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<p>We derive a compact analytic formula for a complete basis of conformally invariant tensor structures for three-point functions of conserved operators in arbitrary 4D Lorentz representations. The construction follows directly from a novel constraint equivalent to applying conservation conditions at each point: the leading terms in all OPE limits appear as symmetric traceless tensors. We derive this by lifting to a unified SU(<i>m</i>, <i>m</i>|2<i>n</i>) analytic superspace framework, where the conservation conditions are automatically solved and then reducing back to 4D CFT. The same method is also used for cases involving one non-conserved operator. This formalism further reveals a map of the counting of CFT tensor structures to that of finite-dimensional SU(2<i>n</i>) representations, solved by Littlewood-Richardson coefficients. All results can be directly re-interpreted as three-point <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 2 and <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = 4 superconformal tensor structures via the unified analytic superspace.</p>

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A compact formula for conserved three-point tensor structures in 4D CFT

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月23日

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  • Paul Heslop,
  • Hector Puerta-Ramisa

摘要

We derive a compact analytic formula for a complete basis of conformally invariant tensor structures for three-point functions of conserved operators in arbitrary 4D Lorentz representations. The construction follows directly from a novel constraint equivalent to applying conservation conditions at each point: the leading terms in all OPE limits appear as symmetric traceless tensors. We derive this by lifting to a unified SU(m, m|2n) analytic superspace framework, where the conservation conditions are automatically solved and then reducing back to 4D CFT. The same method is also used for cases involving one non-conserved operator. This formalism further reveals a map of the counting of CFT tensor structures to that of finite-dimensional SU(2n) representations, solved by Littlewood-Richardson coefficients. All results can be directly re-interpreted as three-point N \( \mathcal{N} \) = 2 and N \( \mathcal{N} \) = 4 superconformal tensor structures via the unified analytic superspace.