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# Lagrangian intersection Floer theory: anomaly and obstruction. Part I

AMS/IP Studies in Advanced Mathematics, v.46,1 (2009)
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• "The perspective on orienting the relevant moduli spaces taken in [2] [25] [7] [6] is heavily influenced by the approach in the open GW-theory going back to the late 1990s and the initial version of [5]. It works well in genus 0 because a splitting of a smooth genus 0 symmetric surface pΣ, σq into two bordered surfaces interchanged by σ is unique up to homotopy and the one-nodal transitions between the (two) different involution types in genus 0 preserve such a splitting. "
##### Article: Real Orientations, Real Gromov-Witten Theory, and Real Enumerative Geometry
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ABSTRACT: The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.
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• "The difficulty in obtaining a global Kuranishi theory lies at the delicate question of how to define coordinate changes between Kuranishi charts. In literature, there are quite a few similar, yet different definitions of this notion [4],[5],[7],[9]. On the other hand, in [13], the notion of homotopy L ∞ spaces which is a global version of L ∞ spaces is already defined. "
##### Article: Casson invariants via virtual counting
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ABSTRACT: This is a research announcement on an alternative definition of the Casson invariants by means of virtual counting of the moduli space of irreducible representations of the fundamental group into $\SU(2)$. Along the way, by using derived differential geometry, we propose a general framework to obtain invariants from Chern-Simons type gauge theories.
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• "Furthermore, a monotone Lagrangian torus L ⊂ (CP n , ω FS ) is extremal, as follows by elementary topological considerations together with the fact that there exists a representative of π 2 (CP 2 , L) having Maslov index two and positive symplectic area by [6, Theorems 1.1, 1.2]. (For previous related results, consider [23], [19], [18], [11], [4], and [8].) In [6] the author learned about the following two conjectures, the first one originally due to L. Lazzarini: Conjecture 1.2. "
##### Article: Uniqueness of extremal Lagrangian tori in the four-dimensional disc
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ABSTRACT: The following interesting quantity was introduced by K. Cieliebak and K. Mohnke for a Lagrangian submanifold $L$ of a symplectic manifold: the minimal positive symplectic area of a disc with boundary on $L$. They also showed that this quantity is bounded from above by $\pi/n$ for a Lagrangian torus inside the $2n$-dimensional unit disc equipped with the standard symplectic form. A Lagrangian torus for which this upper bound is attained is called extremal. We show that an extremal Lagrangian torus inside the four-dimensional unit disc is contained in the boundary $\partial D^4=S^3$, and is hence Hamiltonian isotopic to the product torus $S^1_{1/\sqrt{2}} \times S^1_{1/\sqrt{2}} \subset S^3$. This provides an answer to a question by L. Lazzarini in the four-dimensional case.
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