Vafa-Witten invariants for projective surfaces I: stable case
On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $ \mathbb{C}^*$ action with compact fixed locus. Applying virtual localisation we defin...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
American Mathematical Society
2019
|
| _version_ | 1797108907647172608 |
|---|---|
| author | Tanaka, Y Thomas, RP |
| author_facet | Tanaka, Y Thomas, RP |
| author_sort | Tanaka, Y |
| collection | OXFORD |
| description | On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $ \mathbb{C}^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten. |
| first_indexed | 2024-03-07T07:34:36Z |
| format | Journal article |
| id | oxford-uuid:38a2db78-54db-414a-8161-23e14c3ef2d3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:34:36Z |
| publishDate | 2019 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | oxford-uuid:38a2db78-54db-414a-8161-23e14c3ef2d32023-02-24T09:43:12ZVafa-Witten invariants for projective surfaces I: stable caseJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:38a2db78-54db-414a-8161-23e14c3ef2d3EnglishSymplectic Elements at OxfordAmerican Mathematical Society2019Tanaka, YThomas, RPOn a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $ \mathbb{C}^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten. |
| spellingShingle | Tanaka, Y Thomas, RP Vafa-Witten invariants for projective surfaces I: stable case |
| title | Vafa-Witten invariants for projective surfaces I: stable case |
| title_full | Vafa-Witten invariants for projective surfaces I: stable case |
| title_fullStr | Vafa-Witten invariants for projective surfaces I: stable case |
| title_full_unstemmed | Vafa-Witten invariants for projective surfaces I: stable case |
| title_short | Vafa-Witten invariants for projective surfaces I: stable case |
| title_sort | vafa witten invariants for projective surfaces i stable case |
| work_keys_str_mv | AT tanakay vafawitteninvariantsforprojectivesurfacesistablecase AT thomasrp vafawitteninvariantsforprojectivesurfacesistablecase |