Elliptic Loci of SU(3) Vacua

Abstract

The space of vacua of many four-dimensional, ${\mathcal {N}}=2$ supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group SU(2), this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group SU(3) parametrizes an intricate family of genus two curves. We analyse this family using the so-called Rosenhain form for these curves. We demonstrate that two natural one-dimensional subloci of the space of SU(3) vacua, ${\mathcal {E}}_u$ and ${\mathcal {E}}_v$, each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus ${\mathcal {E}}_u$ contains the points where mutually local dyons become massless and is a fundamental domain for a classical congruence subgroup. Moreover, the locus ${\mathcal {E}}_v$ contains the superconformal Argyres–Douglas points and is a fundamental domain for a Fricke group.

Appendices

Automorphic Forms

In this appendix, we collect examples of modular forms that are used in the text above and discuss some general structures related to these. For further reading, see [45, 54, 55, 81,82,83].

1.1 Elliptic Modular Forms

1.1.1 Modular Groups and Fundamental Domains

We first recall the notion of the congruence subgroups $\Gamma _0(n)$ and $\Gamma ^0(n)$ of $SL(2,{\mathbb {Z}})$. They are defined as

$$\begin{aligned} \begin{aligned} \Gamma _0(n) = \left\{ \begin{pmatrix}a&{} \quad b\\ c&{} \quad d\end{pmatrix}\in SL(2,{\mathbb {Z}})\big | \, c\equiv 0 \; \mod n\right\} ,\\ \Gamma ^0(n) = \left\{ \begin{pmatrix}a&{} \quad b\\ c&{} \quad d\end{pmatrix}\in SL(2,{\mathbb {Z}})\big | \, b\equiv 0 \; \mod n\right\} , \end{aligned} \end{aligned}$$

(A.1)

and are related by conjugation with the matrix $\text {diag}(n,1)$. We furthermore define $\Gamma (n)$ as the subgroup of $SL(2,{\mathbb {Z}})\ni A$ with $A\equiv {\mathbb {1}}\mod n$.

The modular groups of n|h-type are defined in the following way [40]. Consider matrices of the form

$$\begin{aligned} \begin{pmatrix}ae&{} \quad b/h\\ cn&{} \quad de\end{pmatrix} \end{aligned}$$

(A.2)

with determinant e, where $a,b,c,d,e,h,n\in {\mathbb {Z}}$, and h is the largest integer for which $h^2|N$ and h|24 with $n=N/h$. These matrices are also referred to as Atkin–Lehner involutions.

In the case that n is a positive integer and h|n, we define $\Gamma _0(n|h)$ as the set of above matrices with $e=1$. For any positive integer e which satisfies e|n/h and $(e,n/eh)=1$ (e is called an exact divisor of n/h), one can include also matrices of the above form with $e>1$, forming a group denoted by $\Gamma _0(n|h)+e$. In fact, this construction works for any choice $\{e_1,e_2,\dots \}$ of exact divisors of n/h, resulting in the group $\Gamma _0(n|h)+e_1,e_2,\dots$. If $h=1$, the |h is omitted in the notation, and in case that all the possible $e_i$ are included, the group is simply denoted by $\Gamma _0(n|h)+$.

In the $\Gamma ^0$ convention, the notation simplifies, since $\Gamma ^0(n|h)=\Gamma ^0(\tfrac{n}{h})$. This can be checked by conjugating (A.2) with $\text {diag}(n,1)$. The extension by non-unity determinant matrices follows by analogy.

A key concept of the theory of modular forms is the fundamental domain. A fundamental domain for a group $\Gamma \subset SL(2,{\mathbb {R}})$ is an open subset ${\mathcal {F}}\subset {\mathbb {H}}$ with the property that no two distinct points of ${\mathcal {F}}$ are equivalent under the action of $\Gamma$ and every point in ${\mathbb {H}}$ is mapped to some point in the closure of ${\mathcal {F}}$ by the action of an element in $\Gamma$. The quotient $\Gamma \backslash {\mathbb {H}}$ can be compactified by adding finitely many points called cusps. Cusps are $\Gamma$-equivalence classes of ${\mathbb {Q}}\cup \{ i \infty \}$. Special points in the fundamental domain are the elliptic fixed points, which are points in ${\mathbb {H}}$ that have a non-trivial $\Gamma$-stabiliser. There, the quotient $\Gamma \backslash {\mathbb {H}}$ becomes singular. Elliptic points can always be mapped to the boundary of the fundamental domain. They furthermore contribute non-trivially to the order of vanishing, which determines the dimension of the spaces of modular forms for fixed weight.

1.1.2 Examples of Modular Forms

The Eisenstein series $E_k:{\mathbb {H}}\rightarrow {\mathbb {C}}$ for even $k\ge 2$ are defined as the q-series

$$\begin{aligned} E_{k}(\tau )=1-\frac{2k}{B_k}\sum _{n=1}^\infty \sigma _{k-1}(n)\,q^n, \quad q=e^{2\pi i \tau }, \end{aligned}$$

(A.3)

with $B_k$ the Bernoulli numbers and $\sigma _k(n)=\sum _{d|n} d^k$ the divisor sum. For $k\ge 4$ even, $E_{k}$ is a modular form of weight k for ${\text {SL}}(2,{\mathbb {Z}})$. With this normalization, the j-invariant can be written as

$$\begin{aligned} j=1728\frac{E_4^3}{E_4^3-E_6^2}. \end{aligned}$$

(A.4)

The Jacobi theta functions $\vartheta _j:{\mathbb {H}}\rightarrow {\mathbb {C}}$, $j=2,3,4$, are defined as

$$\begin{aligned} \begin{aligned}&\vartheta _2(\tau )= \sum _{r\in {\mathbb {Z}}+\frac{1}{2}}q^{r^2/2},\\&\vartheta _3(\tau )= \sum _{n\in {\mathbb {Z}}}q^{n^2/2},\\&\vartheta _4(\tau )= \sum _{n\in {\mathbb {Z}}} (-1)^nq^{n^2/2}, \end{aligned} \end{aligned}$$

(A.5)

with $q=e^{2\pi i\tau }$. These functions transform under the generators T and S of $SL(2,{\mathbb {Z}})$ as

$$\begin{aligned} S:\quad&\vartheta _2(-1/\tau )=\sqrt{-i\tau }\vartheta _4(\tau ),\quad&\vartheta _3(-1/\tau )= \sqrt{-i\tau }\vartheta _3(\tau ),\quad&\vartheta _4(-1/\tau )=\sqrt{-i\tau } \vartheta _2(\tau )\nonumber \\ T:\quad&\vartheta _2(\tau +1)=e^{\frac{\pi i}{4}}\vartheta _2(\tau ),\quad&\vartheta _3(\tau +1)=\vartheta _4(\tau ),&\vartheta _4(\tau +1)=\vartheta _3(\tau ). \end{aligned}$$

