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We study decays of $D^0$, $D^+$, and $D_s^+$ mesons into two pseudoscalarmesons by expressing the decay amplitudes in terms of topological amplitudes.Including consistently SU(3)$_F$ breaking to linear order, we show how thetopological-amplitude decomposition can be mapped onto the standard expansionusing reduced amplitudes characterized by SU(3) representations. The tree andannihilation amplitudes can be calculated in factorization up to correctionswhich are quadratic in the color-counting parameter $1/N_c$. We find new sumrules connecting $D^+\rightarrow K_SK^+$, $D_s^+\rightarrow K_S\pi^+$ and$D^+\rightarrow K^+\pi^0$, which test the quality of the $1/N_c$ expansion.Subsequently, we determine the topological amplitudes in a global fit to thedata, taking the statistical correlations among the various measurements intoaccount. We carry out likelihood ratio tests in order to quantify the role ofspecific topological contributions. While the SU(3)$_F$ limit is excluded witha significance of more than five standard deviations, a good fit (with $\Delta\chi^2 <1$) can be obtained with less than $28\%$ of SU(3)$_F$ breaking in thedecay amplitudes. The magnitude of the penguin amplitude $P_{\mathrm{break}}$,which probes the Glashow Iliopoulos Maiani (GIM) mechanism, is consistent withzero; the hypothesis $P_{\mathrm{break}}=0$ is rejected with a significance ofjust $0.7\sigma$. We obtain the Standard-Model correlation between$\mathcal{B}(D^0\rightarrow K_L \pi^0)$ and $\mathcal{B}(D^0\rightarrowK_S\pi^0)$, which probes doubly Cabibbo-suppressed amplitudes, and find that$\mathcal{B}(D^0\rightarrow K_L \pi^0)<\mathcal{B}(D^0\rightarrow K_S\pi^0)$holds with a significance of more than $4\sigma$. We finally predict$\mathcal{B}(D_s^+\rightarrow K_L K^+)= 0.012^{+0.006}_{-0.002}$ at $3\sigma$CL.

Topological amplitudes in $D$ decays to two pseudoscalars: a global analysis with linear $SU(3)_F$ breaking