Weakly Renormalized Near 1/16 SUSY Fermi Liquid Operators in N = 4 SYM

TL;DR

This work identifies weakly renormalized Fermi-liquid–like operators in ${\cal N}=4$ SYM, showing that a 1-dim Fermi-sea built from the fermionic $\mathfrak{su}(1,1)$ sector is an eigenstate of the full dilatation operator with anomalous dimensions that are parametrically small at large filling $K$, yielding a natural $1/E_{\rm f}$–type expansion. Using oscillator representation and spin-chain techniques, the authors compute the dilatation operator up to two loops ${\cal O}(g^4)$, demonstrating a near-BPS cancelation that suppresses corrections by $1/K$ and enabling a Landau Fermi-liquid–like description of quasi-particles near the Fermi surface. They extend the analysis to a 2D Fermi-sea, find similar cancellations at leading order but note mixing at $\mathcal{O}(g^3)$, and discuss non-planar contributions given the large $N$ scaling of the operator. The results motivate a potential bridge from weak to strong coupling, suggestive connections to $1/16$ BPS degeneracies and AdS$_5$ black holes, and point to further algebraic, bosonization, and higher-dimensional generalizations to explore approximately 1/16-BPS sectors and their holographic duals.

Abstract

We discuss a class of Fermi Liquid Operators in N = 4 SYM. We show that these operators are eigenstates of the full quantum dilatation operator. We compute their 1 and 2 loop anomalous dimensions, and show that, similar to Fermi liquids in condensed matter systems, these corrections are suppressed by an arbitrarily small parameter, which is the equivalent of one over the Fermi energy. These operators are, at the classical level, descendants of 1/16 BPS operators, with some scaling properties similar to those of the 1/16 Black Holes in AdS_5.

Figures (2)

• Figure 1: The continuous, dashed, doted lines stands for $\psi_{(n)}$, $\phi^i_{(n)}$ and $\bar{\psi}_{(n)}$ respectively. The arrows indicate the flow of momentum. For the conjugated diagrams ${\bar{{\mathfrak T}}}^-_1$ and ${{\mathfrak T}}^+_1$ the vertices structure do not change, only the arrows changes sign. Note that these are not Feynman diagrams, but rather they only describe the how the various terms in these operators act.
• Figure 2: Three typical diagrams for ${\delta\mathfrak D}_4$, which keep track of how the partons change when acting on them with the sequence of operators in (3.26). The vertices are ordered in the vertical axis by the order the operators acts on the state. Since ${\mathfrak T}^-~,~\bar{{\mathfrak T}}^+_1$ appear in the inner commutators only, the corresponding vertices are inserted sequently, as a result they form a either a 1-1 and 2-2 fermionic operator, any none fermionic line emerging from the inner commutator will continue to the outer legs (irrelevant for our purpose).