(A.6)

Some special values that we use are

$$\begin{aligned} \vartheta _2(i)=\vartheta _4(i)= \root 4 \of {\tfrac{\pi }{2}}/\Gamma (\tfrac{3}{4}), \qquad \vartheta _3(i) = \root 4 \of {\pi }/\Gamma (\tfrac{3}{4}). \end{aligned}$$

(A.7)

The Dedekind eta function $\eta : {\mathbb {H}}\rightarrow {\mathbb {C}}$ is defined as the infinite product

$$\begin{aligned} \eta (\tau )=q^{\frac{1}{24}}\prod _{n=1}^{\infty }(1-q^n), \quad q=e^{2\pi i\tau }. \end{aligned}$$

(A.8)

It transforms under the generators of $SL(2,{\mathbb {Z}})$ as

$$\begin{aligned} \begin{aligned} S: \quad&\eta (-1/\tau )=\sqrt{-i\tau }\, \eta (\tau ),\\ T: \quad&\eta (\tau +1)=e^{\frac{\pi i}{12}}\, \eta (\tau ). \end{aligned} \end{aligned}$$

(A.9)

Quotients of $\eta$ functions are frequently used to generate bases for the spaces of modular forms for congruence subgroups of $SL(2,{\mathbb {Z}})$. We use the following

Theorem 1 [54, 55]: Let $f(\tau )=\prod _{\delta |N}\eta (\delta \tau )^{r_\delta }$ be an eta-quotient with $k=\frac{1}{2}\sum _{\delta |N}r_\delta \in {\mathbb {Z}}$ and $\sum _{\delta |N}\delta r_\delta \equiv \sum _{\delta |N}\frac{N}{\delta } r_\delta \equiv 0\mod 24$. Then, f is a weakly holomorphic modular form for $\Gamma _0(N)$ with weight k. In particular, f transforms as $f(\tau |_\gamma )=\chi (d)(c\tau +d)^k f(\tau )$ under $\gamma =\begin{pmatrix}a&{}b\\ c&{}d\end{pmatrix}\in \Gamma _0(N)$ with character $\chi (d)=\left( \frac{(-1)^k s}{d}\right)$, where $s=\prod _{\delta |N}\delta ^{r_\delta }$.

1.2 Siegel Modular Forms

Ordinary modular forms are constructed by the action of an $SL(2,{\mathbb {Z}})$ Möbius transformation on the upper half-plane ${\mathbb {H}}$. Siegel modular forms [81, 83] generalize this notion by introducing an action of $Sp(2g,{\mathbb {Z}})$ on the so-called Siegel upper half-plane ${\mathbb {H}}_g$, which works for any genus $g\in {\mathbb {N}}$.

Define the Siegel modular group of genus g as

$$\begin{aligned} Sp(2g,{\mathbb {Z}}) = \{M\in \text {Mat}(2g;{\mathbb {Z}})\,|\, M^TJ M =J\} \quad \text {with } J = \left( {\begin{matrix} 0&{} {\mathbb {1}}_g\\ -{\mathbb {1}}_g&{}0\end{matrix}}\right) . \end{aligned}$$

(A.10)

The group $Sp(4,{\mathbb {Z}})$ can be generated [81] by the elements J and $T=\left( {\begin{matrix}{}{} {\mathbb {1}}_g &{} s \\ 0 &{} \mathbb {1}_g \end{matrix}}\right)$ with $s=s^T$. The Siegel upper half-plane

$$\begin{aligned} {\mathbb {H}}_g =\{\Omega \in \text {Mat}(g;{\mathbb {C}})\,|\, \Omega ^T=\Omega , \, \text {Im}\Omega >0\} \end{aligned}$$

(A.11)

consists of complex symmetric $g\times g$ matrices whose (componentwise) imaginary part is positive definite. This generalizes the ordinary upper half-plane ${\mathbb {H}}={\mathbb {H}}_1$. For example, for $g=2$ this means that

$$\begin{aligned} \Omega =\begin{pmatrix}\tau _{11}&{}\tau _{12}\\ \tau _{12}&{}\tau _{22}\end{pmatrix},\quad \text {Im}\tau _{11}>0, \quad \text {Im}\tau _{11}\text {Im}\tau _{22}-(\text {Im}\tau _{12})^2>0. \end{aligned}$$

(A.12)

An element $\gamma =\left( {\begin{matrix} A&{}B\\ C&{}D\end{matrix}}\right) \in Sp(2g,{\mathbb {Z}})$ acts on the Siegel upper half-plane by

$$\begin{aligned} \Omega \longmapsto \gamma (\Omega )=(A\Omega +B)(C\Omega +D)^{-1}. \end{aligned}$$

(A.13)

A (classical) Siegel modular form of weight k and genus g is then a holomorphic function $f:{\mathbb {H}}_g\rightarrow {\mathbb {C}}$ satisfying

$$\begin{aligned} f(\gamma (\Omega ))=\det (C\Omega +D)^kf(\Omega ) \qquad \forall \gamma =\begin{pmatrix}A&{}B\\ C&{}D\end{pmatrix}\in Sp(2g,{\mathbb {Z}}), \end{aligned}$$

(A.14)

where for $g=1$ holomorphicity at $i\infty$ is required in addition.

