\subsection{Effective low-energy Hamiltonian of multiterminal Josephson junctions}
We consider $n \in \mathbb{N}$ superconducting (SC) terminals connected to a normal conducting region [see Fig.~\ref{fig:sfig1}] that consists of a single-level, noninteracting quantum dot described by the Hamiltonian $H_\mathrm{D} =  \varepsilon_0 \sum_{\sigma} d_\sigma^\dag d_\sigma^{\phantom\dag}$, where $d_\sigma^\dag$ creates an electron in the dot level with spin $\sigma = \uparrow,\downarrow$ at energy $\varepsilon_0$. 
In the absence of microwave drive, the Hamiltonian of the $n$-terminal junction reads 
\begin{align}
H &=  H_\mathrm{D} + \sum_{j=1}^n \left( H_{\mathrm{S}}^{(j)} +   H_\mathrm{S-D}^{(j)} +  H_\mathrm{S-S}^{(j,j+1)} \right) ,
\end{align}
with
\begin{subequations}
	\begin{align}
	H_{\mathrm{S}}^{(j)} &= \sum\nolimits_{\boldsymbol{k}\sigma} \xi_{\boldsymbol{k}} \,  c_{j \boldsymbol{k} \sigma}^{\dag} c_{j \boldsymbol{k} \sigma}^{\phantom\dag}  
	\nonumber \\
	&\quad + \Delta  \sum\nolimits_{\boldsymbol{k}}  \Bigl[ \, e^{i \varphi_j} c_{j \boldsymbol{k}\uparrow}^\dag c_{j (-\boldsymbol{k})\downarrow}^{\dag} + \mathrm{H.c.} \Bigr] ,
	\\
	H_\mathrm{S-D}^{(j)} &= \sum\nolimits_{\boldsymbol{k}\sigma} w \Bigl[ c_{j \boldsymbol{k}\sigma}^\dag d_{ \sigma}^{\phantom\dag} + d_{ \sigma}^{\dag} 
	c_{j \boldsymbol{k} \sigma}^{\phantom\dag} \Bigl] ,
	\\
	H_\mathrm{S-S}^{(j,j+1)} &= 
	\sum\nolimits_{\boldsymbol{k} \sigma} t \Bigl[ e^{-i\alpha} c_{j \boldsymbol{k} \sigma}^\dag c_{(j+1) \boldsymbol{k} \sigma}^{\phantom\dag} + \mathrm{H.c.} \Bigl],
	\end{align}
\end{subequations}
where $H_{\mathrm{S}}^{(j)}$ is the Hamiltonian of the $j$-th SC lead, $H_\mathrm{S-D}^{(j)}$ is the coupling between the $j$-th SC lead and the dot with coupling strength $w$ and $H_\mathrm{S-S}^{(j,j+1)}$ describes the coupling between neighboring SC leads (periodic: $j = n 
\Rightarrow j+1 = 1$) which is assumed to be weak, i.e., $t \ll w$.
We furthermore include the effect of a magnetic flux $\Phi = n\alpha\Phi_0$ in the normal region with $\Phi_0 = \hbar / 2e$ being the flux quantum leading to a hopping phase $\alpha$ between two neighboring leads.
Moreover, $c_{j \boldsymbol{k} \sigma}^\dag$ creates an electronic state in lead 
$j$ with quasi-momentum $\boldsymbol{k}$ and spin $\sigma = \uparrow,\downarrow$ at energy $\xi_{\boldsymbol{k}} = \hbar^2\boldsymbol{k}^2 / 2 m_\mathrm{e} - \mu$, where $m_\mathrm{e}$ is the mass of the electrons and $\mathrm{H.c.}$ denotes the Hermitian conjugate.
We assume that all leads have the same absolute value of the SC gap $\Delta$, the same chemical potential $\mu$, but they differ in the SC phase denoted by $\varphi_j$. 
By introducing spinors in Nambu space $\Psi_{j \boldsymbol{k}}^{\dag} = (c_{j \boldsymbol{k} \uparrow}^\dag, c_{j (-\boldsymbol{k}) \downarrow})$ for the leads ($j = 1,\ldots,n$) and $\Psi_\mathrm{D}^{\dag} = (d_\uparrow^\dag, d_\downarrow^{\phantom\dag})$, we rewrite the unperturbed Hamiltonians as $H_{\mathrm{S}}^{(j)} = \sum_{\boldsymbol{k}} \Psi_{j \boldsymbol{k}}^{\dag} \hat{H}_{\mathrm{S},j\boldsymbol{k}} \Psi_{j \boldsymbol{k}}^{\phantom\dag}$ and $H_{\mathrm{D}}= \Psi_\mathrm{D}^{\dag} \hat{H}_\mathrm{D} \Psi_\mathrm{D}^{\phantom\dag}$, respectively, with $\hat{H}_{\mathrm{S},j\boldsymbol{k}} = \xi_{\boldsymbol{k}} \tau_3 + \Delta e^{i \varphi_j \tau_3} \tau_1$ and $\hat{H}_\mathrm{D} = \varepsilon_0 \tau_3$ and the Pauli matrices  $\tau_1,\tau_2$ and $\tau_3$ in Nambu space. 
