TY - JOUR

T1 - Interpolating between a and F

AU - Giombi, Simone

AU - Klebanov, Igor R.

N1 - Funding Information:
We thank L. Fei, S. Pufu, B. Safdi, G. Tarnopolsky and I. Yaakov for useful discussions. The work of SG was supported in part by the US NSF under Grant No. PHY-1318681. The work of IRK was supported in part by the US NSF under Grant No. PHY-1314198.
Publisher Copyright:
© 2015, The Author(s).

PY - 2015/3/1

Y1 - 2015/3/1

N2 - 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/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 (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 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 (Formula presented.) 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.

AB - 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/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 (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 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 (Formula presented.) 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.

KW - Field Theories in Higher Dimensions

KW - Field Theories in Lower Dimensions

KW - Renormalization Group

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U2 - 10.1007/JHEP03(2015)117

DO - 10.1007/JHEP03(2015)117

M3 - Article

AN - SCOPUS:84940971764

SN - 1126-6708

VL - 2015

SP - 1

EP - 34

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 3

M1 - 117

ER -