Conformality or confinement: (IR)relevance of topological excitations

What distinguishes two asymptotically-free non-abelian gauge theories on R-4, one of which is just below the conformal window boundary and confines, while the other is slightly above the boundary and flows to an infrared conformal field theory? In this work, we aim to answer this question for non-supersymmetric Yang-Mills theories with fermions in arbitrary chiral or vectorlike representations. We use the presence or absence of mass gap for gauge fluctuations as an identifier of the infrared behavior. With the present-day understanding of such gauge theories, the mass gap for gauge fluctuations cannot be computed on R-4. However, recent progress allows its non-perturbative computation on R-3 x S-1 by using either the twisted partition function or deformation theory, for a range of sizes of S-1 depending on the theory. For small number of fermions, N-f, we show that the mass gap increases with increasing radius, due to the non-dilution of monopoles and bions - the topological excitations relevant for confinement on R-3 x S-1. For sufficiently large Nf, we show that the mass gap decreases with increasing radius. In a class of theories, we claim that the decompactification limit can be taken while remaining within the region of validity of semiclassical techniques, giving the first examples of semiclassically solvable Yang-Mills theories at any size S1. For general non-supersymmetric vectorlike or chiral theories, we conjecture that the change in the behavior of the mass gap on R-3 x S-1 as a function of the radius occurs near the lower boundary of the conformal window and give non-perturbative estimates of its value. For vectorlike theories, we compare our estimates of the conformal window with existing lattice results, truncations of the Schwinger-Dyson equations, NSVZ beta function-inspired estimates, and degree of freedom counting criteria. For multi-generation chiral gauge theories, to the best of our knowledge, our estimates of the conformal window are the only known ones.