Semi-universality of CFT$_d$ entropy at large spin
Abstract
<jats:p> The thermal partition function, <jats:inline-formula> <jats:alternatives> <jats:tex-math>Z</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Z</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> , of a <jats:inline-formula> <jats:alternatives> <jats:tex-math>CFT_d</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>F</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <jats:tex-math>S^{d-1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is parameterized by the inverse temperature <jats:inline-formula> <jats:alternatives> <jats:tex-math>\beta</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> along with <jats:inline-formula> <jats:alternatives> <jats:tex-math>\lfloor d/2\rfloor</jats:tex-math> <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>d</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌋</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> angular velocities <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> . In this paper, we investigate the behaviour of this partition function when <jats:inline-formula> <jats:alternatives> <jats:tex-math>n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> of the <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> are scaled to unity (the largest allowed value) at fixed values of the other <jats:inline-formula> <jats:alternatives> <jats:tex-math>(\lfloor d/2\rfloor-n)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo stretchy="false" form="prefix">⌊</mml:mo> <mml:mi>d</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">⌋</mml:mo> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> angular velocities. We argue that <jats:inline-formula> <jats:alternatives> <jats:tex-math>\ln Z</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">ln</mml:mi> <mml:mo></mml:mo> </mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> develops a simple pole in <jats:inline-formula> <jats:alternatives> <jats:tex-math>(1-\omega_i)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> for each <jats:inline-formula> <jats:alternatives> <jats:tex-math>\omega_i</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> </jats:alternatives> </jats:inline-formula> that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of <jats:inline-formula> <jats:alternatives> <jats:tex-math>\beta</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While <jats:inline-formula> <jats:alternatives> <jats:tex-math>n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> </jats:alternatives> </jats:inline-formula> special angular momenta <jats:inline-formula> <jats:alternatives> <jats:tex-math>J_1\ldots J_n</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>…</mml:mi>