Kefeng Liu

A mathematical theory of the topological vertex

We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing Gromov-Witten invariants of smooth toric Calabi-Yau threefolds derived from duality between open string theory of smooth Calabi-Yau threefolds and Chern-Simons theory on three manifolds.

A formula of two-partition Hodge integrals

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]:

On elliptic genera and theta-functions

The main purpose of this paper is to give a simple and unified new proof of the Witten rigidity theorems, which were conjectured by Witten and first proved by Taubes [T], Bott-Taubes [BT], Hirzebruch [H] and Krichever [Kr]. Our proof shows that the modular invariance, which is the intrinsic symmetry of elliptic genera, actually implies their rigidity. Some new properties of elliptic genera and their relationships with theta-functions are also discussed. We remark that our proof makes essential uses of the new feature of loop groups and loop spaces, the modular invariance. We note that, with the help of the modular group, we can catch the topological information on loop space by simply working on finite dimensional manifold. By developing this idea further, in [Liu1] we have proved the rigidity of the Dirac operator on loop space twisted by higher level loop group representations, while the Witten rigidity theorems are the special cases of level 1. Many topological vanishing theorems are also derived in [Liu1] by refining the argument in this paper, especially an Â-vanishing theorem for loop space. In [Liu2] modular invariance is used again to establish a general miraculous cancellation formula, relating the Hirzebruch L-form to certain twisted Â-forms, which has many interesting topological results as consequences. These results were anounced in [Liu3]. Let X be a smooth compact spin manifold admitting a circle action and P be an elliptic operator on X commuting with the action. Then both the kernal and the cokernal of P are S-modules. The Lefschetz number, or the

Modular invariance and characteristic numbers

We prove that a general miraculous cancellation formula, the divisibility of certain characteristic numbers, and some other topological results related to the generalized Rochlin invariant, the η-invariant and the holonomies of certain determinant line bundles, are consequences of the modular invariance of elliptic operators on loop space.

ON FAMILY RIGIDITY THEOREMS, I

Note that Ind(P ) is a virtual G-representation. Let chg(Ind(P )) with g ∈G be the equivariant Chern character of Ind(P ) evaluated at g. In this paper, we first prove a family fixed-point formula that expresses chg(Ind(P )) in terms of the geometric data on the fixed points X of the fiber of π . Then by applying this formula, we generalize the Witten rigidity theorems and several vanishing theorems proved in [Liu3] for elliptic genera to the family case. Let G = S1. A family elliptic operator P is called rigid on the equivariant Chern character level with respect to this S1-action, if chg(Ind(P )) ∈H ∗(B) is independent of g ∈ S1. When the base B is a point, we recover the classical rigidity and vanishing theorems. When B is a manifold, we get many nontrivial higher-order rigidity and vanishing theorems by taking the coefficients of certain expansion of chg . For the history of the Witten rigidity theorems, we refer the reader to [T], [BT], [K], [L2], [H], [Liu1], and [Liu4]. The family vanishing theorems that generalize those vanishing theorems in [Liu3], which in turn give us many higher-order vanishing theorems in the family case. In a forthcoming paper, we extend our results to general loop group representations and prove much more general family vanishing theorems that generalize the results in [Liu3]. We believe there should be some applications of our results to topology and geometry, which we hope to report on a later occasion. This paper is organized as follows. In Section 1, we prove the equivariant family index theorem. In Section 2, we prove the family rigidity theorem. In the last part of Section 2, motivated by the family rigidity theorem, we state a conjecture. In Section 3, we generalize the family rigidity theorem to the nonzero anomaly case. As corollaries, we derive several vanishing theorems.

GEOMETRY OF HERMITIAN MANIFOLDS

On Hermitian manifolds, the second Ricci curvature tensors of various metric connections are closely related to the geometry of Hermitian manifolds. By refining the Bochner formulas for any Hermitian complex vector bundle (Riemannain real vector bundle) with an arbitrary metric connection over a compact Hermitian manifold, we can derive various vanishing theorems for Hermitian manifolds and complex vector bundles by the second Ricci curvature tensors. We will also introduce a natural geometric flow on Hermitian manifolds by using the second Ricci curvature tensor.

Mariño-Vafa formula and Hodge integral identities

Based on string duality Marino and Vafa [10] conjectured a closed formula on certain Hodge integrals in terms of representations of symmetric groups. This formula was first explicitly written down by the third author in [13] and proved in joint work [8] of the authors of the present paper. For a different approach see [12]. Our proof follows the strategy of proving both sides of the equation satisfy the same cut-and-join equation and have the same initial values. In this note we will describe a proof of the ELSV formula relating Hurwitz numbers and Hodge integrals:

Some Remarks on Circle Action on Manifolds

This paper contains several results concerning circle action on almost com- plex and smooth manifolds. More precisely, we show that, for an almost-complex mani- fold M 2mn (resp. a smooth manifold N 4mn ), if there exists a partition = ( 1,··· , u) of weight m such that the Chern number (c 1 ···c u) n (M) (resp. Pontrjagin number (p 1 ···p u) n (N)) is nonzero, then any circle action on M 2mn (resp. N 4mn ) has at least n + 1 fixed points. When an even-dimensional smooth manifold N 2n admits a semi-free action with isolated fixed points, we show that N 2n bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with nonempty isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Botts residue formula and rigidity theorem. 1. Introduction and main results Unless otherwise stated, all the manifolds (smooth or almost-complex) mentioned in the paper are closed, connected and oriented. For almost-complex manifolds, we take the canonical orientations induced from the almost-complex structures. We denote by superscripts the corresponding real dimensions of such manifolds. When M is a smooth (resp. almost-complex) manifold, we say M has an S 1 -action if M admits a circle action which preserves the smooth (resp. almost-complex) structure. Given a manifold M and an S 1 -action, the study of the fixed point set M S 1 is an important topic in geometry and topology. In ((13), p.338), Kosniowski proposed the following conjecture, which relates the number of fixed points to the dimension of the manifold. Conjecture 1.1 (Kosniowski). Suppose that M 2n is a unitary S 1 -manifold with isolated fixed points. If M is not a boundary then this action has at least ( n ) + 1 fixed points. Remark 1.2. A weakly almost-complex structure on a manifold M 2n is determined by a complex structure in the vector bundle (M 2n ) R 2k for some k, where (M 2n ) is the tangent bundle of M 2n and R 2k denotes a trivial real 2k-dimensional vector bundle over M 2n . A unitary S 1 -manifold means that M 2n has a weakly almost- complex structure and S 1 acts on M 2n preserving this structure. Recently, Pelayo and Tolman showed that ((19), Theorem 1), if a symplectic man- ifold (M 2n ,!) has a symplectic S 1 -action and the weights induced from the isotropy

Time-periodic solutions of the Einstein's field equations I: general framework

In this paper, we develop a new algorithm to find the exact solutions of the Einstein’s field equations. Time-periodic solutions are constructed by using the new algorithm. The singularities of the time-periodic solutions are investigated and some new physical phenomena, such as degenerate event horizon and time-periodic event horizon, are found. The applications of these solutions in modern cosmology and general relativity are expected.

Time-periodic solutions of the Einstein's field equations II:geometric singularities

In this paper, we construct several kinds of new time-periodic solutions of the vacuum Einstein’s field equations whose Riemann curvature tensors vanish, keep finite or take the infinity at some points in these space-times, respectively. The singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.