# Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory

## TL;DR

This paper reveals that the quantum simplicity constraint in 4d spinfoam models with cosmological constant can be represented as a surface defect in complex Chern-Simons theory, enabling new amplitude constructions that recover Einstein-Regge action semiclassically.

## Contribution

It introduces a novel surface defect interpretation of the simplicity constraint within Chern-Simons theory, leading to new spinfoam amplitudes with correct semiclassical limits.

## Key findings

- Constructed new spinfoam amplitude with cosmological constant for arbitrary complexes.
- Demonstrated semiclassical limit reproduces Einstein-Regge action.
- Established surface defect as realization of simplicity constraint in Chern-Simons theory.

## Abstract

The simplicity constraint is studied in the context of 4d spinfoam models with cosmological constant. We find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2,$\mathbb{C}$) Chern-Simons theory in the construction of spinfoam amplitudes. By this realization of simplicity constraint in Chern-Simons theory, we are able to construct the new spinfoam amplitude with cosmological constant for arbitrary simplicial complex (with many 4-simplices). The semiclassical asymptotics of the amplitude is shown to reproduce correctly the 4-dimensional Einstein-Regge action with cosmological constant term.