Duality defect of the monster CFT
Ying-Hsuan Lin, Shu-Heng Shao
Abstract
Abstract We show that the fermionization of the Monster CFT with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>A</mml:mi> </mml:mrow> </mml:msub> </mml:math> orbifold, i.e. it enjoys the Kramers–Wannier duality. The Kramers–Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay – Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay–Thompson series is invariant under the genus-zero congruence subgroup 16 D 0 of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> .