Generalized Witten Genus and Vanishing Theorems

Qingtao Chen, Fei Han, Weiping Zhang

Type: Article

Publication Date: 2011-05-01

Citations: 20

DOI: https://doi.org/10.4310/jdg/1317758867

Abstract

We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalizedWitten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.

Locations

Journal of Differential Geometry - View - PDF
arXiv (Cornell University) - View - PDF

Similar Works

Action	Title	Year	Authors
+	Generalized Witten Genus and Vanishing Theorems	2010	Qingtao Chen Fei Han Weiping Zhang
+ PDF Chat	Witten genus and vanishing results on complete intersections	2010	Qingtao Chen Fei Han Weiping Zhang
+ PDF Chat	Witten genera of complete intersections	2024	Michael Wiemeler
+ PDF Chat	Witten genus and string complete intersections	2008	Qingtao Chen Fei Han
+	String$\mathbf{^c}$ Structures and Modular Invariants	2019	Haibao Duan Fei Han Ruizhi Huang
+	String$\mathbf{^c}$ Structures and Modular Invariants	2019	Haibao Duan Fei Han Ruizhi Huang
+ PDF Chat	Correlated Gromov-Witten invariants	2024	Thomas Blomme Francesca Carocci
+	Witten Genus and String Complete Intersections	2006	Qingtao Chen Fei Han
+	Gromov-Witten Theory of Spin Curves and Orbifolds	2006	Tyler J. Jarvis Takashi Kimura Arkady Vaintrob
+	The mod 2 Seiberg-Witten invariants of spin structures and spin families	2023	David Baraglia
+	-INVARIANT AND MODULAR FORMS	2014	Fei Han Weiping Zhang
+ PDF Chat	Topological Elliptic Genera I -- The mathematical foundation	2024	Ying-Hsuan Lin Mayuko Yamashita
+	Tensor products of Frobenius manifolds and moduli spaces of higher spin curves	1999	Tyler J. Jarvis Takashi Kimura Arkady Vaintrob
+ PDF Chat	The Witten genus and equivariant elliptic cohomology	2002	Matthew Ando Maria Basterra
+	Witten genus of generalized complete intersections in products of Grassmannians	2014	Jian Zhou Xiaobo Zhuang
+	Topological recursion relations and Gromov-Witten invariants in higher genus	2003	Andreas Gathmann
+ PDF Chat	Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local	2021	Pierrick Bousseau Honglu Fan Shuai Guo Longting Wu
+ PDF Chat	Spin Gromov-Witten Invariants	2005	Tyler J. Jarvis Takashi Kimura Arkady Vaintrob
+	$η$-invariant and Modular Forms	2013	Fei Han Weiping Zhang
+	Universal equations for higher genus Gromov-Witten invariants from Hodge integrals	2024	Felix Janda Xin Wang

Works That Cite This (21)

Action	Title	Year	Authors
+	String^{𝑐} structures and modular invariants	2020	Ruizhi Huang Fei Han Haibao Duan
+	Vanishing theorems of generalized Witten genus for generalized complete intersections in flag manifolds	2016	Xiaobo Zhuang
+	Six-dimensional gauge theories and (twisted) generalized cohomology	2019	Hisham Sati
+	A vanishing theorem for elliptic genera under a Ricci curvature bound	2020	Xiaoyang Chen Fei Han
+ PDF Chat	Variations of rational higher tangential structures	2018	Hisham Sati Matthew W. Wheeler
+	Differential cohomology in a cohesive infinity-topos	2013	Urs Schreiber
+ PDF Chat	Cubic forms, anomaly cancellation and modularity	2021	Fei Han Ruizhi Huang Kefeng Liu Weiping Zhang
+ PDF Chat	GEOMETRIC AND TOPOLOGICAL STRUCTURES RELATED TO M-BRANES II: TWISTED STRING AND STRING<sup><i>C</i></sup>STRUCTURES	2011	Hisham Sati
+	SL(2, ℤ) modular forms and anomaly cancellation formulas for almost complex manifolds	2024	Yong Wang
+ PDF Chat	Twistorial cohomotopy implies Green–Schwarz anomaly cancellation	2022	Domenico Fiorenza Hisham Sati Urs Schreiber

Works Cited by This (22)

Action	Title	Year	Authors
+ PDF Chat	On modular invariance and rigidity theorems	1995	Kefeng Liu
+	Manifolds and Modular Forms	1992	Friedrich Hirzebruch Thomas Berger Rainer Jung
+	Principles of Algebraic Geometry	1978	Phillip Griffiths Joseph Harris
+	Algebraic topology and modular forms	2002	Michael J. Hopkins
+	The index of the dirac operator in loop space	1988	Edward Witten
+	The Index of Elliptic Operators: III	1968	Michael Atiyah I. M. Singer
+	Note on the Landweber-Stong elliptic genus	1988	Don Zagier
+ PDF Chat	Witten genus and vanishing results on complete intersections	2010	Qingtao Chen Fei Han Weiping Zhang
+	Cobordism and rokhlin congruences	2009	Weiping Zhang
+ PDF Chat	Orientation and string structures on loop space	1992	D. A. McLaughlin