Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1)
By conformal welding, there is a pair of univalent functions (f, g) associated to every point of the complex Kahler manifold Mob(S(1))\Diff(+)(S(1)). For every integer n >= 1, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic (1 - n)-and n-differentia...
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| Format: | Article |
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ELSEVIER SCIENCE BV
2008
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| Online Access: | http://shdl.mmu.edu.my/2156/ |
| _version_ | 1848789978765590528 |
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| author | TEO, L |
| author_facet | TEO, L |
| author_sort | TEO, L |
| building | MMU Institutional Repository |
| collection | Online Access |
| description | By conformal welding, there is a pair of univalent functions (f, g) associated to every point of the complex Kahler manifold Mob(S(1))\Diff(+)(S(1)). For every integer n >= 1, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic (1 - n)-and n-differentials associated to the pair (f, g). Using these bases, we generalize the definition of Grunsky matrices to define matrices whose columns are the coefficients of the differentials with respect to standard bases of differentials on the unit disc and the exterior unit disc. We derive some identities among these matrices which are reminiscent of the Grunsky equality. By using these identities, we showed that we can define the Fredholm determinants of the period matrices of holomorphic n-differentials N(n), which are the Gram matrices of the canonical bases of holomorphic n-differentials with respect to the inner product given by the hyperbolic metric. Finally we proved that det N(n) = (det N(1))(6n2-6n+1) and partial derivative(partial derivative) over bar log der N(n) is - (6n(2) - 6n +1)/(6 pi i) of the Weil-Petersson symplectic form. (C) 2008 Elsevier B.V. All rights reserved. |
| first_indexed | 2025-11-14T18:05:19Z |
| format | Article |
| id | mmu-2156 |
| institution | Multimedia University |
| institution_category | Local University |
| last_indexed | 2025-11-14T18:05:19Z |
| publishDate | 2008 |
| publisher | ELSEVIER SCIENCE BV |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | mmu-21562011-09-19T08:27:09Z http://shdl.mmu.edu.my/2156/ Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) TEO, L T Technology (General) QC Physics By conformal welding, there is a pair of univalent functions (f, g) associated to every point of the complex Kahler manifold Mob(S(1))\Diff(+)(S(1)). For every integer n >= 1, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic (1 - n)-and n-differentials associated to the pair (f, g). Using these bases, we generalize the definition of Grunsky matrices to define matrices whose columns are the coefficients of the differentials with respect to standard bases of differentials on the unit disc and the exterior unit disc. We derive some identities among these matrices which are reminiscent of the Grunsky equality. By using these identities, we showed that we can define the Fredholm determinants of the period matrices of holomorphic n-differentials N(n), which are the Gram matrices of the canonical bases of holomorphic n-differentials with respect to the inner product given by the hyperbolic metric. Finally we proved that det N(n) = (det N(1))(6n2-6n+1) and partial derivative(partial derivative) over bar log der N(n) is - (6n(2) - 6n +1)/(6 pi i) of the Weil-Petersson symplectic form. (C) 2008 Elsevier B.V. All rights reserved. ELSEVIER SCIENCE BV 2008-11 Article NonPeerReviewed TEO, L (2008) Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1). Journal of Geometry and Physics, 58 (11). pp. 1540-1570. ISSN 03930440 http://dx.doi.org/10.1016/j.geomphys.2008.07.004 doi:10.1016/j.geomphys.2008.07.004 doi:10.1016/j.geomphys.2008.07.004 |
| spellingShingle | T Technology (General) QC Physics TEO, L Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title | Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title_full | Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title_fullStr | Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title_full_unstemmed | Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title_short | Universal Index Theorem on Möb(S1)∖Diff+(S1)Möb(S1)∖Diff+(S1) |
| title_sort | universal index theorem on möb(s1)∖diff+(s1)möb(s1)∖diff+(s1) |
| topic | T Technology (General) QC Physics |
| url | http://shdl.mmu.edu.my/2156/ http://shdl.mmu.edu.my/2156/ http://shdl.mmu.edu.my/2156/ |