Semi-universality of CFT$_d$ entropy at large spin

Abstract

The thermal partition function, Z <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Z</mml:mi> </mml:math> , of a CFT_d <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> on S^{d-1} <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> is parameterized by the inverse temperature \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> along with \lfloor d/2\rfloor <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> angular velocities \omega_i <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> . In this paper, we investigate the behaviour of this partition function when n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> of the \omega_i <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> are scaled to unity (the largest allowed value) at fixed values of the other (\lfloor d/2\rfloor-n) <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> angular velocities. We argue that \ln Z <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> develops a simple pole in (1-\omega_i) <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> for each \omega_i <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> that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> 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 n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> special angular momenta J_1\ldots J_n <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>