},n}= \,\mathrm{Sym}\, \left\{ \sum _{T\in \mathbb {T}_n^{\mathrm{S}}}(-1)^{n_T-1} \left( g^{(0)}_{v_0}-{\mathcal {E}}^{(0)}_{v_0}\right) \prod _{v\in V_T{\setminus }{\{v_0\}}}{\mathcal {E}}^{(0)}_{v}\right\} , \end{aligned}$$

(F.43)

following from (5.33), all contributions due to marked trees cancel leaving only contributions of trees without marks, i.e with $m=0$. There are several possible situations which we need to analyze.

Fig. 11

Combination of two Schröder trees ensuring the cancellation of contributions generated by marked trees

First, let us consider the contributions generated by non-trivial marked trees, i.e. trees having more than one vertex and at least one mark. Let us focus on the contribution corresponding to a vertex ${\mathfrak {v}}$ with $m_{\mathfrak {v}}>0$ marks of a tree ${\mathcal {T}}$, which appears in the sum over marked unrooted trees living at a vertex v of a Schröder tree T. Let $k=n_v$ be the number of children of the vertex v and $\gamma _i$ ($i=1,\ldots ,k$) their charges so that $\gamma _s$ ($s=1,\ldots ,2m_{\mathfrak {v}}+1$) are the charges labelling the marked vertex ${\mathfrak {v}}$. Note that $k\ge 2m_{\mathfrak {v}}+2$ because the tree ${\mathcal {T}}$ has at least one additional vertex except ${\mathfrak {v}}$. Then the contribution we described is cancelled by the contribution coming from another Schröder tree, which is obtained from T by adding an edge connecting the vertex v to a new vertex $v'$, whose children are the $2m_{\mathfrak {v}}+1$ children of v in T carrying charges $\gamma _s$ (see Fig. 11).³⁵ Indeed, choosing the same tree ${\mathcal {T}}$ as before in the sum over marked trees at vertex v, but now with $m_{\mathfrak {v}}=0$, and in the sum at vertex $v'$ the trivial tree having one vertex and $m_{\mathfrak {v}}$ marks, one gets exactly the same contribution as before, but now with an opposite sign due to the presence of an additional vertex in the Schröder tree. Thus, all contributions from non-trivial marked trees are cancelled.

As a result, we remain only with the contributions generated by trivial marked trees, i.e. having only one vertex and $m_{\mathfrak {v}}$ marks. One has to distinguish two cases: either the corresponding vertex v of the Schröder tree is the root or not. In the former case, this contribution is trivially cancelled in the difference $g^{(0)}_{v_0}-{\mathcal {E}}^{(0)}_{v_0}$ in (F.43). In the latter case, this is precisely the contribution used above to cancel the contributions from non-trivial marked trees. This exhausts all possibilities and we arrive at the formula (5.40).

G. Explicit Results Up to 4th Order

In this appendix we provide explicit expressions for various functions appearing in our construction up to the forth order. To write them down, we will use the shorthand notation $\gamma _{i+j}=\gamma _i+\gamma _j$, $c_{i+j}=c_i+c_j$, etc. as well as indicate the arguments of functions through their indices, for instance, ${\mathcal {E}}_{i_1\cdots i_n}={\mathcal {E}}_n({{\check{\gamma }}}_{i_1},\ldots ,{{\check{\gamma }}}_{i_n})$. These expressions are obtained by simple substitutions using the results found in the main text and the sets of trees shown in Fig. 1. For $n=2$ they all agree with the results of [14].

The results (5.33) and (5.34) generate the following expansions

$$\begin{aligned} h^{\mathrm{DT}}_{p,q}= & {} {\widehat{h}}_{p,\mu }+ \sum _{{{\check{\gamma }}}_1+{{\check{\gamma }}}_2={{\check{\gamma }}}} \left[ g^{(0)}_{12}-{\mathcal {E}}_{12}\right] e^{\pi \mathrm {i}\tau Q_2(\{{{\check{\gamma }}}_i\})} {\widehat{h}}_{p_1,\mu _1}{\widehat{h}}_{p_2,\mu _2} \nonumber \\&+\sum _{\sum _{i=1}^3 {{\check{\gamma }}}_i={{\check{\gamma }}}} \left[ g^{(0)}_{123}-{\mathcal {E}}_{123}-2\left( g^{(0)}_{1+2,3}-{\mathcal {E}}_{1+2,3}\right) {\mathcal {E}}_{12}\right] e^{\pi \mathrm {i}\tau Q_3(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^3 {\widehat{h}}_{p_i,\mu _i} \nonumber \\&+\sum _{\sum _{i=1}^4 {{\check{\gamma }}}_i={{\check{\gamma }}}} \left[ g^{(0)}_{1234}-{\mathcal {E}}_{1234}-2\left( g^{(0)}_{1+2+3,4}-{\mathcal {E}}_{1+2+3,4}\right) \left( {\mathcal {E}}_{123} -2{\mathcal {E}}_{1+2,3}{\mathcal {E}}_{12}\right) \right. \nonumber \\&\left. -3\left( g^{(0)}_{1+2,34}-{\mathcal {E}}_{1+2,34}\right) {\mathcal {E}}_{12} +\left( g^{(0)}_{1+2,3+4}-{\mathcal {E}}_{1+2,3+4}\right) {\mathcal {E}}_{12}{\mathcal {E}}_{34} \right] e^{\pi \mathrm {i}\tau Q_4(\{{{\check{\gamma }}}_i\})} \nonumber \\&\quad \times \prod _{i=1}^4 {\widehat{h}}_{p_i,\mu _i}+\cdots , \end{aligned}$$

