Green’s functions for Vladimirov derivatives and Tate’s thesis

Shing-Tung Yau (Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts, U.S.A.; and Dept. of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Xiao Zhong (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Abstract

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.

Keywords

Vladimirov derivative, Tate’s thesis, Green’s function, adelic product formula

2010 Mathematics Subject Classification

Primary 11M06, 47S10. Secondary 81T40.

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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