Note that due to isotropic symmetry $\xi_{\boldsymbol{k}} = \xi_{-\boldsymbol{k}}$ holds, which we have already used.
\begin{figure}
	\includegraphics[width=0.7\columnwidth]{Figure4}
	\caption{Microscopic model of the $n$-terminal junction. $n$ superconducting leads, each with a superconducting phase $\varphi_j$, are coupled to a normal dot with energy level $\varepsilon_0$ via the couplings $w$. The normal region is subjected a magnetic flux $\Phi = n\alpha \Phi_0$ (flux quantum $\Phi_0 = \hbar / 2 e$) such that the couplings between the superconductors $t$ are modified by a phase factor $e^{i \alpha}$.}
	\label{fig:sfig1}
\end{figure}

From $\hat{g}_{j\boldsymbol{k}} = (\varepsilon - \hat{H}_{\mathrm{S},j\boldsymbol{k}})^{-1}$, we obtain the bare Green's function (GF) of the $j$-th lead
\begin{align}
\hat{g}_{j\boldsymbol{k}} = \frac{\varepsilon - \xi_{\boldsymbol{k}} \tau_3 + \Delta e^{i \varphi_j \tau_3} \tau_1}{\varepsilon^2 - \xi_{\boldsymbol{k}}^2 - \Delta^2}.
\end{align}
Since the tunneling matrix elements $t$ and $w$ do not depend on quasi-momentum $\boldsymbol{k}$, we can already sum over all quasi-momenta to obtain the new GF of the lead $\hat{g}_{j} = \sum\nolimits_{\boldsymbol{k}} \hat{g}_{j\boldsymbol{k}}$. 
Turning the summation over $\boldsymbol{k}$ into an integration over energies in the wide-band limit via $\sum_{\boldsymbol{k}} \to N_0 \int \mathrm{d}\xi_{\boldsymbol{k}}$, where $N_0$ is the normal state density of states at the Fermi energy, we obtain
\begin{align}
\hat{g}_{j} &= - \pi N_0 \frac{\varepsilon  + \Delta e^{i \varphi_j \tau_3} \tau_1}{ \sqrt{\Delta^2 - \varepsilon^2 } } \, \stackrel{\Delta \to \infty}{\longrightarrow} \, - \pi N_0  e^{i \varphi_j \tau_3} \tau_1 
\end{align}
defined in the local basis $\Psi_{j}^{\dag} = (c_{j  \uparrow}^\dag, c_{j \downarrow})$, where we assume $\Delta$ to be larger than all relevant energies in the system. 
The GF of the bare dot is given by $\hat{g}_\mathrm{D} = 1 / (\varepsilon - \varepsilon_0 \tau_3)$. 
From the Hamiltonian describing the hopping between the $j$-th lead and the dot we obtain $\hat{V}_{j\mathrm{D}} = w \tau_3$, while we obtain for the coupling between the $j$-th and $(j+1)$-th lead $\hat{V}_{j(j+1)} = t \tau_3 e^{i \alpha \tau_3}$ (periodic: $j = n$ $\Rightarrow$ $j+1 = 1$).
By means of Dyson's equation $G = g + g V G$, where $G$ is the dressed GF of the total system, we find the system of coupled equations
\begin{subequations}
	\begin{align}
	\hat{G}_\mathrm{D} &= \hat{g}_\mathrm{D} + w \, \hat{g}_\mathrm{D} \tau_3 
	\sum_{j=1}^{n}  \hat{G}_{j \mathrm{D}} ,
	\\
	\hat{G}_{j \mathrm{D}} &= \hat{g}_\mathrm{D} 
	\bigl( \hat{V}_{j\mathrm{D}}  \hat{G}_\mathrm{D} 
	+ \hat{V}_{j,j-1}  \hat{G}_\mathrm{j-1,D} 
	+ \hat{V}_{j,j+1}  \hat{G}_\mathrm{j+1,D} 
	\bigr) ,
	\\
	\hat{G}_{j\pm 1,\mathrm{D}} &= \hat{g}_{j \pm 1,j \pm 1} \sum_k \hat{V}_{j \pm 1,k}  \hat{G}_{k\mathrm{D}} ,
	\end{align}
\end{subequations}
which we solve for the dressed GF of the dot to linear order in $t$ and obtain
\begin{align}
\hat{G}_\mathrm{D}^{-1} = \hat{g}_\mathrm{D}^{-1} -  \hat{\Sigma}_\mathrm{D},
\end{align}
with the self-energy
\begin{align}
\hat{\Sigma}_\mathrm{D} &= \sum_{j=1}^n \Bigl( 
\hat{V}_{\mathrm{D} j} \hat{g}_{jj}  \hat{V}_{j \mathrm{D} } 
+
\hat{V}_{\mathrm{D} j} \hat{g}_{jj}  \hat{V}_{j,j-1 } \hat{g}_{j-1,j-1 } \hat{V}_{j-1,\mathrm{D} }
\nonumber \\
&\quad +
\hat{V}_{\mathrm{D} j} \hat{g}_{jj}  \hat{V}_{j,j+1 } \hat{g}_{j+1,j+1 } \hat{V}_{j+1,\mathrm{D} }
\Bigr) + \mathcal{O}(t^2) .
