Enumerative invariants for local Calabi-Yau threefolds

Choi, Jinwon

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https://hdl.handle.net/2142/34220 Copy

Description

Title

Enumerative invariants for local Calabi-Yau threefolds

Author(s)

Choi, Jinwon

Issue Date

2012-09-18T21:06:28Z

Director of Research (if dissertation) or Advisor (if thesis)

Katz, Sheldon

Doctoral Committee Chair(s)

Haboush, William J.

Committee Member(s)

• Katz, Sheldon
• Schenck, Henry K.
• Nevins, Thomas A.

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

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

Abstract

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.

Graduation Semester

2012-08

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

Copyright and License Information

Copyright 2012 Jinwon Choi

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