(G.1)

$$\begin{aligned} {\widehat{h}}_{p,q}= & {} h_{p,\mu }+ \sum _{{{\check{\gamma }}}_1+{{\check{\gamma }}}_2={{\check{\gamma }}}} {\mathcal {E}}^{(+)}_{12}\,e^{\pi \mathrm {i}\tau Q_2(\{{{\check{\gamma }}}_i\})} h_{p_1,\mu _1}h_{p_2,\mu _2} \nonumber \\&+\sum _{\sum _{i=1}^3 {{\check{\gamma }}}_i={{\check{\gamma }}}} \Bigl [{\mathcal {E}}^{(+)}_{123}-2{\mathcal {E}}^{(+)}_{1+2,3}{\mathcal {E}}^{(0)}_{12}\Bigr ] \,e^{\pi \mathrm {i}\tau Q_3(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^3 h_{p_i,\mu _i} \nonumber \\&+\sum _{\sum _{i=1}^4 {{\check{\gamma }}}_i={{\check{\gamma }}}} \Bigl [{\mathcal {E}}^{(+)}_{1234}-2{\mathcal {E}}^{(+)}_{1+2+3,4}\left( {\mathcal {E}}^{(0)}_{123} -2{\mathcal {E}}^{(0)}_{1+2,3}{\mathcal {E}}^{(0)}_{12}\right) -3{\mathcal {E}}^{(+)}_{1+2,34}{\mathcal {E}}^{(0)}_{12} \Bigr . \nonumber \\&\Bigl . +{\mathcal {E}}^{(+)}_{1+2,3+4}{\mathcal {E}}^{(0)}_{12}{\mathcal {E}}^{(0)}_{34} \Bigr ]\, e^{\pi \mathrm {i}\tau Q_4(\{{{\check{\gamma }}}_i\})} \prod _{i=1}^4 h_{p_i,\mu _i}+\cdots , \end{aligned}$$

(G.2)

where the functions $g^{(0)}_n$ and ${\mathcal {E}}_n$ can be read off from (5.27), (5.32) and (5.31),

$$\begin{aligned} \begin{aligned} g^{(0)}_2=&\, \frac{(-1)^{1+\gamma _{12}}}{4}\, \gamma _{12}\,\text{ sgn }(c_1), \\ g^{(0)}_3=&\, \frac{(-1)^{1+\gamma _{12}+\gamma _{1+2,3}}}{8}\,\,\mathrm{Sym}\, \biggl \{ \gamma _{12}\,\gamma _{23}\, \text{ sgn }(c_1)\, \text{ sgn }(c_3)+\frac{1}{3}\,\gamma _{12}\,\gamma _{23} \biggr \}, \\ g^{(0)}_4=&\, \frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}+\gamma _{1+2+3,4}}}{16}\, \,\mathrm{Sym}\, \biggl \{\gamma _{12}\gamma _{23}\gamma _{34}\,\text{ sgn }(c_1)\text{ sgn }(c_{1+2})\text{ sgn }(c_4) \\&\, \qquad -\frac{1}{3}\,\gamma _{12}\gamma _{23}\gamma _{24}\,\text{ sgn }(c_1)\text{ sgn }(c_3)\text{ sgn }(c_4) -\frac{1}{3}\, \gamma _{12}\gamma _{23} \gamma _{1+2+3,4}\text{ sgn }(c_4) \biggr \}, \end{aligned} \end{aligned}$$

(G.3)

$$\begin{aligned} {\mathcal {E}}_{12}= & {} \frac{(-1)^{\gamma _{12}}}{4\sqrt{2\tau _2}}\, {\widetilde{\Phi }}_{1}^E\bigl ({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}\bigr ) \nonumber \\= & {} \frac{(-1)^{\gamma _{12}}}{4}\left[ \gamma _{12}\, E_1\left( \frac{\sqrt{2\tau _2}\gamma _{12} }{\sqrt{(pp_1p_2)}}\right) +\frac{\sqrt{(pp_1p_2)}}{\pi \sqrt{2\tau _2}}\, e^{-\frac{2\pi \tau _2\gamma _{12}^2 }{(pp_1p_2)}} \right] , \nonumber \\ {\mathcal {E}}_{123}= & {} \frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}}}{8}\, \,\mathrm{Sym}\, \left\{ \frac{1}{2\tau _2}\,{\widetilde{\Phi }}_{2}^E\bigl (({{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2,3}),({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr ) \right. \nonumber \\&\left. \qquad -\frac{1}{3}\left( \gamma _{12}\gamma _{23}-\frac{(p_1p_2p_3)}{4\pi \tau _2} \right) \right\} , \nonumber \\ {\mathcal {E}}_{1234}= & {} \frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}+\gamma _{1+2+3,4}}}{16\sqrt{2\tau _2}}\,\nonumber \\&\times \,\mathrm{Sym}\, \biggl \{\frac{1}{2\tau _2}\biggl ({\widetilde{\Phi }}_{3}^E\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2+3,4}),({{\varvec{v}}}_{12},{{\varvec{v}}}_{23},{{\varvec{v}}}_{34})\bigr ) \nonumber \\&+\frac{1}{3}\,{\widetilde{\Phi }}_{3}^E\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2+4,3},{{\varvec{v}}}_{1+2+3,4}),({{\varvec{v}}}_{12},{{\varvec{v}}}_{23},{{\varvec{v}}}_{24})\bigr )\biggr ) \nonumber \\&-\frac{1}{3}\left( \gamma _{12}\gamma _{23}-\frac{(p_1p_2p_3)}{4\pi \tau _2}\right) {\widetilde{\Phi }}_{1}^E\bigl ({{\varvec{v}}}_{1+2+3,4},{{\varvec{v}}}_{1+2+3,4}\bigr ) \biggr \}, \end{aligned}$$

