ar X iv : h ep - t h / 05 09 01 8 v 1 1 Se p 20 05 The geometry of wrapped M 5 - branes in Calabi - Yau 2 - folds

We study the geometry of M5-branes wrapping a 2-cycle which is Special Lagrangian with respect to a specific complex structure in a CalabiYau two-fold. Using methods recently applied to the three-fold case, we are again able find a characterization of the geometry, in terms of a nonintegrable almost complex structure and a (2,0) form. This time, however, due to the hyper-Kähler nature of the underlying 2-fold we also have the freedom of choosing a different almost complex structure with respect to which the wrapped 2-cycle is holomorphic. We show that this latter almost complex structure is integrable. We then relate our geometry to previously found geometries of M5-branes wrapping holomophic cycles and go further to prove some previously unknown results for M5-branes on holomorphic cycles. HUTP-05/A0040 BCCUNY-HEP /05-04 hep-th/xxxxxxx email: Ansar [email protected] email: [email protected] [email protected]