Branes Wrapped on Coassociative Cycles

We obtain a supergravity solution arising when D6-branes are wrapped on coassociative four-cycles of constant curvature in seven manifolds of G2 holonomy. The solutions preserve two supercharges and thus represent supergravity duals of three dimensional Yang-Mills with N = 1 supersymmetry. When uplifted to eleven dimensions our solution describes M-theory on the background of an eight manifold with Spin(7) holonomy. † e-mail address: [email protected] An interesting possibility to construct gravity duals of field theories with low supersymmetry is that provided by branes wrapped on supersymmetric cycles. As cycles will not in general have covariantly constant spinors, supersymmetry will only be preserved after an identification of the spin connection on the cycle with some external R-symmetry gauge fields; this identification defines a topologically twisted supersymmetric field theory [1] (a detailed classification of different twists can be found in [2]). The way the cycle is embedded in a higher dimensional manifold determines the amount of preserved supersymmetry. When the number of branes is large, the uplifts to ten or eleven dimensions of the solutions, found in an adequate gauged supergravity, represent a gravity dual description of field theories with reduced supersymmetry. The case originally considered by Maldacena and Núñez [3, 4] was that of fivebranes and D3-branes wrapped on holomorphic curves, and has been applied in a series of related works to different dimension branes wrapped on diverse supersymmetric cycles [5]-[12]. In [9], Edelstein and Núñez studied a configuration of D6-branes wrapping holomorphic two-cycles and special Lagrangian three-cycles. When the size of the cycles is taken to zero, their solutions represent respectively a supergravity description of the infrared dynamics of five dimensional N = 2 supersymmetric Yang-Mills, or four dimensional Yang-Mills with N = 1 supersymmetry. D6-branes wrapping an S in T S had previously been proposed to be dual through a conifold transition to a type IIA geometry with the D6-branes replaced by RR fluxes on the blown up S [13]. However, a better understanding of this duality came in terms of M-theory on a seven manifold of G2 holonomy [14], where it corresponds to an S flop transition [15] (see also [16]-[26] for further recent developments). These results were extended by Gomis in [18], where it was argued how compactifications of M-theory on manifolds with reduced holonomy arise as the local eleven dimensional description of D6-branes wrapped on supersymmetric cycles in manifolds of lower dimension and with a different holonomy group. The authors of [9] explicitly reproduced the geometry of a manifold with G2 holonomy and of the small resolution of the conifold when uplifting to eleven dimensions the solutions they found in eight dimensional maximal gauged supergravity, which is the natural arena to perform twisting for D6-branes. The purpose of this letter is to use the approach of [9] to study one of the lifts considered in [18], namely D6-branes wrapped on a coassociative four-cycle in a seven manifold