Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories

We establish a linear relation between the $a$-type Weyl anomaly and the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies in six-dimensional $(1,0)$ superconformal field theories. For RG flows onto the tensor branch, where conformal symmetry is spontaneously broken, supersymmetry relates the anomaly mismatch $\Delta a$ to the square of a four-derivative interaction for the dilaton. This establishes the $a$-theorem for all such flows. The four-derivative dilaton interaction is in turn related to the Green-Schwarz-like terms that are needed to match the 't Hooft anomalies on the tensor branch, thus fixing their relation to $\Delta a$. We use our formula to obtain exact expressions for the $a$-anomaly of $N$ small $E_8$ instantons, as well as $N$ M5-branes probing an orbifold singularity, and verify the $a$-theorem for RG flows onto their Higgs branches. We also discuss aspects of supersymmetric RG flows that terminate in scale but not conformally invariant theories with massless gauge fields.