Enumerative invariants for local Calabi-Yau threefolds

This thesis consists of three parts. In the first part, we compute the topological Euler characteristics of the moduli spaces of stable sheaves of dimension one on the total space of rank 2 bundle on P1 whose determinant is O(−2). We count the torus fixed stable sheaves of low degrees and show the results verify the predictions in physics and the local Gromov-Witten theory. In the second part, we compute the Poincar´e polynomial of the moduli space of stable sheaves with Hilbert polynomial 4n + 1 on P2. This is done by classifying all torus fixed points in the moduli space and computing the torus representation of their tangent spaces. The result is also in agreement with a computation in physics. In the third part, we propose an algorithm to compute the Euler characteristics of the moduli spaces of stable sheaves of dimension one on P2 by means of Joyce’s wall crossing formula. The wall crossing takes place over the moduli spaces of α-stable pairs as the stability parameter α varies. The results verify a conjecture in the theory of curve counting invariants motivated by physics.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2012

Author dc:creator

• Choi, Jinwon

Contributors dc:contributor

• Katz, Sheldon
• Haboush, William J.
• Schenck, Henry K.
• Nevins, Thomas A.

• Bogomol'nyi-Prasad-Sommerfeld (BPS) invariant
• moduli space
• equivariant sheaf
• toric variety
• wall crossing

Statement dc:rights

• Copyright 2012 Jinwon Choi

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/34220

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/34220