(G.4)

where all generalized error functions are evaluated at ${{\varvec{x}}}=\sqrt{2\tau _2}({{\varvec{q}}}+{{\varvec{b}}})$. The last terms appearing in the above quantities for $n=3$ and $n=4$ correspond to contributions of trees with one mark ($m=1$). In fact, at these orders these results can be rewritten in a simpler form, which coincides with the representations (5.42) and (5.48) (in the latter formula one should drop the sum over partitions and take $d_T=d_n$):

$$\begin{aligned} g^{(0)}_3= & {} \frac{(-1)^{1+\gamma _{12}+\gamma _{1+2,3}}}{12}\,\,\mathrm{Sym}\, \Bigl \{ \gamma _{12}\,\gamma _{1+2,3}\, \text{ sgn }(c_1)\, \text{ sgn }(c_3) \Bigr \}, \nonumber \\ g^{(0)}_4= & {} \textstyle {\frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}+\gamma _{1+2+3,4}}}{96}} \,\mathrm{Sym}\, \!\Bigl \{\! \Bigl (2\,\gamma _{23}\gamma _{1,2+3}\gamma _{1+2+3,4}+\gamma _{12}\gamma _{34}\gamma _{1+2,3+4}\Bigr )\nonumber \\&\times \text{ sgn }(c_1)\text{ sgn }(c_{1+2})\text{ sgn }(c_4) \Bigr \}, \end{aligned}$$

(G.5)

$$\begin{aligned} {\mathcal {E}}_{123}= & {} \frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}}}{24\tau _2}\, \,\mathrm{Sym}\, \left\{ {\widetilde{\Phi }}_{2}^E\bigl (({{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2,3}),({{\varvec{v}}}_{12},{{\varvec{v}}}_{1+2,3})\bigr ) \right\} , \nonumber \\ {\mathcal {E}}_{1234}= & {} \frac{(-1)^{\gamma _{12}+\gamma _{1+2,3}+\gamma _{1+2+3,4}}}{96(2\tau _2)^{3/2}}\, \,\mathrm{Sym}\, \biggl \{2\,{\widetilde{\Phi }}_{3}^E\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2+3,4}),\nonumber \\&\times ({{\varvec{v}}}_{23},{{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2+3,4})\bigr ) \nonumber \\&+{\widetilde{\Phi }}_{3}^E\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2+3,4}),({{\varvec{v}}}_{12},{{\varvec{v}}}_{34},{{\varvec{v}}}_{1+2,3+4})\bigr )\biggr ) \biggr \}. \end{aligned}$$

(G.6)

The simplest way to prove the equality of the two representations of $g^{(0)}_n$ is to expand the DSZ products appearing in (G.5) into elementary $\gamma _{ij}$’s and then, using symmetrization, bring all their products to the form appearing in (G.3). These products are multiplied by combinations of sign functions which can be recombined with the help of the identity (D.12). As an example, let us perform these manipulations for $n=3$:

$$\begin{aligned} \begin{aligned}&\, \,\mathrm{Sym}\, \Bigl \{ \gamma _{12}\,\gamma _{1+2,3}\,\text{ sgn }(c_1)\, \text{ sgn }(c_3) \Bigr \} =\,\mathrm{Sym}\, \Bigl \{ \gamma _{12}\,\gamma _{23}\bigl (\text{ sgn }(c_1)-\text{ sgn }(c_2) \bigr )\,\text{ sgn }(c_3)\Bigr \} \\&\quad = \, \,\mathrm{Sym}\, \Bigl \{ \gamma _{12}\,\gamma _{23}\Bigl (\text{ sgn }(c_1)\,\text{ sgn }(c_3)+{1\over 2}\, \text{ sgn }(c_{1+3})\bigl (\text{ sgn }(c_1)+\text{ sgn }(c_3) \bigr )\Bigr )\Bigr \} \\&\quad = \, {1\over 2}\,\,\mathrm{Sym}\, \Bigl \{ \gamma _{12}\,\gamma _{23}\Bigl (3\, \text{ sgn }(c_1)\,\text{ sgn }(c_3)+1\Bigr )\Bigr \}, \end{aligned} \end{aligned}$$

(G.7)

where we used that $c_2=-(c_1+c_3)$. This identity then shows the equality of the two forms of $g^{(0)}_3$ given above. For $g^{(0)}_4$ the manipulations are very similar, but a bit more cumbersome. The equality of the two forms of ${\mathcal {E}}_n$ follows from the equality of their asymptotics ${\mathcal {E}}^{(0)}_n$, which is in turn ensured by the same identities as for $g^{(0)}_n$.

