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2015 | OriginalPaper | Chapter

# Lectures on BCOV Holomorphic Anomaly Equations

Authors: Atsushi Kanazawa, Jie Zhou

Publisher: Springer New York

## Abstract

The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky et al. (Nucl Phys B 405:279–304, 1993; Comm Math Phys 165:311–428, 1994). It grew from a series of lectures the authors gave at the Fields Institute in the Thematic Program of Calabi–Yau Varieties in the fall of 2013.
Footnotes
1
We take a universal covering of $$\mathcal{M}$$ if necessary but most of what follows works in a local setting.

2
We use the Einstein summation convention.

3
We have $$N_{g}(0) =\int _{\overline{M}_{ g}\times X^{\vee }}c_{top}(Ob) = (-1)^{g}\frac{\chi (X^{\vee })} {2} \int _{\overline{M}_{ g}}c_{g-1}^{3}(\mathcal{H}_{g})$$, where $$Ob \rightarrow \overline{M}_{g,0}(X, 0)\mathop{\cong}\overline{M}_{g} \times X^{\vee }$$ is the obstruction bundle and $$\mathcal{H}_{g} \rightarrow \overline{M}_{g}$$ is the Hodge bundle .

4
See  which proposes a rigorous definition for the $$\mathcal{F}_{g}$$’s.

5
This case is somewhat misleading because an elliptic curve is a self-mirror manifold. However, we believe this is still a good example the reader should keep in mind.

6
We have to take care of the first term of the second line, see .

7
Computationally, for genus one amplitude, we need to take its derivative to get rid of the anti-holomorphic terms. Also the generating function of genus one Gromo-Witten invariants with one insertion, which is given by the first derivative of $$\mathcal{F}_{1}$$, is more natural due to stability reasons.

