Gradient flows and the curvature of theory space

A bstract The metric and potential associated with the gradient property of renormalisation group flow in multiscalar models in d = 4 − ε dimensions are studied. The metric is identified with the Zamolodchikov metric of nearly marginal operators on the sphere. An explicit form for the associated Ricci scalar in d = 4 − ε is derived, which shows that the space of multiscalar field theories is curved. The potential is identified with a quantity $$ \overset{\sim }{F} $$ <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> that was previously proposed as a weakly monotonic function interpolating between the a -theorem in four dimensions and the F -theorem in three dimensions. This implies that the $$ \overset{\sim }{F} $$ <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> -theorem can be extended perturbatively to a theorem about gradient flow in d = 4 − ε .