Book ChapterDOI
Mirror Symmetry and Elliptic Curves
Robbert Dijkgraaf
- Iss: 129, pp 149-163
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In this article, the authors review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are quasimodular forms for the mirror family; they can be computed by a summation over trivalent Feynman graphs.Abstract:
I review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: (1) counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are ‘quasimodular forms’ for the mirror family; (2) they can be computed by a summation over trivalent Feynman graphs.read more
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Journal ArticleDOI
Tropical Mirror Symmetry for Elliptic Curves
TL;DR: In this paper, a tropical generalization of mirror symmetry for elliptic curves is presented, i.e., a statement relating certain labeled Gromov-Witten invariants of a tropical elliptic curve to more refined Feynman integrals.
Dissertation
Arithmetic Properties of Moduli Spaces and Topological String Partition Functions of Some Calabi-Yau Threefolds
TL;DR: In this article, the moduli spaces of complex structures of some special mirror Calabi-Yau three-folds (B-model) were studied and the corresponding topological string partition functions defined from them which are closely related to the generating functions of Gromov-Witten invariants of their mirror CCA threefolds by the mirror symmetry conjecture.
Journal ArticleDOI
Orbitwise countings in H(2) and quasimodular forms
Samuel Lelièvre,Emmanuel Royer +1 more
TL;DR: In this article, the authors prove formulae for countings by orbit of square-tiled surfaces of genus two with one singularity, and show that these countings admit quasimodular forms as generating functions.
Posted Content
Hurwitz numbers and BKP hierarchy
Sergei M. Natanzon,A. Yu. Orlov +1 more
TL;DR: In this article, the authors considered a special series in ratios of the Schur functions which are defined by integers and also by the set of $3k parameters, and presented in form of matrix integrals.
Posted Content
BCOV ring and holomorphic anomaly equation
TL;DR: In this paper, the authors studied certain differential rings over the moduli space of Calabi-Yau manifolds, and observed a close relation to the differential ring of quasi-modular forms due to Kaneko and Zagier.
References
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