Theta series provide an explicit class of classical Siegel modular forms. For a, $b \in {\mathbb {Q}}^2$ and $\Omega \in {\mathbb {H}}_2$, define

$$\begin{aligned} \Theta \begin{bmatrix}a\\ b\end{bmatrix}(\Omega ) = \sum _{k\in {\mathbb {Z}}^2}\exp \left( \pi i(k+a)^T\Omega (k+a)+2\pi i(k+a)^T\, b \right) . \end{aligned}$$

(A.15)

We are especially interested in the case where the entries of these column vectors take values in the set $\{0,\frac{1}{2} \}$. The corresponding theta functions are usually referred to as the theta characteristics. We call $\gamma = \left[ {\begin{matrix}a\\ b\end{matrix}}\right]$ an even (odd) characteristic if $4a^T b$ is even (odd). In the case of genus two there are ten even theta constants [82],

$$\begin{aligned} \begin{aligned}&\Theta _{1} = \Theta \begin{bmatrix}0&{}0\\ 0&{}0\end{bmatrix},\quad \Theta _{2} = \Theta \begin{bmatrix}0&{}0\\ \frac{1}{2}&{}\frac{1}{2}\end{bmatrix}, \quad \Theta _{3} = \Theta \begin{bmatrix}0&{}0\\ \frac{1}{2}&{}0\end{bmatrix}, \quad \Theta _{4} = \Theta \begin{bmatrix}0&{}0\\ 0&{}\frac{1}{2}\end{bmatrix}, \quad \Theta _{5} = \Theta \begin{bmatrix}\frac{1}{2}&{}0\\ 0&{}0\end{bmatrix}, \\&\Theta _{6} = \Theta \begin{bmatrix}\frac{1}{2}&{}0\\ 0&{}\frac{1}{2}\end{bmatrix}, \quad \Theta _{7} = \Theta \begin{bmatrix}0&{}\frac{1}{2}\\ 0&{}0\end{bmatrix},\quad \Theta _{8} = \Theta \begin{bmatrix}\frac{1}{2}&{}\frac{1}{2}\\ 0&{}0\end{bmatrix},\quad \Theta _{9} = \Theta \begin{bmatrix}0&{}\frac{1}{2}\\ \frac{1}{2}&{}0\end{bmatrix},\quad \Theta _{10} = \Theta \begin{bmatrix}\frac{1}{2}&{}\frac{1}{2}\\ \frac{1}{2}&{}\frac{1}{2}\end{bmatrix}. \end{aligned} \end{aligned}$$

(A.16)

All even theta constants can be related through algebraic identities to four fundamental ones, $\Theta _1$, $\Theta _2$, $\Theta _3$, $\Theta _4$ [82].

The above theta functions are weight $\frac{1}{2}$ Siegel modular forms for a subgroup of $Sp(4,{\mathbb {Z}})$. Their transformation properties under the Siegel modular group can be found in [83].

Picard–Fuchs Solution

In the limit of large u and small v, reference [12] determines the $a_I$ and $a_{D,I}$ non-perturbatively in terms of the fourth Appell hypergeometric function $F_4(a,b,c,d;x,y)$. For $\sqrt{|x|}+\sqrt{|y|}<1$, this function is given by

$$\begin{aligned} F_4(a,b,c,d;x,y)=\sum _{m,n\ge 0}\frac{(a)_{m+n}\,(b)_{m+n}}{m!\,n!\,(c)_m(d)_n}\,x^m\,y^n, \end{aligned}$$

(B.1)

where $(a)_m=\frac{\Gamma (a+m)}{\Gamma (a)}$ is the Pochhammer symbol. We will also need expansions of $F_4$ for large y, which can be achieved by replacing the sum over n by the hypergeometric series $_2F_1$,

$$\begin{aligned} F_4(a,b,c,d;x,y)=\sum _{m\ge 0}\frac{(a)_{m}\,(b)_{m}}{m!\,(c)_m}\, {_2F_1}(a+m,b+m,d;y)\, x^m. \end{aligned}$$

(B.2)

While analytic continuations are known for $_2F_1$, they are not well established for $F_4$.

1.1 Classical Roots

In order to match the Picard–Fuchs solutions with the periods, we need to expand the periods around the classical solutions in (3.4). We therefore need to find the roots of these two cubics.

The general formula for the roots of a depressed cubic equation, $ax^3+bx+c=0$, is given by

$$\begin{aligned} \xi _k=-\frac{1}{3a}\left( \alpha ^k C+\frac{\Delta _0}{\alpha ^k C}\right) , \quad k\in \{0,1,2 \}, \end{aligned}$$

(B.3)

where $\alpha =e^{2\pi i/3}$, $C^3=\frac{\Delta _1\pm \sqrt{\Delta _1^2-4\Delta _0}}{2}$, $\Delta _0=-3ab$ and $\Delta _1=27a^2c$ [84]. The choice of sign in front of the square root in C is arbitrary, in the sense that it only corresponds to a permutation of the roots.

It is, however, important to fix the ambiguities in taking the square and cubic root. We fix the ambiguity in the square root by the following choice for the branch of the logarithm: for any complex number $z\in {\mathbb {C}}^*$, we set $\log (z)=\log \!|z|+i \mathrm {Arg}(z)$ with $-\pi <\mathrm {Arg(z)}\le \pi$. The ambiguity in the cubic root of a complex number z is fixed by demanding that the real part of $\root 3 \of {z}$ has the largest absolute value among the three solutions to $\rho ^3=z$. Thus, $\root 3 \of {1}=1$ and $\root 3 \of {-1}=-1$. Two of the cube roots of i and $-i$ have equal real parts. We fix the remaining ambiguity by setting $\root 3 \of {i}=e^{\pi i/6}=\frac{\sqrt{3}}{2}+\frac{i}{2}$ and $\root 3 \of {-i}=e^{-\pi i/6}=\frac{\sqrt{3}}{2}-\frac{i}{2}$.

To list the roots of our two equations, we define

$$\begin{aligned} s_\pm (a,b)=\root 3 \of {\frac{b}{2}\pm \sqrt{\frac{b^2}{4}-\frac{a^3}{27}}}. \end{aligned}$$

(B.4)

Using Eq. (B.3), we then find that the roots of (3.4) for $a_1$ are given by

$$\begin{aligned} \begin{aligned} \xi _{1}(u,v)&=s_+(u,v)+s_-(u,v),\\ \xi _{2}(u,v)&=\alpha \,s_+(u, v)+\alpha ^2\,s_-(u,v),\\ \xi _{3}(u,v)&=\alpha ^2\,s_+(u, v)+\alpha \,s_-(u, v),\\ \end{aligned} \end{aligned}$$

(B.5)

and the roots for $a_2$ by $-\xi _j(u,v)$. This gives the $3\times 3=9$ solutions to the equations in (3.4). However, (3.3) is supposed to have only $2\times 3 =6$ solutions. Let us determine the 6 solutions in one of the regimes of interest for SU(3) Yang–Mills theory: we assume u is large and close to the positive axis: $u=\lambda -i\epsilon \lambda$ with $\lambda$ real and very large and $0< \epsilon \ll 1$. Note that in this regime

$$\begin{aligned} \begin{aligned}&s_\pm (u,v)=\root 3 \of {\frac{v}{2}\pm i\sqrt{\frac{u^3}{27}-\frac{v^2}{4}}}. \end{aligned} \end{aligned}$$

