INTRODUCTION xv A-model log geometry B-model geometry Tropical As this program with Siebert is ongoing, with much work still to be done, my lectures at the CBMS regional conference in Manhattan, Kansas were intended to give a snapshot of the current state of this program. This monograph closely follows the outline of those lectures. The basic goal is threefold. First, I wish to explain explicitly, at least in special cases, all the worlds sug- gested in the above diagram: the tropical world, the “classical” world of the A- and B-model, and log geometry. Second, I would like to explain one very concrete case where the full picture has been worked out for both the A- and B-models. This is the case of P2. For the A-model, curve counting is the result of Mikhalkin, and here I will give a proof of his result adapted from a more general result of Nishinou and Siebert [86], as that approach is more in keeping with the philosophy of the program. For the B-model, I will explain my own recent work [42] which shows how period integrals extract tropical information. Third, I wish to survey some of the results obtained by Siebert and myself in the Calabi-Yau case, outlining how this approach can be expected to yield a proof of mirror symmetry. While for P2 I give complete details, this third part is intended to be more of a guide for reading the original papers, which unfortunately are quite long and technical. I hope to at least convey an intuition for this approach. I will take a very ahistorical approach to all of this, starting with the basics of tropical geometry and working backwards, showing how a study of tropical geom- etry can lead naturally to other concepts which first arose in the study of mirror symmetry. In a way, this may be natural. To paraphrase Witten’s statement about string theory, mirror symmetry often seems like a piece of twenty-first century math- ematics which fell into the twentieth century. Its initial discovery in string theory represents some of the more difficult aspects of the theory. Even an explanation of the calculations carried out by Candelas et al. can occupy a significant portion of a course, and the theory built up to define and compute Gromov-Witten invariants is even more involved. On the other hand, the geometry that now seems to underpin mirror symmetry, namely tropical geometry, is very simple and requires no partic- ular background to understand. So it makes sense to develop the discussion from the simplest starting point. The prerequisites of this volume include a familiarity with algebraic geometry at the level of Hartshorne’s text [57] as well as some basic differential geometry. In addition, familiarity with toric geometry will be very helpful the text will recall many of the basic necessary facts about toric geometry, but at least some previ- ous experience will be useful. For a more in-depth treatment of toric geometry, I
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