Evidence for C-theorems in 6D SCFTs

Using the recently established classification of 6D SCFTs we present evidence for the existence of families of weak C-functions, that is, quantities which decrease in a flow from the UV to the IR. Introducing a background R-symmetry field strength R, and a nontrivial tangent bundle T on the 6D spacetime, we consider C-functions given by the linear combinations C = m(1)alpha + m(2)beta+ m(3)gamma, where the cx, are the anomaly polynomial coefficients for the formal characteristic classes c(2)(R)(2), c(2)(R)p(1)(T) and p(1)(T)(2). By performing a detailed sweep over many theories, we determine the shape of the unbounded region in "in-space" compatible with both Higgs branch flows and tensor branch flows. We also verify that - as expected - the Euler density conformal anomaly falls in the admissible region.