(B.6)

Furthermore, $s_+(u,v)\,s_-(u,v)=u/3$ and $s_{-}(u,-v)=e^{-\pi i/3}s_{+}(u,v)=-\alpha s_{+}$(u, v) hold. For $v=0$, we have $s_+(u,0)=e^{\pi i/6} \sqrt{u/3}$ and $s_-(u,0)=e^{-\pi i/6}$$\sqrt{u/3}$, and thus

$$\begin{aligned} \begin{aligned} \xi _{1}(u,0)&=\sqrt{u}, \\ \xi _{2}(u,0)&=-\sqrt{u}, \\ \xi _{3}(u,0)&= 0. \end{aligned} \end{aligned}$$

(B.7)

This demonstrates that the solutions to (3.3) for $(a_1,a_2)$ are given by

$$\begin{aligned} (\xi _1,-\xi _2),\ (\xi _1,-\xi _3),\ (\xi _2,-\xi _1),\ (\xi _2,-\xi _3),\ (\xi _3,-\xi _1),\ (\xi _3,-\xi _2). \end{aligned}$$

(B.8)

1.2 Picard–Fuchs System for Large u

To express $a_I$ and $a_{D,I}$ in terms of u and v, we will start by working in the patch with large u and small v and use the variables $x=\frac{27v^2}{4u^3}$ and $y=\frac{27\Lambda ^6}{4u^3}$. In [12], the authors use the notation $P_3$ for this patch and, similarly, $P_2$ for the patch where v is large and u is small and we will adopt this notation in the following. We have four solutions [12, Eq. (6.1)] to the Picard–Fuchs system [12, Eq. (5.11)] for SU(3),

$$\begin{aligned} \begin{aligned} \omega _1^{P_3}&= \sqrt{3}\, 2^{\frac{2}{3}}\Lambda \,y^{-\frac{1}{6}}\,F_4\!\left( -\tfrac{1}{6},\tfrac{1}{6},\tfrac{1}{2},1;x,y\right) , \\ \omega _2^{P_3}&= \frac{2^{\frac{2}{3}}\Lambda }{3} \sqrt{x}\,y^{-\frac{1}{6}}\,F_4\!\left( \tfrac{1}{3},\tfrac{2}{3},\tfrac{3}{2},1;x,y\right) , \\ \Omega _1^{P_3}&= 36\pi \,e^{-\pi i/6}\, 2^{2/3}\Lambda \,\frac{\Gamma (\tfrac{1}{3})}{\Gamma (\tfrac{1}{6})^2}\,F_4\!\left( -\tfrac{1}{6},-\tfrac{1}{6},\tfrac{1}{2},\tfrac{2}{3};\tfrac{x}{y},\tfrac{1}{y}\right) + \beta _1^{P_3}\,\omega _1^{P_3}, \\ \Omega _2^{P_3}&= - e^{\frac{\pi i}{3}} \frac{2^{\frac{2}{3}}\Lambda }{\sqrt{3}\,2\pi }\,\Gamma (\tfrac{1}{3})^3\,\sqrt{\frac{x}{y}}\,F_4\!\left( \tfrac{1}{3},\tfrac{1}{3},\tfrac{3}{2},\tfrac{2}{3};\tfrac{x}{y},\tfrac{1}{y}\right) +\beta _2^{P_3}\,\omega _2^{P_3}, \end{aligned} \end{aligned}$$

(B.9)

where $\beta _1^{P_3}=(i-\sqrt{3})\pi +4\log (2)+3\log (3)-5$ and $\beta _2^{P_3}=1+(i+\frac{1}{\sqrt{3}})\pi +3\log (3)$.⁴ The $a_I$ and $a_{D,I}$ are linear combinations of these periods found by comparing the expansions of these solutions with the classical and semi-classical solutions in the previous section for large u. Using the classical solutions $(a_1,a_2)=(\xi _1,-\xi _2)$ one finds [12, Eq. 6.4],

$$\begin{aligned} \begin{aligned} a_{D,1}(u,v)&= -\frac{i}{4\pi }(\Omega _1^{P_3}+3\Omega _2^{P_3})-\frac{1}{\pi }(\alpha _1\omega _1^{P_3}-\alpha _2\omega _2^{P_3}) \\&=-\frac{i}{2\pi } \left( \sqrt{u}+\frac{3}{2}\frac{v}{u}\right) \log \! \left( \frac{27\Lambda ^6}{4u^3} \right) -\frac{1}{\pi } \left( \frac{i}{2}+2\alpha _1\right) \sqrt{u}+O(u^{-1}),\\ a_{D,2}(u,v)&=-\frac{i}{4\pi }(\Omega _1^{P_3}-3\Omega _2^{P_3})-\frac{1}{\pi }(\alpha _1\omega _1^{P_3}+\alpha _2\omega _2^{P_3})=a_{D,1}(u,-v)\\ a_1(u,v)&=\frac{1}{2}(\omega _1^{P_3}+\omega _2^{P_3}) \sim \sqrt{u} +\frac{1}{2} \frac{v}{u}+\dots ,\\ a_2(u,v)&=\frac{1}{2}(\omega _1^{P_3}-\omega _2^{P_3}) \sim \sqrt{u} -\frac{1}{2} \frac{v}{u}+\dots , \end{aligned} \end{aligned}$$

(B.10)

with $\alpha _1 = \frac{5i}{4}-i\log (2)-\frac{3i}{4}\log (3)$ and $\alpha _2=\frac{3i}{4}+\frac{9i}{4}\log (3)$. The chain rule then allows to compute the coupling matrix,

$$\begin{aligned} \Omega (u,v)=\begin{pmatrix}\partial _ua_1 &{} \partial _u a_2\\ \partial _va_1&{}\partial _va_2\end{pmatrix}^{-1}\, \begin{pmatrix}\partial _ua_{D,1}&{}\partial _ua_{D,2}\\ \partial _va_{D,1}&{}\partial _va_{D,2}\end{pmatrix}. \end{aligned}$$

(B.11)

