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JMg,n where tpi = ci(L?) is the first Chern class of L?, and Xj = Cj(E) is the j-th Chern class of the Hodge bundle. The study of Hodge integrals is an important part of intersection theory on Mg,n> Hodge integrals also naturally arise when one computes Gromov-Witten invariants by localization techniques. For example, the following generating series of Hodge integrals arises when one computes local invariants of a toric Fano surface in a Calabi-Yau 3-fold by virtual localization [33]

We derive some Hodge integral identities by taking various limits of the Mariño-Vafa formula using the cut-and-join equation. These identities include the formula of general λ g \lambda _g -integrals, the formula of λ g − 1 \lambda _{g-1} -integrals on M ¯ g , 1 {\overline {\mathcal {M}}}_{g,1} , the formula of cubic λ \lambda integrals on M ¯ g {\overline {\mathcal {M}}}_g , and the ELSV formula relating Hurwitz numbers and Hodge integrals. In particular, our proof of the MV formula by the cut-and-join equation leads to a new and simple proof of the λ g \lambda _g conjecture. We also present a proof of the ELSV formula completely parallel to our proof of the Mariño-Vafa formula.

A spacetime is a four-manifold with a pseudo-metric of signature (+,+,-+-,?). A hypersurface or a 2-surface in a spacetime is spacelike if the induced metric is pos itive definite. A quasi-local energy-momentum vector is a vector in R3'1 associated to a spacelike 2-surface which depends on its first and second fundamental forms and the connection to its normal bundle in the spacetime. The time component of the four-vector is called quasi-local energy (mass). Similar to [10, 8], we require the quasi-local energy-momentum vector to satisfy the following properties. (1) It should be zero for the flat spacetime. (2) The quasi-local mass should be equivalent to the standard definition if the spacetime is spherically symmetric and the quasi-local mass is evaluated on the spheres [7]. (We say that two masses mi and m2 are equivalent if there is a universal constant c > 0 such that c~lm\ /Area(E)/167r. (6) The quasi-local energy-momentum vector should be nonspacelike and the quasi-local mass should be nonnegative.