Line Operators in $U(1|1)$ Chern-Simons Theory

Niklas Garner; Wenjun Niu

Home/Archive/arXiv

arXiv · 2304.05414

Line Operators in $U(1|1)$ Chern-Simons Theory

Niklas Garner·Wenjun Niu

View original source ↗Read source document ↗Export metadata ↓

Abstract

We analyze the non-semisimple category of line operators in Chern-Simons gauge theories based off the Lie superalgebra $\mathfrak{gl}(1|1)$. Our proposal is that the category of line operators $\mathcal{C}$ can be identified with the derived category of modules for a boundary vertex operator algebra $\mathcal{V}$ realized as a certain infinite-order simple current extension of the affine current algebra $V(\mathfrak{gl}(1|1))$ by boundary monopole operators. By translating this simple current extension of $V(\mathfrak{gl}(1|1))$ to the unrolled, restricted quantum group $\overline{U}^E(\fgl(1|1))$, we show that our category of line operators admits a second description in terms of a quasi-quantum group $\mathcal{A}$ realized by uprolling. We also compare our results across an expected physical duality with the cyclic orbifold of a free, $B$-twisted hypermultiplet and find a slight discrepancy at the level of braiding and associator. We end with a detailed analysis of coupling to background flat $GL(1, \C)$ connections and the resulting category of non-genuine line operators.

Explore related subjects

hep-th math.QA math.RT

Keep this discovery

BibTeX RIS

Niklas Garner, Wenjun Niu. 2023-04-11. Line Operators in $U(1|1)$ Chern-Simons Theory. https://arxiv.org/abs/2304.05414

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Timelike Entanglement from Spacetime Density Matrices: A Lattice Realization

We investigate timelike entanglement in quantum field theory using spacetime density matrices and provide a microscopic lattice realization. For a two-dimensional free real scalar field, we extend Gaussian diagonalization methods to the generally non-Hermitian reduced spacetime density matrix and determine its complete nonzero spectrum in the generic regular case, together with all integer R\'enyi moments. The real-time replica construction identifies these moments with Lorentzian branch-point twist-operator correlation functions. We test this identification against the full four-point function on a circle, boundary two-point functions with Dirichlet and Neumann boundary conditions, and massive form-factor predictions, finding quantitative agreement in both magnitude and phase across distinct causal regimes. The boundary setup exhibits a finite causally connected window in which every integer R\'enyi entropy is real, showing that reality is not equivalent to causal disconnection. These results provide a microscopic lattice foundation for timelike entanglement and for Lorentzian twist-operator methods beyond equal-time regions.

hep-th↗

arXiv2026

Landau-Ginzburg description of an exceptional ${\mathcal N}=1$ minimal model

The $\mathcal N=1$ superconformal minimal model with $m=12$ and the exceptional modular invariant $(E_6,D_8)$ is the unitary minimal model of the super-$W_3$ algebra. We propose its Landau-Ginzburg description using two real scalar superfields with the cubic superpotential ${\cal W}=g_1 XY^2/2 + g_2X^3/6$. For $g_1=g_2$, this superpotential is known to describe a product of two $m=3$ $\mathcal N=1$ superconformal minimal models, which is the $m=10$ model with the $(D_6,E_6)$ modular invariant. The exceptional $m=12$ superconformal minimal model is realized at a different fixed point of the same theory. Testing this Landau-Ginzburg description requires the fusion ring of the minimal model, which we obtain from the modular data of the extended algebra. The fusion ring has a $\mathbb Z_2$ grading by chiral fermion parity that the ordinary fusion coefficients do not determine. This grading, composed with conjugation, gives the generator of the R-parity $\mathbb Z_2^{R}$ of the Landau-Ginzburg theory. We then treat the theory with superpotential $\cal W$ as a Gross-Neveu-Yukawa model in $d=4-\epsilon$ and find a weakly coupled infrared fixed point with $g_1/g_2=3/2+\mathcal O(\epsilon)$, at which supersymmetry emerges. We also describe the renormalization group flow from this fixed point to the decoupled fixed point with $g_1=g_2$. The operator dimensions at the coupled fixed point, continued to $d=2$, agree approximately with their values in the $m=12$ superconformal minimal model. Finally, we estimate the scaling dimensions in the new interacting $d=3$ $\mathcal N=1$ superconformal field theory.

hep-th↗

arXiv2026

Detecting one-dimensional bosonic SPT phases via twisted entropic order parameter

Entanglement asymmetry, introduced by F. Ares, S. Murciano and P. Calabrese, provides a density-matrix diagnostic of symmetry breaking and successfully captures the Landau data associated with a broken symmetry pattern. However, it is by now well established that gapped quantum many-body systems can exhibit phases which are not characterized solely by Landau symmetry breaking. A fundamental example is a symmetry-protected topological (SPT) phase, and the ordinary definition of entanglement asymmetry is insensitive to this topological information. In this work we introduce a refined quantity, which we call the twisted entropic order parameter, designed to detect SPT phases from reduced density matrices, particularly focusing on one-dimensional bosonic systems. The key ingredient in our construction is an ancilla degrees of freedom that coherently records the untwisted state and the twisted state associated to a one-ended topological defect of unbroken symmetry, so that the enlarged density matrix retains the charge carried by the defect endpoint. We demonstrate our proposal in concrete lattice models and further generalize it beyond ordinary group symmetries, establishing its ability to diagnose SPT phases. This provides a first step toward a unified entanglement-asymmetry framework for diagnosing quantum phases of matter.

hep-th↗