We use holography to study $d=4$, $mathcal{N}=4$, SU($N_{rm tiny{c}}$) super Yang-Mills coupled to $N_{rm tiny{F}} ll N_{rm tiny{c}}$ quark flavors. We place the theory at finite isospin density $n_{rm tiny{I}}$ by turning on an isospin chemical potential $mu_{rm tiny{I}}=M_{rm tiny{q}}$, with $M_{rm tiny{q}}$ the quark mass. We also turn on two R-symmetry charge densities $n_1=n_2$. We show that the ground state is a supersymmetric, superfluid, color superconductor, namely a finite-density state that preserves a fraction of supersymmetry in which part of the global symmetries and part of the gauge symmetries are spontaneously broken. The holographic description consists of $N_{rm tiny{F}}$ D7-brane probes in $mbox{AdS}_5 times mbox{S}^5$. The symmetry breaking is due to the dissolution of some D3-branes inside the D7-branes triggered by the electric field associated to the isospin charge. The massless spectrum contains Goldstone bosons and their fermionic superpartners. The massive spectrum contains long-lived, mesonic quasi-particles if $n_{rm tiny{I}} ll mu_{rm tiny{I}}^3$, and no quasi-particles otherwise. We discuss the possibility that, despite the presence of mass scales and charge densities in the theory, conformal and relativistic invariance arise as emergent symmetries in the infrared.
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