S-Duality and Refined BPS Indices

Abstract

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson–Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi–Yau threefold $\mathfrak {Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak {Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface S, refined BPS indices are well-defined, and equal to Vafa–Witten invariants of S. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak {Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for U(2) and U(3) Vafa–Witten theory on $\mathbb {P}^2$, and make them explicit for U(4).

Appendices

Theta Series and Modularity

1.1 Theta series and refinement

In this work we consider theta series of the following type

$$\begin{aligned} \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau , {\varvec{v}})=\tau _2^{-\lambda /2} \sum _{{{\varvec{q}}}\in {{\varvec{\Lambda }}}+{\varvec{\mu }}+{1\over 2}{{\varvec{p}}}} (-1)^{{\varvec{q}}\cdot {{\varvec{p}}}}\,\Phi (\sqrt{2\tau _2}({\varvec{q}}+{\varvec{b}}))\, \mathbf{e}\left( - \tfrac{\tau }{2}\,{\varvec{q}}^2+{\varvec{q}}\cdot {\varvec{v}}\right) , \end{aligned}$$

(A.1)

where ${\varvec{v}}={\varvec{c}}-\tau {\varvec{b}}$. Here ${{\varvec{\Lambda }}}$ is 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. ${\varvec{q}}^2\equiv {\varvec{q}}\cdot {\varvec{q}}\in \mathbb {Z}$ for ${\varvec{q}}\in {{\varvec{\Lambda }}}$. Furthermore, ${{\varvec{p}}}$ is a characteristic vector²⁸, ${\varvec{\mu }}\in {{\varvec{\Lambda }}}^*/{{\varvec{\Lambda }}}$ a glue vector, and $\lambda$ an arbitrary integer. Provided the kernel $\Phi ({\varvec{x}})$ satisfies the following differential equation

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

(A.2)

which we call Vignéras equation, and subject to suitable decay conditions on $\Phi ({\varvec{x}}) e^{\pi {\varvec{x}}^2}$, then under $SL(2,\mathbb {Z})$ transformations

$$\begin{aligned} \tau \mapsto \frac{a \tau +b}{c \tau + d} \, , \qquad \begin{pmatrix} {\varvec{c}}\\ {\varvec{b}}\end{pmatrix} \mapsto \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \begin{pmatrix} {\varvec{c}}\\ {\varvec{b}}\end{pmatrix} , \qquad {\varvec{v}}\mapsto \frac{{\varvec{v}}}{c\tau +d} \end{aligned}$$

(A.3)

the theta series transforms as a vector-valued Jacobi form of weight $(\lambda +d/2,0)$ and index $-1/2$ [83]. Namely,

$$\begin{aligned} \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}\left( \Phi ,\lambda ; -\frac{1}{\tau }, \frac{{\varvec{v}}}{\tau }\right)= & {} \frac{(-\mathrm {i}\tau )^{\lambda +\frac{n}{2}}}{\sqrt{|{{\varvec{\Lambda }}}^*/{{\varvec{\Lambda }}}|}}\, \mathbf{e}\left( \frac{1}{4} \,{{\varvec{p}}}^2-\frac{{\varvec{v}}^2}{2\tau } \right) \sum _{{\varvec{\nu }}\in {{\varvec{\Lambda }}}^*/{{\varvec{\Lambda }}}} \mathbf{e}\left( {\varvec{\mu }}\cdot {\varvec{\nu }}\right) \vartheta _{{{\varvec{p}}},{\varvec{\nu }}}(\Phi ,\lambda ;\tau , {\varvec{v}}), \nonumber \\ \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau +1, {\varvec{v}})= & {} \mathbf{e}\left( -\tfrac{1}{2} \,({\varvec{\mu }}+\tfrac{1}{2} {{\varvec{p}}})^2\right) \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau , {\varvec{v}}). \end{aligned}$$

(A.4)

Note that (A.1) differs from the definition used in [12,13,14,15] by the factor

$$\begin{aligned} V_{{\varvec{p}}}=\mathbf{e}\left( {1\over 2}\, {\varvec{v}}\cdot {\varvec{b}}\right) . \end{aligned}$$

(A.5)

It changes the index of the theta series so that $V_{{\varvec{p}}}\vartheta _{{{\varvec{p}}},{\varvec{\mu }}}$ transforms as a standard modular form with vanishing index.

We are interested in the case where ${{\varvec{\Lambda }}}=\oplus _{i=1}^n \Lambda _i$ and $\Lambda _i$ are charge lattices associated to divisors $\mathcal {D}_i$. Thus, the charges appearing in the description of the theta series (A.1) are of the type ${\varvec{q}}=(q_1^a,\dots ,q_n^a)$, 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 $i=1,\dots , n$, where $b^a$ and $c^a$ are the integrals of the Neveu-Schwarz and Ramond-Ramond fields on the basis of two-cycles $\omega ^a\in H_2(\mathfrak {Y},\mathbb {Z})$. The lattices $\Lambda _i$ carry the bilinear forms $\kappa _{i,ab}=\kappa _{abc}p_i^c$ which are all of signature $(1,b_2(\mathfrak {Y})-1)$. This induces a natural bilinear form on ${{\varvec{\Lambda }}}$

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

(A.6)

of signature $(n,nb_2-n)$.

Let us now study the effect of inserting a factor $y^{\sum _{i<j} \gamma _{ij}}$ into the summand of the theta series. Defining $y=\mathbf{e}\left( w\right)$ and the $nb_2$-dimensional vector $\mathfrak {p}$ with components

$$\begin{aligned} \mathfrak {p}_i^a=-\sum _{j<i}p_j^a+\sum _{j>i}p_j^a, \end{aligned}$$

(A.7)

this factor can be rewritten as $\mathbf{e}\left( w {\varvec{q}}\cdot \mathfrak {p}\right)$. Hence, it can be absorbed into a redefinition of the Jacobi variable, $\hat{{\varvec{v}}}={\varvec{v}}+w\mathfrak {p}$,

$$\begin{aligned} y^{\sum _{i<j} \gamma _{ij}}\,\mathbf{e}\left( {\varvec{q}}\cdot {\varvec{v}}\right) = \mathbf{e}\left( {\varvec{q}}\cdot \hat{{\varvec{v}}}\right) . \end{aligned}$$

(A.8)

This observation suggests that under $SL(2,\mathbb {Z})$ transformations the parameter w must transform like ${\varvec{v}}$, i.e.

$$\begin{aligned} w\rightarrow \frac{w}{c\tau +d}\, . \end{aligned}$$

(A.9)

Equivalently, upon decomposing w as $w=\alpha -\tau \beta$, the real parameters $\alpha$ and $\beta$ must transform as a doublet

$$\begin{aligned} \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \mapsto \begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \begin{pmatrix} \alpha \\ \beta \end{pmatrix}, \end{aligned}$$

(A.10)

and similarly for the vectors

$$\begin{aligned} \hat{{\varvec{b}}}={\varvec{b}}+\beta \mathfrak {p}, \qquad \hat{{\varvec{c}}}={\varvec{c}}+\alpha \mathfrak {p}. \end{aligned}$$

(A.11)

Indeed, it follows from the theorem stated at the beginning of this section that provided the kernel $\Phi ({\varvec{x}})$ satisfies Vignéras equation (A.2) and suitable decay conditions, then the following refined theta series

$$\begin{aligned}&\vartheta ^{\mathrm{ref}}_{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau , {\varvec{v}},y)= \vartheta _{{{\varvec{p}}},{\varvec{\mu }}}(\Phi ,\lambda ;\tau , \hat{{\varvec{v}}}) \nonumber \\&\quad =\tau _2^{-\lambda /2} \sum _{{{\varvec{q}}}\in {{\varvec{\Lambda }}}+{\varvec{\mu }}+{1\over 2}{{\varvec{p}}}} (-1)^{{\varvec{q}}\cdot {{\varvec{p}}}}\,\Phi \left( \sqrt{2\tau _2}\left( {\varvec{q}}+{\varvec{b}}+\frac{\,\mathrm{Re}\,(\log y )}{2\pi \tau _2}\,\mathfrak {p}\right) \right) \,y^{\sum _{i<j} \gamma _{ij}}\, \mathbf{e}\left( - \tfrac{\tau }{2}\,{\varvec{q}}^2+{\varvec{q}}\cdot {\varvec{v}}\right) \nonumber \\ \end{aligned}$$

(A.12)

transforms as a vector-valued Jacobi form of weight $(\lambda +nb_2(\mathfrak {Y})/2,0)$ and index $-1/2$. Note that modularity requires the y-dependent shift in the argument of the kernel which leads to various important consequences.

1.2 Generalized error functions

In this appendix we recall the definition and some properties of the generalized error functions introduced in [18, 64] and revisited from a more conceptual viewpoint in [84], and prove a new identity which is used in the study of the instanton generating potential.

First, we define the generalized (complementary) error functions

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

(A.13)

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

(A.14)

where $\mathbb {z}=(z_1,\dots ,z_n)$ and ${\mathbb {u}}=(u_1,\dots ,u_n)$ 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 [64]. Here we just note that the information carried by $\mathcal {M}$ is highly redundant. For instance, the generalized error functions are invariant (up to sign) under rescaling of its columns. As a result, for $n=1$ the dependence on $\mathcal {M}$ drops out, whereas for $n=2$ (respectively $n=3$) they can always be expressed in terms of functions parametrized only by one (respectively 3) parameters, e.g.

$$\begin{aligned} E_2(\alpha ;u_1,u_2):= E_2\left( \left( \begin{array}{cc} 1 &{} 0 \\ \alpha &{} 1 \end{array}\right) ; {\mathbb {u}}\right) , \quad \ E_3(\alpha ,\beta ,\gamma ;u_1,u_2,u_3):= E_3\left( \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ \alpha &{} 1 &{} 0 \\ \beta &{} \gamma &{} 1 \end{array}\right) ; {\mathbb {u}}\right) , \end{aligned}$$

(A.15)

and similarly for $M_{2}$ and $M_3$. This parametrization will be used to express the explicit results for modular completions in “Appendix F”. In the case of vanishing arguments one has [13, Eq.(3.23)]

$$\begin{aligned} E_2(\alpha ;0,0)=\frac{2}{\pi }\, \mathrm{Arctan}(\alpha ). \end{aligned}$$

(A.16)

Next, we define the boosted versions of the generalized error functions. To write them down, let us consider $d\times n$ matrix $\mathcal {V}$ which can be viewed as a collection of n vectors, $\mathcal {V}=({\varvec{v}}_1,\dots ,{\varvec{v}}_n)$. 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

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

(A.17)

Both types of generalized error functions can be shown to be independent of $\mathcal {B}$ and solve Vignéras equation (A.2) with $\lambda =0$. However, whereas $\Phi _n^E(\{{\varvec{v}}_i\};{\varvec{x}})$ are smooth functions of ${\varvec{x}}\in \mathbb {R}^{n b_2(\mathfrak {Y})}$, asymptotic to $\prod _{i=1}^n \mathrm{sgn}({\varvec{v}}_i,{\varvec{x}})$, the complementary functions $\Phi _n^M(\{{\varvec{v}}_i\};{\varvec{x}})$ are smooth only away from the real-codimension-1 loci on $\mathbb {R}^{n b_2}$, and are exponentially suppressed for $|{\varvec{x}}|\rightarrow \infty$. The $\Phi _n^E$’s provide the kernels for modular completions of indefinite theta series, while the $\Phi _n^M$’s provide the non-holomorphic terms that must be added to reach that modular completions.

