Formal Formality of the Hypercommutative Algebras of Low Dimensional Calabi–Yau Varieties

Abstract

There is a homotopy hypercommutative algebra structure on the cohomology of a Calabi–Yau variety. The truncation of this homotopy hypercommutative algebra to a strict hypercommutative algebra is well-known as a mathematical realization of the genus zero B-model. It is shown that this truncation loses no information for some cases, including all Calabi–Yau 3-folds.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Barannikov S., Kontsevich M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. 4, 201–215 (1998)

    Article  MathSciNet  Google Scholar 

  2. 2

    Drummond-Cole G.C., Vallette B.: The minimal model for the Batalin–Vilkovisky operad. Selecta Math. 19, 1–47 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3

    Loday, J.-L., Vallette, B.: Algebraic operads. In: Grundlehren der mathematischen Wissenschaften, vol. 346. Springer, Berlin

  4. 4

    Getzler, E.: Operads and moduli spaces of genus 0 Riemann surfaces. In: The Moduli Space of Curves. Progress in Mathematics, vol. 129, pp. 199–230. Birkhäuser, Boston (1995)

  5. 5

    Getzler E.: Batalin–Vilkovisky algebras and two-dimensional topological field theories. Comm. Math. Phys. 159, 265–285 (1994)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  6. 6

    Galvez-Carrillo I., Tonks A., Vallette B.: Homotopy Batalin–Vilkovisky algebras. J. Noncommut. Geom. 6, 539–602 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7

    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978)

    Google Scholar 

  8. 8

    Merkulov S.: Strongly homotopy algebras of a Kähler manifold. Int. Math. Res. Not. 1999, 153–164 (1998)

    Article  Google Scholar 

  9. 9

    Park J.-S.: Semi-classical quantum fields theories and Frobenius manifolds. Lett. Math. Phys. 81, 41–59 (2007)

    ADS  Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gabriel C. Drummond-Cole.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Drummond-Cole, G.C. Formal Formality of the Hypercommutative Algebras of Low Dimensional Calabi–Yau Varieties. Commun. Math. Phys. 327, 433–441 (2014). https://doi.org/10.1007/s00220-014-2018-9

Download citation

Keywords

  • Modulus Space
  • Internal Edge
  • Harmonic Form
  • Formal Unit
  • Diagonal Part