Abstract
We review some classical and quantum geometry of Calabi-Yau moduli related to B-model aspects of closed string mirror symmetry. This note comes out of the author’s lectures in the workshop “B-model aspects of Gromov-Witten theory” held at University of Michigan in 2013.
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References
Barannikov, S.: Extended moduli spaces and mirror symmetry in dimensions n > 3, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.), University of California, Berkeley (1999)
Barannikov, S.: Quantum periods, I. Semi-infinite variations of Hodge structures. Int. Math. Res. Not. 23, 1243–1264 (2001)
Barannikov, S.: Non-commutative periods and mirror symmetry in higher dimensions. Commun. Math. Phys. 228(2), 281–325 (2002)
Barannikov, S.: Semi-infinite Hodge structures and mirror symmetry for projective spaces (2001). arXiv:math.AG/0010157
Barannikov, S., Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvectorfields. Int. Math. Res. Not. 4, 201–215 (1998)
Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165(2), 311–427 (1994). MR1301851(95f:32029)
Coates, T., Givental, A.B.: Quantum Riemann-Roch, Lefschetz and Serre. Ann. Math. (2) 165(1), 15–53 (2007)
Costello, K.: Topological conformal field theories and Calabi-Yau categories. Adv. Math. 210(1), 165–214 (2007)
Costello, K.: The partition function of a topological field theory. J. Topol. 2(4), 779–822 (2009)
Costello, K.: Renormalization and Effective Field Theory. Mathematical Surveys and Monographs, vol. 170. American Mathematical Society, Providence (2011). MR2778558
Costello, K., Li, S.: Quantum BCOV theory on Calabi-Yau manifolds and the higher genus Bmodel (2011). arXiv:1201.4501 [math.QA]
Douai, A., Sabbah, C.: Gauss-Manin systems, Brieskorn lattices and Frobenius structures, I. In: Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002), pp. 1055–1116 (2003)
Douai, A., Sabbah, C.: Gauss-Manin systems, Brieskorn lattices and Frobenius structures, II. In: Frobenius Manifolds. Aspects of Mathematics, vol. E36, pp. 1–18. Vieweg, Wiesbaden (2004)
Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993). Lecture Notes in Mathematics, vol. 1620, pp. 120–348. Springer, Berlin (1996)
Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry, and quantum singularity theory. Ann. Math. (2) 178(1), 1–106 (2013)
Givental, A.B.: A mirror theorem for toric complete intersections. In: Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996). Progress in Mathematics, vol. 160, pp. 141–175. Birkhäuser, Boston (1998)
Givental, A.B.: A tutorial on quantum cohomology. In: Symplectic Geometry and Topology (Park City, UT, 1997). IAS/Park City Mathematics Series, vol. 7, pp. 231–264. American Mathematical Society, Providence (1999)
Givental, A.B.: Gromov-Witten invariants and quantization of quad ratic Hamiltonians. Mosc. Math. J. 1(4), 551–568, 645 (2001). Dedicated to the memory of I. G. Petrovskii on the occasion of his100th anniversary
Givental, A.B.: Symplectic geometry of Frobenius structures. In: Frobenius Manifolds. Aspects of Mathematics, vol. E36, pp. 91–112. Vieweg, Wiesbaden (2004)
He, W., Li, S., Shen, Y., Webb, R.: Landau-Ginzburg mirror symmetry conjecture (2015). arXiv:math.AG/1503.01757
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The Modulispace of Curves (Texel Island, 1994). Progress in Mathematics, vol. 129, pp. 165–172. Birkhäuser, Boston (1995)
Kontsevich, M., Soibelman, Y.: Notes on A∞-algebras, A∞-categories and non-commutative geometry. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol. 757, pp. 153–219. Springer, Berlin (2009). MR2596638
Li, S.: Calabi-Yau geometry and higher genus mirror symmetry. Thesis, Harvard (2011)
Li, S.: Feynman graph integrals and almost modular forms. Commun. Number Theory and Phys. 6, 129–157 (2012)
Li, S.: BCOV theory on the elliptic curve and higher genus mirror symmetry (2011). arXiv:1112.4063 [math.QA]
Li, S.: Vertex algebras and quantum master equation (2016). arXiv:1612.01292 [math.QA]
Li, C., Li, S., Saito, K., Shen, Y.: Mirror symmetry for exceptional unimodular singularities. J. Eur. Math. Soc. 19(4), 1189–1229 (2017)
Li, C., Li, S., Saito, K.: Primitive forms via polyvector fields (2013). arXiv: math.AG/1311.1659
Lian, B.H., Liu, K.F., Yau, S.T.: Mirror principle I. Asian J. Math. 1(4), 729–763 (1997)
Losev, A., Shadrin, S., Shneiberg, I.: Tautological relations in Hodge field theory. Nucl. Phys. B 786(3), 267–296 (2007)
Okounkov, A., Pandharipande, R.: Virasoro constraints for target curves. Invent. Math. 163(1), 47–108 (2006)
Saito, K.: The higher residue pairings \(K_{F}^{(k)}\) for a family of hypersurface singular points. In: Singularities, Part 2. Proceedings of Symposia in Pure Mathematics (Arcata, Calif., 1981), pp. 441–463 (1983)
Saito, K.: Period mapping associated to a primitive form. Publ. Res. Inst. Math. Sci. 19(3), 1231–1264 (1983)
Saito, M.: On the structure of Brieskorn lattice. Ann. Inst. Fourier (Grenoble) 39(1), 27–72 (1989)
Saito, K.: From primitive form to mirror symmetry (2014). arXiv: math.AG/1408.4208
Saito, M.: On the structure of Brieskorn lattices, II (2013). arXiv: math.AG/1312.6629
Takahashi, A.: Primitive forms, topological LG models coupled to gravity and mirror symmetry (1998). arXiv: math.AG/9802059
Zwiebach, B.: Closed string field theory: quantum action and the Batalin-Vilkovisky master equation. Nucl. Phys. B 390(1), 33–152 (1993)
Acknowledgements
The authors would like to thank the organizers and participants of the workshop on B-model aspects of Gromov-Witten theory, and the hospitality of the mathematics department at University of Michigan.
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Li, S. (2018). Some Classical/Quantum Aspects of Calabi-Yau Moduli. In: Clader, E., Ruan, Y. (eds) B-Model Gromov-Witten Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-94220-9_4
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DOI: https://doi.org/10.1007/978-3-319-94220-9_4
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