Shklyarov's normalization conjecture for the higher residue pairing

About 7 years old · traced to 2019

Let $Q$ be the polynomial ring and $f$ a polynomial as in the source, let $n$ be the relevant degree, and let $c_f\in\mathbb{C}$ be the constant such that the canonical pairing on $HN_n(mf(Q,f))$ corresponds under the Hochschild–Kostant–Rosenberg isomorphism $I_f$ to $c_f u^n$ times the higher residue pairing $K_f$ on $H_f^{(0)}$ . Shklyarov's conjecture. For any $f$ ,

c_f = (-1)^{n(n+1)/2}.

The prediction specifies the universal normalization constant relating the canonical pairing and the higher residue pairing under the HKR isomorphism. The supplied text gives no resolution status for this prediction.

References

Primary source

Michael K. Brown and Mark E. Walker, “A proof of a conjecture of Shklyarov”, arXiv:1909.04088 (2020).

higher-residue-pairings hochschild-homology matrix-factorizations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2019 paper claims to prove the predicted universal normalization constant, and later work repeats the claim, but this report does not independently verify it.

Shklyarov conjectured that the canonical pairing differs from the higher residue pairing by the universal factor $c_f=(-1)^{n(n+1)/2}$ for every polynomial $f$ in the stated setting.

September 2019 claimed proof

M. Brown and M. Walker’s preprint A proof of a conjecture of Shklyarov states in Theorem $1.8$ that the formula holds for every permitted $f$ , including the polynomial-ring-over- $\mathbb{C}$ case. A 2023 paper attributes the first proof to Brown and Walker, notes an independent proof by B. Kim, and gives another proof. These are reported as proofs in the sources but remain unverified by this report.

Current status (as of September 2026): The formula $c_f=(-1)^{n(n+1)/2}$ is claimed proved by Brown and Walker, with independent and later proofs reported, but this automated report does not certify the proofs.

Sources

No solutions have been posted yet.