Some results of the Mariño-Vafa formula (Q862062)

Let \(\overline{\mathcal M}_{g,n}\) is the Deligne-Mumford moduli stack of stable genus \(g\) curves with \(n\) marked points, and denote by \({\mathbb L}_i\) the line bundle whose fibre at the point \([(C,x_1,\dots,x_n)]\) is \(T^*_{x_i}C\). Also denote by \({\mathbb E}\) the Hodge bundle, i.e. the rank \(g\) bundle with fibre \(H^0(C,\omega_C)\). The cohomology classes \(\psi_i\) and \(\lambda_j\) are defined respectively as \(\psi_i=c_1({\mathbb L}_i)\) and \(\lambda_j=c_j({\mathbb E})\). An integral of the form \[ \int_{\overline{\mathcal M}_{g,n}}\lambda_1^{j_1}\cdots \lambda_g^{j_g}\psi_1^{k_1}\cdots \psi_n^{k_n}, \] is called an Hodge integral. Based on string duality, \textit{M.~Mariño} and \textit{C.~Vafa} [Contemp. Math. 310, 185--204 (2002; Zbl 1042.81071)] conjectured a closed formula for certain Hodge integrals; this formula has then been given a rigorous mathematical proof in [\textit{C.-C.~M.~Liu}, \textit{K.~Liu} and \textit{J.~Zhou}, J. Differ. Geom. 65, No. 2, 289-340 (2003; Zbl 1077.14084)]. The author uses results from [\textit{C.-C.~M.~Liu}, \textit{K.~Liu} and \textit{J.~Zhou}, J. Algebr. Geom. 15, No. 2, 379--398 (2006; Zbl 1102.14018)] to derive some new Hodge integral identities from the Mariño-Vafa formula. More precisely, the following identity is proven: if \(g\geq 2\) and \(1\leq m\leq 2g-3\), then \[ \begin{multlined} -(2g-2-m)!(-1)^{2g-3-m}\int_{\overline{\mathcal M}_{g,1}}\lambda_g\text{ch}_{2g-2-m}({\mathbb E})\psi_1^{m}\\ =b_g\sum_{k=0}^{m-1}\frac{(-1)^{2g-1-k}}{2g-1-k}\binom{2g-1}{k} \binom{2g-1-k}{2g-1-m}B_{2g-1-m}+\\ \frac{1}{2}\sum_{\underset{g_1+g_2=g}{g_1,g_2>0}}b_{g_1}b_{g_2} \sum_{k=0}^{\mu(g_2,m)}\frac{(-1)^{2g_2-1-k}}{2g-1-k}\binom{2g_2-1}{k} \binom{2g-1-k}{2g-1-m}B_{2g-1-m}, \end{multlined} \] where the \(B_n\)'s are the Bernoulli numbers, \(b_g=\frac{|(2^{2g-1}-1)B_{2g}|}{2^{2g-1}(2g)!}\) and \(\mu(g,m)=\min(2g-1,m-1)\). From this formula, the Hodge integral identity \[ \int_{\overline{\mathcal M}_{g,1}}\lambda_1\lambda_g\psi_1^{2g-3}=\frac{1}{12}\bigl(g(2g-3)b_g+b_1b_{g-1}\bigr),\qquad g\geq 2, \] and the vanishing result \[ \int_{\overline{\mathcal M}_{g,1}}\lambda_g\text{ch}_{2t}({\mathbb E})\psi_1^{2(g-1-t)}=0,\qquad g\geq 2,\quad 1\leq t\leq g-1, \] are obtained. The author also derives a recursion formula for \(\lambda_{g-1}\)-integrals and gives two simple proofs of the \(\lambda_g\)- conjecture, \[ \int_{\overline{\mathcal M}_{g,n}}\lambda_g\psi_1^{k_1}\cdots \psi_n^{k_n}=\binom{2g+n-3}{k_1,\dots,k_n}b_g. \] In the final part of the paper some low genus examples of the main identity are computed.