Symplectic quotients of unstable Morse strata for normsquares of moment maps
Let K be a compact Lie group and fix an invariant inner product on its Lie algebra k. Given a Hamiltonian action of K on a compact symplectic manifold X with moment map µ: X → k , the normsquare ||µ|| of µ defines a Morse stratification {S : β &#...
| Egile nagusia: | |
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| Formatua: | Journal article |
| Hizkuntza: | English |
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International Press
2020
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| _version_ | 1826303556749099008 |
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| author | Kirwan, F |
| author_facet | Kirwan, F |
| author_sort | Kirwan, F |
| collection | OXFORD |
| description | Let K be a compact Lie group and fix an invariant inner product on its Lie algebra k. Given a Hamiltonian action of K on a compact symplectic manifold X with moment map µ: X → k , the normsquare ||µ|| of µ defines a Morse stratification {S : β ∈ B} of X by locally closed symplectic submanifolds of X such that the stratum to which any x ∈ X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow of ||µ|| with respect to a suitably compatible Riemannian metric on X. The open stratum S retracts K-equivariantly via this gradient flow to the minimum µ (0) of ||µ|| (if this is not empty). If β ≠ 0 the usual ‘symplectic quotient’ (S ∩ µ (0))/K for the action of K on the stratum S is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural ‘symplectic quotients’ for the unstable strata with β ≠ 0. There is an analogous infinite-dimensional picture for the Yang–Mills functional over a Riemann surface with strata determined by Harder–Narasimhan type. ∗ 2 2 −1 2 −1 β 0 β β |
| first_indexed | 2024-03-07T06:04:30Z |
| format | Journal article |
| id | oxford-uuid:ed5bea9e-db8b-4a28-8158-9cf32d90a2f1 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:04:30Z |
| publishDate | 2020 |
| publisher | International Press |
| record_format | dspace |
| spelling | oxford-uuid:ed5bea9e-db8b-4a28-8158-9cf32d90a2f12022-03-27T11:24:27ZSymplectic quotients of unstable Morse strata for normsquares of moment mapsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:ed5bea9e-db8b-4a28-8158-9cf32d90a2f1EnglishSymplectic ElementsInternational Press2020Kirwan, FLet K be a compact Lie group and fix an invariant inner product on its Lie algebra k. Given a Hamiltonian action of K on a compact symplectic manifold X with moment map µ: X → k , the normsquare ||µ|| of µ defines a Morse stratification {S : β ∈ B} of X by locally closed symplectic submanifolds of X such that the stratum to which any x ∈ X belongs is determined by the limiting behaviour of its downwards trajectory under the gradient flow of ||µ|| with respect to a suitably compatible Riemannian metric on X. The open stratum S retracts K-equivariantly via this gradient flow to the minimum µ (0) of ||µ|| (if this is not empty). If β ≠ 0 the usual ‘symplectic quotient’ (S ∩ µ (0))/K for the action of K on the stratum S is empty. Nonetheless, motivated by recent results in non-reductive geometric invariant theory, we find that the symplectic quotient construction can be modified to provide natural ‘symplectic quotients’ for the unstable strata with β ≠ 0. There is an analogous infinite-dimensional picture for the Yang–Mills functional over a Riemann surface with strata determined by Harder–Narasimhan type. ∗ 2 2 −1 2 −1 β 0 β β |
| spellingShingle | Kirwan, F Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title | Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title_full | Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title_fullStr | Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title_full_unstemmed | Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title_short | Symplectic quotients of unstable Morse strata for normsquares of moment maps |
| title_sort | symplectic quotients of unstable morse strata for normsquares of moment maps |
| work_keys_str_mv | AT kirwanf symplecticquotientsofunstablemorsestratafornormsquaresofmomentmaps |