Finally, we provide expressions for the kernels ${\widehat{\Phi }}^{\mathrm{tot}}_n$ of the indefinite theta series appearing in the expansion (5.38) of ${\mathcal {G}}$ in powers of ${\widehat{h}}_{p,\mu }$. For the first two orders, one has

$$\begin{aligned} \begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_1=&\,\Phi ^{\scriptscriptstyle \,\int }_1, \\ {\widehat{\Phi }}^{\mathrm{tot}}_2=&\, \Phi ^{\scriptscriptstyle \,\int }_2+\Phi ^{\scriptscriptstyle \,\int }_1 \Phi ^{\,{\widehat{g}}}_2 =\frac{1}{4}\, \Phi ^{\scriptscriptstyle \,\int }_1\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{12},{{\varvec{v}}}_{12})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{12},{{\varvec{v}}}_{12})\right) , \end{aligned} \end{aligned}$$

(G.8)

where $\Phi ^{\scriptscriptstyle \,\int }_1(x)$ is defined in (4.13). At the next order,

$$\begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_3=\Phi ^{\scriptscriptstyle \,\int }_3+2\,\mathrm{Sym}\, \Bigl \{\Phi ^{\scriptscriptstyle \,\int }_2(x_{1+2},x_3)\Phi ^{\,{\widehat{g}}}_2(x_1,x_2) \Bigr \}+\Phi ^{\scriptscriptstyle \,\int }_1 \Phi ^{\,{\widehat{g}}}_3. \end{aligned}$$

(G.9)

To get an explicit expression in terms of smooth solutions of Vignéras’ equation, one should use the relation (D.15). Applying it to the case $n=2$ with ${\mathcal {V}}=({{\varvec{u}}}_{1,2+3},{{\varvec{u}}}_{1+2,3})$, using the orthogonality properties

$$\begin{aligned} {{\varvec{u}}}_{(1,2+3)\perp (1+2,3)}={{\varvec{u}}}_{12}, \qquad {{\varvec{u}}}_{(1+2,3)\perp (1,2+3)}={{\varvec{u}}}_{23}, \end{aligned}$$

(G.10)

and acting by the operator ${\mathcal {D}}({{\varvec{v}}}_{12}){\mathcal {D}}({{\varvec{v}}}_{23})$, one can show that

$$\begin{aligned}&\,\mathrm{Sym}\, \Bigl \{{\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{u}}}_{1,2+3},{{\varvec{u}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr )\Bigr \}\nonumber \\&\quad = \,\mathrm{Sym}\, \biggl \{{\widetilde{\Phi }}^M_{2}\bigl (({{\varvec{u}}}_{1,2+3},{{\varvec{u}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr ) \nonumber \\&\qquad +({{\varvec{v}}}_{12},{{\varvec{x}}})\,\text{ sgn }({{\varvec{u}}}_{12},{{\varvec{x}}})\, {\widetilde{\Phi }}^M_1\bigl ({{\varvec{u}}}_{1+2,3},{{\varvec{v}}}_{1+2,3}\bigr )\nonumber \\&\qquad +({{\varvec{v}}}_{12},{{\varvec{x}}})\,({{\varvec{v}}}_{23},{{\varvec{x}}})\,\text{ sgn }({{\varvec{u}}}_{1,2+3},{{\varvec{x}}})\,\text{ sgn }({{\varvec{u}}}_{1+2,3},{{\varvec{x}}}) \nonumber \\&\qquad -\frac{(p_1p_2p_3)}{6\pi } \biggr \}. \end{aligned}$$

(G.11)

This result allows to obtain the following representation for the kernel

$$\begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_3= & {} \frac{1}{8}\,\Phi ^{\scriptscriptstyle \,\int }_1 \,\mathrm{Sym}\, \biggl \{{\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{u}}}_{1,2+3},{{\varvec{u}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr ) - {\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr ) \nonumber \\&-\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{1+2,3},{{\varvec{v}}}_{1+2,3})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{1+2,3})\right) {\widetilde{\Phi }}^E_1({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}) \biggr \}. \end{aligned}$$

(G.12)

Using the identity between functions ${\widetilde{\Phi }}^E_{2}$ implied by the identity (G.7), the kernel can also be rewritten as

$$\begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_3= & {} \frac{1}{8}\,\Phi ^{\scriptscriptstyle \,\int }_1 \,\mathrm{Sym}\, \biggl \{\frac{2}{3}\, \Bigl ({\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{u}}}_{1,2+3},{{\varvec{u}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{1+2,3})\bigr )\nonumber \\&- {\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{1+2,3})\bigr )\Bigr ) \nonumber \\&-\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{1+2,3},{{\varvec{v}}}_{1+2,3})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{1+2,3})\right) {\widetilde{\Phi }}^E_1({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}) \biggr \}, \end{aligned}$$

(G.13)

so that the vectors appearing in the second argument of ${\widetilde{\Phi }}_2^E$ are now mutually orthogonal.

