Homological algebra of knots and BPS states.

*(English)*Zbl 1296.57014
Kirby, Rob (ed.) et al., Proceedings of the Freedman Fest. Based on the conference on low-dimensional manifolds and high-dimensional categories, Berkeley, CA, USA, June 6–10, 2011 and the Freedman symposium, Santa Barbara, CA, USA, April 15–17, 2011 dedicated to Mike Freedman on the occasion of his 60th birthday. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 18, 309-367 (2012).

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 \(P^{\mathfrak{g},R}(K;q)\).

Note that \(P^{\mathfrak{sl}_2(\mathbb{C}),V}(K;q)\) is the Jones polynomial [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 [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 [P. Freyd et al., Bull. Am. Math. Soc., New Ser. 12, 239–246 (1985; Zbl 0572.57002)], [J. H. Przytycki and P. Traczyk, Kobe J. Math. 4, No. 2, 115–139 (1987; Zbl 0655.57002)].

Motivated 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, \[ P^{S^r}(K;a,q)=\sum_{i,j,k}(-1)^ka^iq^j\dim\mathcal{H}^{S^r}_{i,j,k}(K), \] 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 \(\mathcal{H}^{S^m}_{i,j,k}(K)\).

The 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” \(\mathcal{H}^{S^r}_{i,j,\ast}(K)\cong\mathcal{H}^{\Lambda^r}_{i,-j,\ast}(K)\).

For the entire collection see [Zbl 1253.00022].

Note that \(P^{\mathfrak{sl}_2(\mathbb{C}),V}(K;q)\) is the Jones polynomial [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 [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 [P. Freyd et al., Bull. Am. Math. Soc., New Ser. 12, 239–246 (1985; Zbl 0572.57002)], [J. H. Przytycki and P. Traczyk, Kobe J. Math. 4, No. 2, 115–139 (1987; Zbl 0655.57002)].

Motivated 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, \[ P^{S^r}(K;a,q)=\sum_{i,j,k}(-1)^ka^iq^j\dim\mathcal{H}^{S^r}_{i,j,k}(K), \] 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 \(\mathcal{H}^{S^m}_{i,j,k}(K)\).

The 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” \(\mathcal{H}^{S^r}_{i,j,\ast}(K)\cong\mathcal{H}^{\Lambda^r}_{i,-j,\ast}(K)\).

For the entire collection see [Zbl 1253.00022].

Reviewer: Hitoshi Murakami (Tokyo)

##### MSC:

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |