Bounds on 4D Conformal and Superconformal Field Theories

Abstract

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. In any CFT containing a scalar primary phi of dimension d we show that crossing symmetry of <phi phi phi phi> implies a completely general lower bound on the central charge c >= f_c(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients tau^{IJ} and flavor charges. We extend these bounds to N=1 superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Phi and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Phi x Phi* OPE, and show that there is an upper bound on the dimension of Phi* Phi when dim(Phi) is close to 1. We also present even more stringent bounds on c and tau^{IJ}. In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.

47 pages, 9 figures; V2: small corrections and clarifications

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• Central Charge Bounds in 4D Conformal Field Theory
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• Bootstrapping phase transitions in QCD and frustrated spin systems
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• A Semidefinite Program Solver for the Conformal Bootstrap
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• JuliBootS: a hands-on guide to the conformal bootstrap
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• Conformal Bootstrap Approach to O(N) Fixed Points in Five Dimensions
• Numerical determination of OPE coefficients in the 3D Ising model from off-critical correlators
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