Built on

S. Giombi and I. R. Klebanov, “Interpolating between a a and F F ,” JHEP
Original
1937
Earlier work this paper cites.
K. G. Wilson and M. E. Fisher, “Critical exponents in 3.99 dimensions,” Phys. Rev. Lett
1972
Earlier work this paper cites.
K. G. Wilson and J. B. Kogut, “The Renormalization group and the epsilon expansion,” Phys. Rept
1974
Earlier work this paper cites.
J. Wess and B. Zumino, “Supergauge Transformations in Four-Dimensions,” Nucl. Phys
1974
Earlier work this paper cites.
A. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett
1986
Earlier work this paper cites.
J. L. Cardy, “Is There a c Theorem in Four-Dimensions?,” Phys.Lett
1988
Earlier work this paper cites.
E. Witten, “Topological Quantum Field Theory,” Commun. Math. Phys
1988
Earlier work this paper cites.
E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys
1989
Earlier work this paper cites.
I. Affleck and A. W. W. Ludwig, “Universal noninteger ’ground state degeneracy’ in critical quantum systems,” Phys. Rev. Lett
1991
Earlier work this paper cites.
I. Affleck and A. W. W. Ludwig, “Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity,” Phys. Rev
1993
Earlier work this paper cites.
J. R. Quine and J. Choi, “Zeta regularized products and functional determinants on spheres,” Rocky Mountain J. Math
1996
Earlier work this paper cites.
I. R. Klebanov and A. A. Tseytlin, “Entropy of near extremal black p-branes,” Nucl. Phys
1996
Earlier work this paper cites.
K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett
1996
Earlier work this paper cites.
P. M. Ferreira, I. Jack, and D. R. T. Jones, “The Quasiinfrared fixed point at higher loops,” Phys. Lett
1997
Earlier work this paper cites.
O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler, “Aspects of 𝒩 = 2 {\cal N}=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys
1997
Earlier work this paper cites.
P. M. Ferreira and J. A. Gracey, “The Beta function of the Wess-Zumino model at O ⁡ ( 1 / N 2 ) O(1/N^{2}) ,” Nucl. Phys
1998
Earlier work this paper cites.
P. M. Ferreira and J. A. Gracey, “Nonzeta knots in the renormalization of the Wess-Zumino model?,” Phys. Lett
1998
Earlier work this paper cites.
I. Jack, D. R. T. Jones, and A. Pickering, “The soft scalar mass beta function,” Phys. Lett
1998
Earlier work this paper cites.
H. Kumagai, “The determinant of the Laplacian on the n n -sphere,” Acta Arith
1999
Earlier work this paper cites.
I. Jack and D. R. T. Jones, “Quasiinfrared fixed points and renormalization group invariant trajectories for nonholomorphic soft supersymmetry breaking,” Phys. Rev
2000
Earlier work this paper cites.
V. Borokhov, A. Kapustin, and X.-k. Wu, “Monopole operators and mirror symmetry in three-dimensions,” JHEP
2002
Earlier work this paper cites.
V. Borokhov, A. Kapustin, and X.-k. Wu, “Topological disorder operators in three-dimensional conformal field theory,” JHEP
2002
Earlier work this paper cites.
K. A. Intriligator and B. Wecht, “The Exact superconformal R symmetry maximizes a,” Nucl.Phys
2003
Earlier work this paper cites.
J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP
2004
Earlier work this paper cites.
H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys
2007
Earlier work this paper cites.
D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-Matter theories,” JHEP
Original
2007
Earlier work this paper cites.
O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “ 𝒩 = 6 {\cal N}=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP
Original
2008
Earlier work this paper cites.
R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” Phys.Rev
Original
2010
Earlier work this paper cites.
A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,” JHEP
Original
2010
Earlier work this paper cites.
F. Benini, C. Closset, and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4 singularities,” JHEP
Original
2010
Earlier work this paper cites.
A. Kapustin, B. Willett, and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP
Original
2010
Earlier work this paper cites.
R. Jackiw and S. Y. Pi, “Tutorial on Scale and Conformal Symmetries in Diverse Dimensions,” J. Phys
Original
2011
Earlier work this paper cites.

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Then

D. Gang, C. Hwang, S. Kim, and J. Park, “Tests of AdS 4 /CFT 3 correspondence for 𝒩 = 2 \mathcal{N}=2 chiral-like theory,” JHEP
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2012
Later among the works it cites.
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D. Martelli, A. Passias, and J. Sparks, “The gravity dual of supersymmetric gauge theories on a squashed three-sphere,” Nucl. Phys
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I. R. Klebanov, T. Nishioka, S. S. Pufu, and B. R. Safdi, “Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?,” JHEP
Original
2012
Later among the works it cites.
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2016

The F-Theorem and F-Maximization

Pufu, Silviu S.

Understand

This contribution contains a review of the role of the three-sphere free energy F in recent developments related to the F-theorem and F-maximization.

