Math 615

Quantum Invariants of Calabi-Yau threefolds

2013/2014 Term 2

Instructor: Jim Bryan 

Time: MWF 10:30am -- 11:50  

Place: Math Room 225

Course Description:

In this course I will discuss quantum invariants of Calabi-Yau threefolds. "Quantum invariants" in this context is a catch-all phrase referring to deformation invariants constructed in algebraic geometry which are mathematical analogs of quantities arising in string theory. The invariants we will consider are Gromov-Witten invariants, Donaldson-Thomas invariants, Pandharipande-Thomas invariants, and Gopakumar-Vafa invariants. They can all be regarded as theories which provide virtual counts of curves on a Calabi-Yau threefold. We will study the structure which underlies these invariants and the various relationships (many of which are conjectural) between the invariants as well as techniques for computing these invariants.

Course Outline:

  1. Introduction. Calabi-Yau manifolds.

  2. Invariants from virtual curve counting

  3. Motivic DT theory

  4. Computations

Reference:

A good overview is the paper 13/2 ways to count curves by Pandharipande and Thomas. The references within this paper provide an excellent guide to the literature.

Homework:

Homework Assignment 1 , due Friday, January 24th.

Homework Assignment 2 , due Wednesday, February 5th.

Homework Assignment 3 , due Wednesday, March 7th.

Homework Assignment 4 , due Friday, April 4th.



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