Note that the generalized error functions can be lifted to solutions of Vignéras equation with $\lambda \in \mathbb {N}$ by using the differential operator

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

(A.18)

acting on the functions on $\mathbb {R}^{n b_2(\mathfrak {Y})}$. Its main feature is that it maps solutions of Vignéras equation with parameter $\lambda$ to another solution with $\lambda +1$.

An important fact is that the functions $\Phi _n^E$ can be expressed as linear combinations of products of $\Phi _m^M$ and $n-m$ sign functions with $0\le m\le n$, and vice-versa [64]:

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

(A.19)

where the sum goes over all possible subsets (including the empty set) of the set $\mathscr {Z}_{n}=\{1,\dots ,n\}$, $|\mathcal {I}|$ 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 $\{{\varvec{v}}_i\}_{i\in \mathcal {I}}$. However, in the derivation of the integral form of the instanton generating potential we will need a slightly modified version of this decomposition where ${\varvec{x}}$ in the argument of the sign functions is shifted by a certain vector. To state the corresponding result, let us introduce a modification of the boosted complementary generalized error function replacing the function $M_n$ in its definition by a similar function with an additional shift of the integration contour

$$\begin{aligned} \hat{\Phi }_n^M(\mathcal {V};{\varvec{x}},{\varvec{\chi }})= & {} \hat{M}_n(\mathcal {B}\cdot \mathcal {V};\mathcal {B}\cdot {\varvec{x}},\mathcal {B}\cdot {\varvec{\chi }}), \end{aligned}$$

(A.20)

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

(A.21)

Then we have

Proposition 1

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

(A.22)

Proof

First, we note that the dependence on ${\varvec{\chi }}$ of the right-hand side of (A.22) is locally constant because the integration contours can be safely deformed provided they do not cross the poles of the integrands. Next we note that the smoothness of $\Phi _n^E$ implies that all discontinuities due to signs and contour integrals in (A.19) cancel. Then the same should be true for the right-hand side of (A.22) as well. Indeed, the shift induced by ${\varvec{\chi }}$ just changes the position of the discontinuities of both signs and integrals in the same way, and it does not affect the jumps of individual terms since the integrands are independent of ${\varvec{\chi }}$. It is clear that smoothness in ${\varvec{x}}$ also implies the smoothness in ${\varvec{\chi }}$. Combined with the above fact that the dependence on ${\varvec{\chi }}$ is locally constant, one obtains that the resulting function is actually constant in ${\varvec{\chi }}$ and hence coincides with $\Phi _n^E$. $\quad \square$

1.3 Matrix of parameters

The main building blocks of the construction proposed in this work are the boosted generalized error functions $\Phi ^E_{n-1}(\{ {\varvec{v}}_{\ell }\};{\varvec{x}})$ where the vectors ${\varvec{v}}_\ell$ are defined in (3.11). In this appendix we express them through the original generalized error functions (A.13).

According to (A.17), we should find the matrix $\mathcal {B}$ representing an orthonormal basis in the subspace spanned by ${\varvec{v}}_\ell$ and evaluate the scalar products $\mathcal {B}\cdot \mathcal {V}$ and $\mathcal {B}\cdot {\varvec{x}}$. Note that the vectors ${\varvec{v}}_\ell$ coincide with the vectors ${\varvec{v}}_e$ (B.8) computed for the trivial unrooted tree $\bullet \text{--- }\bullet \text{-- }\cdots \text{-- }\bullet \text{--- }\bullet$, with vertices labelled by charges $\gamma _1,\dots ,\gamma _n$ consecutively. In [15, Appendix E] it was shown that for any set of vectors defined by an unrooted tree, an orthonormal basis can be constructed from a rooted ordered binary tree with leaves labelled by the charges, which is derived in a certain way from the initial unrooted tree. Namely [15, Lemma 2]:²⁹

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

(A.23)

where

$$\begin{aligned} \tilde{{\varvec{v}}}_k=&\,\sum _{i\in \mathcal {I}_{L(v_k)}}\sum _{j\in \mathcal {I}_{R(v_k)}}{\varvec{v}}_{ij}, \nonumber \\ \Delta _k=&\,\tilde{{\varvec{v}}}_k^2=(p_{v_k} p_{L(v_k)}p_{R(v_k)}), \end{aligned}$$

(A.24)

$L(v)$, $R(v)$ are the two children of the vertex v, and $\mathcal {I}_v$ is the set of leaves which are descendants of v. In our case, one can choose the binary tree to be as in Fig. 2. Then one finds

$$\begin{aligned} \tilde{{\varvec{v}}}_k\cdot {\varvec{v}}_\ell =&\,\left\{ \begin{array}{cc} \sum _{j=1}^{\ell }(p p_j p_{k+1}), &{}\ k\ge \ell , \\ 0, &{} \ k<\ell , \end{array}\right. \nonumber \\ \Delta _k=&\,\sum _{i=1}^k \sum _{j=1}^{k+1}(p_i p_j p_{k+1}), \end{aligned}$$

(A.25)

while the matrix of parameters is lower triangular, given by

$$\begin{aligned} \mathcal {M}_{k\ell }=\left\{ \begin{array}{cc} \sqrt{\frac{\Delta _\ell }{\Delta _k}} \, \tilde{{\varvec{v}}}_k\cdot {\varvec{v}}_\ell , &{}\ k\ge \ell , \\ 0, &{} \ k<\ell . \end{array}\right. \end{aligned}$$

(A.26)

For the special case of equal charges where $p_i=p_0$ for $i=1,\dots ,n$, relevant for the discussion around (3.22), the matrix reduces to

$$\begin{aligned} \mathcal {M}^{(0)}_{k\ell }=\left\{ \begin{array}{cc} n \ell \sqrt{\frac{\ell (\ell +1)}{k(k+1)}}\, p_0^3, &{}\ k\ge \ell , \\ 0, &{} \ k<\ell . \end{array}\right. \end{aligned}$$

(A.27)

Fig. 2

The rooted binary tree encoding the orthonormal basis $\mathcal {B}$ and corresponding to the unrooted tree of trivial topology shown at the bottom

Relevant Functions

In this section, we provide the definition of various functions determining the completion $\widehat{h}_{p,\mu }$ and the theta series decomposition of the instanton generating potential $\mathcal {G}$.

In fact, all these functions are uniquely determined by one set of functions $g^{(0)}_n(\{\check{\gamma }_i,c_i\})$. To define those, we take $\mathbb {T}_{n,m}^\ell$ to be the set of marked unrooted labelled trees with n vertices and m marks assigned to vertices. Let $m_\mathfrak {v}\in \{0,\dots m\}$ be the number of marks carried by the vertex $\mathfrak {v}$, so that $\sum _\mathfrak {v}m_\mathfrak {v}=m$. Furthermore, the vertices are decorated by charges from the set $\{\gamma _1,\dots ,\gamma _{n+2m}\}$ such that a vertex $\mathfrak {v}$ with $m_\mathfrak {v}$ marks carries $1+2m_\mathfrak {v}$ charges $\gamma _{\mathfrak {v},s}$, $s=1,\dots ,1+2m_\mathfrak {v}$ and we set $\gamma _\mathfrak {v}=\sum _{s=1}^{1+2m_\mathfrak {v}}\gamma _{\mathfrak {v},s}$. Given a tree $\mathcal {T}\in \mathbb {T}_{n,m}^\ell$, we denote the set of its edges by $E_{\mathcal {T}}$, the set of vertices by $V_{\mathcal {T}}$, and the source and target vertex of an edge e by s(e) and t(e), respectively.³⁰ Then $g^{(0)}_n$ is given by a sum over marked unrooted labelled trees as follows [15, Eq.(5.27)]

$$\begin{aligned} g^{(0)}_n(\{\check{\gamma }_i,c_i\})=\frac{(-1)^{n-1+\sum _{i<j} \gamma _{ij} }}{2^{n-1} n!} \sum _{m=0}^{[(n-1)/2]}\sum _{\mathcal {T}\in \, \mathbb {T}_{n-2m,m}^\ell } \prod _{\mathfrak {v}\in V_\mathcal {T}}\tilde{\mathcal {V}}_{m_\mathfrak {v}}(\{\check{\gamma }_{\mathfrak {v},s}\}) \prod _{e\in E_{\mathcal {T}}}\gamma _{s(e) t(e)}\,\mathrm{sgn}(S_e), \end{aligned}$$

(B.1)

where $S_k=\sum _{i=1}^k c_i$ and

$$\begin{aligned} \tilde{\mathcal {V}}_{m}(\{\check{\gamma }_{s}\})=\sum _{\mathcal {T}'\in \, \mathbb {T}_{2m+1}^\ell } a_{\mathcal {T}'}\prod _{e\in E_{\mathcal {T}'}}\gamma _{s(e)t(e)}. \end{aligned}$$

(B.2)

Here for each tree $\mathcal {T}$ we introduced rational coefficients $a_\mathcal {T}$ determined recursively by the relation

$$\begin{aligned} a_\mathcal {T}=\frac{1}{n_\mathcal {T}}\sum _{\mathfrak {v}\in V_\mathcal {T}} \epsilon _\mathfrak {v}\prod _{s=1}^{n_\mathfrak {v}} a_{\mathcal {T}_s(\mathfrak {v})}, \end{aligned}$$

(B.3)

where $n_\mathcal {T}$ is the number of vertices, $n_\mathfrak {v}$ is the valency of the vertex $\mathfrak {v}$, $\mathcal {T}_s(\mathfrak {v})$ are the trees obtained from $\mathcal {T}$ by removing this vertex, and $\epsilon _\mathfrak {v}$ is the sign determined by the choice of orientation of edges, $\epsilon _\mathfrak {v}=(-1)^{n_\mathfrak {v}^+}$ with $n_\mathfrak {v}^+$ being the number of incoming edges at the vertex. Finally, we use the following definition of the sign function

$$\begin{aligned} \text{ sgn }(x)=\left\{ \begin{array}{cl} -1, &{}\quad x<0, \\ 0, &{} \quad x=0, \\ 1, &{}\quad x>0. \end{array} \right. \end{aligned}$$

(B.4)

Given $g^{(0)}_n$, all other functions can be obtained via the following procedure:

• Setting the stability parameters to the attractor values, $c_i=\beta _{ni}$, one obtains moduli-independent functions $\mathcal {E}^{(0)}_n(\{\check{\gamma }_i\})$, see (2.20).

