Surprising One-Loop Finiteness of 6D Half-Maximal Supergravities

In four dimensions, it has long been established that gravity coupled to matter exhibits ultraviolet divergences at one loop, irrespective of supersymmetry. Notably, the four-matter one-loop amplitudes of half-maximal supergravity coupled to Maxwell multiplets were shown in the 1970s to be divergent. Surprisingly, we demonstrate in this Letter that half-maximal theories can nevertheless become one-loop finite when uplifted to higher dimensions, contrary to naive expectations. Specifically, we study the ultraviolet properties of the four-matter and two-matter two-graviton amplitudes in six-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi mathvariant="script">N</a:mi> <a:mo>=</a:mo> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo>,</a:mo> <a:mn>0</a:mn> <a:mo stretchy="false">)</a:mo> </a:mrow> </a:math> and <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mrow> <f:mi mathvariant="script">N</f:mi> <f:mo>=</f:mo> <f:mo stretchy="false">(</f:mo> <f:mn>1</f:mn> <f:mo>,</f:mo> <f:mn>1</f:mn> <f:mo stretchy="false">)</f:mo> </f:mrow> </f:math> supergravities, coupled to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>n</k:mi> <k:mi>T</k:mi> </k:msub> </k:math> tensor and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>n</m:mi> <m:mi>V</m:mi> </m:msub> </m:math> vector multiplets, respectively. We find that the one-loop amplitudes are finite for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>n</o:mi> <o:mi>T</o:mi> </o:msub> <o:mo>=</o:mo> <o:mn>21</o:mn> </o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi>n</q:mi> <q:mi>V</q:mi> </q:msub> <q:mo>=</q:mo> <q:mn>20</q:mn> </q:math> . This finiteness is unexpected, as symmetry-preserving counterterms do exist. Interestingly, both values exactly correspond to low-energy limits of type II string theories compactified on K3, which hints at possible origins to the surprising cancellations.