\end{align}
From the self-energy, we obtain the effective low-energy Hamiltonian $H_0$ of the dot as
\begin{align}
H_0 = \varepsilon - \hat{G}_\mathrm{D}^{-1} = \varepsilon - \hat{g}_\mathrm{D}^{-1} + \hat{\Sigma}_\mathrm{D} = \varepsilon_0 \tau_3 + \hat{\Sigma}_\mathrm{D} ,
\end{align}
which can be written as $H_0 = \boldsymbol{d} \cdot \boldsymbol{\tau}$, with the pseudo-spin $\boldsymbol{\tau}$ (set of Pauli matrices in Nambu space) and the effective magnetic field 
\begin{align}
\boldsymbol{d} =  \begin{pmatrix}
\Gamma \sum_{j=1}^n \cos\varphi_j \\
- \Gamma \sum_{j=1}^n \sin\varphi_j \\
\varepsilon_{0}   - 2 t_0 \Gamma \sum_{j=1}^n \cos(\varphi_j - \varphi_{j+1} -\alpha  )
\end{pmatrix} ,
\label{eq:magField}
\end{align}
where we defined $\Gamma = \pi N_0 w^2$ and $t_0 = \pi N_0 t$. The case $n=4$ and $\alpha = 0$ is discussed in the main text, see Eq.~\eqref{eq:vectorD}.
This two-level Hamiltonian has a ground state $\left| \mathrm{g} \right\rangle $ and an excited state $\left| \mathrm{e} \right\rangle$ which satisfy $H_0 \left| \mathrm{e/g} \right\rangle = \varepsilon_{\mathrm{e/g}} \left| \mathrm{e/g} \right\rangle$, with the eigenenergies $\varepsilon_{\mathrm{e/g}} = \pm d$ and the absolute value of the effective field $d = \sqrt{d_1^2 + d_2^2 + d_3^2}$. 
The eigenstates are given by
\begin{align}
\left| \mathrm{e/g} \right\rangle 
&= \frac{1}{\sqrt{2d (d\mp d_3 )}} 
\begin{pmatrix}
d_1 - i d_2  \\
\pm d - d_3
\end{pmatrix}
\nonumber \\
&= \frac{1}{\sqrt{2 \mp 2 \cos\theta}} 
\begin{pmatrix}
e^{-i \varphi } \sin\theta \\
\pm 1 - \cos\theta
\end{pmatrix} ,
\label{eq:GroundExcitedState}
\end{align}
where the spherical angles $\theta \in [0,\pi)$ and $\varphi \in [0,2\pi)$ are defined by the set of equations
\begin{align}
\cos\varphi &= \frac{d_1}{\sqrt{d_1^2 + d_2^2}} , 
\quad  \cos\theta = \frac{d_3}{d } , 
\nonumber \\
\sin\varphi &= \frac{d_2}{\sqrt{d_1^2+ d_2^2}}, 
\quad \sin\theta = \frac{\sqrt{d_1^2 + d_2^2}}{d } ,
\label{eq:relationsAnglesToParameters}
\end{align}
which parametrize the Bloch sphere. 
This explicitly shows that the eigenstates are independent of the absolute value $d$ of the effective field.
\subsection{Topological charge of Weyl points}
Here, we describe how to obtain the sign of the topological charge of the four Weyl points. 
We focus on the case $n=4$ and $\alpha = 0$ as discussed in the main text. 
In the vincinity of one of the points of degeneracy $\boldsymbol{\varphi}_\mathrm{W}^{(s)} = (\varphi_{\mathrm{W},1}^{(s)},\varphi_{\mathrm{W},2}^{(s)},\varphi_{\mathrm{W},3}^{(s)})$ for $s = 1,2,3,4$, we linearize the spectrum via the replacement $\varphi_{k} = \varphi_{\mathrm{W},k}^{(s)} + \delta\varphi_{k}$ for $k = 1,2,3$, where $\delta\varphi_{k}$ is a small deviation from the Weyl point in the direction of $\varphi_{k}$. 
