$A=10$ nuclei and $^{12}$C with SRG evolved chiral three-nucleon interactions*

We investigate selected static and transition properties of $A=10$ nuclei and $^{12}$C using {\it ab initio} No-Core Shell Model (NCSM) methods with chiral two- and SRG-evolved three-nucleon interactions. We examine the dependences of observables on the SRG evolution scale and on the model-space parameters. We obtain nearly converged low-lying excitation spectra for $^{12}$C. We compare results of the full NCSM with the Importance Truncated NCSM in large model spaces for benchmarking purposes in $^{12}$C. The agreement of some observables with experiment is improved significantly by the inclusion of 3N interactions, e.g., the B(M1) from the first $(J^\pi,T)=(1^+,1)$ state to the ground state of $^{12}$C. However, in some cases the agreement deteriorates, e.g., for the excitation energy of the first $(1^+,0)$ state, leaving room for improved next-generation chiral Hamiltonians. On the other hand, the excitation energies of $^{10}$C, $^{10}$B,and $^{10}$Be are not as well converged as those of $^{12}$C. In particular the lowest two $(1^+,0)$ states of $^{10}$B are sensitive to both the basis truncation parameters and the 3N interaction.