Huai-Liang Chang

Virtual Residue and an integral formalism

We generalize Grothendieck’s residues

Witten’s top Chern class via cosection localization

Res\frac{\psi }{s}

Torus Localization and Wall Crossing for Cosection Localized Virtual Cycles

Resψs to virtual cases, namely cases when the zero loci of the section s has dimension larger than the expected dimension (zero). We also provide an exponential-type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai–Quillen formalism for localized Euler classes.

Mixed-Spin-P fields of Fermat quintic polynomials

The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables. It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants. Then it brie y discusses the master space technique and its application to the set-up of the MSP moduli. Some key results in MSP theory are explained and some examples are provided.