Tautological classes and symmetry in Khovanov-Rozansky homology (Q6620664)

In 2005 \textit{N. M. Dunfield} et al. [Exp. Math. 15, No. 2, 129--159 (2006; Zbl 1118.57012)] proposed a remarkable conjecture concerning the structure of triply graded Khovanov-Rozansky link homology [\textit{M. Khovanov}, Int. J. Math. 18, No. 8, 869--885 (2007; Zbl 1124.57003); \textit{M. Khovanov} and \textit{L. Rozansky}, Geom. Topol. 12, No. 3, 1387--1425 (2008; Zbl 1146.57018)] categorifying HOMFLY-PT polynomial, which goes as follows.\N\NConjecture. Let \(K\)\ be a knot, and let\N\[\N\overline{\mathrm{HHH}}\left( K\right) =\bigoplus\overline{\mathrm{HHH}}_{i,j,k}\left( K\right)\N\]\Nbe its reduced triply graded homology, where \(i\)\ is the \(a\)-grading, \(j\)\ is the quantum grading, and \(k\)\ is the homological grading. Then\N\[\N\dim \overline{\mathrm{HHH}}_{i,-2j,k}\left( K\right) =\dim \overline {\mathrm{HHH}}_{i,2j,k+2j}\left( K\right)\N\]\N\NThe conjecture was motivated by the well-known symmetry of the HOMFLY-PT polynomial\N\[\N\overline{P}_{K}\left( a,q\right) =\overline{P}_{K}\left( a,q^{-1}\right)\N\]\Nwhere\N\[\N\overline{P}_{K}\left( a,q\right) =\sum\limits_{i,j,k}a^{i}q^{j}\left( -1\right) ^{k}\dim\,\overline{\mathrm{HHH}}_{i,j,k}\left( K\right)\N\]\NThe conjecture was verified in numerous examples in [\textit{N. M. Dunfield} et al., Exp. Math. 15, No. 2, 129--159 (2006; Zbl 1118.57012)]. and it was later related to deep results about algebraic geometry of compactified Jacobians [\textit{A. Oblomkov} et al., Geom. Topol. 22, No. 2, 645--691 (2018; Zbl 1388.14087); \textit{A. Oblomkov} and \textit{V. Shende}, Duke Math. J. 161, No. 7, 1277--1303 (2012; Zbl 1256.14025)], Hilbert schemes of points on the plane [\textit{E. Gorsky} and \textit{M. Hogancamp}, Geom. Topol. 26, No. 2, 587--678 (2022; Zbl 1508.14005); \textit{E. Gorsky} and \textit{A. Neguţ}, J. Math. Pures Appl. (9) 104, No. 3, 403--435 (2015; Zbl 1349.14012); \textit{E. Gorsky} et al., Adv. Math. 378, Article ID 107542, 116 p. (2021; Zbl 1459.57018); \textit{A. Oblomkov} and \textit{L. Rozansky}, Sel. Math., New Ser. 24, No. 3, 2351--2454 (2018; Zbl 1404.57018); \textit{A. Oblomkov} and \textit{L. Rozansky}, ``Soergel bimodules and matrix factorizations'', Preprint, \url{arXiv:2010.14546}], representation theory of rational Cherednik algebras [\textit{E. Gorsky} et al., Duke Math. J. 163, No. 14, 2709--2794 (2014; Zbl 1318.57010)], and combinatorics of \(q,t\)-Catalan numbers [\textit{B. Elias} and \textit{M. Hogancamp}, Compos. Math. 155, No. 1, 164--205 (2019; Zbl 1477.57013); \textit{E. A. Gorsky}, Contemp. Math. 566, 213--232 (2012; Zbl 1294.57007); \textit{E. Gorsky} and \textit{M. Mazin}, J. Comb. Theory, Ser. A 120, No. 1, 49--63 (2013; Zbl 1252.05009); \textit{M. Hogancamp}, ``Khovanov-Rozansky homology and higher Catalan sequences'', Preprint, \url{arXiv:1704.01562}; \textit{M. Hogancamp} and \textit{A. Mellit}, ``Torus link homology'', Preprint, \url{arXiv:1909.00418}].\N\N\textit{A. Oblomkov} and \textit{L. Rozansky} [Sel. Math., New Ser. 24, No. 3, 2351--2454 (2018; Zbl 1404.57018); Transform. Groups 24, No. 2, 531--544 (2019; Zbl 1439.57031); Transform. Groups 28, No. 3, 1245--1275 (2023; Zbl 07773336); ``Dualizable link homology'', Preprint, \url{arXiv:1905.06511}; ``Soergel bimodules and matrix factorizations'', Preprint, \url{arXiv:2010.14546}] resolved the conjecture in general, while \textit{P. Galashin} and \textit{T. Lam} [Sémin. Lothar. Comb. 85B, Article 54, 12 p. (2021; Zbl 1503.14044)] proved it for a class of knots related to positroid varieties. Both approaches used highly heavy machinery from geometric representation theory such as matrix factorizations on Hilbert schemes of points and graded Koszul duality for category \(\mathcal{O}\).\N\NThis paper gives a more direct algebraic proof of the conjecture. The first and second authors [Geom. Topol. 26, No. 2, 587--678 (2022; Zbl 1508.14005)] proposed a solution to the conjecture by introducing \(y\)-ified link homology \(\mathrm{HY}\left( L\right) \), which is naturally a module over \(\mathbb{C}\left[ x_{1},\ldots,x_{c},y_{1},\ldots,y_{c}\right] \), where \(c\)\ is the number of components of \(L\). This paper proves (Theorem 7.7) that the \(y\)-ified link homology is indeed symmetric in the sense of the conjecture, and this symmetry exchanges the actions of \(x_{i}\)\ and \(y_{i}\).\N\NThe key idea of the proof comes from the recent paper of the curious hard Lefshetz property for character varieties by the third author [``Cell decompositions of character varieties'', Preprint, \url{arXiv:1905.10685}], in which, to a positive braid \(\beta\)\ on \(n\)\ strands one can associate a character variety \(X_{\beta}\), whose homology is closedly related to the Khovanov-Rozansky homology of the closure of \(\beta\) [\textit{B. Webster} and \textit{G. Williamson}, Geom. Topol. 12, No. 2, 1243--1263 (2008; Zbl 1198.20037)]. Given a symmetric function \(Q\left( x_{1},\ldots,x_{n}\right) \) of degree \(r\), one can define a closed algebraic \(\left( 2r-2\right) \)-form \(u_{Q}\)\ on the character variety \(X_{\beta}\), which represents a certain tautological homology class. The main result of [\textit{A. Mellit}, ``Cell decompositions of character varieties'', Preprint, \url{arXiv:1905.10685}] claims that cup product with certain powers of \(u_{Q}\)\ satisfies the curious hard Lefshetz property. The proof goes by using a geometric analogue of the skein relation to decompose the varieties into strata, and verifying the Lefschetz property on each stratum by a direct computation.