Communications in Number Theory and Physics
Volume 15 (2021)
Green’s functions for Vladimirov derivatives and Tate’s thesis
Pages: 315 – 361
Given a number field $K$ with a Hecke character $\chi$, for each place $\nu$ we study the free scalar field theory whose kinetic term is given by the regularized Vladimirov derivative associated to the local component of $\chi$. These theories appear in the study of $p$‑adic string theory and $p$‑adic AdS/CFT correspondence. We prove a formula for the regularized Vladimirov derivative in terms of the Fourier conjugate of the local component of $\chi$ We find that the Green’s function is given by the local functional equation for Zeta integrals. Furthermore, considering all places $\nu$, the field theory two-point functions corresponding to the Green’s functions satisfy an adelic product formula, which is equivalent to the global functional equation for Zeta integrals. In particular, this points out a role of Tate’s thesis in adelic physics.
Vladimirov derivative, Tate’s thesis, Green’s function, adelic product formula
2010 Mathematics Subject Classification
Primary 11M06, 47S10. Secondary 81T40.
In memory of Steven Gubser and John Tate
The work of A. Huang and S.-T. Yau was supported in part by a grant from the Simons Foundation in Homological Mirror Symmetry.
The work of A. Huang, B. Stoica, and X. Zhong was supported in part by a grant from the Brandeis University Provost Office.
B. Stoica was supported in part by the U.S. Department of Energy under grant DE-SC-0009987, and by the Simons Foundation through the It from Qubit Simons Collaboration on Quantum Fields, Gravity and Information.
Received 17 March 2020
Accepted 28 December 2020
Published 18 June 2021