Recent $mu$SR measurements on SrPtAs revealed time-reversal-symmetry breaking with the onset of superconductivity [Biswas et al., Phys. Rev. B 87, 180503(R) (2013)], suggesting an unconventional superconducting state. We investigate this possibility via functional renormalization group and find a chiral $(d+mathrm{i}d)$-wave order parameter favored by the multiband fermiology and hexagonal symmetry of SrPtAs. This $(d+mathrm{i}d)$-wave state exhibits significant gap anisotropies as well as gap differences on the different bands, but only has point nodes on one of the bands at the Brillouin zone corners. We study the topological characteristics of this superconducting phase, which features Majorana-Weyl nodes in the bulk, protected surface states, and an associated thermal Hall response. The lack of extended nodes and the spontaneously broken time-reversal symmetry of the $(d+mathrm{i}d)$-wave state are in agreement with the $mu$SR experiments. Our theoretical findings together with the experimental evidence thus suggests that SrPtAs is the first example of chiral $d$-wave superconductivity.
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