• Dividing both $g^{(0)}_n$ and $\mathcal {E}^{(0)}_n$ by a factor $\frac{(-1)^{\sum _{i<j} \gamma _{ij} }}{(\sqrt{2\tau _2})^{n-1}}$, one finds that the resulting functions depend on D2-brane charges, $\tau _2$ and the real part $b^a$ of the Kähler moduli only through the combinations

  $$\begin{aligned} x_i^a=\sqrt{2\tau _2}(\kappa _i^{ab} q_{i,b}+b^a) , \end{aligned}$$

  (B.5)

  where $\kappa _i^{ab}$ is the inverse of the quadratic form $\kappa _{i,ab} = \kappa _{abc} p_i^c$. Therefore, they can be viewed as kernels of theta series of the type considered in “Appendix A.1” with $x_i^a$ being the components of $nb_2(\mathfrak {Y})$-dimensional vector ${\varvec{x}}$.

• By adding contributions exponentially suppressed at large ${\varvec{x}}$, these kernels can be promoted to smooth solutions of Vignéras equation (A.2) with $\lambda =0$, which we call $\widetilde{\Phi }^{\,\mathcal {E}}_n$ and $\Phi ^{\,\mathcal {E}}_n$, respectively.

• Finally, restoring the factor $\frac{(-1)^{\sum _{i<j} \gamma _{ij} }}{(\sqrt{2\tau _2})^{n-1}}$, one defines $\mathcal {E}_n$ by

  $$\begin{aligned} \mathcal {E}_n(\{\check{\gamma }_i\},\tau _2)= \frac{(-1)^{\sum _{i<j} \gamma _{ij} }}{(\sqrt{2\tau _2})^{n-1}}\, \Phi ^{\,\mathcal {E}}_n({\varvec{x}}). \end{aligned}$$

  (B.6)

To present the results for $\widetilde{\Phi }^{\,\mathcal {E}}_n$ and $\Phi ^{\,\mathcal {E}}_n$ following from (B.1), we have to define several sets of $nb_2(\mathfrak {Y})$-dimensional vectors. The two basic sets are given by

$$\begin{aligned} ({\varvec{v}}_{ij})_k^a=&\, \delta _{ki} p_j^a-\delta _{kj} p_i^a \qquad \qquad \qquad \quad \text{ such } \text{ that } \quad {\varvec{v}}_{ij}\cdot {\varvec{x}}=(p_ip_j(x_i-x_j)), \nonumber \\ ({\varvec{u}}_{ij})_k^a=&\, \delta _{ki}(p_jt^2)t^a-\delta _{kj} (p_it^2)t^a \quad \text{ such } \text{ that } \quad {\varvec{u}}_{ij}\cdot {\varvec{x}}=(p_jt^2)(p_ix_it)-(p_it^2)(p_jx_jt), \end{aligned}$$

(B.7)

where $t^a=\,\mathrm{Im}\,z^a$ and the bilinear form is defined in (A.6). For ${\varvec{x}}$ as in (B.5), ${\varvec{v}}_{ij}\cdot {\varvec{x}}=\sqrt{2\tau _2}\gamma _{ij}$ and ${\varvec{u}}_{ij}\cdot {\varvec{x}}=-2\sqrt{2\tau _2}\,\mathrm{Im}\,\left[ Z_{\gamma _i}\bar{Z}_{\gamma _j}\right]$. Furthermore, for a tree $\mathcal {T}\in \mathbb {T}_{n,m}^\ell$, denote by $\mathcal {T}_e^s$ and $\mathcal {T}_e^t$ the two disconnected trees obtained from $\mathcal {T}$ by removing the edge e. Then we introduce another two sets of vectors

$$\begin{aligned} {\varvec{v}}_e=\sum _{i\in V_{\mathcal {T}_e^s}}\sum _{j\in V_{\mathcal {T}_e^t}}{\varvec{v}}_{ij}, \qquad {\varvec{u}}_e=\sum _{i\in V_{\mathcal {T}_e^s}}\sum _{j\in V_{\mathcal {T}_e^t}}{\varvec{u}}_{ij}. \end{aligned}$$

(B.8)

With these definitions, one has [15, Eq.(5.32)]

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

(B.9)

where

$$\begin{aligned} \mathcal {D}_{m}(\{\check{\gamma }_s\})=\sum _{\mathcal {T}'\in \, \mathbb {T}_{2m+1}^\ell } a_{\mathcal {T}'}\prod _{e\in E_{\mathcal {T}'}}\mathcal {D}({\varvec{v}}_{s(e)t(v)}) \end{aligned}$$

(B.10)

is a differential operator constructed from (A.18) and $\Phi ^E_n$ are (boosted) generalized error functions reviewed in “Appendix A.2”. The functions $\widetilde{\Phi }^{\,\mathcal {E}}_n({\varvec{x}})$ are given by the same expression with the vectors ${\varvec{v}}_e$ appearing as parameters in $\Phi ^E_{n-2m-1}$ replaced by ${\varvec{u}}_e$.

Note that in [15] it was shown that all contributions of trees with a non-zero number of marks remarkably cancel in the sum over Schröder trees like (2.13) or (2.16). Therefore, we could omit them from the very beginning arriving at a simpler set of functions $g^{(0)}_n$, $\mathcal {E}_n$, $\widetilde{\Phi }^{\,\mathcal {E}}_n$ and $\Phi ^{\,\mathcal {E}}_n$. However, it is the function (B.1) with contributions of marks included that is reproduced in the limit $y\rightarrow 1$ of $g^{\mathrm{ref}}_n$ defined in (3.3).

Refined Instanton Generating Potential

In this appendix we rewrite the theta series decomposition (3.35) of the refined instanton generating potential as a sum of iterated integrals of the same type which arise in the unrefined case. To this end, we retrace the steps taken in [15], which allowed to rewrite the unrefined potential as in (3.27).

First, we rewrite $\mathcal {G}^\mathrm{ref}$ as an expansion in the holomorphic generating functions $h^\mathrm{ref}_{p_i,\mu _i}$. This changes the kernels of the theta series, which now fail to be modular due to the modular anomaly of $h^\mathrm{ref}_{p,\mu }$. The result (proven below in §C.1)³¹ is given by

$$\begin{aligned} \mathcal {G}^\mathrm{ref}=\frac{1}{2\pi \sqrt{\tau _2}}\sum _{n=1}^\infty \left[ \prod _{i=1}^{n} \sum _{p_i,\mu _i}\sigma _{p_i}h^\mathrm{ref}_{p_i,\mu _i}\right] e^{-S^\mathrm{cl}_p}y^{-\beta m(p)} V_{{\varvec{p}}}\,\vartheta ^{\mathrm{ref}}_{{{\varvec{p}}},{\varvec{\mu }}}\bigl (\Phi ^\mathrm{ref}_{n},-1\bigr ), \end{aligned}$$

(C.1)

where

$$\begin{aligned} \Phi ^{\mathrm{ref}}_n({\varvec{x}})=\sum _{n_1+\cdots n_m=n \atop n_k\ge 1} \Phi ^{\scriptscriptstyle \,\int }_m({\varvec{x}}') \prod _{k=1}^m \Phi ^{\mathrm{tr}}_{n_k}(x_{j_{k-1}+1},\dots ,x_{j_k}). \end{aligned}$$

(C.2)

The first factor in (C.2),

$$\begin{aligned} \Phi ^{\scriptscriptstyle \,\int }_n({\varvec{x}})= \frac{\Phi ^{\scriptscriptstyle \,\int }_1(x)}{2^{n-1}}\, \hat{\Phi }^M_{n-1}(\{ {\varvec{u}}_\ell \};{\varvec{x}},\sqrt{2\tau _2}\beta \mathfrak {p}), \end{aligned}$$

(C.3)

is constructed from the function (3.32) and the modified version of the complementary error functions introduced in appendix A.2. In the above formula it appears with the index m equal to the number of parts in the partition of n. Correspondingly, it depends on ${\varvec{x}}'$ and ${{\varvec{p}}}'$ (the later dependence is not indicated explicitly) which are both $mb_2(\mathfrak {Y})$-dimensional vectors with components

$$\begin{aligned} p'^a_k=\sum _{i=j_{k-1}+1}^{j_k}p^a_i, \qquad x'^a_k=\kappa '^{ab}_k\sum _{i=j_{k-1}+1}^{j_k} \kappa _{i,bc} x^c_i, \end{aligned}$$

(C.4)

where $\kappa '_{k,ab}=\kappa _{abc}p'^c_k$ and $j_k$ are defined below (3.4). The other factors in (C.2) are given by

$$\begin{aligned} \Phi ^{\mathrm{tr}}_{n}({\varvec{x}}) = (-y)^{-\sum _{i<j} \gamma _{ij} } \sum _{T\in \mathbb {T}_n^\mathrm{S}}(-1)^{n_T-1} \left( g^{\mathrm{ref}(0)}_{v_0}-\mathcal {E}^{\mathrm{ref}(0)}_{v_0}\right) \prod _{v\in V_T\setminus {\{v_0\}}}\mathcal {E}^{\mathrm{ref}(0)}_{v}, \end{aligned}$$

(C.5)

where ${\varvec{x}}$ is related to charges via (3.7). Comparing with (2.19), (the symmetrization of) these functions can be recognized as a rescaled version of the refined tree index relating the refined DT and MSW invariants. Namely,

$$\begin{aligned} \bar{\Omega }(\gamma ,z^a,y) = \sum _{\sum _{i=1}^n \gamma _i=\gamma } \frac{(-y)^{\sum _{i<j} \gamma _{ij} }}{(y-y^{-1})^{n-1}}\,\Phi ^{\mathrm{tr}}_{n}({\varvec{x}})\, \prod _{i=1}^n {\bar{\Omega }}^\mathrm{MSW}(\gamma _i,y), \end{aligned}$$

(C.6)

where the symmetrization is ensured by the sum over charges. Note that the power of $y-y^{-1}$ disappears once this relation is rewritten in terms of the generating functions (3.5) and (3.6).

