# The One-Loop Spectral Problem of Strongly Twisted $\mathcal{N}$=4 Super Yang-Mills Theory

## TL;DR

This paper studies the one-loop spectral problem of a simplified, non-unitary version of planar $
4$ SYM theory with a focus on integrability and the complex structure of operator mixing.

## Contribution

It derives one-loop Bethe equations for certain sectors of the twisted $
4$ SYM, revealing the complexity of non-diagonalizable operators and Jordan blocks in the non-unitary model.

## Key findings

- Derived one-loop Bethe equations for multiple sectors.
- Identified the presence of large Jordan blocks with unknown generalized eigenvalues.
- Highlighted the challenges in describing non-diagonalizable operators via integrability methods.

## Abstract

We investigate the one-loop spectral problem of $\gamma$-twisted, planar $\mathcal{N}$=4 Super Yang-Mills theory in the double-scaling limit of infinite, imaginary twist angle and vanishing Yang-Mills coupling constant. This non-unitary model has recently been argued to be a simpler version of full-fledged planar $\mathcal{N}$=4 SYM, while preserving the latter model's conformality and integrability. We are able to derive for a number of sectors one-loop Bethe equations that allow finding anomalous dimensions for various subsets of diagonalizable operators. However, the non-unitarity of these deformed models results in a large number of non-diagonalizable operators, whose mixing is described by a very complicated structure of non-diagonalizable Jordan blocks of arbitrarily large size and with a priori unknown generalized eigenvalues. The description of these blocks by methods of integrability remains unknown.