The F-theorem states that for any Lorentz-invariant RG trajectory connecting a conformal field theory CFT_UV in the ultraviolet to a conformal field theory CFT_IR, the F-coefficient decreases: F_UV > F_IR.
I provide many examples of CFTs where one can compute F, approximately or exactly, and discuss various checks of the F-theorem.
F-maximization is the principle that in an N=2 SCFT, viewed as the deep IR limit of an RG trajectory preserving N=2 supersymmetry, the superconformal R-symmetry maximizes F within the set of all R-symmetries preserved by the RG trajectory.

Built on

S. Giombi and I. R. Klebanov, “Interpolating between a a and F F ,” JHEP
Original
1937
Earlier work this paper cites.
K. G. Wilson and M. E. Fisher, “Critical exponents in 3.99 dimensions,” Phys. Rev. Lett
1972
Earlier work this paper cites.
K. G. Wilson and J. B. Kogut, “The Renormalization group and the epsilon expansion,” Phys. Rept
1974
Earlier work this paper cites.
J. Wess and B. Zumino, “Supergauge Transformations in Four-Dimensions,” Nucl. Phys
1974
Earlier work this paper cites.
A. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett
1986
Earlier work this paper cites.
J. L. Cardy, “Is There a c Theorem in Four-Dimensions?,” Phys.Lett
1988
Earlier work this paper cites.
E. Witten, “Topological Quantum Field Theory,” Commun. Math. Phys
1988
Earlier work this paper cites.
E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys
1989
Earlier work this paper cites.
I. Affleck and A. W. W. Ludwig, “Universal noninteger ’ground state degeneracy’ in critical quantum systems,” Phys. Rev. Lett
1991
Earlier work this paper cites.
I. Affleck and A. W. W. Ludwig, “Exact conformal-field-theory results on the multichannel Kondo effect: Single-fermion Green’s function, self-energy, and resistivity,” Phys. Rev
1993
Earlier work this paper cites.
J. R. Quine and J. Choi, “Zeta regularized products and functional determinants on spheres,” Rocky Mountain J. Math
1996
Earlier work this paper cites.
I. R. Klebanov and A. A. Tseytlin, “Entropy of near extremal black p-branes,” Nucl. Phys
1996
Earlier work this paper cites.
K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett
1996
Earlier work this paper cites.
P. M. Ferreira, I. Jack, and D. R. T. Jones, “The Quasiinfrared fixed point at higher loops,” Phys. Lett
1997
Earlier work this paper cites.
O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg, and M. J. Strassler, “Aspects of 𝒩 = 2 {\cal N}=2 supersymmetric gauge theories in three-dimensions,” Nucl. Phys
1997
Earlier work this paper cites.
P. M. Ferreira and J. A. Gracey, “The Beta function of the Wess-Zumino model at O ⁡ ( 1 / N 2 ) O(1/N^{2}) ,” Nucl. Phys
1998
Earlier work this paper cites.
P. M. Ferreira and J. A. Gracey, “Nonzeta knots in the renormalization of the Wess-Zumino model?,” Phys. Lett
1998
Earlier work this paper cites.
I. Jack, D. R. T. Jones, and A. Pickering, “The soft scalar mass beta function,” Phys. Lett
1998
Earlier work this paper cites.
H. Kumagai, “The determinant of the Laplacian on the n n -sphere,” Acta Arith
1999
Earlier work this paper cites.
I. Jack and D. R. T. Jones, “Quasiinfrared fixed points and renormalization group invariant trajectories for nonholomorphic soft supersymmetry breaking,” Phys. Rev
2000
Earlier work this paper cites.
V. Borokhov, A. Kapustin, and X.-k. Wu, “Monopole operators and mirror symmetry in three-dimensions,” JHEP
2002
Earlier work this paper cites.
V. Borokhov, A. Kapustin, and X.-k. Wu, “Topological disorder operators in three-dimensional conformal field theory,” JHEP
2002
Earlier work this paper cites.
K. A. Intriligator and B. Wecht, “The Exact superconformal R symmetry maximizes a,” Nucl.Phys
2003
Earlier work this paper cites.
J. H. Schwarz, “Superconformal Chern-Simons theories,” JHEP
2004
Earlier work this paper cites.
H. Casini and M. Huerta, “A c-theorem for the entanglement entropy,” J. Phys
2007
Earlier work this paper cites.
D. Gaiotto and X. Yin, “Notes on superconformal Chern-Simons-Matter theories,” JHEP
Original
2007
Earlier work this paper cites.
O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “ 𝒩 = 6 {\cal N}=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP
Original
2008
Earlier work this paper cites.
R. C. Myers and A. Sinha, “Seeing a c-theorem with holography,” Phys.Rev
Original
2010
Earlier work this paper cites.
A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,” JHEP
Original
2010
Earlier work this paper cites.
F. Benini, C. Closset, and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4 singularities,” JHEP
Original
2010
Earlier work this paper cites.
A. Kapustin, B. Willett, and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities,” JHEP
Original
2010
Earlier work this paper cites.
R. Jackiw and S. Y. Pi, “Tutorial on Scale and Conformal Symmetries in Diverse Dimensions,” J. Phys
Original
2011
Earlier work this paper cites.