Topological 4D gravity and gravitational defects

Abstract

Abstract Using the Chern-Simons formulation of AdS 3 gravity as well as the Costello-Witten-Yamazaki (CWY) theory for quantum integrability, we construct a novel topological 4D gravity given by equation (5.1) with observables based on gravitational gauge field holonomies. The field action <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="italic">grav</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> of this gravity has a gauge symmetry <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and reads also as the difference <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic">CWY</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic">CWY</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:math> with 4D Chern-Simons field actions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">S</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>D</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic">CWY</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msubsup> </mml:math> given by left/right CWY theory equation (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.