Fermionic and parafermionic CFTs with $$ \hat{su}(2) $$ and $$ \hat{su}(3) $$ symmetry

A bstract We investigate two-dimensional conformal field theories (CFTs) with affine $$ \hat{su}(2) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mn>2</mml:mn> </mml:mfenced> </mml:math> and $$ \hat{su}(3) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> </mml:math> algebra symmetry. Their bosonic modular-invariant partition functions have been fully classified based on the ADE classification. In this work, we extend the classification to include fermionic and parafermionic CFTs with the same affine symmetries, utilizing techniques of fermionization and parafermionization. We find that the fermionic and parafermionic $$ \hat{su}(2) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mn>2</mml:mn> </mml:mfenced> </mml:math> models are related to non-simply laced Dynkin diagrams.