We provide a construction of equivariant Lagrangian Floer homology , for a compact Lie group acting on a symplectic manifold in a Hamiltonian fashion, and a pair of -Lagrangian submanifolds . We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of . Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent of the auxiliary choices involved in their construction, and are -bimodules. In the case when , we show that their chain complex is homotopy equivalent to the equivariant Morse complex of . Furthermore, if zero is a regular value of the moment map and if acts freely on , we construct two ‘Kirwan morphisms’ from to (respectively, from to ). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat -connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah–Floer conjecture.

中文翻译：

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通过 EGN 余切束的等变拉格朗日弗洛尔同调

我们提供等变拉格朗日弗洛尔同调的构造，对于紧李群作用于辛流形以哈密顿式的方式，以及一对-拉格朗日子流形。我们通过使用涉及近似值的余切丛的辛同伦商来做到这一点。我们的构造依赖于韦尔海姆和伍德沃德的被子理论以及望远镜构造。我们证明这些群体独立于其构建中涉及的辅助选择，并且是-双模块。在这种情况下，当，我们表明他们的链复杂是同伦等价于等变莫尔斯复数。此外，如果零是矩图的常规值而如果自由地行动于，我们构造两个“Kirwan 态射”到（分别从到）。我们的构造适用于精确且单调的设置，以及平面的扩展模空间的设置-黎曼曲面的连接，在 Manolescu 和 Woodward 的工作中考虑过。应用于后一种情况，我们的构造为阿提亚-弗洛尔猜想提供了等变辛边。