For $n=4$, the kernel is given by

$$\begin{aligned} \begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_4=&\, \Phi ^{\scriptscriptstyle \,\int }_4+\,\mathrm{Sym}\, \Bigl \{3\Phi ^{\scriptscriptstyle \,\int }_3(x_{1+2},x_3,x_4)\Phi ^{\,{\widehat{g}}}_2(x_1,x_2) \\&+\Phi ^{\scriptscriptstyle \,\int }_2(x_{1+2},x_{3+4})\Phi ^{\,{\widehat{g}}}_2(x_1,x_2)\Phi ^{\,{\widehat{g}}}_2(x_3,x_4) \\&\, +2\Phi ^{\scriptscriptstyle \,\int }_2(x_{1+2+3},x_4)\Phi ^{\,{\widehat{g}}}_3(x_1,x_2,x_3)\Bigr \}+\Phi ^{\scriptscriptstyle \,\int }_1 \Phi ^{\,{\widehat{g}}}_4. \end{aligned} \end{aligned}$$

(G.14)

Proceeding in the same way as for $n=3$, obtaining a generalization of (G.11) to $n=4$, one arrives at

$$\begin{aligned} {\widehat{\Phi }}^{\mathrm{tot}}_4= & {} \frac{1}{16}\, \Phi ^{\scriptscriptstyle \,\int }_1 \,\mathrm{Sym}\, \biggl \{{\widetilde{\Phi }}^E_{3}\bigl (({{\varvec{u}}}_{1,2+3+4},{{\varvec{u}}}_{1+2,3+4},{{\varvec{u}}}_{1+2+3,4}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23},{{\varvec{v}}}_{34})\bigr ) \nonumber \\&-{\widetilde{\Phi }}^E_{3}\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2+3,4}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23},{{\varvec{v}}}_{34})\bigr ) \nonumber \\&+\frac{1}{3}\left( {\widetilde{\Phi }}^E_{3}\bigl (({{\varvec{u}}}_{1,2+3+4},{{\varvec{u}}}_{1+2+4,3},{{\varvec{u}}}_{1+2+3,4}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{24},{{\varvec{v}}}_{34})\bigr ) \right. \nonumber \\&\left. -{\widetilde{\Phi }}^E_{3}\bigl (({{\varvec{v}}}_{1,2+3+4},{{\varvec{v}}}_{1+2+4,3},{{\varvec{v}}}_{1+2+3,4}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{24},{{\varvec{v}}}_{34})\bigr )\right) \nonumber \\&-\left( {\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{u}}}_{1+2,3+4},{{\varvec{u}}}_{1+2+3,4}), ({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{34})\bigr )\nonumber \right. \\&-{\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2+3,4}), ({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{34})\bigr ) \nonumber \\&+{1\over 2}\left( {\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{u}}}_{1+2+4,3},{{\varvec{u}}}_{1+2+3,4}), ({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{1+2,4})\bigr ) \right. \nonumber \\&\left. \left. -{\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{v}}}_{1+2+4,3},{{\varvec{v}}}_{1+2+3,4}), ({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{1+2,4})\bigr ) \right) \right) {\widetilde{\Phi }}^E_1({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}) \nonumber \\&-\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{1+2+3,4},{{\varvec{v}}}_{1+2+3,4})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{1+2+3,4},{{\varvec{v}}}_{1+2+3,4})\right) \nonumber \\&\times {\widetilde{\Phi }}^E_{2}\bigl (({{\varvec{v}}}_{1,2+3},{{\varvec{v}}}_{1+2,3}), ({{\varvec{v}}}_{12},{{\varvec{v}}}_{23})\bigr ) \nonumber \\&+\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{1+2+3,4},{{\varvec{v}}}_{1+2+3,4})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{1+2+3,4},{{\varvec{v}}}_{1+2+3,4})\right) \nonumber \\&\times {\widetilde{\Phi }}^E_1({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}){\widetilde{\Phi }}^E_1({{\varvec{v}}}_{1+2,3},{{\varvec{v}}}_{1+2,3}) \nonumber \\&+\frac{1}{4}\left( {\widetilde{\Phi }}_1^E({{\varvec{u}}}_{1+2,3+4},{{\varvec{v}}}_{1+2,3+4})-{\widetilde{\Phi }}_1^E({{\varvec{v}}}_{1+2,3+4},{{\varvec{v}}}_{1+2,3+4})\right) \nonumber \\&\times {\widetilde{\Phi }}^E_1({{\varvec{v}}}_{12},{{\varvec{v}}}_{12}){\widetilde{\Phi }}^E_1({{\varvec{v}}}_{34},{{\varvec{v}}}_{34}) \biggr \}. \end{aligned}$$

(G.15)

It is possible also to rewrite this expression in terms of generalized error functions ${\widetilde{\Phi }}^E_n({\mathcal {V}},{\tilde{{\mathcal {V}}}})$ where the vectors entering the second argument are mutually orthogonal, as in (G.13). The reader can easily guess the result by comparing (G.4) and (G.6).

The explicit results for ${\widehat{\Phi }}^{\mathrm{tot}}_n$, $n\le 4$, presented above are the basis for the conjectural formula (5.39). Note that all terms in these expressions have the sum of ranks of the generalized error functions equal to $n-1$. This shows that all contributions due to trees with non-zero number of marks cancel in the sum over Schröder trees.

