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Signs of the time: Melonic theories over diverse number systems

Steven S. Gubser, Matthew Heydeman, Christian Jepsen, Sarthak Parikh, Ingmar Saberi, Bogdan Stoica, Brian Trundy

TL;DR

The paper extends melonic tensor models to both real and $p$-adic number systems via sign characters, unifying fermionic and bosonic Klebanov–Tarnopolsky constructions under either $O(N)^3$ or $Sp(N)^3$ symmetry. It develops bilocal kinetic terms, analyzes the Schwinger–Dyson equation in the leading melonic limit, and demonstrates universal infrared scaling of the two-point function of the form $G(t)=b\,\mathrm{sgn}(t)/|t|^{1/2}$ across realizations, with the IR amplitude dictated by generalized $\Gamma$-function data of multiplicative characters. For direction-dependent ultrametric signs, the SD equation becomes local and admits an exact quartic-solution interpolation between UV and IR; a Wilsonian non-renormalization perspective explains why the bilocal kinetic term remains undeformed at leading order while the IR physics is controlled by a strongly coupled quartic interaction. The work provides a coherent framework that bridges Archimedean and non-Archimedean melonic theories, highlights the role of sign characters in determining symmetry and statistics, and outlines several directions for extending the analysis to broader $p$-adic extensions and deeper renormalization-group insights.

Abstract

Melonic field theories are defined over the $p$-adic numbers with the help of a sign character. Our construction works over the reals as well as the $p$-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.

Signs of the time: Melonic theories over diverse number systems

TL;DR

The paper extends melonic tensor models to both real and -adic number systems via sign characters, unifying fermionic and bosonic Klebanov–Tarnopolsky constructions under either or symmetry. It develops bilocal kinetic terms, analyzes the Schwinger–Dyson equation in the leading melonic limit, and demonstrates universal infrared scaling of the two-point function of the form across realizations, with the IR amplitude dictated by generalized -function data of multiplicative characters. For direction-dependent ultrametric signs, the SD equation becomes local and admits an exact quartic-solution interpolation between UV and IR; a Wilsonian non-renormalization perspective explains why the bilocal kinetic term remains undeformed at leading order while the IR physics is controlled by a strongly coupled quartic interaction. The work provides a coherent framework that bridges Archimedean and non-Archimedean melonic theories, highlights the role of sign characters in determining symmetry and statistics, and outlines several directions for extending the analysis to broader -adic extensions and deeper renormalization-group insights.

Abstract

Melonic field theories are defined over the -adic numbers with the help of a sign character. Our construction works over the reals as well as the -adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.

Paper Structure

This paper contains 18 sections, 107 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The kinetic term
  3. The full model
  4. Diagrammatics and the Schwinger-Dyson equation
  5. Multiplicative characters
  6. Sign characters in finite extensions of $\mathbb{Q}_p$ for odd $p$
  7. Sign characters over $\mathbb{Q}_2$
  8. An alternative parametrization of sign characters
  9. The generalized $\Gamma$ and ${\rm B}$ functions
  10. Generalized $\Gamma$ functions for $\mathbb{R}$ and $\mathbb{Q}_p$
  11. Solving the Schwinger-Dyson equation
  12. Infrared limit
  13. An exact solution of the Schwinger-Dyson equation
  14. A Wilsonian perspective
  15. $p$-adic non-renormalization
  16. ...and 3 more sections

Figures (2)

  • Figure 1: The hierarchical model, with an Ising spin $s_n = \pm 1$ at every integer point $n$, and successively weaker ferromagnetic couplings between pairs of spins, pairs of pairs, and so on.
  • Figure 2: Solving the quartic equation (\ref{['QPagain']}). We use the value ${g^2 N^3 \@@over p}=1$ as an example in order to be able to draw a definite curve of $1/\hat{f}_v$ versus $\hat{g}_v$. Other positive values of ${g^2 N^3 \@@over p}$ give qualitatively similar results. Suppose we pick a value of $\hat{f}_v$: small if we're in the UV, and large in the IR. If $\sigma_\psi = -1$, as in the left-hand plot, then there is always a unique positive solution $\hat{g}_v$ corresponding to the chosen value of $\hat{f}_v$. But if $\sigma_\psi = +1$, as in the right-hand plot, then sufficiently far into the IR we have no such solution, and it appears therefore that the interacting theory is ill-defined.