1.3 Picard–Fuchs System for Large v

We can run a similar analysis as in the previous section for the patch $P_2$, i.e. for large v and small u. This is not done explicitly in [12], but the authors hint at how it should be done. Here, we use the variables $x=\frac{4u^3}{27v^2}$ and $y=\frac{\Lambda ^6}{v^2}$ to express the solutions of the Picard–Fuchs equations as

$$\begin{aligned} \begin{aligned} \omega _1^{P_2}&=2y^{-1/6}F_4\left( -\frac{1}{6},\frac{1}{3},\frac{2}{3},1;x,y\right) ,\\ \omega _2^{P_2}&=2^{1/3}x^{1/3}y^{-1/6}F_4\left( \frac{1}{6},\frac{2}{3}, \frac{4}{3},1;x,y\right) ,\\ \Omega ^{P_2}_1&= -\frac{\alpha ^2}{2}\pi ^{-3/2}\Gamma \left( -\tfrac{1}{6}\right) \Gamma \left( \tfrac{2}{3}\right) F_4\left( -\tfrac{1}{6},-\tfrac{1}{6},\tfrac{2}{3}, \tfrac{1}{2};\tfrac{x}{y},\tfrac{1}{y}\right) +\beta _1^{P_2}\omega _1^{P_2},\\ \Omega _2^{P_2}&= -\frac{\alpha }{3}\pi ^{-3/2}\root 3 \of {\tfrac{x}{y}}\Gamma \left( -\tfrac{2}{3}\right) \Gamma \left( \tfrac{1}{6}\right) F_4\left( \tfrac{1}{6},\tfrac{1}{6},\tfrac{4}{3},\tfrac{1}{2};\tfrac{x}{y},\tfrac{1}{y}\right) +\beta _2^{P_2}\omega _2^{P_2}, \end{aligned} \end{aligned}$$

(B.12)

with

$$\begin{aligned} \begin{aligned} \beta _1^{P_2}&=-\frac{i}{4\pi }\left( 2\log 2+3\log 3-6+\pi (i-2/\sqrt{3})\right) , \\ \beta _2^{P_2}&=-\frac{i}{2^{4/3}\pi }\left( 2\log 2+3\log 3+\pi (i+2/\sqrt{3})\right) . \end{aligned} \end{aligned}$$

(B.13)

Comparing the expansions of these solutions with the asymptotic expansions of $a_{(D),I}$ for the semi-classical contributions fixes the coefficients. For this, one needs to match the $F_4$ expansions with the leading coefficients of the (differentiated) prepotential [10]

$$\begin{aligned} {\mathcal {F}}= \frac{\tau _0}{6}\sum _{i=1}^{3}Z_i^2+{\mathcal {F}}_{1-\text {loop}}+{\mathcal {F}}_{inst.}, \end{aligned}$$

(B.14)

where⁵

$$\begin{aligned} \tau _0 =\frac{9-\log 4}{2\pi i}. \end{aligned}$$

(B.15)

From this, one finds

$$\begin{aligned} \begin{aligned} a_{D,1}&= -i\sqrt{3}\alpha \left( \Omega _1^{P_2}-2^{-2/3}\alpha \Omega _2^{P_2}\right) +\left( \alpha c_1-\tfrac{i\sqrt{3}}{2}\right) \omega _1^{P_2} +\left( \alpha ^2c_2+\tfrac{i\sqrt{3}}{2}\right) \omega _2^{P_2}, \\ a_{D,2}&=-i\sqrt{3}\left( \Omega _1^{P_2}-2^{-2/3}\Omega _2^{P_2}\right) +\left( c_1+\tfrac{i\sqrt{3}}{2}\right) \omega _1^{P_2} +\left( c_2-\tfrac{i\sqrt{3}}{2}\right) \omega _2^{P_2}, \\ a_1&= \frac{1}{2}\left( \omega ^{P_2}_1+\omega ^{P_2}_2\right) , \\ a_2&= -\frac{\alpha }{2}\left( \omega ^{P_2}_1+\alpha \omega ^{P_2}_2\right) , \end{aligned} \end{aligned}$$

(B.16)

where $c_1 = \frac{\sqrt{3}}{4\pi }\left( 2\log 2+3\log 3+\frac{\pi }{\sqrt{3}}-6\right)$ and $c_2=-\frac{\sqrt{3}}{4\pi }\left( 2\log 2+3\log 3 \right.$$\left. -\frac{\pi }{\sqrt{3}}\right)$. We note that for $u=0$, we find $a_2=-\alpha a_1$.

1.4 The ${\mathbb {Z}}_2$ Vacua and Massless States

In deriving the above results for the large v regime, we have used a different symplectic basis than what is used in for example [12, 77]. In this subsection, we briefly comment on how the two bases relate. The basis chosen in [12, 77] is more natural to use when comparing such quantities as the strong-coupling periods for the two different loci, and in this basis, we also compute the periods for all the points of interest. The change of basis is done by interchanging the roots $\xi _2\leftrightarrow \xi _3$ as given in (B.5). Quantum mechanically, the singular branch of the classical theory splits into two branches separated by the scale $\Lambda$. Therefore, we must also interchange $r_2\leftrightarrow r_3$ and $r_5\leftrightarrow r_6$. One finds that this symplectic change of basis is given by the semi-classical version of the second Weyl reflection of the $A_2$ root lattice,

$$\begin{aligned} {\mathcal {R}}_2=\begin{pmatrix}1&{} \quad 1&{} \quad 0&{} \quad 0\\ 0&{} \quad -1&{} \quad 0&{} \quad 0\\ 0&{} \quad 0&{} \quad 1&{} \quad 0\\ 0&{} \quad 0&{} \quad 1&{} \quad -1\end{pmatrix} \in Sp(4,{\mathbb {Z}}). \end{aligned}$$

(B.17)

This merely changes some prefactors of the solution (B.16). The change of roots modifies the cross-ratios in a trivial way, and they agree asymptotically with the theta quotients (3.19) computed from the new periods, as expected. One can show that the algebraic relations (5.3) for $u=0$ take the same form. However, on this locus we now find

$$\begin{aligned} \tau _{12} = \frac{1-\tau _{11}}{2}, \qquad \tau _{22}=\tau _{11}-2, \end{aligned}$$

(B.18)

from which it follows that

$$\begin{aligned} \begin{aligned} 2i\sqrt{27}\,v&=-\alpha ^2q^{-\frac{1}{6}}+33\alpha q^{\frac{1}{6}}+153 q^{\frac{1}{2}}+713\alpha ^2q^{\frac{5}{6}}+{\mathcal {O}}(q^{\frac{7}{6}})\\&=m\left( -\alpha q^{\frac{1}{6}}\right) =m\left( \tfrac{\tau }{6}-\tfrac{1}{6}\right) , \end{aligned} \end{aligned}$$

(B.19)

which is identical to (5.8) up to phases.

