Finite NN from resurgent large NN

Due to instanton effects, gauge-theoretic large NN expansions yield asymptotic series, in powers of 1/N21/N2. The present work shows how to generically make such expansions meaningful via their completion into resurgent transseries, encoding both perturbative and nonperturbative data. Large NN resurgent transseries compute gauge-theoretic finite NN results nonperturbatively (no matter how small NN is). Explicit calculations are carried out within the gauge theory prototypical example of the quartic matrix model. Due to integrability in the matrix model, it is possible to analytically compute (fixed integer) finite NN results. At the same time, the large NN resurgent transseries for the free energy of this model was recently constructed. Together, it is shown how the resummation of the large NN resurgent transseries matches the analytical finite NN results up to remarkable numerical accuracy. Due to lack of Borel summability, Stokes phenomena has to be carefully taken into account, implying that instantons play a dominant role in describing the finite NN physics. The final resurgence results can be analytically continued, defining gauge theory for any complex value of NN.