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


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.


  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



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).

Download citation


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