We can use the new solution to analyse the ${\mathbb {Z}}_3$ symmetry $u\mapsto \alpha u$. This leads to the matrix

$$\begin{aligned} {\tilde{\sigma }}_v = \alpha ^2\begin{pmatrix} 0&{} \quad 1&{} \quad -1&{} \quad 2\\ -1&{} \quad -1&{} \quad 2&{} \quad -1\\ 0&{} \quad 0&{} \quad -1&{} \quad 1\\ 0&{} \quad 0&{} \quad -1&{} \quad 0 \end{pmatrix}. \end{aligned}$$

(B.20)

It can also be obtained from the previous result (7.8) by conjugation with ${\mathcal {R}}_2$. It satisfies ${\tilde{\sigma }}_v^3=\mathbb {1}$, and we can use it to generate the charges of the states that become massless at the ${\mathbb {Z}}_2$ points. To this end, we introduce the purely integral matrix $U=\alpha ^2{\tilde{\sigma }}_v^{-1}\in Sp(4,{\mathbb {Z}})$, which is the matrix used in [12, 77], and act with this on the monopole basis,

$$\begin{aligned} {\begin{array}{ll} {\tilde{\nu }}_1=(1,0,0,0), &{}\qquad {\tilde{\nu }}_2 =\, (0,1,0,0), \\ {\tilde{\nu }}_3 ={\tilde{\nu }}_1 U= (-1,-1,1,-2), &{}\qquad {\tilde{\nu }}_4 ={\tilde{\nu }}_2 U= \,(1,0,-2,1), \\ {\tilde{\nu }}_5 ={\tilde{\nu }}_1 U^{-1}= (0,1,-1,2), &{}\qquad {\tilde{\nu }}_6 ={\tilde{\nu }}_2 U^{-1}= \,(-1,-1,2,-1). \end{array}} \end{aligned}$$

(B.21)

Using the periods from Table 1, we can confirm that ${\tilde{\nu }}_{\{1,3,5\}}$ become massless at the AD point (0, 1) and ${\tilde{\nu }}_{\{2,3,6\}}$ at the AD point $(0,-1)$. Furthermore, the charges in row $k+1$ in (B.21) become massless at the ${\mathbb {Z}}_2$ point $({\underline{u}},v)=(\alpha ^{k},0)$. It can be checked that the charges in each row are mutually local with respect to the symplectic inner product induced by J, given in (A.10). The charges in both columns, however, are mutually non-local. This is a crucial observation that leads to the discovery of new superconformal theories [22,23,24].

Table 1 Periods at the ${\mathbb {Z}}_3$, ${\mathbb {Z}}_2$ points and the origin, computed from the analytic continuation of the large v PF solution and appropriately normalized

The matrix (B.20) conjugates the strong-coupling matrices [12] as well as the semi-classical matrices according to

$$\begin{aligned} {\tilde{\sigma }}_v^{-1} M^{(r_1)}{\tilde{\sigma }}_v=M^{(r_2)}, \quad {\tilde{\sigma }}_v^{-1} M^{(r_2)}{\tilde{\sigma }}_v=M^{(r_3)},\quad {\tilde{\sigma }}_v^{-1} M^{(r_3)}{\tilde{\sigma }}_v=M^{(r_1)}.\nonumber \\ \end{aligned}$$

(B.22)

The same equations hold for the ${\mathbb {Z}}_2$ symmetry

$$\begin{aligned} {\tilde{\rho }}_v=\left( \begin{array}{cccc} 1 &{} \quad 1 &{} \quad -2 &{} \quad 1 \\ -1 &{} \quad 0 &{} \quad 4 &{} \quad -2 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{} \quad -1 &{} \quad 1 \\ \end{array} \right) , \end{aligned}$$

(B.23)

as is also the case for large u. As a consistency check, the pair $({\tilde{\sigma }}_v,{\tilde{\rho }}_v)$ again satisfies the relation (7.7), and ${\tilde{\rho }}_v^2$ is a non-trivial monodromy. The matrix ${\tilde{\rho }}_v$ maps $\{{\tilde{\nu }}_2,{\tilde{\nu }}_4,{\tilde{\nu }}_6\}$ to $\{-{\tilde{\nu }}_1,-{\tilde{\nu }}_3,-{\tilde{\nu }}_5\}$ and therefore exchanges the AD points $v=\pm 1$.

The periods in Table 1 obtain different values depending on the direction from which the various points are approached.⁶ On the locus ${\mathcal {E}}_u$, where $v=0$, we have three singularities located at ${\underline{u}}=1,\, \alpha ,\, \alpha ^2$. Reference [61] argues that one finds consistent values if the points are approached from the negative real axis. In this way, we can go from weak to strong coupling without crossing walls of the second kind.⁷ On ${\mathcal {E}}_v$, with $u=0$, we instead have two singularities on the real line at $v=\pm 1$, analogous to the u-plane in the SU(2) theory. There, we find a consistent picture by taking the limits from the lower half-plane in order to avoid the singular points (see discussion in [41]).

The two patches with large u and large v (from this subsection), respectively, are connected by a simple change of basis. It is given by

$$\begin{aligned} {\mathcal {M}}= {\mathcal {M}}_{{\tilde{\nu }}_2}=\left( \begin{array}{cccc} 1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad -1 &{} \quad 0 &{} \quad 1 \\ \end{array} \right) . \end{aligned}$$

(B.24)

This matrix is the strong-coupling monodromy (7.12) associated with the magnetic monopole ${\tilde{\nu }}_2=(0,1,0,0)$.

1.5 The ${\mathbb {Z}}_3$ Vacua

With the explicit result (7.11) for the coupling matrix, the charges of the massless states (B.21) and the periods from Table 1 we can revisit the results of [22]. Starting from the three states ${\tilde{\nu }}_{\{1,3,5\}}$ which become massless at $({\underline{u}},v)=(0,1)$, we aim to find a symplectic projection such that the massless states are charged only under the first U(1) factor. Following the logic of [22, 77], in this basis the coupling matrix becomes diagonal ($\tau _{12}=0$) and the curve splits into a small and a large torus, parametrized by $\tau _{11}$ and $\tau _{22}$, respectively. The modulus of the large torus is fixed by the ${\mathbb {Z}}_3$ symmetry to be $\tau _{22}=-\alpha ^2$. The small torus $\tau _{11}=\tau (\rho )$ depends on the direction $\rho$ from which the AD point is approached, where $\delta v = 2\varepsilon ^3$, $\delta u=3\varepsilon ^3 \rho$. The small torus near the ${\mathbb {Z}}_3$ point takes the form $w^2=z^3-3\rho z-2$. This curve degenerates if $\rho ^3=1$, has a ${\mathbb {Z}}_2$ symmetry at $\rho ^3=\infty$ and a ${\mathbb {Z}}_3$ symmetry at $\rho ^3=0$.

