** Math 615**

** Quantum Invariants of Calabi-Yau threefolds**

**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:

Introduction. Calabi-Yau manifolds.

Invariants from virtual curve counting

- Gromov-Witten invariants
- Donaldson-Thomas invariants
- the GW/DT correspondence
- Pandharipande-Thomas theory
- the PT/DT correspondence
- Gopakumar-Vafa theory
- The Gopakumar-Vafa conjecture

Motivic DT theory

- Motivic stack functions, the Hall algebra of a CY category
- Motivic Milnor fiber and the motivic Thom-Sabastiani theorem.
- Motivic DT invariants a la Kontsevich-Soibelman

Computations

- Hall algebra proof of the PT/DT correspondence
- Motivic Donaldson-Thomas invariants of 0-dimensional sheaves
- Power structures on motives
- DT/PT invariants of local curves

#### 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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