Branes wrapped on orbifolds and their gravitational blocks
Federico Faedo, Alessio Fontanarossa, Dario Martelli
Abstract
Abstract We construct new supersymmetric $$\textrm{AdS}_2\times \mathbb {M}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> solutions of $$D=6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> gauged supergravity, where $$\mathbb {M}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> are certain four-dimensional orbifolds. After uplifting to massive type IIA supergravity these correspond to the near-horizon limit of a system of N D4-branes and $$N_f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> D8-branes wrapped on $$\mathbb {M}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> . In one class of solutions, "Equation missing" is a spindle fibered over a smooth Riemann surface of genus $$\textrm{g}>1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>g</mml:mtext> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , while in another class "Equation missing" is a spindle fibered over another spindle. Both classes can be thought of as orbifold generalizations of Hirzebruch surfaces and, in the second case, we describe the solutions in terms of toric geometry. We show that the entropy associated with these solutions is reproduced by extremizing an entropy function obtained by gluing gravitational blocks, using a general recipe for orbifolds that we propose. We also discuss how our prescription can be used to define an off-shell central charge whose extremization reproduces the gravitational central charge of analogous $$\textrm{AdS}_3\times \mathbb {M}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mtext>AdS</mml:mtext> <mml:mn>3</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> </mml:math> solutions of $$D=7$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:math> gauged supergravity, arising from wrapping M5-branes on $$\mathbb {M}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> .