Kuranishi Structures and Virtual Fundamental Chains

Abstract

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Chain level loop bracket and pseudo‐holomorphic disks

“…Here, we summarize some conventions on signs. Our sign conventions for direct/fiber products follow [, Section 8.2], and for pushout of differential forms we follow [, Section 7.1]. Note that these conventions are different from those in .…”

“…Another key idea in is to systematically use (Moore) loops with arbitrarily many marked points, which enables us to define the chain model of the free loop space as a geometric analogue of the Hochschild complex of the space of differential forms. The aim of this paper is to combine the construction in with the theory of Kuranishi structures to carry out details of the argument sketched above. In particular, we give the first (as far as the author knows) rigorous proofs of Corollaries and , which are definitely important in symplectic topology.…”

“…Remark In , technical details of the proof were deferred to forthcoming (as of the time was written) works. Technical works on moduli spaces of pseudo‐holomorphic curves were carried out in subsequent papers by Fukaya–Oh–Ohta–Ono, culminating in . On the other hand, technical works on chain level string topology (at least those relevant to the argument outlined above) had not been worked out until .…”

“…On the other hand, technical works on chain level string topology (at least those relevant to the argument outlined above) had not been worked out until . As explained above, the aim of this paper is to combine techniques of and , which causes new technical difficulties; the main two difficulties (one algebraic and the other geometric/topological) are briefly explained at the beginnings of Sections 6 and 7, respectively.…”

“…Our proof uses the theory of Kuranishi structures on moduli spaces of (perturbed) pseudo‐holomorphic disks. In particular, our arguments heavily rely on by Fukaya–Oh–Ohta–Ono. Section 10 very briefly explains some notions in the theory of Kuranishi structures, mainly to fix notations.…”