If we approach the AD point from the $\rho =0$ plane, we find that $\tau _{11}=\alpha$. By an $Sp(4,{\mathbb {Z}})$ transformation, we can go to a basis where the mutually non-local states ${\tilde{\nu }}_1$, ${\tilde{\nu }}_3$ and ${\tilde{\nu }}_5$ are mapped to an electron, a monopole and a dyon, all charged with respect to the first U(1) factor only. This is done, for example, by the transformation

$$\begin{aligned} {\mathcal {A}}= \left( \begin{array}{cccc} -1 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 1 &{} \quad 0 &{} \quad 0 &{} \quad -1 \\ 1 &{} \quad 1 &{} \quad -1 &{} \quad 2 \\ 1 &{} \quad 1 &{} \quad 0 &{} \quad 1 \\ \end{array} \right) \in Sp(4,{\mathbb {Z}}). \end{aligned}$$

(B.25)

This furthermore diagonalizes the coupling matrix

$$\begin{aligned} {\mathcal {A}}: \Omega (0,1)\mapsto \begin{pmatrix} \alpha &{} 0\\ 0 &{} -\alpha ^2 \end{pmatrix}, \end{aligned}$$

(B.26)

as anticipated. The periods $\pi (0,1)=(0,*,0,*)$ depend on the exact transformation, but the relations $a_1=\alpha ^2 a_{D,1}\rightarrow 0$ and $a_2=-\alpha ^2 a_{D,2}$ are fixed.

Proofs of Modular Identities

In this section, we collect some rigorous proofs of exact statements made in the sections above.

1.1 The Origin of the Moduli Space

The zeros of $u(\tau )$ for both the SU(2) and SU(3) theory can be derived from the properties of the Jacobi theta functions.

1.1.1 The SU(2) Theory

The moduli space of the pure SU(2) theory is parametrized by the modular function

$$\begin{aligned} u(\tau ) = \frac{\vartheta _2(\tau )^4+\vartheta _3(\tau )^4}{2\vartheta _2(\tau )^2 \vartheta _3(\tau )^2} = 1+\frac{1}{8}\left( \frac{\eta (\tfrac{\tau }{4})}{\eta (\tau )}\right) ^8. \end{aligned}$$

(C.1)

The Jacobi theta functions $\vartheta _j$ and their transformation properties are given in “Appendix A.1”. The zeros of u are given by the $\Gamma ^0(4)$-orbit of $1+i$. To prove this, it suffices to observe that

$$\begin{aligned} \vartheta _2(1+i)^4+\vartheta _3(1+i)^4= \left( e^{\frac{\pi i}{4}}\vartheta _2(i)\right) ^4+\vartheta _4(i)^4=- \vartheta _2(i)^4+\vartheta _2(i)^4=0,\nonumber \\ \end{aligned}$$

(C.2)

where we have used the T-transformation in the first equation and the S-transformation of $\vartheta _4$ in the second equation. Using the result (A.7), we know that the denominator is nonzero. Therefore, we have proven that $u(1+i)=0$.

1.1.2 The SU(3) Theory

Let us prove that (4.21) is a root of (4.13). Notice that

$$\begin{aligned} b_{3,0}(\tau ) = \vartheta _3(2\tau )\vartheta _3(6\tau )+\vartheta _2(2\tau )\vartheta _2(6\tau ). \end{aligned}$$

(C.3)

Without computing any of these sums, we can simplify the terms in $b_{3,0}\left( \frac{\tau _0}{3}\right)$ by making use of the transformation identities in Sect. A.1,

$$\begin{aligned} \begin{aligned} \vartheta _3(1+\tfrac{i}{\sqrt{3}})&= \vartheta _4(\tfrac{i}{\sqrt{3}}), \\ \vartheta _3(3+\sqrt{3} i)&= \vartheta _4(\sqrt{3} i), \\ \vartheta _2(1+\tfrac{i}{\sqrt{3}})&= e^{\frac{\pi i}{4}}\vartheta _2(\tfrac{i}{\sqrt{3}}) =\root 4 \of {3}\, e^{\frac{\pi i}{4}}\vartheta _4(\sqrt{3} i)\\ \vartheta _2(3+\sqrt{3} i)&= e^{\frac{3\pi i}{4}}\vartheta _2(\sqrt{3} i)=\tfrac{1}{\root 4 \of {3}}\,e^{\frac{3\pi i}{4}} \vartheta _4(\tfrac{i}{\sqrt{3}}). \end{aligned} \end{aligned}$$

(C.4)

We thus find that $b_{3,0}(\frac{\tau _0}{3})=b_{3,0}(\frac{1}{2}+\frac{i}{2\sqrt{3}})=0$. The denominator

$$\begin{aligned} b_{3,1}(\tau ) = 3\, \frac{\eta (3\tau )^3}{\eta (\tau )} \end{aligned}$$

(C.5)

vanishes nowhere on ${\mathbb {H}}$, as $\eta ^{24}(\tau )=\Delta (\tau )$ is a holomorphic cusp form of weight 12 for $SL(2,{\mathbb {Z}})$. This proves that indeed $u(\tau _0)=0$.

1.2 The Function v

Since on the locus ${\mathcal {E}}_v$ the relations (5.5) among the $\tau _{IJ}$ are exact, it is possible to prove the step from (5.6) to (5.11) by computing the theta constants analytically instead of perturbatively (as done on ${\mathcal {E}}_u$). First, note that $C_1=\lambda _3=\tfrac{\Theta _8^2}{\Theta _{10}^2}$, since $\Theta _1=\Theta _2$ due to (5.5). This lets us simplify,

$$\begin{aligned} v=- \frac{i}{\sqrt{27}}\frac{(\Theta _8^2-2\Theta _{10}^2)(\Theta _8^2 +\Theta _{10}^2)(2\Theta _8^2-\Theta _{10}^2)}{\Theta _8^2 \Theta _{10}^2(\Theta _8^2-\Theta _{10}^2)}. \end{aligned}$$

(C.6)

