{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/34220"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/34220","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Enumerative invariants for local Calabi-Yau threefolds","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Choi, Jinwon"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Katz, Sheldon","Haboush, William J.","Schenck, Henry K.","Nevins, Thomas A."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-09-18T21:06:28Z","date_published":"2012-09-18T21:06:28Z","updated_at":"2026-07-22T22:25:30Z","subjects":["Bogomol'nyi-Prasad-Sommerfeld (BPS) invariant","moduli space","equivariant sheaf","toric variety","wall crossing"],"languages":["en"],"rights":["Copyright 2012 Jinwon Choi"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/34220","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Katz, Sheldon","Haboush, William J.","Schenck, Henry K.","Nevins, Thomas A."]},{"key":"dc:creator","label":"Author","values":["Choi, Jinwon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-09-18T21:06:28Z","2012-08"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Bogomol'nyi-Prasad-Sommerfeld (BPS) invariant","moduli space","equivariant sheaf","toric variety","wall crossing"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 Jinwon Choi"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/34220"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-03T13:56:39Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Choi_Jinwon.tex: 237201 bytes, checksum: 0a9526a53640d29bb33bbf61afe3efa0 (MD5) Choi_Jinwon.pdf: 705606 bytes, checksum: 277d3cd8d4541bae15511ebfa7a0c583 (MD5)","Made available in DSpace on 2012-09-18T21:06:28Z (GMT). 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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. 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