Dimensional reduction of the M-theory Chern-Simons term at order ℓ p 6

The dimensional reduction of M-theory couplings at order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msubsup> <a:mo>ℓ</a:mo> <a:mi>p</a:mi> <a:mn>6</a:mn> </a:msubsup> </a:math> is known to produce one-loop <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mrow> <c:mi>α</c:mi> </c:mrow> <c:mrow> <c:mo>′</c:mo> <c:mn>3</c:mn> </c:mrow> </c:msup> </c:math> corrections in type IIA string theory. In this paper, we perform the Kaluza-Klein reduction of the M-theory Chern-Simons coupling <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>t</e:mi> <e:mn>8</e:mn> </e:msub> <e:msub> <e:mi>ε</e:mi> <e:mn>11</e:mn> </e:msub> <e:mi>A</e:mi> <e:msup> <e:mi>R</e:mi> <e:mn>4</e:mn> </e:msup> </e:math> at this order. By meticulously accounting for nongauge-invariant total derivative terms, we derive the complete set of corresponding one-loop, gauge-invariant couplings in the type IIA effective action. Our results not only reproduce the standard Chern-Simons term <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>t</g:mi> <g:mn>8</g:mn> </g:msub> <g:msub> <g:mi>ε</g:mi> <g:mn>10</g:mn> </g:msub> <g:mi>B</g:mi> <g:msup> <g:mi>R</g:mi> <g:mn>4</g:mn> </g:msup> </g:math> —which is gauge invariant up to total derivatives—but also unveil a new set of gauge-invariant couplings involving Ramond-Ramond and Neveu-Schwarz–Neveu-Schwarz field strengths. To validate our findings, we test their consistency under string dualities. We dimensionally reduce the derived type IIA couplings on a K3 manifold and show that the resulting one-loop <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>α</i:mi> <i:mo>′</i:mo> </i:msup> </i:math> corrections in six dimensions transform under S-duality into the dimensional reduction of the tree-level heterotic string Chern-Simons coupling <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>H</k:mi> <k:mrow> <k:mi>μ</k:mi> <k:mi>ν</k:mi> <k:mi>α</k:mi> </k:mrow> </k:msub> <k:msup> <k:mi mathvariant="normal">Ω</k:mi> <k:mrow> <k:mi>μ</k:mi> <k:mi>ν</k:mi> <k:mi>α</k:mi> </k:mrow> </k:msup> </k:math> on <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mi>T</n:mi> <n:mn>4</n:mn> </n:msup> </n:math> . This nontrivial agreement provides strong evidence for the correctness of both the M-theory Chern-Simons term and its reduction to type IIA.