Given the relation (C.6), $\mathcal {G}^\mathrm{ref}$ can be rewritten as an expansion in the generating functions of refined DT invariants $h^\mathrm{DT,ref}_{p_i,q_i}$. It is easy to see that such expansion is given by

$$\begin{aligned} \mathcal {G}^\mathrm{ref}=\frac{1}{2\pi }\sum _{n=1}^\infty \left[ \prod _{i=1}^{n} \sum _{p_i,q_i}\sigma _{p_i,q_i}h^\mathrm{DT,ref}_{p_i,q_i} \mathcal {X}^{\,\theta }_{p_i,q_i} \right] y^{-\beta m(p)+\sum _{i<j} \gamma _{ij} }\Phi ^{\scriptscriptstyle \,\int }_n({\varvec{x}}), \end{aligned}$$

(C.7)

where $\sigma _{p,q}\equiv \sigma _\gamma =\mathbf{e}\left( {1\over 2}\, p^a q_a\right) \sigma _p$ is the quadratic refinement specified for our set of charges and

$$\begin{aligned} \mathcal {X}^{\,\theta }_{p,q} = e^{-S^\mathrm{cl}_p}\, \mathbf{e}\left( - \frac{\tau }{2}\,(q+b)^2+c^a (q_a +\textstyle {1\over 2}(pb)_a)\right) \end{aligned}$$

(C.8)

is a combination of three contributions evaluated for a single charge: the classical D3-brane action, the exponential defining the theta series (A.1), and the factor $V_p$ (A.5).

Next, we use the result proven in “Appendix E” of [15] which states that for any unrooted labelled tree $\mathcal {T}$ one has the following identity

$$\begin{aligned} \frac{\Phi ^{\scriptscriptstyle \,\int }_1(x)}{2^{n-1}}\, \Phi ^M_{n-1}(\{ {\varvec{u}}_e\};{\varvec{x}})= \frac{\mathrm {i}^{n-1}}{(2\pi )^n}\left[ \prod _{i=1}^n\int _{z_i^\star +\mathbb {R}}\mathrm {d}z_i \, W_{p_i}(x_i,z_i)\right] \frac{1}{\prod _{e\in E_\mathcal {T}}\left( z_{s(e)}-z_{t(e)}\right) }\, , \end{aligned}$$

(C.9)

where

$$\begin{aligned} W_{p}(x,z)=e^{-2\pi \tau _2 z^2(pt^2)-2\pi \mathrm {i}\sqrt{2\tau _2}\, z\, (pxt)} \end{aligned}$$

(C.10)

and $z_i^\star =- \frac{\mathrm {i}\,(p_ix_i t)}{\sqrt{2\tau _2}(p_it^2)}$ is the saddle point governing the integral over $z_i$. On the left hand side the data about $\mathcal {T}$ are encoded in the set of vectors ${\varvec{u}}_e$ (B.8). In our case this set is given by ${\varvec{u}}_\ell$ defined in (3.34) which can be seen as vectors ${\varvec{u}}_e$ for the trivial unrooted tree $\bullet \text{--- }\bullet \text{-- }\cdots \text{-- }\bullet \text{--- }\bullet \,$. Furthermore, it is easy to see that if one replaces $\Phi ^M_{n-1}$ by its modified version $\hat{\Phi }^M_{n-1}$ appearing in (C.3), on the right-hand side one simply changes $z_i^\star$ by

$$\begin{aligned} z_{\gamma _i}=- \frac{\mathrm {i}\,(p_i(x_i-\sqrt{2\tau _2}\beta \mathfrak {p}_i) t)}{\sqrt{2\tau _2}(p_it^2)} =-\mathrm {i}\,\frac{(q_a+(pb)_a)\,t^a}{(pt^2)}\, . \end{aligned}$$

(C.11)

Therefore, we conclude that the functions $\Phi ^{\scriptscriptstyle \,\int }_n$ can be represented in the following integral form

$$\begin{aligned} \Phi ^{\scriptscriptstyle \,\int }_n({\varvec{x}}) = \left[ \prod _{i=1}^n\int _{z_{\gamma _i}+\mathbb {R}}\frac{\mathrm {d}z_i}{2\pi } \, W_{p_i}(x_i,z_i)\right] \frac{\mathrm {i}^{n-1}}{\prod _{i=1}^{n-1}(z_i-z_{i+1})}\, . \end{aligned}$$

(C.12)

Besides, due to the shift of the b-field produced by the refinement, one has

$$\begin{aligned} \prod _{i=1}^n W_{p_i}(\sqrt{2\tau _2}(q_i+\hat{b}_i),z_i)=\mathbf{e}\left( -2\tau _2\beta \sum _{i<j} (p_ip_jt)(z_i-z_j)\right) \prod _{i=1}^n W_{p_i}(\sqrt{2\tau _2}(q_i+ b_i),z_i). \end{aligned}$$

(C.13)

Furthermore, since we work in the large volume limit $t^a\rightarrow \infty$, the contours $z_{\gamma _i}+\mathbb {R}$ can be deformed into the standard BPS rays $\ell _{\gamma _i}$ [10, 28] which in the z-plane go along the arcs running from $-1$ to 1 and passing through $z_{\gamma _i}$. Thus, substituting (C.12) and (C.13) into (C.7) and taking into account the definition of $h^\mathrm{DT,ref}_{p,q}$, one obtains the representation (3.39), where we decomposed the $\beta$-dependent power of y as in the original formula (3.35).

1.1 Proof of Eq.(C.1)

Substituting the explicit formula for the completion of the generating function (3.12) into the refined potential (3.35) and rewriting it as an expansion in powers of $h^\mathrm{ref}_{p_i,\mu _i}$, it is easy to see that one gets (C.1) where the kernel $\Phi ^{\mathrm{ref}}_n$ is given by

$$\begin{aligned} \Phi ^{\mathrm{ref}}_n({\varvec{x}})=\sum _{n_1+\cdots n_m=n \atop n_k\ge 1} \widehat{\Phi }^\mathrm{ref}_{m}({\varvec{x}}') \prod _{k=1}^m \left[ \sum _{T_k\in \mathbb {T}_{n_k}^\mathrm{S}}(-1)^{n_{T_k}-1} \Phi ^{(+)}_{v_0} \prod _{v\in V_{T_k}\setminus {\{v_0\}}} \Phi ^{(0)}_{v}\right] , \end{aligned}$$

(C.14)

where we introduced the rescaled versions of $\mathcal {E}^{\mathrm{ref}(0)}_{n}$ and $\mathcal {E}^{{\mathrm{ref}(+)}}_{n}$³²

$$\begin{aligned} \Phi ^{(0)}_{n}=\frac{1}{2^{n-1}}\prod _{k=1}^{n-1}\text{ sgn }({\varvec{v}}_k,{\varvec{x}}-\sqrt{2\tau _2}\,\beta \mathfrak {p}), \qquad \Phi ^{(+)}_{n}=\Phi _{n}-\Phi ^{(0)}_{n}, \end{aligned}$$

(C.15)

the trees $T_k$ are labelled by the charges $\check{\gamma }_{j_{k-1}+1},\dots ,\check{\gamma }_{j_{k}}$, whereas the first factor depends on the sums of charges in each subset $\check{\gamma }'_k=\check{\gamma }_{j_{k-1}+1}+\cdots +\check{\gamma }_{j_{k}}$, or equivalently on the $mb_2$-dimensional vectors (C.4). Taking into account the explicit form of $\widehat{\Phi }^\mathrm{ref}_{n}$ (3.31), we see that the kernel (C.14) can be visualized as a sum over Schröder trees, the leaves of which themselves sprouting further Schröder trees. Regarding all these trees as parts of one big tree, one arrives at the following representation

$$\begin{aligned} \Phi ^{\mathrm{ref}}_n= \Phi ^{\scriptscriptstyle \,\int }_1\sum _{T\in \mathbb {T}_n^\mathrm{S}}(-1)^{n_{T}-1}\left( \widetilde{\Phi }_{v_0}-\Phi _{v_0}\right) \sum _{T'\subseteq T} \prod _{v\in V_{T'}\setminus {\{v_0\}}}\Phi _{v} \prod _{v\in L_{T'}\setminus L_T} (-\Phi ^{(+)}_{v}) \prod _{v\in V_T\setminus (V_{T'}\cup L_{T'})} \Phi ^{(0)}_{v}, \end{aligned}$$

(C.16)

where the second sum goes over all subtrees $T'$ of T containing its root³³ and $L_{T}$ denotes the set of leaves of T.

Let us now fix a tree T and a vertex v whose only children are leaves of T, i.e. v has height 1. Then for each subtree $T'$ with $v\in L_{T'}$ one can put into a correspondence another subtree for which the children of v are added to the subtree, i.e. now $v\in V_{T'}$ and the rest of $T'$ is the same. The contributions of two such subtrees in (C.16) differ only by the factor assigned to the vertex v: it is $-\Phi ^{(+)}_{v}$ in the first contribution, whereas $\Phi _{v}$ in the second. Thus, the two contribution recombine giving $\Phi ^{(0)}_{v}$ as the weight assigned to the vertex.

After performing such recombination for all vertices of height 1, one moves to the next level and considers v of height 2. Here again one picks up pairs of subtrees with $v\in L_{T'}$ and $v\in V_{T'}$, respectively. But now the contribution of the latter is already the one after the recombination done at the first step. As a result, the two contributions again differ only by the factor assigned to the vertex v and the result of their recombination is the same as above: the new factor is $\Phi ^{(0)}_{v}$.

In this way one covers all vertices of T up to the root. At the root one again compares two contributions: one of the trivial subtree (see footnote 33) and another one from all previous recombinations. Their sum leads to the weight $\widetilde{\Phi }_{v_0}-\Phi ^{(0)}_{v_0}$ assigned to the root. As a result, one remains with the following kernel

$$\begin{aligned} \Phi ^{\mathrm{ref}}_n= \Phi ^{\scriptscriptstyle \,\int }_1\sum _{T\in \mathbb {T}_n^\mathrm{S}}(-1)^{n_{T}-1}\left( \widetilde{\Phi }_{v_0}-\Phi ^{(0)}_{v_0}\right) \prod _{v\in V_{T}\setminus {\{v_0\}}}\Phi ^{(0)}_{v}. \end{aligned}$$

(C.17)

It it worth noting that this formula makes it manifest that the kernel $\Phi ^{\mathrm{ref}}_n$ is smooth across walls of marginal stability, since all moduli dependence comes from $\widetilde{\Phi }_{v_0}$.

