Interpolating between a and F

By S. Giombi, I. Klebanov. 2014

160 citations recorded by the source corpus.

A bstractWe study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension d we define the quantity F˜$$ \tilde{F} $$ =sin(πd/2) log Z, where Z is the path integral of the Euclidean CFT on the d-dimensional round sphere. F˜$$ \tilde{F} $$ smoothly interpolates between (−1)d/2π/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 F˜$$ \tilde{F} $$ 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 ϵ expansion. For all these examples F˜$$ \tilde{F} $$ is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate F˜$$ \tilde{F} $$ in the Wilson-Fisher fixed point of the O(N) vector model in d = 4 − ϵ to order ϵ4. 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 F˜$$ \tilde{F} $$ may be calculated exactly using an appropriate version of localization on Sd. Our approach provides an interpolation between the a-maximization in d = 4 and the F-maximization in d = 3.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/ca9e2f26255eb448dfc6145d6cf63be2647a14f4

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In Nuclear and High Energy Physics

Citations in the corpus

Bootstrapping SCFTs with four supercharges (2015, methodology): “It could also be interesting to see if CT can be derived using localization in continuous d, in the spirit of [61], and matched with our results for the cWZ model or used as a tool to probe the other kinks.”
Three loop analysis of the critical O (N) models in 6-ε dimensions (2014, background): “3), the leading order terms are found to be [1, 33]”
Notes on the complex Sachdev-Ye-Kitaev model (2019, methodology): “We note that this procedure is similar to that used to compute the influence of double trace operators on the free energy in the AdS/CFT correspondence [40–45].”
Entanglement entropy: Holography and renormalization group (2018, background): “The entropic C-theorems in d ≥ 4 dimensions 47 F. Holographic RG flow 48 1.”