McDuff D., Salamon D., J-Holomorphic Curves and Quantum Cohomology
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The theory of J-holomorphic curves has been of great importance to symplectic topologists ever since its inception in Gromov's paper  of 1985. Its applications include many key results in symplectic topology: see, for example, Gro-mov , McDuff [42, 45], Lalonde-McDufF , and the collection of articles in Audin-Lafontaine . It was also one of the main inspirations for the creation of Floer homology [18, 19, 73], and recently has caught the attention of mathematical physicists through the theory of quantum cohomology: see Vafa  and Aspinwall-Morrison .Because of this increased interest on the part of the wider mathematical community, it is a good time to write an expository account of the field, which explains the main technical steps in the theory. Although all the details are available in the literature in some form or other, they are rather scattered. Also, some improvements in exposition are now possible. Our account is not, of course, complete, but it is written with a fair amount of analytic detail, and should serve as a useful introduction to the subject. We develop the theory of the Gromov-Witten invariants as formulated by Ruan in  and give a detailed account of their applications to quantum cohomology. In particular, we give a new proof of Ruan-Tian's theorem [67, 68] that the quantum cup-product is associative.Many people have made useful comments which have added significantly to our understanding. In particular, we wish to thank Givental for explaining quantum cohomology, Ruan for several useful discussions and for pointing out to us the connection between associativity of quantum multiplication and the WDVV-equation, Taubes for his elegant contribution to Section 3.4, and especially Gang Liu for pointing out a significant gap in an earlier version of the gluing argument. We are also grateful to Lalonde for making helpful comments on a first draft of this manuscript. The first author wishes to acknowledge the hospitality of the University of California at Berkeley, and the grant GER-9350075 under the NSF Visiting Professorship for Women program which provided partial support during some of the work on this book.
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