Next, we should take into account the relation (A.22) between functions $\Phi _n^E$ and $\hat{\Phi }_n^M$ with ${\varvec{\chi }}=\sqrt{2\tau _2}\beta \mathfrak {p}$. In our case $n=k_{v_0}-1$ where $k_{v_0}$ is the number of children of the root vertex, and the vectors ${\varvec{v}}_i$ coincide with ${\varvec{u}}_\ell$ defined in (3.34). For such vectors one has

Proposition 2

Let $\mathcal {I}=\{j_1,\dots ,j_{m-1}\}$ where $0\equiv j_0<j_1<\cdots<j_{m-1}< j_{m}\equiv n$. For $\ell \in \mathscr {Z}_{n-1}\setminus \mathcal {I}$ find k such that $j_{k-1}< \ell <j_k$. Then one has

$$\begin{aligned} {\varvec{u}}_{\ell \perp \mathcal {I}}= \frac{(pt^2)}{\sum _{i=j_{k-1}+1}^{j_k} (p_it^2)}\,{\varvec{u}}^{j_{k-1}+1, j_k}_\ell \qquad \text{ where }\qquad {\varvec{u}}^{ij}_\ell =\sum _{k=i}^\ell \sum _{k'=\ell +1}^j {\varvec{u}}_{kk'}, \quad i\le \ell < j. \end{aligned}$$

Proof

First, we prove the statement for the case of the set $\mathcal {I}$ consisting of one element which we formulate as a Lemma.

Lemma 1

$$\begin{aligned} {\varvec{u}}_{\ell \perp k}= \left\{ \begin{array}{ll} \frac{(pt^2)}{\sum _{i=1}^{k} (p_it^2)}\,{\varvec{u}}^{1k}_\ell , &{} \ell <k, \\ \frac{(pt^2)}{\sum _{i=k+1}^n (p_it^2)}\,{\varvec{u}}^{k+1,n}_\ell ,\quad &{} \ell >k. \end{array} \right. \end{aligned}$$

Proof

First, let us consider the case $n=3$. A straightforward calculation gives

$$\begin{aligned}&{\varvec{u}}_1^2=(pt^2)(p_1t^2)(p_{2+3}t^2), \qquad {\varvec{u}}_2^2=(pt^2)(p_{1+2}t^2)(p_3t^2), \nonumber \\&\qquad \qquad \qquad ({\varvec{u}}_1,{\varvec{u}}_2)=(pt^2)(p_1t^2)(p_3t^2), \end{aligned}$$

(C.18)

where we introduced the convenient notations $p^a_{i+j}=p^a_i+p^a_j$. Using these results, one finds

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

(C.19)

which agrees with the statement of the Lemma since ${\varvec{u}}_1^{12}={\varvec{u}}_{12}$, ${\varvec{u}}_2^{23}={\varvec{u}}_{23}$.

The case of arbitrary n then reduces to the case $n=3$ by identifying

$$\begin{aligned} \tilde{\gamma }_1=\sum _{i=1}^\ell \gamma _i, \qquad \tilde{\gamma }_2=\sum _{i=\ell +1}^{k} \gamma _i, \qquad \tilde{\gamma }_3=\sum _{i=k+1}^n \gamma _i, \end{aligned}$$

(C.20)

for $\ell <k$, or

$$\begin{aligned} \tilde{\gamma }_1=\sum _{i=1}^{k} \gamma _i, \qquad \tilde{\gamma }_2=\sum _{i=k+1}^{\ell } \gamma _i, \qquad \tilde{\gamma }_3=\sum _{i=\ell +1}^n \gamma _i, \end{aligned}$$

(C.21)

in the opposite case. $\quad \square$

If $\mathcal {I}$ has several elements, let us find k as in the statement of the Proposition. We note that the projection on the subspace orthogonal to the span of $\{{\varvec{u}}_i\}_{i\in \mathcal {I}}$ can be equivalently obtained by first projecting with respect to ${\varvec{u}}_{j_k}$ and then with respect to $\{{\varvec{u}}_{i\perp j_k}\}_{i\in \mathcal {I}\setminus \{j_k\}}$. According to the Lemma, the latter set is equivalent to $\{{\varvec{u}}_{j_\ell }^{1j_k}\}_{\ell =1}^{k-1}\cup \{{\varvec{u}}_{j_\ell }^{j_k+1, n}\}_{\ell =k+1}^{m-1}$, whereas the first projection gives us ${\varvec{u}}_{\ell \perp j_k}=\frac{(pt^2)}{\sum _{i=1}^{j_k} (p_it^2)}\,{\varvec{u}}^{1j_k}_\ell$. Since ${\varvec{u}}^{1j_k}_\ell$ is already orthogonal to any ${\varvec{u}}_{j_\ell }^{j_k+1, n}$, it remains only to do the orthogonal projection with respect to $\{{\varvec{u}}_{j_\ell }^{1j_k}\}_{\ell =1}^{k-1}$.³⁴ This projection we again split into two steps: with respect to ${\varvec{u}}_{j_{k-1}}^{1j_k}$ and $\{{\varvec{u}}_{j_\ell \perp j_{k-1}}^{1j_k}\}_{\ell =1}^{k-2}$. The projections can be evaluated using the Lemma provided one replaces n by $j_k$ since all components beyond $j_k$ vanish. As a result, the first projection gives ${\varvec{u}}^{1j_k}_{\ell \perp j_{k-1}}=\frac{\sum _{i=1}^{j_k}(p_it^2)}{\sum _{i=j_{k-1}+1}^{j_k} (p_it^2)}\,{\varvec{u}}^{j_{k-1}+1,j_k}_{\ell }$, whereas the remaining set of vectors is equivalent to the span of $\{{\varvec{u}}_{j_\ell }^{1j_{k-1}}\}_{\ell =1}^{k-2}$. All these vectors are orthogonal to ${\varvec{u}}^{j_{k-1}+1,j_k}_{\ell }$ and therefore there is no need to do any further projection. Combining the prefactors, one recovers the statement of the Proposition. $\quad \square$

Fig. 3

An example of the effect of the partition on the tree for the case where the root has six children and the partition is $6=2+1+3$

Note that each set $\mathcal {I}=\{j_1,\dots ,j_{m-1}\}$ provides an ordered partition of $n=n_1+\cdots n_m$ with $n_k=j_{k}-j_{k-1}$. Then according to the Proposition we have

$$\begin{aligned} \Phi ^{\scriptscriptstyle \,\int }_1\widetilde{\Phi }_n=\sum _{n_1+\cdots n_m=n \atop n_k\ge 1} \Phi ^{\scriptscriptstyle \,\int }_m({\varvec{x}}') \prod _{k=1}^m \widetilde{\Phi }^{(0)}_{n_k}(x_{j_{k-1}+1},\dots ,x_{j_k}), \end{aligned}$$

(C.22)

where

$$\begin{aligned} \widetilde{\Phi }^{(0)}_{n}({\varvec{x}})=\frac{1}{2^{n-1}}\prod _{k=1}^{n-1}\text{ sgn }({\varvec{u}}_k,{\varvec{x}}-\sqrt{2\tau _2}\,\beta \mathfrak {p}) =(-y)^{-\sum _{i<j} \gamma _{ij} }g^{\mathrm{ref}(0)}_n(\{\check{\gamma }_i,c_i\},y). \end{aligned}$$

(C.23)

The contribution of the trivial partition ($m=1$) combined with $\Phi ^{(0)}_{v_0}$ in (C.17) is equivalent to $\Phi ^{\scriptscriptstyle \,\int }_1\Phi ^{\mathrm{tr}}_{n}$ and thus already has the required form (C.2). For non-trivial partitions, the effect of substitution of (C.22) into (C.17) can be interpreted as a replacement of tree T by a new tree $T_{k_1,\cdots ,k_m}\equiv T'$ with $k_1+\cdots k_m=k_{v_0}$, which is constructed as follows. Group all children of the original root $v_0$ according to the decomposition of $k_{v_0}$ under consideration. If $k_j>1$, all children in the jth group are connected to a vertex $v_j$ which is itself connected to the root $v'_0$ of the new tree. Otherwise the corresponding child is connected directly to the root (see Fig. 3). The contribution assigned to the new tree is then given by

$$\begin{aligned} (-1)^{n_{T'}-1}\Phi ^{\scriptscriptstyle \,\int }_m\prod _{v\in \mathrm{Ch}^\mathrm{new}}(-\widetilde{\Phi }^{(0)}_v)\prod _{v\in \mathrm{Ch}^\mathrm{old}}\Phi ^{(0)}_v \prod _{v\in V_{T'}\setminus {\{v'_0\cup \mathrm{Ch}(v'_0)\}}}\Phi ^{(0)}_{v}, \end{aligned}$$

(C.24)

where $\mathrm{Ch}^\mathrm{new}$ denotes the set of the added vertices $v_j$, whereas $\mathrm{Ch}^\mathrm{old}$ is the set of those children of the root which have already been such children before the above operation and are not the leaves. It is clear that the sum over trees T and partitions is equivalent to the sum over trees $T'$ supplemented by the sum over all possible assignments of ‘new’ and ‘old’ to the children of the root. The latter sum can easily be evaluated and one obtains

$$\begin{aligned} \sum _{T'\in \mathbb {T}_n^\mathrm{S}}(-1)^{n_{T'}-1}\Phi ^{\scriptscriptstyle \,\int }_{v'_0}\prod _{v\in Ch(v'_0)\setminus L_{T'}} \left( \Phi ^{(0)}_{v}-\widetilde{\Phi }^{(0)}_{v}\right) \prod _{v\in V_{T'}\setminus {\{v'_0\cup \mathrm{Ch}(v'_0)\}}}\Phi ^{(0)}_{v}. \end{aligned}$$

(C.25)

This result precisely coincides with the contribution of non-trivial partitions to (C.2). To see this, it is enough to split $T'$ into subtrees corresponding to descendants of the root which then correspond to the trees in the formula (C.5), whereas the effect of the root is captured by the sum over partitions. This completes the proof of (C.1).

Proof of the Truncation Theorem

In this appendix we prove Theorem 1 from section 5. Our first step is to establish some useful properties of the orthogonal projections ${\varvec{v}}_{\ell \perp k}$ appearing as parameters of the generalized error functions in (5.5), the factor assigned to the root vertex of each Schröder tree contributing to the anomaly coefficient (5.4).

