Abstract
Let BunG be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, Gaiotto (2016) associated to any symplectic representation of G a Lagrangian subvariety of T∗BunG. We give a simple interpretation of (a generalization of) Gaiotto’s construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Braverman, A., Finkelberg, M., Nakajima, H.: Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories II (2016). arXiv:1601.03586
Calaque, D.: Lagrangian structures on mapping stacks and semi-classical TFTs. Stacks and categories in geometry, topology, and algebra, 1–23. Contemp. Math. 643 (2015). arXiv:1306.3235
Calaque, D.: Shifted cotangent stacks are shifted symplectic (2016). arXiv:1612.08101
Calaque, D., Pantev, T., Toën, B., Vezzosi, G.: Shifted Poisson structures and deformation quantization (2015). arXiv:1506.03699
Gaitsgory, D., Rozenblyum, N.: A study in derived algebraic geometry. Mathematical Surveys and Monographs 221
Gaiotto, D.: S-duality of boundary conditions and the Geometric Langlands program (2016). arXiv:1609.09030
Ginzburg, V.: The global nilpotent variety is Lagrangian. Duke Math. J. 109, 511–519 (2001)
Hitchin, N.: Spinors, Lagrangians and rank 2 Higgs bundles (2016). arXiv:1605.06385
Melani, V., Safronov, P.: Derived coisotropic structures (2016). arXiv:1608.01482
Nakajima, H.: Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories I (2015). arXiv:1503.03676
Pantev, T., Toën, B., Vaquié, M., Vezzosi, G.: Shifted symplectic structures. Publ. Math. Inst. Hautes Études Sci. 117, 271–328 (2013)
Pridham, J.P.: Deformation quantisation for (− 1)-shifted symplectic structures and vanishing cycles (2015). arXiv:1508.07936
Safronov, P.: Quasi-Hamiltonian reduction via classical Chern-Simons theory. Adv. Math. 287, 733–773 (2016). arXiv:1311.6429
Spaide, T.: Shifted Symplectic and Poisson Structures on Spaces of Framed Maps (2016). arXiv:1607.03807
Acknowledgments
The authors are grateful to Davide Gaiotto, Kevin Costello, and Li Yu for inspiring discussions. The first author was supported in part by the NSF grant DMS-1303462.
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by: Valentin Ovsienko
To Alexander Alexandrovich Kirillov on his 80th Birthday.
Rights and permissions
About this article
Cite this article
Ginzburg, V., Rozenblyum, N. Gaiotto’s Lagrangian Subvarieties via Derived Symplectic Geometry. Algebr Represent Theor 21, 1003–1015 (2018). https://doi.org/10.1007/s10468-018-9801-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-9801-9