sissa, Trieste, july 3-7 2017

Lecture 1: Moduli of stable maps and Gromov-Witten invariants

Abstract: We will introduce stable maps and their moduli, and define Gromov-Witten (GW) invariants

of non-singular complex projective varieties

in terms of the fundamental class (in nice cases)

or the Behrend-Fantechi virtual fundamental class (in the general case).

We will give explicit examples when the target is a projective space or a hypersurface in a projective space.

Lecture 2: Stable maps with fields

Abstract: We will introduce moduli of stable maps to a projective space with fields,

and describe an alternative definition of GW invariants of Calabi-Yau hypersurfaces

in projective spaces in terms of the Kiem-Li cosection localized virtual cycle.

This is an exposition of "Gromov-Witten invariants of stable maps with fields"

by Huai-Liang Chang and Jun Li.

Lecture 3: Witten's top Chern class and Fan-Jarvis-Ruan-Witten invariants

Abstract: The Fan-Jarvis-Ruan-Witten (FJRW) invariants of a quasi-homogeneous polynomial

are virtual counts of solutions to the Witten's equation associated to the quasi-homogeneous polynomial.

We will describe an algebraic definition of FJRW invariants (in the "narrow" sector) of a Fermat polynomial

x_1^n+ ... + x_n^n in terms of the Kiem-Li cosection localized virtual cycle.

This is a special case of the general construction in "Witten's top Chern class via cosection localization"

by Huai-Liang Chang, Jun Li, and Wei-Ping Li.

Lecture 4: Landau-Ginzburg/Calabi-Yau correspondence

Abstract: We will describe the Landau-Ginzburg/Calabi-Yau correspondence relating the FJRW invariants

the Fermat polynomial W_n = x_1^n+... + x_n^n and GW invariants of the Calabi-Yau hypersurface {W_n=0} in P^n.

Lecture 5: Mixed-Spin-P Fields

Abstract: We will describe the theory of Mixed-Spin-P (MSP) fields which interpolates

the FJRW theory of the Fermat polynomial W_n and GW theory of the Calabi-Yau hypersurface {W_n=0} in P^n.

We will present some computations of GW invariants and FJRW invariants using localization

on MSP moduli spaces.

This is based on joint work with H.-L. Chang, J. Li, and W.-P. Li.