Black hole microstate counting from the gravitational path integral

<!--HTML--><p><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">Reproducing the integer count&nbsp;</span></span><span style="color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">of</span></span><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">&nbsp;black hole micro-states from the gravitational path integral is an important problem in quantum gravity. In this paper, we show that, by using supersymmetric localization, the gravitational path integral for 1/16-BPS black holes in supergravity reproduces the index obtained in the string theory construction&nbsp;</span></span><span style="color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">of</span></span><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">&nbsp;such black holes, including all non-perturbatively suppressed geometries. A more refined argument then shows that not only the black hole index but also the total number&nbsp;</span></span><span style="color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">of</span></span><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">&nbsp;black hole microstates within an energy window above extremality that is polynomially suppressed in the charges also matches this string theory index. To achieve such a match, we compute the one-loop determinant arising in the localization calculation for all N=2 supergravity supermultiplets in the N=8 gravity supermultiplet. Furthermore, we carefully account for the contribution&nbsp;</span></span><span style="color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">of</span></span><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">&nbsp;boundary zero-modes, which can be seen as arising from the zero-temperature limit&nbsp;</span></span><span style="color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">of</span></span><span style="background-color:rgb(255,255,255);color:rgb(0,0,0);"><span style="-webkit-tap-highlight-color:rgba(26, 26, 26, 0.3);-webkit-text-size-adjust:auto;-webkit-text-stroke-width:0px;caret-color:rgb(0, 0, 0);display:inline !important;float:none;font-family:UICTFontTextStyleBody;font-size:17px;font-style:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;orphans:auto;text-align:start;text-decoration:none;text-indent:0px;text-transform:none;white-space:normal;widows:auto;word-spacing:0px;">&nbsp;the N=4 super-Schwarzian, and show that performing the exact path integral over such modes provides a critical contribution needed for the match to be achieved.</span></span></p>