Lagrangian correspondences and Donaldson’s TQFT construction of the Seiberg–Witten invariants of 3–manifolds

Nguyen, Timothy¹

¹Simons Center for Geometry and Physics, State University of New York, Stony Brook, NY 11794-3636, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 863-923

Résumé

Using Morse–Bott techniques adapted to the gauge-theoretic setting, we show that the limiting boundary values of the space of finite energy monopoles on a connected 3–manifold with at least two cylindrical ends provides an immersed Lagrangian submanifold of the vortex moduli space at infinity. By studying the signed intersections of such Lagrangians, we supply the analytic details of Donaldson’s TQFT construction of the Seiberg–Witten invariants of a closed 3–manifold.

Détail

DOI : 10.2140/agt.2014.14.863

Classification : 53C05, 53D12
Keywords: Seiberg–Witten invariants, Lagrangian correspondences

Bibliographie

[1] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43

[2] B Booß-Bavnbek, M Lesch, C Zhu, The Calderón projection: New definition and applications, J. Geom. Phys. 59 (2009) 784

[3] S K Donaldson, Topological field theories and formulae of Casson and Meng–Taubes, from: "Proceedings of the Kirbyfest" (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2 (1999) 87

[4] S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge Univ. Press (2002)

[5] S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press (1990)

[6] O García-Prada, A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994) 88

[7] M Hutchings, Y J Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999) 369

[8] P Kronheimer, T S Mrowka, Monopoles and three-manifolds, New Mathematical Monographs 10, Cambridge Univ. Press (2007)

[9] C Kutluhan, Y J Lee, C Taubes, $\mathit{HF}{=}\mathit{HM}$ I: Heegaard Floer homology and Seiberg–Witten Floer homology

[10] M Lipyanskiy, A semi-infinite cycle construction of Floer homology, PhD thesis, MIT (2008)

[11] R B Lockhart, R C Mcowen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985) 409

[12] T Mark, Torsion, TQFT, and Seiberg–Witten invariants of $3$–manifolds, Geom. Topol. 6 (2002) 27

[13] G Meng, C H Taubes, Seiberg–Witten equations and the Milnor torsion, Math. Res. Lett. 3 (1996) 661

[14] J W Morgan, T S Mrowka, D Ruberman, The $L^2$–moduli space and a vanishing theorem for Donaldson polynomial invariants, Monographs in Geometry and Topology, II, International Press (1994)

[15] J W Morgan, T S Mrowka, Z Szabó, Product formulas along $T^3$ for Seiberg–Witten invariants, Math. Res. Lett. 4 (1997) 915

[16] J W Morgan, Z Szabó, C H Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706

[17] T S Mrowka, P Ozsváth, B Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997) 685

[18] T Nguyen, The Seiberg–Witten equations on manifolds with boundary, PhD thesis, MIT (2011)

[19] T Nguyen, The Seiberg–Witten equations on manifolds with boundary I: The space of monopoles and their boundary values, Comm. Anal. Geom. 20 (2012) 565

[20] L I Nicolaescu, The Maslov index, the spectral flow, and decompositions of manifolds, Duke Math. J. 80 (1995) 485

[21] L I Nicolaescu, Notes on Seiberg–Witten theory, Graduate Studies in Mathematics 28, Amer. Math. Soc. (2000)

[22] C H Taubes, Casson's invariant and gauge theory, J. Differential Geom. 31 (1990) 547

[23] V Turaev, A combinatorial formulation for the Seiberg–Witten invariants of $3$–manifolds, Math. Res. Lett. 5 (1998) 583