Since for all Weyl points $d(\boldsymbol{\varphi}_\mathrm{W}^{(s)}) = 0$ holds, we obtain $\boldsymbol{d} = \boldsymbol{M}^{(s)} \boldsymbol{\delta\varphi}$ with $\boldsymbol{\delta\varphi} = (\delta\varphi_1 , \delta\varphi_2 , \delta \varphi_3)^\mathrm{T}$ and the matrix elements 
\begin{subequations}
	\begin{align}
	M_{1k}^{(s)} &= -  \Gamma \sin  \varphi_{\mathrm{W},k}^{(s)} ,  \\
	M_{2k}^{(s)} &= - \Gamma \cos  \varphi_{\mathrm{W},k}^{(s)} , \\
	M_{31}^{(s)} &= 2\Gamma t_0 [ \sin(\varphi_{\mathrm{W},1}^{(s)} - \varphi_{\mathrm{W},2}^{(s)}) +  \sin(\varphi_{\mathrm{W},1}^{(s)}) ] , \\
	M_{32}^{(s)} &=2 \Gamma t_0 [  \sin(\varphi_{\mathrm{W},2}^{(s)} - \varphi_{\mathrm{W},3}^{(s)}) - \sin(\varphi_{\mathrm{W},1}^{(s)} - \varphi_{\mathrm{W},2}^{(s)}) ] , \\
	M_{33}^{(s)} &= 2 \Gamma t_0 [ \sin(\varphi_{\mathrm{W},3}^{(s)}) -  \sin(\varphi_{\mathrm{W},2}^{(s)} - \varphi_{\mathrm{W},3}^{(s)})   ] ,
	\end{align}
\end{subequations}
of the transformation matrix $\boldsymbol{M}^{(s)} = (M_{jk}^{(s)})$ which define the {\textquotedblleft}velocities{\textquotedblright} $\boldsymbol{v}_k^{(s)} = (M_{k1}^{(s)},M_{k2}^{(s)},M_{k3}^{(s)})^\mathrm{T}$. 
The Weyl-Hamiltonian can then be written as
\begin{align}
H_\mathrm{W}^{(s)} &=  \sum\nolimits_{kl}  \tau_l M_{lk}^{(s)} \, \delta\varphi_k   = \sum\nolimits_{k}   (\boldsymbol{v}_k^{(s)} \cdot \boldsymbol{\delta\varphi}  ) \tau_k .
\end{align}
The topological charge $c_s$ (or chirality) of a Weyl point is defined via $c_s = \mathrm{sgn}[\boldsymbol{v}_1^{(s)} \cdot (\boldsymbol{v}_2^{(s)} \times \boldsymbol{v}_3^{(s)})] = \mathrm{sgn}[\mathrm{det}(\boldsymbol{M}^{(s)})]$, where the triple product of the velocities is
\begin{align}
&\mathrm{det}(\boldsymbol{M}^{(s)}) =  2 t_0 \Gamma^3 \sin(\varphi_{\mathrm{W},1}^{(s)} - \varphi_{\mathrm{W},3}^{(s)})  
\nonumber \\
& \quad \bigl[ \sin(\varphi_{\mathrm{W},1}^{(s)} - \varphi_{\mathrm{W},2}^{(s)}) + \sin(\varphi_{\mathrm{W},2}^{(s)})  - \sin(\varphi_{\mathrm{W},2}^{(s)} - \varphi_{\mathrm{W},3}^{(s)}) \bigr] .
\label{eq:topCharge}
\end{align}
From Eq.~\eqref{eq:WeylNodes} in the main text, we see that we have four Weyl points in the 3D parameter space, where $\boldsymbol{\varphi}_\mathrm{W}^{(1,2)}$ are located in the $\varphi_3 = \pi$ plane and $\boldsymbol{\varphi}_\mathrm{W}^{(3,4)}$
are located in the $\varphi_1 = \pi$ plane and recall that $-8 < m < 0$ and, therefore, $\mathrm{sgn}(m) = -1$. 
For the four points we find $c_{1,2} =  \mathrm{sgn}(t_0) $ and $c_{3,4} = - \mathrm{sgn}(t_0)$ since $\Gamma > 0$.
This also follows from time-reversal symmetry which links $\boldsymbol{\varphi}_\mathrm{W}^{(1)} = - \boldsymbol{\varphi}_{\mathrm{W}}^{(2)} (+ 2 \pi \boldsymbol{z})$, $\boldsymbol{z} \in \mathbb{Z}^3$, and $\boldsymbol{\varphi}_{\mathrm{W}}^{(3)} = - \boldsymbol{\varphi}_{\mathrm{W}}^{(4)} (+ 2 \pi \boldsymbol{z})$, $\boldsymbol{z} \in \mathbb{Z}^3$. 
