Twisted geometric parametrization of holonomy-flux phase space in all dimensional loop quantum gravity

Abstract The regularization of the scalar constraint and the Fermion coupling problem indicate that it is necessary to consider some kind of gauge fixing methods to deal with the simplicity constraint in all dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> loop quantum gravity (LQG). The coherent state with well-behaved peakedness property is an essential ingredient to carry out the gauge fixing method. To provide the basic tool for constructing such kind of coherent state, we generalize the twisted geometry parametrization of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> holonomy-flux phase space of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>D</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> -dimensional LQG from the edge simplicity constraint surface to the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> holonomy-flux phase space. The symplectic structure on the twisted geometric parameter space and the Poisson structure in terms of the twisted geometric variables are analyzed. Besides, we discuss the relation between the two twisted geometry parametrizations constructed respectively on the edge simplicity constraint surface and the full <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> holonomy-flux phase space. Our result shows that these two type of parametrizations are equivalent to each other by carrying out the gauge reduction with respect to the edge simplicity constraint.