On orbifold elliptic genus

Torus orbifolds, slice-maximal torus actions and rational ellipticity

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjectur…

Positive curvature and the elliptic genus

We prove several results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard's improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.

椭圆亏格的几个新Witten刚性定理

Several new Witten rigidity theorems for elliptic genus

本文利用Liu和Han-Yu的方法，分别证明了偶数维和奇数维spin流形在圆作用下扭曲Dirac算子与Toplitz算子椭圆亏格的新Witten刚性定理，并获得了其他类似结论。

轨形消解上的狄拉克算子：一致椭圆理论

Dirac Operators on Orbifold Resolutions: Uniform Elliptic Theory

本文通过将轨形视为锥纤维奇异空间并用渐近锥纤维化进行消解，发展了一致椭圆理论，从而得到轨形消解上狄拉克算子的Fredholm性质、一致有界性及指标分解公式。

An equivariant orbifold index for proper actions

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal comp…

Indefinite theta functions arising in Gromov-Witten Theory of elliptic orbifolds

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim…

On the essential spectrum of differential operators over geometrically finite orbifolds

We discuss the essential spectrum of essentially self-adjoint elliptic differential operators of first order and of Laplace type operators on Riemannian vector bundles over geometrically finite orbifolds.

Nonnegative curvature, elliptic genus and unbounded Pontryagin numbers

We discuss the cobordism type of spin manifolds with nonnegative sectional curvature. We show that in each dimension $4k \geq 12$, there are infinitely many cobordism types of simply connected and nonnegatively curved spin manifolds. Moreover, we raise and analyze a question about possible cobordism obstructions to nonnegative curvature.

紧局部对称轨形上解析挠率的完整渐近性

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

本文研究紧局部对称轨形上平坦向量丛序列的渐近 Ray-Singer 解析挠率，利用 Selberg 迹公式并借助几何局部化公式将椭圆轨道积分化为中心化子的恒等轨道积分。

-1-Phenomena for the pluri $χ_y$-genus and elliptic genus

Several independent articles have observed that the Hirzebruch $χ_y$-genus has an important feature, which the author calls -1-phenomenon and tells us that the coefficients of the Taylor expansion of the $χ_y$-genus at $y=-1$ have explicit expressions. Hirzebruch's original $χ_y$-genus can be extended towards two directions: the pluri-case and the case of elliptic genus. This paper contains two parts in which we investigate the -1-phenomena in these two generalized cases respectively and show that in each case there exists a -1-phenomenon in a suitable sense. Our main results in the first part…

Witten Genus and Elliptic genera for proper actions

In this paper, we construct for the first time, the Witten genus and elliptic genera on noncompact manifolds with a proper cocompact action by an almost connected Lie group and prove vanishing and rigidity results that generalise known results for compact group actions on compact manifolds. We also compute our genera for some interesting examples, and study the non-spinc case in an appendix.

维度$24$的椭圆亏格与string配边

Elliptic genus and string cobordism at dimension $24$

本文证明了维度24的string配边可由高阶指标不变量椭圆亏格确定，计算了其在椭圆亏格下的像，并由此得出特定曲率条件下紧致24维string流形必为边缘的结论。

轨形（Orbifold）的环化

Cyclification of Orbifolds

本文在凝聚高阶拓扑斯理论中将轨形环化严格构造为模叠上的基变换，证明Ganter/Huan扩展惯性群胚是其模型，并应用于ADE-轨形等变Tate-椭圆上同调的扭曲及其与椭圆M5-膜亏格的联系。

Twisted index theory on good orbifolds, I: noncommutative Bloch theory

This paper, together with Part II, expands the results of math.DG/9803051. In Part I we study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompa…

The trace of the heat kernel on a compact hyperbolic 3-orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.