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# Communications in Number Theory and Physics

## Volume 16 (2022)

### Number 2

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

Pages: 353 – 400

DOI: https://dx.doi.org/10.4310/CNTP.2022.v16.n2.a3

#### Authors

#### 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 $[E]_\mathbb{C}$ ($E$ as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kähler parameter of the target space $[E]_\mathbb{C}$ in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.

#### 2010 Mathematics Subject Classification

11G05, 11G15, 11G40, 81T30, 81T40

Received 17 September 2020

Accepted 1 March 2022

Published 27 April 2022