Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 5

General relativity from $p$-adic strings

Pages: 1203 – 1237



An Huang (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Bogdan Stoica (Department of Physics & Astronomy, Northwestern University, Evanston, Illinois, U.S.A.)

Shing-Tung Yau (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that spherical vectors of unramified principal series representations of $PGL (2,\mathbb{Q}_p)$ are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the padic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann Zeta function.

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Published 30 March 2023