$T$-equivariant disc potential and SYZ mirror construction

Abstract

We develop a G-equivariant Lagrangian Floer theory by counting pearly trees in the Borel construction L G . We apply the construction to smooth moment-map fibers of toric semi-Fano varieties and obtain the T-equivariant Landau-Ginzburg mirrors. We also apply this to the typical S 1 -invariant SYZ singular fiber, which is the single-pinched torus, and compute its S 1 -equivariant disc potential. Contents 1. Introduction 1 2. A Morse model for equivariant Lagrangian Floer theory 4 2.1. The non-equivariant singular chain model 5 2.2. The non-equivariant Morse model with a strict unit 8 2.3. The equivariant Morse model 15 2.4. Homotopy partial units and the equivariant disc potentials 20 2.5. Equivariant Floer theory for Lagrangian immersions 25 3. The T-equivariant disc potentials of toric manifolds 26 3.1. Morse theory on the approximation spaces 27 3.2. Computing the T-equivariant disc potentials 31 4. S 1 -equivariant disc potential for the immersed 2-sphere 36 References 40

Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces

“…In this section we first review some heuristics in mirror symmetry, which are well-known, and possibly all established. 12 We then discuss a series of examples giving evidence for the conjectures of Section 1.…”

“…This is the pair-of-pants. The wrapped Fukaya category W(P) can easily be calculated, in particular the endomorphisms of the arc Λ -which is a non-compact Lagrangian in P -is: 1 Other recent studies of G-equivariant Floer theory can be found in [12], [10], [4]. Though, the technical aspects of wrapped Floer theory involving non-compact invariant Lagrangians seems not to have been addressed yet.…”

Equivariant Lagrangian Floer homology via cotangent bundles of EGN$EG_N$

“…• In [30], in the case of a single Lagrangian 𝐿 0 = 𝐿 1 , Kim, Lau and Zheng defined an equivariant Lagrangian Floer homology using a Borel construction similar (but slightly different, see Remark 4.38) to the one in this paper. • In gauge theory, Kronheimer and Mrowka [33] defined several versions of monopole homology, corresponding to U(1)-equivariant theories.…”

Disk potential functions for polygon spaces

“…Givental [Giv98] derived the T -equivariant quantum cohomology on a smooth toric manifold and expressed it in terms of a T -equivariant superpotential. In [KLZ19], for a semi-Fano toric manifold, the present authors defined the T -equivariant disk potential of a toric fiber using T -equivariant Lagrangian Floer theory in Morse model. They computed the equivariant terms in the model, and verified that it agrees with Givental's equivariant superpotential via the mirror map.…”

“…Let us first briefly recall the definition of T -equivariant disk potential introduced in [KLZ19]. We identify T with (S 1 ) n+2 and denote by L T the Borel construction L × T (S ∞ ) n+2 .…”