Proposition 3

For collinear charges $p^a_i=N_i p_0^a$ with $N=\sum _{i=1}^n N_i$, one has

$$\begin{aligned} {\varvec{v}}_{\ell \perp k}= \left\{ \begin{array}{ll} \frac{N}{\sum _{i=1}^{k} N_i}\,{\varvec{v}}^{1k}_\ell , &{} \ell<k, \\ \frac{N}{\sum _{i=k+1}^n N_i}\,{\varvec{v}}^{k+1,n}_\ell ,\quad &{} \ell >k, \end{array} \right. \qquad \text{ where }\qquad {\varvec{v}}^{ij}_\ell =\sum _{k=i}^\ell \sum _{k'=\ell +1}^j {\varvec{v}}_{kk'}, \quad i\le \ell < j. \end{aligned}$$

Proof

This Proposition is a direct analogue of Lemma 1 and their proofs are identical provided one uses the following dictionary: ${\varvec{u}}\rightarrow {\varvec{v}}$, $t^a\rightarrow p_0^a$, $(p_it^2)\rightarrow N_i$. Note however that in contrast to the Lemma this Proposition holds only for collinear charges.

$\square$

This Proposition shows that after the orthogonal projection the set of vectors ${\varvec{v}}_\ell$ is split into two sets of mutually orthogonal vectors, $\{{\varvec{v}}_{\ell \perp k}\}_{\ell <k}$ and $\{{\varvec{v}}_{\ell \perp k}\}_{\ell >k}$ (of course, for $k=1$ and $k=n-1$ one of these sets is empty). At the same time, the generalized error functions are known to possess the property that if the vectors defining them are split into two mutually orthogonal sets, the function is given by a product of two generalized error functions of lower ranks evaluated on the respective sets of vectors. Therefore, we can rewrite (5.5) as

$$\begin{aligned} \partial _{\tau _2}\mathcal {E}^{\mathrm{ref}}_n= & {} \frac{(-y)^{\sum _{i<j} \gamma _{ij} }}{2^{n-1}\tau _2}\sum _{k=1}^{n-1}\frac{({\varvec{v}}_k,\check{\varvec{x}})}{|{\varvec{v}}_k|}\, e^{-\pi \,\frac{({\varvec{v}}_k,{\varvec{x}})^2}{{\varvec{v}}_k^2} } \Phi ^E_{k-1}(\{ {\varvec{v}}^{1k}_{\ell }\}_{\ell =1}^{k-1};{\varvec{x}}) \Phi ^E_{n-k-1}(\{ {\varvec{v}}^{k+1,n}_{\ell }\}_{\ell =k+1}^{n-1};{\varvec{x}}) \nonumber \\= & {} \sum _{k=1}^{n-1} A_k(\{\check{\gamma }_i\}_{i=1}^{n},y)\, \mathcal {E}^{\mathrm{ref}}_{k}(\{\check{\gamma }_i\}_{i=1}^{k},y)\,\mathcal {E}^{\mathrm{ref}}_{n-k}(\{\check{\gamma }_i\}_{i=k+1}^{n},y), \end{aligned}$$

(D.1)

where we set $\Phi ^E_{0}=1$, whereas in the last line we used the definition (3.10) of $\mathcal {E}^{\mathrm{ref}}_n$, the fact that

$$\begin{aligned} \sum _{1\le i<j\le n} \gamma _{ij}=\sum _{1\le i<j\le k} \gamma _{ij}+\sum _{k+1\le i<j\le n} \gamma _{ij}+\sum _{i=1}^k\sum _{j=k+1}^n\gamma _{ij}, \end{aligned}$$

(D.2)

and introduced

$$\begin{aligned} A_k(\{\check{\gamma }_i\}_{i=1}^{n},y)=J\left( \sum _{i=1}^k\check{\gamma }_i,\sum _{i=k+1}^n\check{\gamma }_i;y\right) , \qquad J(\check{\gamma }_1,\check{\gamma }_2;y)=\frac{(-y)^{\gamma _{12}}}{2\tau _2}\, \frac{({\varvec{v}}_{12},\check{\varvec{x}})}{|{\varvec{v}}_{12}|}\,e^{-\pi \,\frac{({\varvec{v}}_{12},{\varvec{x}})^2}{{\varvec{v}}_{12}^2} }. \end{aligned}$$

(D.3)

As a result, each Schröder tree produces a sum of contributions given by a product of two Schröder trees, obtained by cutting the original tree at the root between the kth and the $(k+1)$th children, for which every vertex carries a factor of $\mathcal {E}^{\mathrm{ref}}_{v}$, and of the factor $A_{j_k}$ where $j_k$ is the number of leaves in the first subtree. The latter number can also be expressed as $j_k=n(T_1)+\cdots +n(T_k)$ where n(T) is the number of leaves of a rooted tree T and $T_i$ are the subtrees growing from descendants of the root $v_0$. Then it is easy to see that for each such product there are four Schröder trees which produce it, and the resulting four contributions cancel each other as shown in Fig. 4. In fact, if $k=1$, the second and fourth trees do not exist (they spoil the definition of a Schröder tree) and the cancelation happens just between two trees. Similarly, if $k=k_{v_0}$, the third and fourth do not exist.

Fig. 4

Combination of contributions to the anomaly coming from four Schröder trees. The green dashed lines show how each tree is cut to produce the contribution at the bottom

The only special case when three of the four shown trees do not exist is $n=2$. Then only the first tree contributes and there is no cancelation giving³⁵

$$\begin{aligned} \mathcal {J}^\mathrm{ref}_2=&\,\frac{\mathrm {i}}{2}\, \,\mathrm{Sym}\, J(\check{\gamma }_1,\check{\gamma }_2;y) \nonumber \\ =&\,\frac{\mathrm {i}(-y)^{\gamma _{12}}}{4\sqrt{2\tau _2}}\, \frac{\gamma _{12}-\beta (pp_1p_2)}{\sqrt{(pp_1p_2)}}\, e^{-2\pi \tau _2\,\frac{(\gamma _{12}+\beta (pp_1p_2))^2}{(pp_1p_2)} }+(1\leftrightarrow 2), \end{aligned}$$

(D.4)

where we evaluated all contractions with the bilinear form. For all other n’s, all contributions cancel and $\mathcal {J}^\mathrm{ref}_n$ vanish. This proves the statement of the Theorem.

Geometric Data for Hirzebruch and del Pezzo Surfaces

In this appendix, we provide the geometric data for the complex surfaces used in constructing local Calabi–Yau threefolds in §4.3.

• For the Hirzebruch surface $S=\mathbb {F}_k$ (also known as ruled rational surface), defined as the projectivization of the $\mathcal {O}(k)\oplus \mathcal {O}(0)$ bundle over $\mathbb {P}^1$, one has $b_2(S)=2$ and $\chi (S)=4$. Using the same basis as in [85, §4.1.1] (see also [41, §A.2]), we get

  $$\begin{aligned}&C_{\alpha \beta } =\begin{pmatrix} 0 &{} 1 \\ 1 &{} k \end{pmatrix}, \qquad C^{\alpha \beta } = \begin{pmatrix} -k &{} 1 \\ 1 &{} 0 \end{pmatrix}, \nonumber \\ c_1^\alpha = ( 2-k , 2 ),&\qquad c_{2,e} = 92, \qquad c_{2,\alpha }=12C_{\alpha \beta }c_1^\beta = 12( 2, 2+k ), \end{aligned}$$

  (E.1)

  hence

  $$\begin{aligned} K_S^2 = [S]^3 = 8, \qquad [S] \cap c_2(T\mathfrak {Y}) = -4. \end{aligned}$$

  (E.2)

• For the del Pezzo surface $S=\mathbb {B}_k$, defined as the blow-up of $\mathbb {P}^2$ over k generic points, one has $b_2(S)=k+1$, $\chi (S)=k+3$. Using the same basis as in [85, §4.1.2], we get

  $$\begin{aligned} C_{\alpha \beta } =&\begin{pmatrix} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{}\dots &{} 1 \\ \vdots &{} &{} 0 &{} 1 &{} 1 \\ 1 &{} \dots &{} 1 &{} 0 &{} 1\\ 1 &{} \dots &{} 1 &{} 1 &{} 1 \end{pmatrix}, \qquad C^{\alpha \beta } = \begin{pmatrix} -1 &{} 0 &{} 0 &{} \dots &{} 1 \\ 0 &{} -1 &{} 0 &{}\dots &{} 1 \\ \vdots &{} &{} -1 &{} 0 &{} 1 \\ 0 &{} \dots &{} 0 &{} -1 &{} 1\\ 1 &{} \dots &{} 1 &{} 1 &{} 2-k \end{pmatrix}, \nonumber \\ c_1^\alpha =&\, ( 1,\dots , 1, 3-k ), \quad \ c_{2,e} = 102-10 k, \quad \ c_{2,\alpha }=12C_{\alpha \beta }c_1^\beta = 12( 2,\dots , 2, 3 ), \end{aligned}$$

  (E.3)

  hence

  $$\begin{aligned} K_S^2 = [S]^3 = 9-k, \qquad [S] \cap c_2(T\mathfrak {Y}) = 2k-6. \end{aligned}$$

  (E.4)

Note that $\mathbb {B}_1=\mathbb {F}_1$, whereas $\mathbb {B}_0=\mathbb {P}^2$. Smooth elliptic fibrations for these two cases have been discussed in detail in [57]. For $k=9$, $\mathbb {B}_9$ is almost Fano and known as the rational elliptic surface or half-K3. Vafa–Witten invariants on $\mathbb {B}_9$ were studied in [40, 57, 86].

Modular Completion of Vafa–Witten Invariants on $\mathbb {P}^2$

In this appendix we provide a detailed comparison of the modular completion of the generating function of refined VW invariants on $S=\mathbb {P}^2$ for ranks $N=2$ and 3 known in the literature [25] with the prediction of our general formula (3.12), and spell out prediction for $N=4$.

Applying the general formulae of Sect. 4.3 to the case at hand supplemented by the data in (E.3) with $k=0$, one obtains that the D4-brane charge for the divisor $[\mathbb {P}^2]$ is $p_0^a=(1,-3)$ and $p_0^3=9$, such that the Dirac product (4.30) becomes

$$\begin{aligned} \langle \gamma _1,\gamma _2\rangle = 3 (N_1 q_2 -N_2 q_1). \end{aligned}$$

(F.1)

Since $b_2(\mathbb {P}^2)=1$, the choice of J inside the Kähler cone is irrelevant, and the first Chern class $\mu$ is an integer modulo the rank N. It will be convenient to define

$$\begin{aligned} h^\mathrm{VW,ref}_{N,\mu }(\tau ,w) = g_{N,\mu }(\tau ,w)\, \left( h^\mathrm{VW,ref}_{1,0}(\tau ,w)\right) ^N, \end{aligned}$$

(F.2)

where $h^\mathrm{VW,ref}_{1,0}$ was given in (4.13), and similarly for the modular completion $\widehat{h}^\mathrm{VW,ref}_{N,\mu }$, so that $\widehat{g}_{N,\mu }$ transforms as a vector valued Jacobi form of weight $\frac{1}{2}(N-1)$ and index $-\frac{3}{2}(N^3-N)$. The identification (4.36) implies

$$\begin{aligned} g'_{N,\mu }(\tau ,w)= g_{N,\mu +\frac{1}{2} N(N-1)}(\tau ,w+\textstyle {1\over 2}), \end{aligned}$$

(F.3)

where $g'_{N,\mu }$ is the function defined by $h^\mathrm{ref}_{Np_0,\mu }$ similarly to (F.2). Below we verify that once this relation is satisfied, it continues to hold for the respective modular completions. To this end, we borrow the results for $\widehat{g}_{N,\mu }$ at $N=2$ and $N=3$ from [25, 43].

