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The open KdV conjecture for open gravitational descendents

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Let Fo(s,t0,t1,…;u)F^o(s,t_0,t_1,\ldots;u) be the open free energy, with double brackets denoting its derivatives with respect to the variables tit_i and ss, and let FcF^c denote the closed free energy. Open KdV conjecture. For every ne1n\mathbf{e} 1, the following equations are satisfied:

(2n+1)u−1⟨⟨τn⟩⟩o=u⟨⟨τn−1τ0⟩⟩c⟨⟨τ0⟩⟩o−u2⟨⟨τn−1τ02⟩⟩c+2⟨⟨τn−1⟩⟩o⟨⟨σ⟩⟩o+2⟨⟨τn−1σ⟩⟩o.(2n+1)u^{-1}{\big\langle}{\big\langle} \tau_n {\big\rangle}{\big\rangle}^o=u{\big\langle}{\big\langle} \tau_{n-1} \tau_0{\big\rangle}{\big\rangle}^c {\big\langle}{\big\langle} \tau_0{\big\rangle}{\big\rangle}^o -\frac{u}{2} {\big\langle}{\big\langle} \tau_{n-1}\tau_0^2 {\big\rangle}{\big\rangle}^c+2 {\big\langle}{\big\langle} \tau_{n-1}{\big\rangle}{\big\rangle}^o {\big\langle}{\big\langle}\sigma{\big\rangle}{\big\rangle}^o + 2 {\big\langle}{\big\langle} \tau_{n-1} \sigma {\big\rangle}{\big\rangle}^o.

The open string and open dilaton equations are known in genus zero and, by the cited context, in all genera; the main open KdV system itself was proved in the referenced later work.

References

Primary source

Ran J. Tessler, “The combinatorial formula for open gravitational descendents”, arXiv:1507.04951 (2022).

gravitational-descendentsintegrable-systemsopen-intersection-theory
Posted

Progress summary

Refreshed
Claimed solved

A 2015 paper reports a proof of the conjecture, but independent verification is not recorded in this scan.

The conjecture asserts that the open gravitational-descendant potential satisfies the stated open KdV equations for every n≥1n \ge 1.

Known results

  • Buryak (2014) proved uniqueness of the formal open KdV solution with the prescribed genus-zero initial condition and established its equivalence with the open Virasoro description.
  • A related 2014 work showed that the open KdV equations and initial data uniquely determine the formal open potential, leaving its identification with the geometric potential as the central statement.

2015 claimed proof

The paper Matrix models and a proof of the open analog of Witten’s conjecture states that it proves the open KdV conjecture; a July 2015 catalogue entry likewise records the conjecture as proved. The supplied sources report no later counterexample, correction, withdrawal, or retraction, but this scan does not independently verify the proof.

Current status (as of September 2026): The conjecture is claimed proved in 2015, while independent verification of that proof is not recorded here and no conflicting result was found.

Sources

Solutions 0

No solutions have been posted yet.