Fourier–Mukai and Nahm transforms for holomorphic triples on elliptic curves |
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| Institution: | 1. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 113 bis, 28006 Madrid, Spain;2. Departamento de Matemáticas and Instituto Universitario de Matemáticas y Física Fundamental, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain;3. Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain;1. Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105, United States of America;2. Department of Mathematics, University of Rochester, Rochester, NY, 14627, United States of America;1. ITEP, Moscow 117218, Russia;2. Institute for Information Transmission Problems, Moscow 127994, Russia;3. National Research Nuclear University MEPhI, Moscow 115409, Russia;1. Universidade Federal do Cariri (UFCA), Av. Tenente Raimundo Rocha, Cidade Universitária, Juazeiro do Norte, Ceará, CEP 63048-080, Brazil;2. Universidade Federal do Ceará (UFC), Departamento de Física, Campus do Pici, Fortaleza, CE, C.P. 6030, 60455-760, Brazil |
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| Abstract: | We define a Fourier–Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases, the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple—vortices—and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line. |
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