Weyl points which are linked by time-reversal symmetry must have the same charge.
\subsection{Quantum geometric tensor from an effective magnetic field}
\begin{figure*}
	\includegraphics[width=\textwidth]{Figure5}
	\caption{Oscillator strengths for linearly polarized absorption and resulting elements of the metric tensor [Eqs.~\eqref{eq:FGRonephase} and \eqref{eq:offDiagonalOfMetric} in the main text]. (a) Off-diagonal and (b) diagonal elements in the trivial phase ($\varphi_3 = 0$). (c) Off-diagonal and (d) diagonal elements in the topological phase ($\varphi_3 = 2\pi/3$). The phases are defined by the Chern number, see Fig.~\hyperref[fig:fig2]{2(f)} in the main text. Parameters: $t_0 = 0.1, \varepsilon_0 / \Gamma = -0.2$. }
	\label{fig:sfig3}
\end{figure*}
Here, we provide explicit formulas of how to compute the Berry curvature $F_{jk}$ and the quantum metric tensor $g_{jk}$ of the ground state $\left| \mathrm{g} \right\rangle $ directly from the effective Hamiltonian $H_0 = \boldsymbol{d} \cdot  \boldsymbol{\tau}$. 
In general, the effective field is controlled by a set of $n$ parameters $ \boldsymbol{\lambda}  = (\lambda_1, \ldots , \lambda_n)$, i.e., $ \boldsymbol{d}  =  \boldsymbol{d} (\boldsymbol{\lambda})$. 
Using the explicit expression of $\left| \mathrm{g} \right\rangle $, see Eq.~\eqref{eq:GroundExcitedState}, in the definition of the quantum geometric tensor of the ground state $\chi_{jk} = \left\langle  \partial_j \mathrm{g} \middle| (1 - \left| \mathrm{g} \right\rangle \left\langle \mathrm{g} \right| ) \middle| \partial_k \mathrm{g} \right\rangle$, we find
\begin{align}
\chi_{jk} &= \frac{1}{4} \bigl( \partial_j \theta(\boldsymbol{\lambda})  + i \sin\theta(\boldsymbol{\lambda}) \, \partial_j \varphi(\boldsymbol{\lambda})    \bigr)
\nonumber \\
&\quad \times 
\bigl( \partial_k \theta(\boldsymbol{\lambda}) - i \sin\theta(\boldsymbol{\lambda}) \, \partial_k \varphi(\boldsymbol{\lambda}) \bigr),
\end{align}
where $\partial_j = \partial/\partial \lambda_j$. 
The quantum metric (i.e., the Fubini-Study metric) is given by the real part, $g_{jk} = \mathrm{Re}(\chi_{jk})$, and the Berry curvature is given by the imaginary part, $F_{jk} = - 2  \mathrm{Im}(\chi_{jk})$. 
We obtain
\begin{subequations}
	\begin{align}
	F_{jk} &= \frac{1}{2}  \sin\theta(\boldsymbol{\lambda}) \left(
	\frac{\partial \theta(\boldsymbol{\lambda})}{\partial\lambda_j}  \frac{\partial \varphi(\boldsymbol{\lambda})}{\partial\lambda_k}
	-  \frac{\partial \theta(\boldsymbol{\lambda})}{\partial\lambda_k} \frac{\partial \varphi(\boldsymbol{\lambda})}{\partial\lambda_j} 
	\right) ,
	\label{eq:berryC}
	\\
	g_{jk} &= \frac{1}{4} 
	\left(  
	\frac{\partial \theta(\boldsymbol{\lambda})}{\partial\lambda_j} \frac{\partial \theta(\boldsymbol{\lambda})}{\partial\lambda_k}
	+  \sin^2\theta(\boldsymbol{\lambda}) \,  \frac{\partial \varphi(\boldsymbol{\lambda})}{\partial\lambda_j} \frac{\partial \varphi(\boldsymbol{\lambda})}{\partial\lambda_k}
	\right) . 
