Chongying Dong, Kefeng Liu, Xiaonan Ma
We construct bundles of modules of vertex operator algebras, and prove the rigidity and vanishing theorem for the Dirac operator on loop space twisted by such bundles. This result generalizes many previous results.
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We define orbifold elliptic genus for general orbifolds which generalizes the definition of Borisov and Libgober, and prove their rigidity property.
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.
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。
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.
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…
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.
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。
The two-variable elliptic genus in odd dimensions
本文提出了一种针对奇数维自旋流形的双变量椭圆泛函，并通过Jacobi形式得到模形式，从而推导出异常消除公式和可除性结果。
We provide a differential cocycle model for elliptic cohomology with complex coefficients and use analytic methods to construct a cocycle representative for the Witten class in this language. Our motivation stems from the conjectural connection between 2-dimensional field theories and elliptic cohomology originally due to G. Segal and E. Witten. The specifics of our constructions are informed by the work of S. Stolz and P. Teichner on super Euclidean field theories and K. Costello's construction of the Witten genus using perturbative quantization. As a warm-up, we prove analogous results for s…
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.
We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an ``elliptic end'' (resp. a ``hyperbolic end'') if the monodromy matrix at the end is diagonalizable with eigenvalues in the unit circle (resp. in the reals). Although the existence of the surfaces is numerical, the types of ends are mathematically determined.
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.
We compute the genus one family Gromov-Witten invariants of K3 surfaces for non-primitive classes. These calculations verify Gottsche-Yau-Zaslow formula for non-primitive classes with index two. Our approach is to use the genus two topological recursion formula and the symplectic sum formula to establish relationships among various generating functions.
Anomaly cancellation formulas and E_8 bundles for almost complex manifolds
本文用E_8及E_8×E_8推广椭圆亏格，证明其为弱Jacobi形式，进而导出SL_2(Z)模形式，得到近复流形的新反常消去公式。
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