Both sides of this equation are functions of $\Omega =\left( {\begin{matrix} \tau _{11} &{} -\tau _{11}/2+1 \\ \ -\tau _{11}/2+1 &{} \tau _{11}-1 \end{matrix}}\right)$. We have that $\tau _-=\frac{3}{2}\tau _{11}-1$ and therefore $\tau =\tau _-+1=\frac{3}{2}\tau _{11}$, as defined in Sect. 5.3. In view of the claim $v\propto m(\tfrac{\tau }{6})$, let us further define $\sigma =\tfrac{\tau }{6}=\frac{\tau _{11}}{4}$ to obtain integral powers of $\mathtt {q}:=e^{2\pi i\sigma }$. This allows to compute the theta constants,

$$\begin{aligned} \begin{aligned} \Theta _8(\Omega )&=e^{\frac{\pi i}{4}}\sum _{k,l\in {\mathbb {Z}}+\frac{1}{2}}(-1)^{k+l} \mathtt {q}^{2(k^2+k l+l^2)},\\ \Theta _{10}(\Omega )&=e^{\frac{\pi i}{4}}\sum _{k,l\in {\mathbb {Z}}+\frac{1}{2}}(-1)^{2k}\mathtt {q}^{2(k^2+k l+l^2)}. \end{aligned} \end{aligned}$$

(C.7)

The $\mathtt {q}$-series on the rhs should be interpreted as functions of $\sigma (\tau _{11})=\frac{\tau _{11}}{4}$; however, it is convenient to consider them as functions of the new variable $\sigma \in {\mathbb {H}}$. On ${\mathcal {E}}_v$, the theta constants $\Theta _8$ and $\Theta _{10}$ collapse to shifted theta functions of the $A_2$ root lattice, as defined in (4.14). In order to see this, define

$$\begin{aligned} f(\tau )=\frac{1}{3}(b_{3,0}(\tau )-b_{3,0}(4\tau ))=\frac{2\eta (4\tau )^2\, \eta (12\tau )^2}{\eta (2\tau )\,\eta (6\tau )}=2(q+q^3+2\,q^7+\dots ).\nonumber \\ \end{aligned}$$

(C.8)

According to Theorem 1 in “Appendix A.1”, f is a modular form of weight 1 for $\Gamma _0(12)$. By splitting the $b_{3,0}$ theta functions into even and odd exponents, it can be easily shown that

$$\begin{aligned} \Theta _{8}(\sigma )=-ie^{\frac{\pi i}{4}} f(\tfrac{\sigma }{2}+\tfrac{1}{4}), \quad \Theta _{10}(\sigma )=-e^{\frac{\pi i}{4}}f(\tfrac{\sigma }{2}), \end{aligned}$$

(C.9)

with the abuse of notation $\Theta _j(\Omega )=\Theta _j(\sigma )$ with $\Omega$ and $\sigma$ as below (C.6). By modularity, it is enough to compare a finite number of coefficients between (C.6) and (5.10), which proves $v=-\tfrac{i}{ 2\sqrt{27}}\, m(\sigma )$.

1.2.1 Special Points of v

The solutions to $v=1$ and $v=-1$ are not straightforward to obtain. Let us start with the point $({\underline{u}},v)=(0,-1)$. In the following, all arguments are those of m. Due to the prefactor in (5.11), $v=-1$ is in fact a quadratic equation with zero discriminant and therefore satisfied if and only if

$$\begin{aligned} \left( \frac{\eta (2 \tau )}{\eta (6 \tau )}\right) ^6 = -\sqrt{27}\, i. \end{aligned}$$

(C.10)

A solution to this equation can be found to be

$$\begin{aligned} \tau _{-1}= \frac{\omega }{2\sqrt{3}}=\frac{1}{4}+\frac{i}{4\sqrt{3}}=\frac{\tau _{\text {AD},2}}{6} \end{aligned}$$

(C.11)

with $\omega = e^{\pi i/6}$ as before and $\tau _{\text {AD},2}$ the argument of v in (5.12). The other AD point can be found using the symmetry of m, and it is given by

$$\begin{aligned} \tau _{+1}= \frac{\omega ^5}{2\sqrt{3}}=-\frac{1}{4}+\frac{i}{4\sqrt{3}}=\frac{\tau _{\text {AD},1}}{6}. \end{aligned}$$

(C.12)

The zero of m (and therefore of v) is given by

$$\begin{aligned} \tau _0 = \frac{i}{2\sqrt{3}}. \end{aligned}$$

(C.13)

Note that all these numbers have the same absolute value $\frac{1}{2\sqrt{3}}$.

Let us prove (C.11) first: in order to compute both the numerator and the denominator, we can resort to the S- and T-transformations of $\eta$ as given in A.1,

$$\begin{aligned} \begin{aligned} \eta (2\tau _{-1})&\overset{S}{=} 3^{\frac{1}{4}}\omega e^{-\frac{\pi i}{12}}\eta (-\tfrac{1}{2}+\tfrac{\sqrt{3}}{2}i) \overset{T}{=} 3^{\frac{1}{4}} e^{\frac{\pi i}{12}}\eta (\alpha )\\ \eta (6\tau _{-1})&=\eta (\tfrac{3}{2}+\tfrac{\sqrt{3}}{2}i) \overset{T}{=} e^{\frac{\pi i}{6}}\eta (\alpha ). \end{aligned} \end{aligned}$$

(C.14)

Equation (C.10) follows immediately.

In order to find the point where $v=+1$, we can make the observation that $m(-\frac{1}{\tau })=-m(\frac{\tau }{12})$. This implies that under the Fricke involution $\left( {\begin{matrix}0&{}-1\\ 12&{}0\end{matrix}}\right)$, the solution receives a minus sign,

$$\begin{aligned} m\left( -\frac{1}{12\tau }\right) =-\,m(\tau ). \end{aligned}$$

(C.15)

Using the T-transformation of $\eta$, one also finds that $m\left( \tau \pm \frac{1}{2}\right) =-m(\tau )$. We can use either of those maps, $\tau _{+1}=\tau _{-1}-\tfrac{1}{2}=-\tfrac{1}{12\tau _{-1}}$ to obtain (C.12).

We can also study the zeros of v. Every root of $m(\tau )$ is given by the equation $\eta (2\tau )^{12}=27\, \eta (6\tau )^{12}$. A solution to this equation is (C.13), which we can prove: using the S-transformation, we find

$$\begin{aligned} \eta (2\tau _0) = \eta (\tfrac{i}{\sqrt{3}})=3^{\frac{1}{4}}\eta (\sqrt{3} i)=3^{\frac{1}{4}}\eta (6\tau _0). \end{aligned}$$

(C.16)

The result follows immediately. Another proof follows simply from the fact that $\tau _0$ is the fixed point under (C.15).