Matrix models and deformations of JT gravity

Recently, it has been found that Jackiw-Teitelboim (JT) gravity, which is a two-dimensional theory with bulk action <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>−</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo>∫</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>x</mml:mi> <mml:msqrt> <mml:mi>g</mml:mi> </mml:msqrt> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> , is dual to a matrix model, that is, a random ensemble of quantum systems rather than a specific quantum mechanical system. In this article, we argue that a deformation of JT gravity with bulk action <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo>−</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo>∫</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>x</mml:mi> <mml:msqrt> <mml:mi>g</mml:mi> </mml:msqrt> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mi>W</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:math> is likewise dual to a matrix model. With a specific procedure for defining the path integral of the theory, we determine the density of eigenvalues of the dual matrix model. There is a simple answer if W (0) = 0, and otherwise a rather complicated answer.