	\label{eq:metric}
	\end{align}
\end{subequations}
We find by differentiating Eq.~\eqref{eq:relationsAnglesToParameters} with respect to $\lambda_j$:
\begin{subequations}
	%
	\label{eq:derivatives}
	\begin{align}
	\frac{\partial \varphi(\boldsymbol{\lambda})}{\partial\lambda_j}   &
	=  \frac{1}{d_1^2 + d_2^2 }
	\sum_{\alpha,\beta=1}^2 \varepsilon_{\alpha\beta} d_\alpha \partial_j d_\beta ,
	,
	\\
	\frac{\partial \theta(\boldsymbol{\lambda})}{\partial\lambda_j}   
	&= \frac{1}{ d^2 \sqrt{d_1^2 + d_2^2 }} \sum_{\alpha=1}^2 d_\alpha   
	\left(  d_3 \partial_j d_\alpha  -  d_\alpha   \partial_j d_3  \right) ,
	\end{align}
	%
\end{subequations}
where $\varepsilon_{\alpha\beta} = - \varepsilon_{\beta\alpha}$ is the total antisymmetric Levi-Civita tensor in two dimensions, with $\varepsilon_{12} = 1$. 
Using Eq.~\eqref{eq:derivatives} in Eq.~\eqref{eq:berryC}, we find that, after some algebra, the Berry curvature can be written as
\begin{align}
F_{jk} 
&=  \frac{ \boldsymbol{d} \cdot [    (\partial_j \boldsymbol{d}) \times (\partial_k \boldsymbol{d})  ]}{2d^3} 
\nonumber \\
&=  \frac{1 }{2} \Bigl(  \boldsymbol{n} \cdot [(\partial_j  \boldsymbol{n}) \times  (\partial_k  \boldsymbol{n})] 
\Bigr) ,
\end{align}
which allows for the computation of the Berry curvature directly from the effective magnetic field $\boldsymbol{d}$. 
For the second equation we introduced the normaized effective field $\boldsymbol{n} = \boldsymbol{d} / d$ which shows explicitly that the Berry curvature is independent of the absolute value $d$. 
A similar formula can be obtained for the quantum metric tensor. 
Using Eq.~\eqref{eq:derivatives} in Eq.~\eqref{eq:metric}, we find after some algebra
\begin{align}
g_{jk} &= \frac{1}{4d^4}  \biggl(     
d^2 \Bigl[(\partial_j \boldsymbol{d}) \cdot (\partial_k \boldsymbol{d}) \Bigr] - \Bigl[ \boldsymbol{d} \cdot (\partial_j \boldsymbol{d})\Bigr] \ \Bigl[\boldsymbol{d} \cdot (\partial_k \boldsymbol{d})\Bigr] 
\biggr) 
\nonumber \\
&= \frac{1}{4} (\partial_j \boldsymbol{n}) \cdot  (\partial_k \boldsymbol{n}) .
\end{align}
\subsection{Quantum geometric tensor and transition rates}
\begin{figure*}
	\includegraphics[width=\textwidth]{Figure6}
	\caption{Model, band structure, Weyl nodes and Chern number in a three-terminal Josephson junction. (a) Microscopic model shown in Fig.~\ref{fig:sfig1} for $n=3$ terminals. Red wiggly arrows indicate that up to two superconducting phases can be modulated at frequency $\omega$ and relative phase difference $\gamma$ to obtain information about the quantum geometric tensor as explained in the main text. (b){\textendash}(f) Energy spectrum $\varepsilon_\mathrm{e/g} = \pm d$ for (b) $\alpha = 0$, (c) $\alpha = \pi/6$, (d) $\alpha = \pi/2$, (e) $\alpha = 5\pi/6$, (f) $\alpha = \pi$,  respectively. (g) Locations of the four Weyl nodes in the FBZ. Blue (red) Weyl nodes carry a topological charge $c = +1$ ($c = -1$). The Chern number becomes nontrivial only if the $(\varphi_1,\varphi_2)$ plane of integration lies between two opposing charges. (h) Chern number $C$ as a function of $\alpha$. The points b, c, d, e, and f correspond to the values of $\alpha$ in panels (b), (c), (d), (e) and (f), respectively. Common parameters for all panels: $t_0 = 0.1$, $\varepsilon_0 / \Gamma = 0$.}
	\label{fig:sfig2}
\end{figure*}
The effective low-energy Hamiltonian of the dot in all the realizations of topological Josephson matter discussed in this work adopts the form $H_{0} = \boldsymbol{d} \cdot \boldsymbol{\tau}$. 
This Hamiltonian describes an effective two-level system with $H_0 \left| \mathrm{e/g} \right\rangle = \varepsilon_\mathrm{e/g} \left| \mathrm{e/g} \right\rangle$, where $\varepsilon_\mathrm{g} = -d$ and $\varepsilon_\mathrm{e} = +d$ are the Andreev bound state energies of the ground state (GS) $\left| \mathrm{g} \right\rangle$ and the excited state $\left| \mathrm{e} \right\rangle$, respectively. 
Here, $d = \sqrt{d_1^2 + d_2^2 + d_3^2}$ denotes the absolute value of $\boldsymbol{d}$.

