Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 101, 17 pages      arXiv:1212.4234      https://doi.org/10.3842/SIGMA.2012.101
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Renormalization Method and Mirror Symmetry

Si Li
Department of mathematics, Northwestern University, 2033 Sheridan Road, Evanston IL 60208, USA

Received May 07, 2012, in final form December 13, 2012; Published online December 18, 2012

Abstract
This is a brief summary of our works [arXiv:1112.4063, arXiv:1201.4501] on constructing higher genus B-model from perturbative quantization of BCOV theory. We analyze Givental's symplectic loop space formalism in the context of B-model geometry on Calabi-Yau manifolds, and explain the Fock space construction via the renormalization techniques of gauge theory. We also give a physics interpretation of the Virasoro constraints as the symmetry of the classical BCOV action functional, and discuss the Virasoro constraints in the quantum theory.

Key words: BCOV; Calabi-Yau; renormalization; mirror symmetry.

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References

  1. Barannikov S., Extended moduli spaces and mirror symmetry in dimensions n>3, Ph.D. thesis, University of California, Berkeley, 1999, math.AG/9903124.
  2. Barannikov S., Non-commutative periods and mirror symmetry in higher dimensions, Comm. Math. Phys. 228 (2002), 281-325.
  3. Barannikov S., Quantum periods. I. Semi-infinite variations of Hodge structures, Int. Math. Res. Not. 2001 (2001), no. 23, 1243-1264, math.AG/0006193.
  4. Barannikov S., Kontsevich M., Frobenius manifolds and formality of Lie algebras of polyvector fields, Int. Math. Res. Not. 1998 (1998), no. 4, 201-215, alg-geom/9710032.
  5. Bershadsky M., Cecotti S., Ooguri H., Vafa C., Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), 311-427, hep-th/9309140.
  6. Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.
  7. Coates T., Givental A.B., Quantum Riemann-Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15-53, math.AG/0110142.
  8. Costello K.J., Renormalization and effective field theory, Mathematical Surveys and Monographs, Vol. 170, American Mathematical Society, Providence, RI, 2011.
  9. Costello K.J., Li S., Open-closed BCOV theory on Calabi-Yau manifolds, in preparation.
  10. Costello K.J., Li S., Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model, arXiv:1201.4501.
  11. Dubrovin B., Zhang Y., Frobenius manifolds and Virasoro constraints, Selecta Math. (N.S.) 5 (1999), 423-466, math.AG/9808048.
  12. Eguchi T., Hori K., Xiong C.S., Quantum cohomology and Virasoro algebra, Phys. Lett. B 402 (1997), 71-80, hep-th/9703086.
  13. Eguchi T., Jinzenji M., Xiong C.S., Quantum cohomology and free-field representation, Nuclear Phys. B 510 (1998), 608-622, hep-th/9709152.
  14. Getzler E., The Virasoro conjecture for Gromov-Witten invariants, in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998), Contemp. Math., Vol. 241, Amer. Math. Soc., Providence, RI, 1999, 147-176, math.AG/9812026.
  15. Givental A.B., A mirror theorem for toric complete intersections, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math., Vol. 160, Birkhäuser Boston, Boston, MA, 1998, 141-175, alg-geom/9701016.
  16. Givental A.B., Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), 551-568, math.AG/0108100.
  17. Givental A.B., Symplectic geometry of Frobenius structures, in Frobenius Manifolds, Aspects Math., Vol. E36, Vieweg, Wiesbaden, 2004, 91-112, math.AG/0305409.
  18. Huang M.X., Klemm A., Quackenbush S., Topological string theory on compact Calabi-Yau: modularity and boundary conditions, in Homological Mirror Symmetry, Lecture Notes in Phys., Vol. 757, Springer, Berlin, 2009, 45-102, hep-th/0612125.
  19. Kaneko M., Zagier D., A generalized Jacobi theta function and quasimodular forms, in The Moduli Space of Curves (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser Boston, Boston, MA, 1995, 165-172.
  20. Li J., Tian G., Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119-174, alg-geom/9602007.
  21. Li S., BCOV theory on the elliptic curve and higher genus mirror symmetry, arXiv:1112.4063.
  22. Li S., Calabi-Yau geometry and higher genus mirror symmetry, Ph.D. thesis, Harvard University, 2011.
  23. Li S., Feynman graph integrals and almost modular forms, Commun. Number Theory Phys. 6 (2012), 129-157, arXiv:1112.4015.
  24. Lian B.H., Liu K., Yau S.T., Mirror principle. I, Asian J. Math. 1 (1997), 729-763, alg-geom/9712011.
  25. Losev A., Shadrin S., Shneiberg I., Tautological relations in Hodge field theory, Nuclear Phys. B 786 (2007), 267-296, arXiv:0704.1001.
  26. Okounkov A., Pandharipande R., Virasoro constraints for target curves, Invent. Math. 163 (2006), 47-108, math.AG/0308097.
  27. Ruan Y., Tian G., A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), 259-367.
  28. Shadrin S., BCOV theory via Givental group action on cohomological fields theories, Mosc. Math. J. 9 (2009), 411-429, arXiv:0810.0725.
  29. Yamaguchi S., Yau S.T., Topological string partition functions as polynomials, J. High Energy Phys. 2004 (2004), no. 7, 047, 20 pages, hep-th/0406078.

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