1.1 Rank 2

For $N=2$, the generating functions of refined VW invariants were computed in [49, 50], generalizing the unrefined case in [87]. They are closely related to the generating function of Hurwitz class numbers [68]. The modular completion is given by [43, 53, 68]

$$\begin{aligned} \widehat{g}_{2,\mu }=g_{2,\mu } +\frac{1}{2}\sum _{\ell \in \mathbb {Z}+\mu /2}\Bigl [E_1(\sqrt{\tau _2}(2\ell +6\beta ))-\text{ sgn }(\ell )\Bigr ]\,\text{ q}^{-\ell ^2}y^{6\ell }. \end{aligned}$$

(F.4)

This should be compared with the result of our general prescription which, after extracting the square of $h^\mathrm{ref}_{1,0}(\tau ,w)$ as in (F.2), reads

$$\begin{aligned} \widehat{g}'_{2,\mu }=g'_{2,\mu }+ \sum _{q_1+q_2=\mu } \frac{(-y)^{\gamma _{12}}}{2}\left( E_1\left( \frac{\sqrt{2\tau _2}(\gamma _{12}+\beta (pp_1p_2)) }{\sqrt{(pp_1p_2)}}\right) -\text{ sgn }(\gamma _{12})\right) e^{\pi \mathrm {i}\tau Q_2(\check{\gamma }_1,\check{\gamma }_2)}, \nonumber \\ \end{aligned}$$

(F.5)

where $\check{\gamma }_1=(p_0^a,(0,q_1))$, $\check{\gamma }_2=(p_0^a,(0,q_2))$. For this set of charges (F.1) gives $\gamma _{12}=3(q_2-q_1)$, whereas $(pp_1p_2)=2p_0^3=18$ and $Q_2(\check{\gamma }_1,\check{\gamma }_2)=-{1\over 2}\, (q_2-q_1)^2$. According to (4.33), both charges are decomposed as $q_i=\epsilon _i+{1\over 2}$ where $\epsilon _i\in \mathbb {Z}$. Therefore, $q_2-q_1=2\epsilon _2-\mu +1$ and if we set $q_2-q_1\equiv 2\ell$, then $\ell \in \mathbb {Z}+(\mu +1)/2$. Substituting all these quantities into (F.5), one finds perfect agreement with (F.4) provided one uses the identification (F.3).

1.2 Rank 3

For $N=3$, the generating functions of refined VW invariants were computed in [25, Eqs.(6.18), (6.22)], generalizing results in the unrefined case in [88, 89]. The modular completion is given by [25, Eqs.(6.20), (6.28)]

$$\begin{aligned} \widehat{g}_{3,0} =&\, g_{3,0}-\sum _{\mu =0,1}g_{2,\mu }\,R_{1,\mu /2}(\tau ,6w)-\frac{1}{4}\,R_{2,0}(\tau ,6w)-\frac{1}{12}\, , \nonumber \\ \widehat{g}_{3,\pm 1}=&\, g_{3,1} -\frac{g_{2,0}}{2} \left[ R_{1,\frac{1}{3}}(\tau ,6w) - R_{1,\frac{1}{3}}(\tau ,-6w) \right] -\frac{g_{2,1}}{2}\left[ R_{1,\frac{1}{6}}(\tau ,6w) - R_{1,\frac{1}{6}}(\tau ,-6w) \right] \nonumber \\&\,-\frac{1}{4}\, R_{2,(-\frac{1}{3},\frac{1}{3})}(\tau ,6w), \end{aligned}$$

(F.6)

where

$$\begin{aligned} R_{1,\alpha }(\tau ,w)= & {} \sum _{\ell \in \mathbb {Z}+\alpha } \Bigl [ \mathrm{sgn}(\ell ) - E_1\left( \sqrt{3\tau _2}(2\ell +\beta )\right) \Bigr ]\, y^{6\ell }\, \text{ q}^{-3\ell ^2}, \end{aligned}$$

(F.7)

$$\begin{aligned} R_{2,\alpha }(\tau ,w)= & {} \sum _{(k_3,k_4)\in \mathbb {Z}^2+\alpha } \biggl [ \mathrm{sgn}( k_3)\, \mathrm{sgn}(k_4)\, - E_2\left( \frac{1}{\sqrt{3}}; \sqrt{\tau _2}(2k_3-k_4+\beta ), \sqrt{3\tau _2}(k_4+\beta )\right) \nonumber \\&- \left( \mathrm{sgn}(k_4) - E_1\left( \sqrt{3\tau _2}(k_4+\beta )\right) \right) \mathrm{sgn}(2k_3-k_4) \nonumber \\&- \left( \mathrm{sgn}(k_3) - E_1\left( \sqrt{3\tau _2}(k_3+\beta )\right) \right) \mathrm{sgn}(2k_4-k_3)\biggr ] y^{k_3+k_4}\, \text{ q}^{-k_3^2-k_4^2+k_3 k_4}, \end{aligned}$$

(F.8)

and we used the parametrization (A.15) for the generalized error function of rank 2. Note the identity

$$\begin{aligned} R_{1,-\alpha }(\tau ,-w)=-R_{1,\alpha }(\tau ,w). \end{aligned}$$

(F.9)

Fig. 5

Schröder trees contributing to the completion for $N=3$. We indicated in red the $N_i$’s assigned to the leaves of the trees

This should be compared with the result of our general prescription, where each term originates from one of the Schröder trees shown in Fig. 5,

$$\begin{aligned} \widehat{g}'_{3,\mu }= & {} g'_{3,\mu } +{1\over 2}\sum _{q_1+q_2=\mu +\frac{3}{2}}g'_{2,\mu _1}\,\mathbf{e}\left( -\frac{\tau }{12}\,(2q_2-q_1)^2\right) \nonumber \\&\ \times \left[ (-y)^{3(2q_2-q_1)} \left( E_1\left( \sqrt{\frac{\tau _2}{3}}(2q_2-q_1+18\beta )\right) -\text{ sgn }(2q_2-q_1)\right) \right. \nonumber \\&\left. \quad +(-y)^{3(q_1-2q_2)} \left( E_1\left( \sqrt{\frac{\tau _2}{3}}(q_1-2q_2+18\beta )\right) -\text{ sgn }(q_1-2q_2)\right) \right] \nonumber \\&\quad +\frac{1}{4} \sum _{q_1+q_2+q_3=\mu +\frac{3}{2}}(-y)^{6(q_3-q_1)} \,\mathbf{e}\left( -\frac{\tau }{6}\left( (q_2-q_1)^2+(q_3-q_2)^2+(q_3-q_1)^2\right) \right) \nonumber \\&\ \times \biggl [\Phi ^E_2({\varvec{v}}_1,{\varvec{v}}_2,{\varvec{x}})- \text{ sgn }(q_2+q_3-2q_1) \, \text{ sgn }(2q_3-q_1-q_2)-\frac{1}{3}\, \delta _{q_1=q_2=q_3} \nonumber \\&\quad -\left( E_1\left( \sqrt{\frac{\tau _2}{3}}(2q_3-q_1-q_2+18\beta )\right) -\text{ sgn }(2q_3-q_1-q_2)\right) \text{ sgn }(q_2-q_1) \nonumber \\&\quad -\left( E_1\left( \sqrt{\frac{\tau _2}{3}}(q_2+q_3-2q_1+18\beta )\right) -\text{ sgn }(q_2+q_3-2q_1)\right) \text{ sgn }(q_3-q_2) \biggr ],\nonumber \\ \end{aligned}$$

(F.10)

where

$$\begin{aligned} \Phi ^E_2({\varvec{v}}_1,{\varvec{v}}_2,{\varvec{x}})=E_2\left( \frac{1}{\sqrt{3}},\sqrt{\tau _2}(q_2-q_1+6\beta ),\sqrt{\frac{\tau _2}{3}}(2q_3-q_1-q_2+18\beta )\right) . \end{aligned}$$

(F.11)

In the second term on the r.h.s. of (F.10) the charges are decomposed as $q_1=2\epsilon _1+\mu _1$ and $q_2=\epsilon _2+{1\over 2}$. Therefore $6\ell \equiv q_1-2q_2=6\epsilon _1-2\mu +3(\mu _1+1)$, which implies that $\ell \in \mathbb {Z}-\mu /3+(\mu _1+1)/2$. The second term then reads

$$\begin{aligned}&\frac{1}{2}\sum _{\mu _1=0,1} g'_{2,\mu _1+1}(\tau ,w) \sum _{\ell \in \mathbb {Z}-\frac{1}{3}\mu +\frac{1}{2}\mu _1} \Bigl [ (-y)^{-18\ell } \left( E_1(\sqrt{3\tau _2}(-2\ell +6\beta )-\text{ sgn }(-\ell )\right) \nonumber \\&\qquad + (-y)^{18\ell } \left( E_1(\sqrt{3\tau _2}(2\ell +6\beta )-\text{ sgn }(\ell )\right) \Bigr ]\,\text{ q}^{-3\ell ^2} \nonumber \\ =&\,\frac{1}{2}\sum _{\mu _1=0,1} g_{2,\mu _1}(\tau ,w+\textstyle {1\over 2})\left[ R_{1,-\frac{1}{3}\mu +\frac{1}{2}\mu _1}(-6(w+\textstyle {1\over 2})) - R_{1,-\frac{1}{3}\mu +\frac{1}{2}\mu _1}(6(w+\textstyle {1\over 2}))\right] \nonumber \\ =&\,-\frac{1}{2} \,g_{2,0}(\tau ,w+\textstyle {1\over 2})\left[ R_{1,\frac{1}{3}\mu }(6(w+\textstyle {1\over 2})) - R_{1,\frac{1}{3}\mu }(-6(w+\textstyle {1\over 2}))\right] \nonumber \\&\, -\frac{1}{2} \,g_{2,1}(\tau ,w+\textstyle {1\over 2}) \left[ R_{1,-\frac{1}{3}\mu +\frac{1}{2}}(6(w+\textstyle {1\over 2}))- R_{1,-\frac{1}{3}\mu +\frac{1}{2}}(-6(w+\textstyle {1\over 2})) \right] , \end{aligned}$$

(F.12)

where we used (F.3) at the second step and (F.9) to get the last line. Taking into account that for $\mu =0$ the functions in the brackets can be added by using again (F.9) and $R_{1,\alpha }=R_{1,\alpha +1}$, this term agrees precisely with the corresponding contributions in (F.6) with shifted w.

