Motivated by the cubic forms and anomaly cancellation formulas of Witten-Freed-Hopkins, we give some new cubic forms on spin, spin^c, spin^{w_2} and orientable 12-manifolds respectively. We relate them to \eta-invariants when the manifolds are with boundary, and mod 2 indices on 10 dimensional characteristic submanifolds when the manifolds are spin^c or spin^{w_2}. Our method of producing these cubic forms is a combination of (generalized) Witten classes and the character of the basic representation of affine $E_8$.

　　Publication: 

　　Advances in Mathematics, Volume 394, 22 January 2022. 

　　Author: 

　　Fei Han 

　　Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076, Singapore 

　　Email: mathanf@nus.edu.sg  

　　Ruizhi Huang 

　　Institute of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China 

　　Email: huangrz@amss.ac.cn  

　　Kefeng Liu 

　　Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA 

　　Mathematical Science Research Center, Chongqing University of Technology, Chongqing, 400054, China 

　　Email: liu@math.ucla.edu  

　　Weiping Zhang 

　　Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, China 

　　Email: weiping@nankai.edu.cn