H. Index of Notations

Symbol	Description	Appears/defined in
${\mathcal {A}}({\mathcal {T}})$	contribution of tree ${\mathcal {T}}$ to the integrand of the multi-instanton expansion of $H_\gamma$ and ${\mathcal {G}}$	(3.18)
$a_{\mathcal {T}}$	coefficient of unrooted labelled tree ${\mathcal {T}}$ in ${\mathcal {D}}_{m}(\{{{\check{\gamma }}}_s\})$	(5.22), (5.24)
$\beta _{k\ell }$	DSZ product $\langle \gamma _1+ \cdots +\gamma _k,\gamma _\ell \rangle$	(2.13)
$b_2=b_2({\mathfrak {Y}})$	second Betti number of ${\mathfrak {Y}}$	p.14
$b^a=\,\mathrm{Re}\,(z^a)$	periods of the Kalb-Ramond field	p.11
$b_n$	rational coefficients in the expansion of $F^{\mathrm{(ref)}}_n$	(5.51), (5.54)
$c_i$	stability parameters	(2.6)
$c_i^{(\ell )}$	stability parameters after attractor flow	(2.15)
$c_{2,a}$	components of the second Chern class of ${\mathfrak {Y}}$	(2.19)
$d=n b_2$	dimension of the lattice ${\varvec{\Lambda }}=\oplus _{i=1}^n \Lambda _i$	p.47
$d_n,d_T$	rational weights in the representation of $g^{(0)}_n$ via flow tree	(5.42), (5.45)
$\Delta (T)$, $\Delta _{\gamma _L\gamma _R}^z$	sign factors assigned to attractor flow tree T	(2.4)
${\mathcal {D}}_{{\mathfrak {h}}}$	Maass raising operator	(3.23)
${\mathcal {D}}({{\varvec{v}}})$	modular-covariant derivative contracted with vector ${{\varvec{v}}}$	(D.16)
${\mathcal {D}}_{m}(\{{{\check{\gamma }}}_s\})$	derivative operator assigning weight to vertices with m marks	(5.22)
$E_n({\mathcal {M}};\mathbb {u})$	generalized error function on $\mathbb {R}^n$	(D.11)
${\mathcal {E}}_n={\mathcal {E}}^{(0)}_n+{\mathcal {E}}^{(+)}_n$	function encoding the modular completion	(5.31), (5.29)
$\phi$	(logarithm of) contact potential on ${\mathcal {M}}_H$	(3.20)
$\Phi _n^E$, $\Phi _n^M$	boosted (complementary) error functions	(D.14)
${\widetilde{\Phi }}_{n,m}^E$, ${\widetilde{\Phi }}_{n,m}^M$	uplifted boosted error functions in the kernel of ${\mathcal {V}}_ m$ (denoted by ${\widetilde{\Phi }}^E_n$, ${\widetilde{\Phi }}^M_n$ when $n=m$)	(D.17)
$\Phi _{\mathcal {T}}$	contribution of tree ${\mathcal {T}}$ to $\Phi ^{\scriptscriptstyle \,\int }_n$	(E.2), (E.17)
$\Phi ^{\scriptscriptstyle \,\int }_n$	kernel defined by twistorial integrals	(4.3), (4.12)
$\Phi ^{\,{\mathcal {E}}}_n$	kernel corresponding to function ${\mathcal {E}}_n$	(5.32)
${\widetilde{\Phi }}^{\,{\mathcal {E}}}_n$	kernel promoting $g^{(0)}_n$ to a solution of Vignéras’ equation	(5.21)
$\Phi ^{\,g}_{n}$	kernel corresponding to the tree index	(4.3), (4.9)
$\Phi ^{\,{\widehat{g}}}_{n}$	kernel corresponding to completed tree index ${\widehat{g}}_n$	(5.6)
$\Phi ^{\mathrm{tot}}_n$	total kernel in the expansion of ${\mathcal {G}}$ in terms of $h_{p,\mu }$	(4.3)
${\widehat{\Phi }}^{\mathrm{tot}}_n$	total kernel in the expansion of ${\mathcal {G}}$ in terms of ${\widehat{h}}_{p,\mu }$	(5.39)
F(X)	holomorphic prepotential	p.11
$F_{\mathrm{tr},n} (\{\gamma _{i}\},z^a)$	partial tree index	(2.12)
$F^{\mathrm{(ref)}}_{n} (\{c_i\})$	partial contribution in $g^{\mathrm{(ref)}}_n$	(5.49)
${\mathcal {F}}$	image of ${\mathcal {G}}$ under Euler operator	(3.28)
$\Gamma$	charge lattice inside $H_{\mathrm{even}}({\mathfrak {Y}},\mathbb {Q})$	(2.19)
$\Gamma _+$	positive cone in the charge lattice	(2.20)
$\Gamma _{k\ell }, \Gamma _e$	sums of DSZ products	(2.13), (5.30)
$\gamma = (0,p^a,q_a,q_0)$	charge vector of a generic D4 (or D3) brane	(2.19)
${{\check{\gamma }}}=(p^a,q_a)$	projection of $\gamma$ on $H_4({\mathfrak {Y}},{\mathbb {Q}})\oplus H_2({\mathfrak {Y}},{\mathbb {Q}})$	(2.31)
$\gamma _{ij}=\langle \gamma _i,\gamma _j\rangle$	Dirac-Schwinger-Zwanziger product, or Euler pairing	(2.24)
$g_{\mathrm{tr},n}(\{\gamma _i\},z^a)$	tree index, also denoted by $g_{\mathrm{tr},n}(\{\gamma _i,c_i\})$	(2.3)
${\widehat{g}}_n(\{{{\check{\gamma }}}_i\},z^a,\tau _2)$	completed tree index, also denoted by ${\widehat{g}}_n(\{{{\check{\gamma }}}_i,c_i\})$	(5.2), (5.33)
$g^{(0)}_n(\{{{\check{\gamma }}}_i,c_i\})$	seed term in recursion for ${\widehat{g}}_n$	(5.9)
$g^{\mathrm{(ref)}}_n(\{{{\check{\gamma }}}_i,c_i\},y)$	refined version of $g^{(0)}_n$	(5.49)
${\mathcal {G}}$	instanton generating function	(3.22)
${\mathcal {G}}_n(\{\gamma _i,z_i\})$	integrand in the n-instanton contribution to ${\mathcal {G}}$	(3.26)
$G_n(\{{{\check{\gamma }}}_i,c_i\};\tau _2)$	large ${{\varvec{x}}}$ limit of ${\widetilde{\Phi }}_n({{\varvec{x}}})$	(5.15), (5.17)
$h^{\mathrm{DT}}_{p,q}(\tau ,z^a)$	generating function of DT invariants	(2.28)
$h_{p,\mu }(\tau )$	generating function of MSW invariants	(2.29)