Next we move to the third term in (F.10) where all charges are decomposed as $q_i=\epsilon _i+{1\over 2}$, $\epsilon _i\in \mathbb {Z}$. We set $k_1=-\epsilon _1+\mu /3$ and $k_2=-(\epsilon _1+\epsilon _2)+2\mu /3$. This term then reads

$$\begin{aligned}&\frac{1}{4} \sum _{k_1\in \mathbb {Z}+\frac{\mu }{3}}\sum _{k_2\in \mathbb {Z}+\frac{2\mu }{3}} (-y)^{6(k_1+k_2)} \,\text{ q}^{-k_1^2-k_2^2+k_1k_2} \biggl [E_2\left( \frac{1}{\sqrt{3}},\sqrt{\tau _2}(2k_1-k_2+6\beta ),\sqrt{3\tau _2}(k_2+6\beta )\right) \nonumber \\&\qquad -\text{ sgn }(k_1) \, \text{ sgn }(k_2) -\left( E_1\left( \sqrt{3\tau _2}(k_2+6\beta )\right) -\text{ sgn }(k_2)\right) \text{ sgn }(2k_1-k_2) \nonumber \\&\qquad -\left( E_1\left( \sqrt{3\tau _2}(k_1+6\beta )\right) -\text{ sgn }(k_1)\right) \text{ sgn }(2k_2-k_1) \biggr ] -\frac{\delta _{\mu =0}}{12}. \end{aligned}$$

(F.13)

Note that the generalized error function $E_2$ appearing in (F.13) is invariant under $k_1\leftrightarrow k_2$ by [18, Corollary 3.10]. Therefore, identifying $(k_1,k_2)$ with $(k_4,k_3)$ in (F.8), one finds that (F.13) equals

$$\begin{aligned} -\frac{1}{4}\, R_{2,\frac{\mu }{3}(-1,1)}(\tau ,6(w+\tfrac{1}{2})) -\frac{\delta _{\mu =0}}{12}, \end{aligned}$$

(F.14)

thus reproducing the last terms in (F.6). Given that for $N=3$ the shift of $\mu$ in (F.3) is irrelevant, we conclude that the completion (F.10) perfectly agrees with the one found in [25].

1.3 Rank 4

For $N=4$, the generating functions of refined VW invariants were computed in [52], as an example of a general procedure valid for arbitrary N. Our general prescription (3.12) predicts that the modular completion should be given by a sum over the trees shown in Fig. 6, where we also indicate the decomposition of charges and the definition of the variables to be summed up. Taking into account the relation (F.3), one then arrives at the following prediction

Fig. 6

Schröder trees contributing to the completion for $N=4$. In the last row all $N_i=1$ and are not indicated. For each group of trees with the same set of $N_i$ we also show the decomposition of charges $q_i$ and the definition of the independent set of summation variables

$$\begin{aligned} \widehat{g}_{4,\mu }= & {} g_{4,\mu } +{1\over 2}\sum _{\mu _1=0,1,2}g_{3,\mu _1}\sum _{\ell \in \mathbb {Z}+\frac{\mu }{4}-\frac{\mu _1}{3}}\text{ q}^{-12\ell ^2} \Bigl [y^{36\ell }\, M(2\sqrt{6},3;\ell ) +y^{-36\ell }\,M(2\sqrt{6},3;-\ell ) \Bigr ] \nonumber \\&+{1\over 2}\sum _{\mu _1=0,1}g_{2,\mu _1}\,g_{2,\mu -\mu _1}\,\sum _{\ell \in \mathbb {Z}+\frac{\mu }{4}-\frac{\mu _1}{2}}\text{ q}^{-2\ell ^2} y^{24\ell }\,M(2\sqrt{2},6;\ell ) \nonumber \\&+\frac{1}{4} \sum _{\mu _1=0,1} g_{2,\mu _1}\sum _{k_1\in \mathbb {Z}+\frac{\mu }{4}-\frac{\mu _1}{2}}\, \sum _{k_2\in \mathbb {Z}+\frac{1}{2}\,(\mu -\mu _1)} \,\text{ q}^{-4k_1^2-k_2^2} \nonumber \\&\times \left\{ y^{6(4k_1+k_2)} \biggl [E_2\left( \frac{1}{\sqrt{2}};2\sqrt{\frac{\tau _2}{3}}\,(2k_1-k_2+9\beta ),2\sqrt{\frac{2\tau _2}{3}}(k_1+k_2+9\beta )\right) -s(k_1,k_1+k_2) \right. \nonumber \\&-M\left( 2\sqrt{\frac{2}{3}}\, ,9;k_1+k_2\right) \text{ sgn }(2k_1-k_2)-M\left( 2\sqrt{2} ,6;k_1\right) \text{ sgn }(2k_1+k_2) \biggr ] \nonumber \\&+y^{18k_2} \biggl [-E_2\left( \frac{1}{2\sqrt{2}};2\sqrt{\frac{\tau _2}{3}}\,(2k_1-k_2-18\beta ),2\sqrt{\frac{2\tau _2}{3}}(k_1+k_2+9\beta )\right) -s(k_2-k_1,k_1+k_2) \nonumber \\&+M\left( 2\sqrt{\frac{2}{3}}\, ,9;k_1+k_2\right) \text{ sgn }(2k_1-k_2) -M\left( 2\sqrt{\frac{2}{3}}\, ,9;k_2-k_1\right) \text{ sgn }(2k_1+k_2) \biggr ] \nonumber \\&+y^{-6(4k_1+k_2)} \biggl [E_2\left( \frac{1}{\sqrt{2}};2\sqrt{\tau _2}(k_2-3\beta ),2\sqrt{2\tau _2}\,(k_1-6\beta )\right) -s(k_1,k_1+k_2) \nonumber \\&\left. \quad -M\left( 2\sqrt{\frac{2}{3}}\, ,-9;k_1+k_2\right) \text{ sgn }(2k_1-k_2) -M\left( 2\sqrt{2},-6;k_1\right) \text{ sgn }(k_2) \biggr ]\right\} \nonumber \\+ & {} \frac{1}{8}\sum _{k_1\in \mathbb {Z}+\frac{\mu }{4}}\, \sum _{k_2\in \mathbb {Z}+\frac{\mu }{2}} \, \sum _{k_3\in \mathbb {Z}-\frac{\mu }{4}} \,\text{ q}^{-\left( k_1^2-k_1k_2+k_2^2-k_2k_3+k_3^2\right) }\,y^{6(k_1+k_2+k_3)} \nonumber \\&\times \biggl \{E_3\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{6}}, \frac{1}{2\sqrt{2}} ; \sqrt{\tau _2}(2k_1-k_2+6\beta ),\sqrt{\frac{\tau _2}{3}}\,(3k_1-k_3+18\beta ), 2\sqrt{\frac{2\tau _2}{3}}\,(k_3+9\beta )\right) \nonumber \\&-\text{ sgn }(k_1) \, \text{ sgn }(k_2)\, \text{ sgn }(k_3) -\frac{1}{3}\bigl (\delta _{k_2=k_3=0}\text{ sgn }(k_1) +\delta _{k_1=k_3=0}\text{ sgn }(k_2) +\delta _{k_1=k_2=0}\text{ sgn }(k_3)\bigr ) \nonumber \\&-\left( E_2\left( \frac{1}{\sqrt{2}};\sqrt{\frac{\tau _2}{3}}\,(3k_2-2k_3+18\beta ),2\sqrt{\frac{2\tau _2}{3}}\,(k_3+9\beta )\right) -s(k_2,k_3)\right) \text{ sgn }(2k_1-k_2) \nonumber \\&-\left( E_2\left( \frac{1}{2\sqrt{2}};\sqrt{\frac{\tau _2}{3}}\,(3k_1-k_3+18\beta ),2\sqrt{\frac{2\tau _2}{3}}\,(k_3+9\beta )\right) -s(k_1,k_3)\right) \text{ sgn }(2k_2-k_3-k_1) \nonumber \\&-\left( E_2\left( \frac{1}{\sqrt{2}};\sqrt{\tau _2}(2k_1-k_2+6\beta ),\sqrt{2\tau _2}\,(k_2+12\beta )\right) -s(k_1,k_2)\right) \text{ sgn }(2k_3-k_2) \nonumber \\&-M\left( 2\sqrt{\frac{2}{3}}\, ,9;k_3\right) \biggl [\text{ sgn }(3k_1-k_3)\text{ sgn }(3k_2-2k_3)+\frac{1}{3}\,\delta _{k_1=\frac{k_2}{2}=\frac{k_3}{3}} \nonumber \\&-\text{ sgn }(2q_3-q_1-q_2)\text{ sgn }(q_2-q_1) -\text{ sgn }(q_2+q_3-2q_1)\text{ sgn }(q_3-q_2)\biggr ] \nonumber \\&-M\left( 2\sqrt{\frac{2}{3}}\, ,9;k_1\right) \biggl [\text{ sgn }(3k_2-2k_1)\text{ sgn }(3k_3-k_1)+\frac{1}{3}\,\delta _{\frac{k_1}{3}=\frac{k_2}{2}=k_3} \nonumber \\&-\text{ sgn }(3k_2-2k_1)\text{ sgn }(2k_3-k_2) -\text{ sgn }(3k_3-k_1)\text{ sgn }(2k_2-k_3-k_1)\biggr ] \nonumber \\&+M\left( \sqrt{2},12;k_2\right) \text{ sgn }(2k_1-k_2)\text{ sgn }(2k_3-k_2) \biggr \}, \end{aligned}$$

(F.15)

where we introduced convenient notations

$$\begin{aligned} M(a,b;\ell )= & {} E_1\left( a\sqrt{\tau _2}(\ell +b\beta )\right) -\text{ sgn }(\ell ), \end{aligned}$$

(F.16)

$$\begin{aligned} s(k_1,k_2)= & {} \text{ sgn }(k_1)\,\text{ sgn }(k_2)+\frac{1}{3}\, \delta _{k_1=k_2=0}\, . \end{aligned}$$

(F.17)