<p>We examine the low-energy spectrum of a four-dimensional near-extremal black hole that arises as a solution to a low energy effective theory of heterotic string theory. The effective two-dimensional gravitational description exhibits features of Lifshitz symmetry, which break the usual SL(2, <i>ℝ</i>) invariance down to U(1). For this effective two-dimensional gravitational description, we derive a one-dimensional Schwarzian-like action that inherits the U(1) symmetry. The Schwarzian-like description allows us to compute a logarithmic correction to the entropy through a saddle-point approximation of the two-dimensional partition function. This logarithmic correction modifies the density of states, lifting the delta-function divergence at extremality, and removes the exponential ground-state degeneracy seen in the semiclassical analysis. Furthermore, the prefactor of the logarithmic term is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25655_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{1}{2} \)</EquationSource> </InlineEquation>, rather than <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_25655_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">\( \frac{3}{2} \)</EquationSource> </InlineEquation> found for the SL(2, <i>ℝ</i>) invariant description, indeed reflecting having fewer symmetries.</p>

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The spectrum of a quantum Lifshitz black hole in two dimensions

  • Matthias Harksen,
  • Watse Sybesma

摘要

We examine the low-energy spectrum of a four-dimensional near-extremal black hole that arises as a solution to a low energy effective theory of heterotic string theory. The effective two-dimensional gravitational description exhibits features of Lifshitz symmetry, which break the usual SL(2, ) invariance down to U(1). For this effective two-dimensional gravitational description, we derive a one-dimensional Schwarzian-like action that inherits the U(1) symmetry. The Schwarzian-like description allows us to compute a logarithmic correction to the entropy through a saddle-point approximation of the two-dimensional partition function. This logarithmic correction modifies the density of states, lifting the delta-function divergence at extremality, and removes the exponential ground-state degeneracy seen in the semiclassical analysis. Furthermore, the prefactor of the logarithmic term is 1 2 \( \frac{1}{2} \) , rather than 3 2 \( \frac{3}{2} \) found for the SL(2, ) invariant description, indeed reflecting having fewer symmetries.