Abstract
Recently, the conformal-bootstrap has been successfully used to obtain generic bounds on the spectrum and OPE coefficients of unitary conformal field theories. In practice, these bounds are obtained by assuming the existence of a scalar operator in the theory and analyzing the crossing-symmetry constraints of its 4-point function. In $\mathcal{N}=1$ superconformal theories with a global symmetry there is always a scalar primary operator, which is the top of the current-multiplet. In this paper we analyze the crossing-symmetry constraints of the 4-point function of this operator for $\mathcal{N}=1$ theories with $SU(N)$ global symmetry. We analyze the current-current OPE, and derive the superconformal blocks, generalizing the work of Fortin, Intrilligator and Stergiou to the non-Abelian case and finding new superconformal blocks which appear in the Abelian case. We then use these results to obtain bounds on the coefficient of the current 2-point function.
Highlights
It would interesting to apply these methods without introducing any assumption on the operator spectrum
Recently, the conformal-bootstrap has been successfully used to obtain generic bounds on the spectrum and OPE coefficients of unitary conformal field theories. These bounds are obtained by assuming the existence of a scalar operator in the theory and analyzing the crossing-symmetry constraints of its 4-point function
One would like to analyze the four-point function of the stress-tensor, which exists for any CFT
Summary
We discuss a specific case of the general bootstrap constraint (2.6) in which φa is a real scalar primary in the adjoint representation of SU(N ). The operators which appear in the φa × φb OPE can be decomposed into any of the 7 irreducible representations in the product of two adjoint representations of SU(N ). Each such representation arises from either the symmetric or anti-symmetric product. Operators in the φa × φb OPE which are in a (anti-)symmetric representation must be of (odd) even spin from Bose symmetry. The reader is referred to appendix A for details regarding the tensor product of two SU(N ) adjoint representations and our notations
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