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Topological 4D gravity and gravitational defects

Y. Boujakhrout, R. Sammani, E. H Saidi

Abstract

Using the Chern-Simons formulation of AdS3 gravity as well as the Costello-Witten-Yamazaki (CWY) theory for quantum integrability, we construct a novel topological 4D gravity given by Eq(5.1) with observables based on gravitational gauge field holonomies. The field action $S^{grav}_{4D}$ of this gravity has a gauge symmetry $SL(2,\mathbb{C})$ and reads also as the difference $S^{CWY_{L}}_{4D}-S^{CWY_{R}}_{4D}$ with 4D Chern-Simons field actions $S^{CWY_{L/R}}_{4D}$ given by left/right CWY theory Eq(3.9). We also use this 4D gravity derivation to build observables describing gravitational topological defects and their interactions. We conclude our study with few comments regarding quantum integrability and the extension of AdS$_{3}$/CFT$_{2}$ correspondence with regard to the obtained topological 4D gravity.

Topological 4D gravity and gravitational defects

Abstract

Using the Chern-Simons formulation of AdS3 gravity as well as the Costello-Witten-Yamazaki (CWY) theory for quantum integrability, we construct a novel topological 4D gravity given by Eq(5.1) with observables based on gravitational gauge field holonomies. The field action of this gravity has a gauge symmetry and reads also as the difference with 4D Chern-Simons field actions given by left/right CWY theory Eq(3.9). We also use this 4D gravity derivation to build observables describing gravitational topological defects and their interactions. We conclude our study with few comments regarding quantum integrability and the extension of AdS/CFT correspondence with regard to the obtained topological 4D gravity.

Paper Structure

This paper contains 13 sections, 62 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Chern-Simons formulation of AdS$_{3}$ gravity
  3. From 3D Chern-Simons to CWY theory
  4. More on 3D Chern-Simons action
  5. CWY theory: building algorithm
  6. From AdS$_{3}$ to topological 4D gravity
  7. Deriving the topological 4D gravity
  8. Action $\mathcal{S}_{4D}^{gravity}$ in terms of CWY fields
  9. Action $\mathcal{S}_{4D}^{gravity}$ in terms of gravity fields
  10. Line defects in 4D topological gravity
  11. Gravitational holonomies
  12. Gravitational integrable equations
  13. Conclusion and comments

Figures (6)

  • Figure 1: On the left, the Lax operator $\mathcal{L}\left( z\right)$ given by the crossing of a 't Hooft line at z=0 (in red) and a Wilson line at z (in blue) with incoming $\left\langle i\right\vert$ and out going $\left\vert j\right\rangle$ states. On the right, the RLL relations encoding the commutation relations between two L-operators at z and w.
  • Figure 2: Propagator $\left\langle E^{a}\left( { \if@compatibility \mathchar"0110 {} \mathchar"0110 } _{1}\right) \Omega ^{b}\left( { \if@compatibility \mathchar"0110 {} \mathchar"0110 } _{2}\right) \right\rangle$ between a vielbein line defects: $\mathbf{E}^{a}$ located at ${ \if@compatibility \mathchar"0110 {} \mathchar"0110 } _{1}$ in $\mathbb{CP}^{1}$; and a spin connection line defect $\mathbf{\Omega }^{b}$ located at ${ \if@compatibility \mathchar"0110 {} \mathchar"0110 } _{2}.$ The wavy line in red represents the exchanged graviton state.
  • Figure 3: Interaction between gravitational line defects. Two blue line defects E interact through a green line defect $\Omega .$ Similarly, two green line defects $\Omega$ couple through a blue line defect E.
  • Figure 4: Yang-Baxter equations for the gravitational line defects $\left( \mathbf{E}_{1},\mathbf{E}_{2}\right)$ and $\left( \mathbf{\Omega }_{1},\mathbf{\Omega }_{2}\right)$: 2+2 crossing gravitational line defects. Because of eq(\ref{['p1']}-\ref{['p3']}), the contributions are given by the crossing of vielbein lines with spin connection lines (red points).
  • Figure 5: The crossing of three line defects with one having a different color.
  • ...and 1 more figures