Abstract
We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map.
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Bottman, N., Wehrheim, K.: Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts. Sel. Math. New Ser., 24, 3381–3443 (2018)
Cieliebak, K., Gaio, A., Mundet i Riera, I., et al.: The symplectic vortex equations and invariants of Hamiltonian group actions. J. Symplectic Geom., 1, 543–645 (2002)
Cieliebak, K., Gaio, A., Salamon, D.: J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions. Int. Math. Res. Not. IMRN, 16, 831–882 (2000)
Cieliebak, K., Mohnke, K.: Symplectic hypersurfaces and transversality in Gromov-Witten theory. J. Symplectic Geom., 5, 281–356 (2007)
Dostoglou, S., Salamon, D.: Self-dual instantons and holomorphic curves. Ann. Math., 139, 581–640 (1994)
Frauenfelder, U.: Floer homology of symplectic quotients and the Arnold-Givental conjecture. Ph.D. thesis, Swiss Federal Institute of Technology (2004)
Frauenfelder, U.: Gromov convergence of pseudoholomorphic disks. J. Fixed Point Theory Appl., 3, 215–271 (2008)
Gaio, A.: J-holomorphic curves and moment maps. Ph.D. thesis, University of Warwick (1999)
Gaio, A., Salamon, D.: Gromov-Witten invariants of symplectic quotients and adiabatic limits. J. Symplectic Geom., 3, 55–159 (2005)
Jaffe, A., Taubes, C.: Vortices and Monopoles, Progress in Physics, Vol. 2, Birkhäuser, Boston (1980)
Ma’u, S., Woodward, C.: Geometric realizations of the multiplihedra. Compos. Math., 146, 1002–1028 (2010)
McDuff, D., Salamon, D.: J-holomorphic Curves and Symplectic Topology, Colloquium Publications, Vol. 52, American Mathematical Society (2004)
Morrison, M., Plesser, N.: Summing the instantons: quantum cohomology and mirror symmetry in toric varieties. Nucl. Phys. B., 440, 279–354 (1995)
Mundet i Riera, I.: Yang-Mills-Higgs theory for symplectic fibrations. Ph.D. thesis, Universidad Autonoma de Madrid (1999)
Mundet i Riera, I.: Hamiltonian Gromov–Witten invariants. Topology, 43, 525–553 (2003)
Nguyen, K., Woodward, C., Ziltener, F.: Morphisms of CohFT algebras and quantization of the Kirwan map, In: Symplectic, Poisson, and Noncommutative Geometry, MSRI Publication, Vol. 62, Cambridge University Press, 2014, 131–170
Schecter, S., Xu, G.: Morse theory for Lagrange multipliers and adiabatic limits. J. Differetial Equation., 257, 4277–4318 (2014)
Stasheff, J.: Homotopy associativity of H-spaces. Trans. Amer. Math. Soc., 108, 275–312 (1963)
Taubes, C.: Arbitrary N-vortex solutions to the first order Ginzburg–Landau equations. Comm. Math. Phy., 72, 277–292 (1980)
Tian, G., Xu, G.: Correlation functions in gauged linear σ-model. Sci. China Math., 59, 823–838 (2016)
Tian, G., Xu, G.: The symplectic approach of gauged linear σ-model. In: Proceedings of the Gökova Geometry-Topology Conference 2016 (Selman Akbulut, Denis Auroux, and Turgut Önder, eds.), Gökova Geometry/Topology Conference, Gökova, 2017, 86–111
Tian, G., Xu, G.: Analysis of gauged Witten equation. J. Reine Angew. Math., 740, 187–274 (2018)
Tian, G., Xu, G.: Virtual fundamental cycles of gauged Witten equation. J. Reine Angew. Math., 2021, 1–64 (2021)
Venugopalan, S., Woodward, C.: Classification of vortices, Duke Math. J., 165, 1695–1751 (2016)
Venugopalan, S., Xu, G.: Local model for the moduli space of affine vortices. Int. J. Math., 29, 1850020, 54 pp. (2018)
Wang, D., Xu, G.: A compactification of the moduli space of disk vortices in adiabatic limit. Math. Z., 287, 405–459 (2017)
Wehrheim, K.: Energy identity for anti-self-dual instantons on ℂ × Σ. Math. Res. Lett., 13, 161–166 (2006)
Wehrheim, K., Woodward, C.: Floer cohomology and geometric composition of Lagrangian correspondences. Adv. Math., 230, 177–228 (2012)
Wehrheim, K., Woodward, C.: Pseudoholomorphic quilts. J. Symplectic Geom., 13, 849–904 (2015)
Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B, 403, 159–222 (1993)
Woodward, C.: Gauged Floer theory for toric moment fibers. Geom.Funct.Anal., 21, 680–749 (2011)
Woodward, C.: Quantum Kirwan morphism and Gromov–Witten invariants of quotients I, II, III. Transform. Groups, 20, 507–556, 881–920, 1155–1193 (2015)
Woodward, C., Xu, G.: An open quantum Kirwan map, ar**v:1806.06717 (2018)
Xu, G.: Moduli space of twisted holomorphic maps with Lagrangian boundary condition: compactness. Adv. Math., 242, 1–49 (2013)
Xu, G.: Classification of U(1)-vortices with target ℂN. Int. J. Math., 26, 1550129, 20 pp. (2015)
Xu, G.: Gauged Hamiltonian Floer homology I: definition of the Floer homology groups. Trans. Amer. Math. Soc., 368, 2967–3015 (2016)
Ziltener, F.: Symplectic vortices on the complex plane and quantum cohomology. Ph.D. thesis, Swiss Federal Institute of Technology Zurich (2005)
Ziltener, F.: The invariant symplectic action and decay for vortices. J. Symplectic Geom., 7, 357–376 (2009)
Ziltener, F.: A quantum Kirwan map: bubbling and Fredholm theory. Mem. Am. Math. Soc., 230, 1–129 (2014)
Acknowledgements
I am greatly indebted to Professor Chris Woodward, who shares numerous interesting ideas with the author. I would also like to thank Professor Gang Tian for long time support and encouragement, and thank Dr. Sushmita Venugopalan for helpful discussions about vortices. I also thank the referee for their time and comments.
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Supported by NSF-DMS 2345030
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Xu, G.B. Gluing Affine Vortices. Acta. Math. Sin.-English Ser. 40, 250–312 (2024). https://doi.org/10.1007/s10114-024-2248-5
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DOI: https://doi.org/10.1007/s10114-024-2248-5