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

SIGMA 15 (2019), 041, 31 pages      arXiv:1807.07703      https://doi.org/10.3842/SIGMA.2019.041
Contribution to the Special Issue on Moonshine and String Theory

Hecke Operators on Vector-Valued Modular Forms

Vincent Bouchard a, Thomas Creutzig ab and Aniket Joshi a
a) Department of Mathematical & Statistical Sciences, University of Alberta, 632 Central Academic Building, Edmonton T6G 2G1, Canada
b) Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Received September 26, 2018, in final form May 13, 2019; Published online May 25, 2019

Abstract
We study Hecke operators on vector-valued modular forms for the Weil representation $\rho_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of type $\rho_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators $\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer $n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, and compare our operators with the alternative construction of Bruinier-Stein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].

Key words: Hecke operators; vector-valued modular forms; Weil representation.

pdf (561 kb)   tex (32 kb)  

References

  1. Ajouz A., Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular form, Ph.D. Thesis, University of Siegen, 2015.
  2. Booker T., Davydov A., Commutative algebras in Fibonacci categories, J. Algebra 355 (2012), 176-204, arXiv:1103.3537.
  3. Borcherds R.E., Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491-562, arXiv:alg-geom/9609022.
  4. Borcherds R.E., Reflection groups of Lorentzian lattices, Duke Math. J. 104 (2000), 319-366, arXiv:math.GR/9909123.
  5. Bouchard V., Creutzig T., Diaconescu D.-E., Doran C., Quigley C., Sheshmani A., Vertical D4-D2-D0 bound states on K3 fibrations and modularity, Comm. Math. Phys. 350 (2017), 1069-1121, arXiv:1601.04030.
  6. Boylan H., Jacobi forms, finite quadratic modules and Weil representations over number fields, Lecture Notes in Math., Vol. 2130, Springer, Cham, 2015.
  7. Bruinier J.H., On the converse theorem for Borcherds products, J. Algebra 397 (2014), 315-342, arXiv:1210.4821.
  8. Bruinier J.H., Stein O., The Weil representation and Hecke operators for vector valued modular forms, Math. Z. 264 (2010), 249-270, arXiv:0704.1868.
  9. Carnahan S., Generalized moonshine, II: Borcherds products, Duke Math. J. 161 (2012), 893-950, arXiv:0908.4223.
  10. Creutzig T., McRae R., Kanade S., Tensor categories for vertex operator superalgebra extensions, arXiv:1705.05017.
  11. Diamond F., Shurman J., A first course in modular forms, Graduate Texts in Mathematics, Vol. 228, Springer-Verlag, New York, 2005.
  12. Eholzer W., Skoruppa N.-P., Conformal characters and theta series, Lett. Math. Phys. 35 (1995), 197-211, arXiv:hep-th/9410077.
  13. Eichler M., Zagier D., The theory of Jacobi forms, Progress in Mathematics, Vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985.
  14. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Mathematical Surveys and Monographs, Vol. 205, Amer. Math. Soc., Providence, RI, 2015.
  15. Gholampour A., Sheshmani A., Donaldson-Thomas invariants of 2-dimensional sheaves inside threefolds and modular forms, Adv. Math. 326 (2018), 79-107, arXiv:1309.0050.
  16. Harvey J.A., Moore G., Algebras, BPS states, and strings, Nuclear Phys. B 463 (1996), 315-368, arXiv:hep-th/9510182.
  17. Harvey J.A., Moore G., On the algebras of BPS states, Comm. Math. Phys. 197 (1998), 489-519, arXiv:hep-th/9609017.
  18. Harvey J.A., Wu Y., Hecke relations in rational conformal field theory, J. High Energy Phys. 2018 (2018), no. 9, 032, 37 pages, arXiv:1804.06860.
  19. Höhn G., Scheithauer N.R., A natural construction of Borcherds' fake Baby Monster Lie algebra, Amer. J. Math. 125 (2003), 655-667, arXiv:math.QA/0312106.
  20. Huang Y.-Z., Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008), 871-911, arXiv:math.QA/0502533.
  21. Huang Y.-Z., Kirillov Jr. A., Lepowsky J., Braided tensor categories and extensions of vertex operator algebras, Comm. Math. Phys. 337 (2015), 1143-1159, arXiv:1406.3420.
  22. Joshi A., Hecke operators on vector-valued modular forms of the Weil representation, MSc. Thesis, University of Alberta, 2018.
  23. Maldacena J., Strominger A., Witten E., Black hole entropy in M-theory, J. High Energy Phys. 1997 (1997), no. 12, 002, 16 pages, arXiv:hep-th/9711053.
  24. Raum M., Computing genus 1 Jacobi forms, Math. Comp. 85 (2016), 931-960, arXiv:1212.1834.
  25. Rössler M., Hecke operators and vector valued modular forms, MSc. Thesis, Technische Universität Darmstadt, 2015.
  26. Scheithauer N.R., Generalized Kac-Moody algebras, automorphic forms and Conway's group. I, Adv. Math. 183 (2004), 240-270.
  27. Scheithauer N.R., Generalized Kac-Moody algebras, automorphic forms and Conway's group. II, J. Reine Angew. Math. 625 (2008), 125-154.
  28. Scheithauer N.R., The Weil representation of ${\rm SL}_2({\mathbb Z})$ and some applications, Int. Math. Res. Not. 2009 (2009), 1488-1545.
  29. Stein O., The Fourier expansion of Hecke operators for vector-valued modular forms, Funct. Approx. Comment. Math. 52 (2015), 229-252.
  30. Stein W., Modular forms, a computational approach, Graduate Studies in Mathematics, Vol. 79, Amer. Math. Soc., Providence, RI, 2007.
  31. Vafa C., Two dimensional Yang-Mills, black holes and topological strings, arXiv:hep-th/0406058.
  32. Weil A., Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211.
  33. Werner F., Vector valued Hecke theory, Ph.D. Thesis, Technische Universität, 2014.
  34. Westerholt-Raum M., Products of vector valued Eisenstein series, Forum Math. 29 (2017), 157-186, arXiv:1411.3877.
  35. Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

Previous article  Next article  Contents of Volume 15 (2019)