Research

Research Interest: Differential Geometry, Topology and Mathematical Physics

Research Papers:

Field Theories, T-duality and Loop Spaces:
● (with Christopher Schommer-Pries, Stephan Stolz and Peter Teichner) Equivariant Cohomology via Gauged Field Theories, in preparation.
● (with Stephan Stolz and Peter Teichner) The Bismut-Chern Character Form as Dimension Reduction, in preparation.
Supersymmetric Field Theories, Super Loop Spaces and Bismut-Chern Charater, arXiv: 0711.3862 (part of this paper will be incorporated into the above paper.)
● (with Daniel Berwick-Evans) The Equivariant Chern Character as Super Holonomy on Loop Stacks, arXiv:1610.02362.
● (with Varghese Mathai) Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut-Chern Character and T-duality, Comm. Math. Phys, Volume 337, Issue 1 (2015), pp 127-150.
● (with Varghese Mathai) T-duality in an H-flux: Exchange of Momentum and Winding, Comm. Math. Phys, Volume 363, no. 1 (2018) pp 333-350.
● (with Varghese Mathai) Exotic Twisted Equivariant K-theory, J. Geom. Phys., to appear.
● (with Varghese Mathai) T-duality, Jacobi forms and Witten Gerbe Modules, arXiv:2001.00322

Some Mathematical Aspects of Anomaly Cancellations:
● (with Kefeng Liu) Gravitational Anomaly Cancellation and Modular Invariance, Alg. Geom. Top. 14 (2014) 91-113.
● (with Kefeng Liu and Weiping Zhang) Modular Forms and Generalized Anomaly Cancellation Formulas, J. Geom. Phys., 62 (2012), pp. 1038-1053.
● (with Kefeng Liu and Weiping Zhang) Anomaly Cancellation and Modularity, Frontiers in Differential Geometry, Partial Differential Equations and Mathematical Physics, In Memory of Gu Chaohao, World Scientific Publishing Co., Inc.
● (with Kefeng Liu and Weiping Zhang) Anomaly Cancellation and Modularity II. The E8 x E8 case, Sci. China Math. 60 (2017), no. 6, 985–994.
● (with Ruizhi Huang, Kefeng Liu and Weiping Zhang) Cubic Forms, Anomaly Cancellation and Modularity, arXiv:2005.02344.

 

Modular Invariants, Projective Elliptic Genera for non-spin Manifolds:
● (with Qingtao Chen) Elliptic Genera, Transgression and Loop Space Chern-Simons Forms, Comm. Anal. Geom., Vol 17, No.1, Jan 2009, 73-106.
● (with Weiping Zhang) eta Invariant and Modular Forms, Quarterly Journal of Mathematics, 66 (2015), 133–142.
● (with Varghese Mathai) Projective Elliptic Genera and Elliptic Pseudodifferential Genera, Advances in Mathematics, 358 (2019), 106860, 25pp.

String and String^c Manifolds:
● (with Qingtao Chen) Witten Genus and String Complete Intersections, Pacific J. of Math, Vol 234, No.2, Feb 2008, 249-260.
● (with Qingtao Chen and Weiping Zhang) Witten Genus and Vanishing Results on Complete Intersections, C. R. Acad. Sci. Paris, Serie I. 348 (2010), 295-298.
● (with Qingtao Chen and Weiping Zhang) Generalized Witten Genus and Vanishing Theorems, J. Diff. Geom. 88 (2011), 1-39.
● (with Haibao Duan and Ruizhi Huang) String^c Structures and Modular Invariants, Trans. A.M.S, to appear.

Rokhlin Type Divisibilities and Congruences:
● (with Weiping Zhang) Spin^c-manifold and Elliptic Genera. C. R. Acad. Sci. Paris, Serie I. 336 (2003), 1011-1014.
● (with Weiping Zhang) Modular Invariance, Characteristic Numbers and eta Invariants, J. Diff. Geom. 67 (2004), 257-288.
● (with Xiaoling Huang) Even Dimensional Manifolds and Generalized Anomaly Cancellation Formulas, Trans. A. M. S., 359 (2007), no. 11, 5365-5381.
● (with Qingtao Chen) Modular Invariance and Twisted Cancellation of Characteristic Numbers, Trans. A. M. S. 361 (2009), no. 3, 1463-1493.
● (with Qingtao Chen) Mod 3 Congruence and Twisted Signature of 24 Dimensional String Manifolds, Trans. A.M.S, 367 (2015), 2959-2577.

Group Actions and Rigidity:
● (with Kefeng Liu and Weiping Zhang) E8 Bundles and Rigidity, arXiv:1307.6137.
● (with Jianqing Yu) On the Witten Rigidity Theorem for Odd Dimensional Manifolds, arXiv:1504.03007.
● (with Junwei Guo) On Certain Multi-variable Rational Identities Derived from the Rigidity of Signature of Manifolds, J. Math. Anal. Appl. 453 (2017) 360–365.
● (with Varghese Mathai) Witten Genus and Elliptic Genera for Proper Actions, arXiv: 1807.06863.

Curvature Conditions and Topological Obstructions:
● (with Xiaoyang Chen) A Vanishing Theorem for Elliptic Genera under a Ricci Curvature Bound, arXiv: 2003.09897.

 

Volumes Edit:

● (jointly edit with Xingwang Xu and Weiping Zhang) Geometric Analysis Around Scalar Curvatures, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore: Vol 31, World Scientific.
● (jointly edit with Varghese Mathai, Guoliang Yu and Weiping Zhang), Index Theory, Duality and Related Fields, Special Issue of Journal of Geometry and Physics.