<p>An explicit analytic formula is presented that computes the conformal (super-)block decomposition of any free scalar or half-BPS diagram in 1d, 2d or 4d CFTs, with various supersymmetries, including none. We prove our formula by exploiting a connection between conformal blocks and symmetric polynomials. Then, we give a direct application of our result to the study of four-point correlators in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{N}=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">N</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> SYM at strong coupling. In particular, we give a CFT proof of the tree-level Witten diagram representation of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\langle \mathcal{O}_2^2\mathcal{O}_2^2 \mathcal{O}_q\mathcal{O}_q\rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msubsup> <mi mathvariant="script">O</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msubsup> <mi mathvariant="script">O</mi> <mn>2</mn> <mn>2</mn> </msubsup> <msub> <mi mathvariant="script">O</mi> <mi>q</mi> </msub> <msub> <mi mathvariant="script">O</mi> <mi>q</mi> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text {AdS}_5\times \text {S}^5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>AdS</mtext> <mn>5</mn> </msub> <mo>×</mo> <msup> <mtext>S</mtext> <mn>5</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, providing new and highly non trivial checks of the AdS/CFT correspondence. Our method works for a more general class of multi-particle correlators and can be used to bootstrap new results at strong coupling.</p>

A formula for the block expansion in free CFTs and applications to $\mathcal{N}=4$ SYM at strong coupling

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发表日期：

2026年3月26日

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F. Aprile,
J. M. Drummond,
P. J. Heslop,
M. Santagata

摘要

An explicit analytic formula is presented that computes the conformal (super-)block decomposition of any free scalar or half-BPS diagram in 1d, 2d or 4d CFTs, with various supersymmetries, including none. We prove our formula by exploiting a connection between conformal blocks and symmetric polynomials. Then, we give a direct application of our result to the study of four-point correlators in $\mathcal{N}=4$ $N = 4$ SYM at strong coupling. In particular, we give a CFT proof of the tree-level Witten diagram representation of $\langle \mathcal{O}_2^2\mathcal{O}_2^2 \mathcal{O}_q\mathcal{O}_q\rangle $ $⟨ O_{2}^{2} O_{2}^{2} O_{q} O_{q} ⟩$ on $\text {AdS}_5\times \text {S}^5$ ${AdS}_{5} \times S^{5}$ , providing new and highly non trivial checks of the AdS/CFT correspondence. Our method works for a more general class of multi-particle correlators and can be used to bootstrap new results at strong coupling.

A formula for the block expansion in free CFTs and applications to \(\mathcal{N}=4\) SYM at strong coupling

摘要