F-extremization determines certain large-N CFTs

A bstract We show that the conformal data of a range of large- N CFTs, the melonic CFTs, are specified by constrained extremization of the universal part of the sphere free energy F = − log $$ {Z}_{S^d} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:msup> <mml:mi>S</mml:mi> <mml:mi>d</mml:mi> </mml:msup> </mml:msub> </mml:math> , called $$ \overset{\sim }{F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> . This family includes the generalized SYK models, the vector models ( O ( N ), Gross-Neveu, etc.), and the tensor field theories. The known F and a -maximization procedures in SCFTs are therefore extended to these non-supersymmetric CFTs in continuous d . We establish our result using the two-particle irreducible (2PI) effective action, and, equivalently, by Feynman diagram resummation. The universal part of $$ \overset{\sim }{F} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>F</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> interpolates in continuous dimension between the known C -functions, so we can interpret this result as an extremization of the number of IR degrees of freedom, in the spirit of the generalized c , F , a -theorems. The outcome is a complete classification of the melonic CFTs: they are the conformal mean field theories which extremize the universal part of the sphere free energy, subject to an IR marginality condition on the interaction Lagrangian.