${\widehat{h}}_{p,\mu }(\tau )$	modular completion of $h_{p,\mu }(\tau )$	(5.1)
$H_\gamma (z)$	generator of contact transformation across $\ell _\gamma$	(3.9)
$H^{\mathrm{cl}}_\gamma (z)$	classical, large volume limit of $H_\gamma (z)$	(3.15)
$\kappa (T)$	weight of attractor flow tree T	(2.7)
$\kappa (x)$	BPS index of two-centered solutions	(2.8)
$\kappa _{abc}$	intersection numbers on a fixed basis of $H_4({\mathfrak {Y}})$	p.11
$K_{\gamma _1\gamma _2}(z_1,z_2)$	integration kernel in large volume limit	(3.16)
${{\hat{K}}}_{ij}(z_i,z_j)$	rescaled integration kernel in large volume limit	(4.7)
${\mathcal {J}}_n(\{{{\check{\gamma }}}_i\},\tau _2)$	coefficient in the formula for the shadow of ${\widehat{h}}_{p,\mu }$	(5.35)
$\Lambda =H_4({\mathfrak {Y}},\mathbb {Z})$	lattice equipped with quadratic form $\kappa _{ab}=\kappa _{abc} p^a$	p.11
$\Lambda _i=H_4({\mathfrak {Y}},\mathbb {Z})$	lattice equipped with quadratic form $\kappa _{i,ab}=\kappa _{abc} p_i^a$	p.47
${\varvec{\Lambda }}$	lattice ${\varvec{\Lambda }}=\oplus _{i=1}^n \Lambda _i$ spanned by n D1-brane charges	p.47
$\lambda$	eigenvalue under Vignéras’ operator	(D.3)
$\ell _\gamma$	BPS ray on the twistor fiber, or its large volume limit	(3.3), p.18
$\mu _a$	residue class of $q_a$ modulo spectral flow	(2.26)
$M_n({\mathcal {M}};\mathbb {u})$	generalized complementary error function on $\mathbb {R}^n$	(D.10)
${\mathcal {M}}_{\alpha \beta }$	matrix of parameters in the generalized error functions	(D.10), (D.11)
${\mathcal {M}}_{{\mathcal {K}}}({\mathfrak {Y}})$	complexified Kähler moduli space of ${\mathfrak {Y}}$	p.11
${\mathcal {M}}_H$	hypermultiplet moduli space in IIB/${\mathfrak {Y}}$, or vector multiplet moduli space in IIA/$({\mathfrak {Y}}\times S^1)=$ M/$({\mathfrak {Y}}\times T^2)$	p.13
$n_T$	number of vertices of rooted tree T excluding the leaves	(5.33)
$n_v(T)$	number of descendants of the vertex v in T plus one	(1.2)
$n_{\mathfrak {v}}$	valency of vertex ${\mathfrak {v}}$ of an unrooted tree	(5.24)
$p^a$	homology class of the divisor wrapped by the D3-brane	(2.20)
${\mathcal {P}}_{m}(\{p_s\})$	weight of a vertex with m marks in $G_n$	(5.19)
${\hat{q}}_0$	invariant D0-brane charge	(2.27)
$Q_n(\{{{\check{\gamma }}}_i\})$	difference of quadratic forms for constituents and the total charge	(2.32)
$R_n(\{{{\check{\gamma }}}_i\},\tau _2)$	non-holomorphic correction in the completion ${\widehat{h}}_{p,\mu }(\tau )$	(5.2), (5.34)
$\sigma _\gamma$	quadratic refinement	(3.4), (D.5)
$S_k, S_e$	sums of stability parameters	(2.13), (5.18)
$S^{\mathrm{cl}}_p$	classical action of a D3-instanton in large volume limit	(3.13)
$\vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\bigl (\Phi ,\lambda )$	indefinite theta series with kernel $\Phi$	(D.1)
$\tau =\tau _1+\mathrm {i}\tau _2$	4D axio-dilaton in IIA, or torus modulus in M theory	p.14
t	complex coordinate on the twistor fiber	p.15
$t^a=\,\mathrm{Im}\,(z^a)$	Kähler moduli on ${\mathfrak {Y}}$	p.11
T	rooted tree with charges assigned to the leaves	footnote 18
${\mathcal {T}}$	tree with charges assigned to vertices	footnote 18
$\mathbb {T}_n, \mathbb {T}_n^\ell$	set of unrooted (labelled) trees with n vertices	(3.26)
$\mathbb {T}_{n,m}, \mathbb {T}_{n,m}^\ell$	set of unrooted (labelled) trees with n vertices and m marks	(5.17)
$\mathbb {T}_n^{\mathrm{r}}$	set of rooted trees with n vertices	(3.17)
$\mathbb {T}_n^{\mathrm{af}}$	set of attractor flow trees with n leaves	(2.3)
$\mathbb {T}_n^{\mathrm{S}}$	set of Schröder trees with n leaves	(5.4)
$\mathbb {T}_{2m+1}^{(3)}$	set of rooted ternary trees with n leaves	(5.19)
$u^\Lambda =(1,u^a)$	complex structure moduli of mirror threefold ${\widehat{{\mathfrak {Y}}}}$	p.15
${{\varvec{u}}}_{ij}$, ${{\varvec{u}}}_e$, ${{\varvec{u}}}_\ell$	vectors in $\mathbb {R}^d$ associated to $-2\,\mathrm{Im}\,[Z_{\gamma _i}{\bar{Z}}_{\gamma _j}]$, $-S_e$ and $-S_\ell$	(4.10), (4.11), (5.8)
${{\varvec{v}}}_{ij}$, ${{\varvec{v}}}_e$, ${{\varvec{v}}}_\ell$	vectors in $\mathbb {R}^d$ associated to $\langle \gamma _i,\gamma _j\rangle$, $\Gamma _e$ and $-\Gamma _{n\ell }$	(4.10), (4.11), (5.8)
$V_T$	set of vertices of rooted tree T excluding leaves	p.8
$V_\lambda$	Vignéras’ operator	(D.3)
${\mathcal {V}}_m$	weight of a vertex with m marks in large ${{\varvec{x}}}$ limit of ${\widetilde{\Phi }}^{\,{\mathcal {E}}}_n$	(5.26)
${\tilde{{\mathcal {V}}}}_m$	weight of a vertex with m marks in $g^{(0)}_n$	(5.28)
$W_n(\{{{\check{\gamma }}}_i\},\tau _2)$	coefficient in the formula for $h_{p,\mu }$ in terms of ${\widehat{h}}_{p_i,\mu _i}$	(5.5)
${\mathcal {X}}_\gamma$	holomorphic Fourier modes on the twistor space of ${\mathcal {M}}_H$	(3.2)

