Nabijou, N. (2019) The fundamental solution matrix and relative stable maps. European Journal of Mathematics, 5(3), pp. 1067-1089. (doi: 10.1007/s40879-018-0299-9)
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Abstract
Givental’s Lagrangian cone LX is a Lagrangian submanifold of a symplectic vector space which encodes the genus-zero Gromov–Witten invariants of X. Building on work of Braverman, Coates has obtained the Lagrangian cone as the push-forward of a certain class on the moduli space of stable maps to Open image in new window . This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. We recast this construction in its natural context, namely the moduli space of stable maps to Open image in new window relative the divisor Open image in new window . We find that the resulting push-forward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental’s quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov–Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.
| Item Type: | Articles |
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| Additional Information: | The author is supported by an EPSRC Standard DTP Scholarship and by the Engineering and Physical Sciences Research Council Grant EP/L015234/1: the EPSRC Centre for Doctoral Training in Geometry and Number Theory at the Interface. |
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Nabijou, Mr Navid |
| Authors: | Nabijou, N. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | European Journal of Mathematics |
| Publisher: | Springer |
| ISSN: | 2199-675X |
| ISSN (Online): | 2199-6768 |
| Published Online: | 23 October 2018 |
| Copyright Holders: | Copyright © 2018 The Author |
| First Published: | First published in European Journal of Mathematics 5(3):1067-1089 |
| Publisher Policy: | Reproduced under a Creative Commons License |
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