Interpolating between \(a\) and \(F\)

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Abstract: We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension

d

we define the quantity

i l d e F = s i n (p i d / 2) l o g Z

, where

Z

is the path integral of the Euclidean CFT on the

d

-dimensional round sphere.

i l d e F

smoothly interpolates between

(- 1)^{d / 2} p i / 2

times the

a

-anomaly coefficient in even

d

, and

(- 1)^{(d + 1) / 2}

times the sphere free energy

F

in odd

d

. We calculate

i l d e F

in various examples of unitary CFT that can be continued to non-integer dimensions, including free theories, double-trace deformations at large

N

, and perturbative fixed points in the

e p s i l o n

expansion. For all these examples

i l d e F

is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate

i l d e F

in the Wilson-Fisher fixed point of the

O (N)

vector model in

d = 4 - e p s i l o n

to order

e p s i l o n^{4}

. We use this result to estimate the value of

F

in the 3-dimensional Ising model, and find that it is only a few percent below

F

of the free conformally coupled scalar field. We use similar methods to estimate the

F

values for the

U (N)

Gross-Neveu model in

d = 3

and the

O (N)

model in

d = 5

. Finally, we carry out the dimensional continuation of interacting theories with 4 supercharges, for which we suggest that

i l d e F

may be calculated exactly using an appropriate version of localization on

S^{d}

. Our approach provides an interpolation between the

a

-maximization in

d = 4

and the

F

-maximization in

d = 3

.

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