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We prove Gamma conjecture I for any flag varieties by following a strategy proposed by Galkin and Iritani. The main ingredient is to prove that the $\widehat{\Gamma}$-class of a flag variety is mirror to the totally positive part of the corresponding Rietsch mirror.

Rietsch constructed a candidate $T$-equivariant mirror LG model for any flag variety $G/P$. In this paper, we prove the following mirror symmetry prediction: the small $T\times\mathbb{G}_m$-equivariant quantum cohomology of $G/P$ equipped with quantum $\hslash$-connection is isomorphic as $D_{\hslash}$-modules to the Brieskorn lattice associated to the LG model equipped with Gauss-Manin $\hslash$-connection.

We prove in full generality that the $T$-equivariant quantum cohomology of any flag variety $G/P$ is isomorphic to the coordinate ring of a stratum of the Peterson scheme associated to the Langlands dual group scheme $G^{\vee}$. This result was discovered by Dale Peterson but remains unpublished. Our isomorphism is constructed using Yun-Zhu's isomorphism and Peterson-Lam-Shimozono's homomorphism.

This paper is the $K$-theoretic analogue of a recent new proof, given by the first named author, of Peterson-Lam-Shimozono's theorem via Savelyev's generalization of Seidel representations. The outcome is a new proof of Lam-Li-Mihalcea-Shimozono's conjecture, including its extension to the parabolic case, which was first verified by Kato., Comment: Due to an unresolved dispute, this paper will not be published in any journal

We give a new proof of an unpublished theorem of Peterson, proved previously by Lam-Shimozono, which identifies explicitly the structure constants for the multiplications of Schubert classes in the $T$-equivariant quantum cohomology $QH_T(G/P)$ of any flag variety $G/P$ with the $T$-equivariant Pontryagin homology $H^T(Gr_G)$ of the affine Grassmannian $Gr_G$, where $G$ is any simple complex Lie group. Our approach is to construct an $H_T(pt)$-algebra homomorphism by Gromov-Witten theory and show that it is equal to Peterson's map. More precisely, the map is defined via Savelyev's generalized Seidel representations which can be interpreted as certain Gromov-Witten invariants with input $H^T(Gr_G)\otimes QH_T(G/P)$. We determine these invariants completely, in a way similar to how Fulton-Woodward did in their proof of quantum Chevalley formula., Comment: 15 pages. The multiplicative property of the localized Savelyev-Seidel homomorphism has been proved by H. Iritani. Thus, our proof, which uses the degeneration formula, is removed

For any coadjoint orbit $G/L$, we determine all useful terms of the associated Savelyev-Seidel morphism defined on $H_{-*}(\Omega G)$. Immediate consequences are: (1) the dimension of the kernel of the natural map $\pi_*(G)\otimes \mathbb{Q}\rightarrow \pi_*(Ham(G/L))\otimes \mathbb{Q}$ is at most the semi-simple rank of $L$, and (2) the Bott-Samelson cycles in $\Omega G$ which correspond to Peterson elements are solutions to the min-max problem for Hofer's max-length functional on $\Omega Ham(G/L)$. The proof is based on Bae-Chow-Leung's recent computation of Ma'u-Wehrheim-Woodward morphism for the moment correspondence associated to $G/T$ where $T$ is a maximal torus, the computation of Abbondandolo-Schwarz isomorphism for $G$, and two theoretical results including the coincidence of the above Savelyev-Seidel and Ma'u-Wehrheim-Woodward morphisms, and a Leray-type spectral sequence relating Savelyev-Seidel morphisms for $G/L$ and $G/T$. These ingredients also allow us to obtain an alternative proof of Peterson-Woodward's comparison formula which relates the quantum cohomology of $G/T$ to that of $G/L$., Comment: 44 pages, comments welcome

We quantize the problem considered by Bott-Samelson who applied Morse theory to any compact symmetric space $G/K$ and the associated real flag manifold $G_{\mathbb{R}}/B$ which is a real locus of a complex partial flag variety $G_{\mathbb{C}}/P_{\sigma}$. We prove that the Pontryagin ring $H_{-*}(\Omega(G/K))$ of the based loop space $\Omega(G/K)$ is isomorphic to the Floer cohomology ring $HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ after localization. When $G/K$ is a Lie group, this is a conjecture of Peterson, proved combinatorially by Lam-Shimozono, in the context of quantum cohomologies of complex flag varieties. Our approach is geometric in nature: we construct a Lagrangian correspondence from $T^*(G/K)$ to $G_{\mathbb{C}}/P_{\sigma}$ which geometrically composes with a cotangent fiber to $G_{\mathbb{R}}/B$, and compute the linear part of the associated Ma'u-Wehrheim-Woodward's $A_{\infty}$ homomorphism from a Floer model of $\Omega(G/K)$ to $CF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$. The crux is to make use of the geometry of $G/K$ to construct specific perturbation data which enables us to reduce the computations to the case when $G/K$ is a torus., Comment: Combined with arXiv:2102.03103, 36 pages, Comments welcome

Let $(X,\omega)$ be a compact symplectic manifold whose first Chern class $c_1(X)$ is divisible by a positive integer $n$. We construct a $\mathbb{Z}_{2n}$-action on its Fukaya category $Fuk(X)$ and a $\mathbb{Z}_n$-action on the local models of its moduli of Lagrangian branes. We show that this action is compatible with the gluing functions for different local models., Comment: The construction of the action on Fukaya category is significantly simplified. A new section about the compatibility of the action on local mirrors with gluing functions is added. Comments are welcome