From $\left| \mathrm{g} \right\rangle$, we calculate the  Berry connection $A_j = i \left\langle \mathrm{g} \middle| \partial_j \mathrm{g} \right\rangle$ and the Berry curvature 
\begin{align}
F_{jk} &= \partial_j A_k - \partial_k  A_j 
= i \left[ \left\langle \partial_j \mathrm{g} \middle| \partial_k \mathrm{g} \right\rangle -  \left\langle \partial_k \mathrm{g} \middle| \partial_j \mathrm{g} \right\rangle \right]
\end{align}
of the GS where now $\partial_j = \partial / \partial \varphi_j$.
Furthermore, we need the relation
\begin{align}
0 =  \left\langle \mathrm{e} \middle| (\partial_j H_0) \middle| \mathrm{g} \right\rangle + 2 d \left\langle \mathrm{e} \middle| \partial_j  \mathrm{g} \right\rangle ,
\label{eq:sm22}
\end{align}
which follows from $\partial_j\left\langle \mathrm{e} \middle|  H_0 \middle| \mathrm{g} \right\rangle=0$. 
First, we proof Eq.~\eqref{eq:FGRonephase} for the oscillator strength in the main text. According to Fermi's golden rule, the transition rate due to a perturbation $2A (\partial_j H_0) \cos(\omega t) / \hbar \omega$ is given by
\begin{align}
R_{jj} = \frac{2\pi}{\hbar} \frac{A^2}{(\hbar\omega)^2} \left| \left\langle \mathrm{e} \right| (\partial_j H_0)  \left| \mathrm{g} \right\rangle \right|^2 \delta(2d - \hbar \omega) .
\end{align}
We are left to evaluate the matrix element which reads
\begin{align}
\left| \left\langle \mathrm{e} \right| (\partial_j H_0)  \left| \mathrm{g} \right\rangle \right|^2
&\stackrel{\eqref{eq:sm22}}{=} 
4 d^2 
\left\langle \partial_j  \mathrm{g} \middle| \mathrm{e}  \right\rangle
\left\langle \mathrm{e} \middle| \partial_j  \mathrm{g} \right\rangle 
= 4 d^2 
g_{jj}
\end{align}
since $\left|\mathrm{g} \right\rangle \left\langle \mathrm{g} \right| + \left|\mathrm{e} \right\rangle \left\langle \mathrm{e} \right| = 1$. 

Now, we proof Eqs.~\eqref{eq:rjkgamma} and \eqref{eq:offDiagonalGeometric} in the main text relating the off-diagonal elements of the quantum geometric tensor to the oscillator strengths. 
According to Fermi's golden rule, the transition rate due to a perturbation $2A (\partial_j H_0) \cos(\omega t) / \hbar \omega + 2A (\partial_k H_0) \cos(\omega t - \gamma) / \hbar \omega$ is given by
\begin{align}
R_{jk}^{(\gamma)}
&= \frac{2\pi }{\hbar } \frac{ A^2 }{ (\hbar\omega)^2}  \left| \left\langle \mathrm{e} \right| \left( \partial_j H_0 + e^{i \gamma} \partial_k H_0 \right) \left| \mathrm{g} \right\rangle \right|^2 \delta(2d - \hbar \omega) \, .
\end{align}
Similarly, the matrix element evaluates to
\begin{align}
&\left| \left\langle \mathrm{e} \right| \left( \partial_j H_0 + e^{i \gamma} \partial_k H_0 \right) \left| \mathrm{g} \right\rangle \right|^2 
\nonumber \\
&\stackrel{\eqref{eq:sm22}}{=} 4d^2
\bigl( 
\left\langle  \partial_j  \mathrm{g} \middle|\mathrm{e} \right\rangle  \left\langle \mathrm{e} \middle| \partial_j  \mathrm{g} \right\rangle 
+ 
e^{i \gamma} \left\langle  \partial_j  \mathrm{g} \middle|\mathrm{e} \right\rangle  \left\langle \mathrm{e} \middle| \partial_k  \mathrm{g} \right\rangle 
\nonumber \\
&\qquad \qquad + 
e^{-i \gamma}  \left\langle \partial_k  \mathrm{g}  \middle| \mathrm{e}\right\rangle \left\langle \mathrm{e} \middle| \partial_j  \mathrm{g} \right\rangle 
+ 
\left\langle \partial_k  \mathrm{g}  \middle| \mathrm{e}\right\rangle  \left\langle \mathrm{e} \middle| \partial_k  \mathrm{g} \right\rangle  \bigr)
\nonumber \\
&= 4d^2
\bigl( 
g_{jj} + g_{kk}
+ 
e^{i \gamma} \chi_{jk}
+ 
e^{-i \gamma} \chi_{jk}^*
\bigr)
\nonumber \\
&= 4d^2
\bigl( 
g_{jj} + g_{kk}
+ 
2 g_{jk} \cos\gamma 
+ 
F_{jk} \sin\gamma
\bigr).
