Modular parametrization as Polyakov path integral: cases with CM elliptic curves as target spaces

KONDO S., Watari T.

Communications in Number Theory and Physics, vol.16, no.2, pp.353-400, 2022 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 16 Issue: 2
  • Publication Date: 2022
  • Doi Number: 10.4310/cntp.2022.v16.n2.a3
  • Journal Name: Communications in Number Theory and Physics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.353-400
  • Middle East Technical University Northern Cyprus Campus Affiliated: Yes

Abstract

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k: K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler parameter of the target space [E]C in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.