${\mathcal {X}}^{\mathrm{sf}}_\gamma$	semi-flat limit of ${\mathcal {X}}_\gamma$	(3.1)
${\mathcal {X}}^{\mathrm{cl}}_\gamma$	classical limit of ${\mathcal {X}}_\gamma$	(3.12)
${\mathcal {X}}^{(\theta )}_{p,q}$	${\hat{q}}_0$-independent part of ${\mathcal {X}}^{\mathrm{cl}}_\gamma$	(3.13)
${{\varvec{x}}}$	d-dimensional vector, argument of kernels of theta series	p.47
z	coordinate on the twistor fiber, after Cayley transf.	(3.6)
$z_\gamma$	saddle point on twistor fiber	(3.10)
$z^a=b^a+\mathrm {i}t^a$	complexified Kähler moduli of ${\mathfrak {Y}}$	p.14
$z^a_*(\gamma )$	attractor moduli for charge $\gamma$	p.7
$z^a_\infty (\gamma )$	large volume attractor point for charge $\gamma$	(1.1)
$Z_\gamma (z^a)$	central charge	(2.21)
$\Omega (\gamma ,z^a)$	generalized Donaldson-Thomas invariant	p.12
${\bar{\Omega }}(\gamma ,z^a)$	rational DT invariant	(2.1)
${\bar{\Omega }}_*(\gamma )$	attractor index	p.7
${{\bar{\Omega }}}^{\mathrm{MSW}}(\gamma )$	MSW invariant, also denoted by ${\bar{\Omega }}_{p,\mu }( {\hat{q}}_0)$	p.12

Note in Proofs In a subsequent work [78], we extend the construction of the modular completion to the case of refined Donaldson-Thomas invariants, by exploiting the observations made in Sect. 5.4.2 of the present paper. While it agrees with the present construction in the unrefined limit, the refined construction turns out to be vastly simpler, with all complications reduced to the final step of taking the limit $y\rightarrow 1$.