\end{align}
The different oscillator strengths of the transition rates for linearly polarized absorption are shown in Fig.~\ref{fig:sfig3}.
\subsection*{Three-terminal junction with magnetic flux: Hamiltonian, Weyl nodes, and Chern number}
We briefly discuss the case of $n = 3$ SC terminals with nonzero magnetic flux $\Phi = n \alpha \Phi_0$.
The system is sketched in Fig.~\hyperref[fig:sfig2]{S3(a)}. 
As discussed above, the Hamiltonian takes the form of a pseudo-spin in an effective magnetic field, i.e., $H_0 = \boldsymbol{d} \cdot \boldsymbol{\tau}$ and the Andreev bound states $\varepsilon_\mathrm{e/g} = \pm d$, where $d = |\boldsymbol{d}|$. The expression of the effective magnetic field is given in Eq.~\eqref{eq:magField}.
Gauge invariance allows us to set one SC phase to zero (we set $\varphi_3 = 0$ in the following) and the magnetic flux plays a similar role as one of the SC phases.
The spectrum is shown in Figs.~\hyperref[fig:sfig2]{S3(b){\textendash}S3(f)} for different values of $\alpha$. 
In this system, we find that Weyl nodes $\boldsymbol{\varphi}_\mathrm{W} = (\varphi_{\mathrm{W},1},\varphi_{\mathrm{W},2},\alpha_\mathrm{W})$, for which $d(\boldsymbol{\varphi}_\mathrm{W})=0$, only exist for $-6 \leq m \leq 6$ with $m = \varepsilon_0 / t_0 \Gamma$.
There are four Weyl nodes located at (modulo $2\pi$ in each direction)
\begin{subequations}
	\begin{align}
	\boldsymbol{\varphi}^{(1)}_\mathrm{W} &= \bigl( -2 \pi / 3, \, 2 \pi / 3 , \, \alpha_+(m) \bigr) ,\\
	\boldsymbol{\varphi}^{(2)}_\mathrm{W} &= \bigl( -2 \pi / 3 , \, 2 \pi / 3 , \, \alpha_-(m) \bigr) ,\\
	\boldsymbol{\varphi}^{(3)}_\mathrm{W} &= \bigl( 2 \pi / 3 , \, -2 \pi / 3 , \, - \alpha_-(m) \bigr) ,\\
	\boldsymbol{\varphi}^{(4)}_\mathrm{W} &= \bigl( 2 \pi / 3 , \, -2 \pi / 3 , \, - \alpha_+(m) \bigr) ,
	\end{align}
\end{subequations}
where
\begin{align}
\alpha_{\pm}(m) &= 2 \arctan\left( \frac{\sqrt{3} \pm 2\sqrt{1 - (m/6)^2}}{m/3 -1} \right) .
\end{align}
Each Weyl node $\boldsymbol{\varphi}^{(s)}_\mathrm{W}$, $s = 1,2,3,4$, carries a topological charge $c_s$ which is obtained in the same way as for the four-terminal junction. 
We find $c_{1,4} = \mathrm{sgn}(t_0)$ and $c_{2,3} = - \mathrm{sgn}(t_0)$.
The locations and the charges of the Weyl points are shown in Fig.~\hyperref[fig:sfig2]{S3(g)}.

We define a first Brillouin zone (FBZ) as $(\varphi_1 , \varphi_2, \alpha) \in [0,2\pi)^3$ and calculate the Chern number $C$ as a function of $\alpha$ from the normalized effective field $\boldsymbol{n} = \boldsymbol{d}/d$ via
\begin{align}
C (\alpha) = \frac{1}{4\pi} \int_{0}^{2\pi} \!\!\! \mathrm{d}\varphi_1 \int_{0}^{2\pi} \!\!\! \mathrm{d}\varphi_2 \, \Bigl( \boldsymbol{n} \cdot [ (\partial_{\varphi_1} \boldsymbol{n}) \times (\partial_{\varphi_2} \boldsymbol{n}) ] \Bigr) .
\end{align}
The Chern number is shown in Fig.~\hyperref[fig:sfig2]{S3(h)} and shows topologically nontrivial regions for certain values of $\alpha$. 
Since we integrate over the phases $\varphi_1$ and $\varphi_2$, the finite jumps of $C$ are associated with the values $\alpha_\pm$ and $-\alpha_\pm$. 
In particular for $m = 0$, the four values of a topological phase transition are $\alpha_\mathrm{W} = \pi/6, 5\pi/6, 7\pi/6, 11\pi/6$ in the FBZ.






\end{document}




















