Maze topiary in supergravity

A bstract We show that the supergravity solutions for $$ \frac{1}{4} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:math> -BPS intersecting systems of M2 and M5 branes are completely characterized by a single “maze” function that satisfies a non-linear “maze” equation similar to the Monge-Ampère equation. We also show that the near-brane limit of certain intersections are AdS 3 ×S 3 ×S 3 solutions warped over a Riemann surface, Σ. There is an extensive literature on these subjects and we construct mappings between various approaches and use brane probes to elucidate the relationships between the M2-M5 and AdS systems. We also use dualities to map our results onto other systems of intersecting branes. This work is motivated by the recent realization that adding momentum to M2-M5 intersections gives a supermaze that can reproduce the black-hole entropy without ever developing an event horizon. We take a step in this direction by adding a certain type of momentum charges that blackens the M2-M5 intersecting branes. The near-brane limit of these solutions is a BTZ extremal ×S 3 ×S 3 × Σ geometry in which the BTZ momentum is a function of the Riemann surface coordinates.