Literature
1.
Aganatic, M., Bouchard, V., Klemm, A.: Topological strings and (almost) modular forms. hep-th/0607100
2.
Alim, M.: Lectures on mirror symmetry and topological string theory. arXiv:1207.0496
3.
Alim, M.: Polynomial Rings and Topological Strings. arXiv:1401.5537 [hep-th]
4.
Alim, M., Länge, J.D.: Polynomial structure of the (open) topological string partition function. JHEP 0710, 045 (2007) CrossRef
5.
Alim, M., Scheidegger, E., Yau, S.-T., Zhou, J.: Special polynomial rings, quasi modular forms and duality of topological strings. arXiv:1306.0002
6.
Antoniadis, I., Gava, E., Narain, K., Taylor, T.: N=2 type II heterotic duality and higher derivative F terms. Nucl. Phys. B455, 109–130 (1995)
7.
Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493–545 (1994)
8.
Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories, (with an appendix by S.Katz). Nucl. Phys. B 405, 279–304 (1993)
9.
Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Comm. Math. Phys. 165, 311–428 (1994)
10.
Bryant, R., Griffiths, P.: Some observations on the infinitesimal period relations for regular threefolds with trivial canonical bundle. Arith. Geom. II, 77–102 (1983); Progr. Math. 36
11.
Candelas, P., de la Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi–Yau manifolds as an exactly solvable superconformal theory. Nucl. Phys. B 359(1), 21–74 (1991)
12.
Cecotti, S., Vafa, C.: Topological anti-topological fusion. Nucl.Phys. B 367, 359–461 (1991)
13.
Chiang, T., Klemm, A., Yau, S.-T., Zaslow, E.: Local mirror symmetry: calculations and interpretations. Adv. Theor. Math. Phys. 3, 495–565 (1999)
14.
Costello, K.J., Li, S.: Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model. arXiv:1201.4501 [math.QA]
15.
Cox, D., Katz, S.: Mirror Symmetry and Algebraic Geometry. Mathematical Surveys and Monographs, vol. 68. American Mathematical Society, Providence (1999)
16.
Dijkgraaf, R.: Mirror Symmetry and Elliptic Curves, the Moduli Space of Curves. Progress in Mathematics, vol. 129, pp. 149–163. Birkhäuser, Boston (1995)
17.
Faber, C., Pandharipande, R.: Hodge integrals and Gromov–Witten theory. Invent. Math. 139(1), 173–199 (2000)
18.
Fang, H., Lu, Z., Yoshikawa, K.-I.: Analytic torsion for Calabi–Yau threefolds. J. Diff. Geom. 80(2), 175–259 (2008)
19.
Freed, D.: Special Kähler manifolds. Comm. Math. Phys. 203(1), 31–52 (1999)
20.
Gerasimov, A.A., Shatashvili, S.L.: Towards integrability of topological strings. I. Three-forms on Calabi–Yau manifolds. JHEP 0411, 074 (2004)
21.
Ghoshal, D., Vafa, C.: C = 1 string as the topological theory of the conifold. Nucl. Phys. B 453, 121 (1995)
22.
Givental, A.: 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)
23.
Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B 338(1), 15–37 (1990)
24.
Hori, K., Vafa, C.: Mirror symmetry. arXiv: hep-th/0002222
25.
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence (2003)
26.
Higashijima, K., Itou, E., Nitta, M.: Normal coordinates in Kähler manifolds and the background field method. Progr. Theor. Phys. 108(1), 185–202
27.
Hosono, S.: BCOV ring and holomorphic anomaly equation. arXiv:0810.4795
28.
Hosono, S., Konishi, Y.: Higher genus Gromov-Witten invariants of the Grassmannian, and the Pfaffian Calabi-Yau 3-folds. Adv. Theor. Math. Phys. 13(2), 463–495 (2009)
29.
Huang, M.-x., Klemm, A.: Holomorphic anomaly in Gauge theories and matrix models. JHEP 0709, 054 (2007)
30.
Huang, M.-x., Klemm, A., Quackenbush, S.: Topological string theory on compact Calabi-Yau, modularity and boundary conditions. hep-th/0612125
31.
Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms, the moduli space of curves. In: Dijkgraaf, R., Faber, C., van der Geer, G. (eds.) The Moduli Space of Curves. Progress in Mathematics, vol. 129, pp. 165–172. Birkhäuser, Boston (1995) CrossRef
32.
Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. arxiv: hep-th/9906046
33.
Kapranov, M.: Rozansky-Witten invariants via Atiyah classes. Compos. Math. 115(1), 71–113 (1999)
34.
Lian, B.H., Liu, K., Yau, S.-T.: Mirror principle. I. Asian J. Math. 1(4), 729–763 (1997)
35.
Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians. J. Am. Math. Soc. 6(1), 223–247 (1993)
36.
Popa, A.: The genus one Gromov–Witten invariants of Calabi–Yau complete intersections. Trans. AMS 365(3), 1149–1181 (2013)
37.
Strominger, A.: Special geometry. Comm. Math. Phys. 133, 163–180 (1990)
38.
Witten, E.: Topological sigma models. Comm. Math. Phys. 118(3), 355–529 (1988)
39.
Witten, E.: Quantum background independence in string theory. arixv: hep-th/9306122
40.
Yamaguchi, S., Yau, S.-T.: Topological string partition functions as polynomials. J. High Energy Phys. 047(7), 20 (2004) MathSciNet
41.
Zhou, J.: Differential rings from special Kähler geometry. arXiv:1310.3555
42.
Zhou, J.: Polynomial Structure of Topological String Partition Functions. arxiv: 1501.00451
43.
Zinger, A.: The reduced genus 1 Gromov–Witten invariants of Calabi–Yau hypersurfaces. J. Am. Math. Soc. 22(3), 691–737 (2009)
44.
Zinger, A.: Standard vs. reduced genus-one Gromov–Witten invariants. Geom. Top. 12(2), 1203–1241 (2008)