Homological algebra of knots and BPS states (Q2845404)

Let \(P^{\mathfrak{g},R}(K;q)\in\mathbb{Z}[q,q^{-1}]\) be the quantum invariant for a knot in the three-dimensional sphere associated with a Lie algebra \(\mathfrak{g}\) and its representation \(R\), normalized so that \(P^{\mathfrak{g},R}(U;q)=1\) for the unknot \(U\). A categorification of \(P^{\mathfrak{g},R}(K;q)\) is a doubly graded homology theory \(\mathcal{H}^{\mathfrak{g},R}_{j,k}(K)\) (over \(\mathbb{Q}\)) such that its graded Euler characteristic \(\displaystyle\sum_{j,k}(-1)^{k}q^j\dim\mathcal{H}^{\mathfrak{g},R}_{j,k}(K)\) equals \(P^{\mathfrak{g},R}(K;q)\).NEWLINENEWLINENote that \(P^{\mathfrak{sl}_2(\mathbb{C}),V}(K;q)\) is the Jones polynomial [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)], where \(V\) is the standard \(2\)-dimensional representation of \(\mathfrak{sl}_2(\mathbb{C})\), and its categorification is known as the Khovanov homology [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)]. If \(S^r\) is the \(r\)-fold symmetric power of the standard representation of \(\mathfrak{sl}_N(\mathbb{C})\), we put \(P^{S^r}_N(K;q):=P^{\mathfrak{sl}_N(\mathbb{C}),S^r}(K;q)\). Let \(P^{S^r}(K;a,q)\) be the \(S^r\)-colored HOMFLY polynomial, that is, it satisfies the equality \(P^{S^r}(K;q^N,q)=P^{S^r}_N(K;q)\). Note that \(P^{S^1}(K;a,q)\) is the original HOMFLY polynomial [\textit{P. Freyd} et al., Bull. Am. Math. Soc., New Ser. 12, 239--246 (1985; Zbl 0572.57002)], [\textit{J. H. Przytycki} and \textit{P. Traczyk}, Kobe J. Math. 4, No. 2, 115--139 (1987; Zbl 0655.57002)].NEWLINENEWLINEMotivated by BPS state theory in physics, the authors conjecture that for any positive integer \(r\) there exists a triply graded homology theory \(\mathcal{H}^{S^r}_{i,j,k}(K)\) that categorifies the \(S^r\)-colored HOMFLY polynomial, that is, NEWLINE\[NEWLINEP^{S^r}(K;a,q)=\sum_{i,j,k}(-1)^ka^iq^j\dim\mathcal{H}^{S^r}_{i,j,k}(K),NEWLINE\]NEWLINE together with two sets of differentials \(\{d^{S^r}_N\}\) (\(N\in\mathbb{Z}\)) and \(\{d_{r\to m}\}\) (\(1\leq m<r\)) such that (1) if \(N>1\), the homology \(\mathcal{H}^{S^r}_{i,j,k}(K)\) with respect to \(d^{S^r}_N\) is isomorphic to \(\mathcal{H}^{\mathfrak{sl}_N(\mathbb{C}),S^r}_{j,k}(K)\), (2) the differentials \(\{d^{S^r}_N\}\) anticommute, that is, \(d^{S^r}_Nd^{S^r}_M=-d^{S^r}_Md^{S^r}_N\), (3) the homology has finite support, that is, \(\dim\mathcal{H}^{S^r}_{i,j,k}(K)<\infty\), (4) the homologies with respect to \(d^{S^r}_1\) and \(d^{S^r}_{-r}\) are one-dimensional, (5) the homology with respect to \(d^{S^r}_{1-k}\) for \(1\leq k\leq r-1\) is isomorphic to the \(S^k\)-colored homology after some regrading, and (6) the homology \(\mathcal{H}^{S^r}_{i,j,k}(K)\) with respect to \(d_{r\to m}\) is isomorphic to the \(S^m\)-colored HOMFLY homology \(\mathcal{H}^{S^m}_{i,j,k}(K)\).NEWLINENEWLINEThe authors give experimental calculations for knots with few crossings and show that the properties (1)--(6) are sufficient to give such a homology for \(r=2,3\). They also propose the existence of a similar homology theory \(\mathcal{H}^{\Lambda^r}_{i,j,k}(K)\) for the antisymmetric power \(\Lambda^r\) and the ``mirror symmetry'' \(\mathcal{H}^{S^r}_{i,j,\ast}(K)\cong\mathcal{H}^{\Lambda^r}_{i,-j,\ast}(K)\).NEWLINENEWLINEFor the entire collection see [Zbl 1253.00022].