Statistical physics of the polarised IKKT matrix model

The polarised IKKT matrix model is the worldpoint theory of N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> D-instantons in a background three-form flux of magnitude \Omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Ω</mml:mi> </mml:math> , and promises to be a highly tractable model of holography. The matrix integral can be viewed as a statistical physics partition function with inverse temperature \Omega^4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>Ω</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> . At large \Omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Ω</mml:mi> </mml:math> the model is dominated by a matrix configuration corresponding to a ‘polarised’ spherical D1-brane. We show that at a critical value of \Omega^2 N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>Ω</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> the model undergoes a first order phase transition, corresponding to tunneling into a collection of well-separated D-instantons. These instantons are the remnant of a competing saddle in the high \Omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Ω</mml:mi> </mml:math> phase corresponding to spherical (p,q) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> fivebranes. We use a combination of numerical and analytical arguments to capture the different regimes of the model.