### Abstract

Abstract: We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension d we define the quantity (Formula presented.) =sin(πd/2) log Z, where Z is the path integral of the Euclidean CFT on the d-dimensional round sphere. (Formula presented.) smoothly interpolates between (−1)^{d/2}π/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 (Formula presented.) 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 ϵ expansion. For all these examples (Formula presented.) is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate (Formula presented.) in the Wilson-Fisher fixed point of the O(N) vector model in d = 4 − ϵ to order ϵ^{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 (Formula presented.) 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.

Original language | English (US) |
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Article number | 117 |

Pages (from-to) | 1-34 |

Number of pages | 34 |

Journal | Journal of High Energy Physics |

Volume | 2015 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2015 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Keywords

- Field Theories in Higher Dimensions
- Field Theories in Lower Dimensions
- Renormalization Group

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## Cite this

*Journal of High Energy Physics*,

*2015*(3), 1-34. [117]. https://doi.org/10.1007/JHEP03(2015)117