An equivalence in random matrix and tensor models via a dually weighted intermediate field representation

TL;DR

The paper develops an exact, general framework to connect complex and self-adjoint random matrix and tensor models via dually weighted intermediate-field representations. By inserting an auxiliary field and a logarithmic dually weighted potential, the authors prove that partition functions and all trace-invariant observables depending on the self-adjoint combinations M†M or φφ† are exactly equivalent across complex and self-adjoint formulations; this holds for both matrices and higher-rank tensors and extends to specific symmetry classes. They further show that with appropriate symmetries, the self-adjoint formulation can reduce the effective tensor/order, offering practical computational advantages. The work also links to causality-inspired constructions from CDT and DWMM, suggesting broad implications for analytic control in discrete quantum gravity models and for unifying various intermediate-field approaches in random geometry. Overall, the results establish a versatile, exact duality framework that broadens the toolbox for studying non-Gaussian matrix and tensor models with nontrivial propagators and interactions.

Abstract

We present novel equivalences in random matrix and tensor models between complex and self-adjoint theories with nontrivial quadratic terms in the action, established through an intermediate field representation. More precisely, we show that the partition functions of certain self-adjoint models and their complex counterparts are different integral representations of the exact same function. A special case of these equivalences takes a form of newly found dually weighted intermediate field representations, which generalize the standard intermediate field representation. We also find indications of an equivalence between real tensor models and self-transpose tensor models.

Figures (23)

• Figure 1: (a) A ribbon graph generated by the causal matrix model \ref{['cdtmm']}. The spacelike ribbon-graph edges are drawn in red and the timelike ribbon-graph edges in blue. (b) Its dual graph corresponds to a CDT triangulation. (c) A CDT triangulation. The parallel horizontal lines correspond to the foliation structure.
• Figure 2: Ribbon graph and dual quadrangulation associated with the complex model \ref{['model1']}. Its chain-like structure shows why the model does not reproduce CDT.
• Figure 3: An example of equivalence between graph elements in the complex and the self-adjoint matrix models. On the rightmost columm, where a quadratic and a cubic vertices are shown, the red shapes represent a factor of ${\rm Tr}(Q^2)$ and ${\rm Tr}(Q^3)$, respectively.
• Figure 4: Left: An example of a graph in the complex matrix theory \ref{['eq:ZG0']} with particular potential $V$ including ${\rm Tr}((M^\dagger M)^2)$ and ${\rm Tr}((M^\dagger M)^3)$. A factor of $Q$ (resp. $P$) in the propagator is represented by blue (resp. red) lines, and interaction vertices ${\rm Tr}((M^\dagger M)^2)$ and ${\rm Tr}((M^\dagger M)^3)$ are represented in black. Right:An example of equivalence between a graph in the complex and the self-adjoint matrix models.
• Figure 5: An example of equivalence between a graph in the complex and the self-adjoint matrix models. Here, the colors red, green, blue and purple represents factors of $P$, $K$, $Q$ and $L$, respectively.

Theorems & Definitions (35)

• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Theorem 4.3
• proof
• Corollary 4.3.1
• Corollary 4.3.2
• proof
• Corollary 4.3.3