Moduli axions, stabilizing moduli, and the large field swampland conjecture in heterotic M-theory

We compute the F- and D-term potential energy for the dilaton, complex structure, and Kähler moduli of realistic vacua of heterotic M-theory compactified on Calabi-Yau threefolds where, for simplicity, we choose <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:msup><a:mrow><a:mi>h</a:mi></a:mrow><a:mrow><a:mn>1</a:mn><a:mo>,</a:mo><a:mn>1</a:mn></a:mrow></a:msup><a:mo>=</a:mo><a:msup><a:mrow><a:mi>h</a:mi></a:mrow><a:mrow><a:mn>2</a:mn><a:mo>,</a:mo><a:mn>1</a:mn></a:mrow></a:msup><a:mo>=</a:mo><a:mn>1</a:mn></a:mrow></a:math>. However, the formalism is immediately applicable to the “universal” moduli of Calabi-Yau threefolds with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:msup><c:mrow><c:mi>h</c:mi></c:mrow><c:mrow><c:mn>1</c:mn><c:mo>,</c:mo><c:mn>1</c:mn></c:mrow></c:msup><c:mo>=</c:mo><c:msup><c:mrow><c:mi>h</c:mi></c:mrow><c:mrow><c:mn>1</c:mn><c:mo>,</c:mo><c:mn>2</c:mn></c:mrow></c:msup><c:mo>></c:mo><c:mn>1</c:mn></c:mrow></c:math> as well. The F-term potential is computed using the nonperturbative complex structure, gaugino condensate and “world sheet instanton” superpotentials in theories in which the hidden sector contains an anomalous <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:mo stretchy="false">)</e:mo></e:math> structure group. The Green-Schwarz anomaly cancellation induces inhomogeneous “axion” transformations for the imaginary components of the dilaton and Kähler modulus—which then produce a D-term potential. <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>V</i:mi><i:mi>D</i:mi></i:msub></i:math> is a function of the real components of the dilaton and Kähler modulus (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>s</k:mi></k:math> and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>t</m:mi></m:math>) that is minimized and precisely vanishes along a unique line in the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mrow><o:mi>s</o:mi><o:mtext>−</o:mtext><o:mi>t</o:mi></o:mrow></o:math> plane. Excitations transverse to this line have a mass <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>m</q:mi><q:mrow><q:mi>anom</q:mi></q:mrow></q:msub></q:math> which is an explicit function of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mi>t</s:mi></s:math>. The F-term potential energy is then evaluated along the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>V</u:mi><u:mi>D</u:mi></u:msub><u:mo>=</u:mo><u:mn>0</u:mn></u:math> line. For values of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>t</w:mi></w:math> small enough that <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>m</y:mi><y:mrow><y:mi>anom</y:mi></y:mrow></y:msub><y:mo>≳</y:mo><y:msub><y:mi>M</y:mi><y:mi>U</y:mi></y:msub></y:math>—where <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>M</ab:mi><ab:mi>U</ab:mi></ab:msub></ab:math> is the compactification scale—we plot <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>V</cb:mi><cb:mi>F</cb:mi></cb:msub></cb:math> for a realistic choice of coefficients as a function of the Pfaffian parameter <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>p</eb:mi></eb:math>. We find values of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>p</gb:mi></gb:math> for which <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msub><ib:mi>V</ib:mi><ib:mi>F</ib:mi></ib:msub></ib:math> has a global minimum at negative or zero vacuum density or a metastable minimum with positive vacuum density. In all three cases, the <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:mi>s</kb:mi></kb:math>, <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>t</mb:mi></mb:math>, and associated “axion” moduli are completely stabilized. Finally, we show that, for any of these vacua, the large <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>t</ob:mi></ob:math> behavior of the potential energy satisfies the “large scalar field” Swampland conjecture. Published by the American Physical Society 2024