Topological Modular Forms and the Absence of All Heterotic Global Anomalies
Yuji Tachikawa, Mayuko Yamashita
Abstract
Abstract We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation $$\text {TMF} ^\bullet \rightarrow (I_\mathbb {Z}\Omega ^{\text {string} })^{\bullet -20}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>TMF</mml:mtext> <mml:mo>∙</mml:mo> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>Z</mml:mi> </mml:msub> <mml:msup> <mml:mi>Ω</mml:mi> <mml:mtext>string</mml:mtext> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>∙</mml:mo> <mml:mo>-</mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , from topological modular forms to the Anderson dual of string bordism groups, using the Segal–Stolz–Teichner conjecture. We will show that this natural transformation vanishes, implying that heterotic global anomalies are always absent. The fact that $$\text {TMF} ^{21}(\text {pt} )=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>TMF</mml:mtext> <mml:mn>21</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mtext>pt</mml:mtext> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> plays an important role in the process. Along the way, we also discuss how the twists of $$\text {TMF} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtext>TMF</mml:mtext> </mml:math> can be described under the Segal–Stolz–Teichner conjecture, by using the result of Freed and Hopkins concerning anomalies of quantum field theories. The paper contains separate introductions for mathematicians and for string theorists, in the hope of making the content more accessible to a larger audience. The sections are also demarcated cleanly into mathematically rigorous parts and those which are not.