Geometric realization via irrelevant deformations induced by the stress-energy tensor

In this paper, we generalize the deformations driven by the stress-energy tensor <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math> and investigate their relation to the flow equation for the background metric at the classical level. For a deformation operator <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="script">O</c:mi></c:math> as a polynomial function of the stress-energy tensor, we develop a formalism that relates a deformed action to a flow equation for the metric in arbitrary spacetime dimensions. It is shown that in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>T</f:mi><f:mover accent="true"><f:mi>T</f:mi><f:mo stretchy="false">¯</f:mo></f:mover></f:math> deformation and the <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="script">O</j:mi><j:mo stretchy="false">(</j:mo><j:mi>T</j:mi><j:mo stretchy="false">)</j:mo><j:mo>=</j:mo><j:mi>tr</j:mi><j:mo stretchy="false">[</j:mo><j:mi mathvariant="bold">T</j:mi><j:msup><j:mo stretchy="false">]</j:mo><j:mi>m</j:mi></j:msup></j:math> deformation, the flow equations for the metric allow us to directly obtain exact solutions in closed forms. We also demonstrate the perturbative approach to find the same results. As several applications of the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi mathvariant="script">O</r:mi><r:mo stretchy="false">(</r:mo><r:mi>T</r:mi><r:mo stretchy="false">)</r:mo><r:mo>=</r:mo><r:mi>tr</r:mi><r:mo stretchy="false">[</r:mo><r:mi mathvariant="bold">T</r:mi><r:msup><r:mo stretchy="false">]</r:mo><r:mi>m</r:mi></r:msup></r:math> deformation, we discuss the relation between the deformations and gravitational models. Besides, we also deform the Lagrangians for scalar field theories.