For generic coupling beta=-epsilon_1/epsilon_2, the four-point conformal block for the SU(2) theory with four fundamental hypermultiplets equals, coefficientwise at every instanton level, the corresponding Nekrasov instanton partition function under the standard AGT parameter identification: F^(4)_c,Delta(Delta_1,Delta_2,Delta_3,Delta_4;q)=Z_inst^SU(2),N_f=4(a,m_1,m_2,m_3,m_4;epsilon_1,epsilon_2;q) for all powers of q.

AGT Conjecture

The conjecture asserts equality, coefficient by coefficient, between a four-point conformal-block expansion and the instanton partition function of the four-flavour $\mathrm{SU}(2)$ theory at generic coupling. The retrieved literature records special-case proofs and earlier all-coupling claims, but no independently verified proof.

Known results

$\beta=1$ , equivalently $\epsilon_1+\epsilon_2=0$ : direct proof for the four-flavour $\mathrm{SU}(2)$ case (Mironov et al., 2010).
A generalized-Jack-polynomial paper claimed the arbitrary- $\beta$ identity and called it a final proof (Mironov et al., 2013), but the retrieved sources provide no independent verification.

August 2026 all-level proof claim

On August 27, 2026, Le-Feng Chen and Kilar Zhang's preprint Proof of the AGT Conjecture at Generic $\beta$ claimed an all-level proof: generalized Jack-polynomial Selberg averages are factorized and matched term by term with Nekrasov fixed-point contributions. The claim would settle the stated conjecture, but remains unverified.

Current status (as of August 2026): special cases are established, while both the 2013 arbitrary- $\beta$ proof claim and the newer all-level claim remain independently unverified.

AGT Conjecture

Primary source

Additional references

Progress summary

Solutions 0