Fortschr. Phys. 50 (2002) 8-9, 802 – 807 BPS 3-brane solution on smooth ALE manifolds with flux M. Bertolini 1 , V. L. Campos 2 , G. Ferretti 2 , P. Fr´ e 3 , P. Salomonson 2 , and M. Trigiante 4,* 1 NORDITA, Blegdamsvej, 17 2100 Copenhagen Ø, Denmark 2 Institute for Theoretical Physics, G¨ oteborg University and Chalmers University of Technology, 412 96 G¨ oteborg, Sweden 3 Dipartimento di Fisica Teorica, Universit´ a di Torino, & INFN, Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italy 4 Spinoza Institute, Leuvenlaan 4, Utrecht, The Netherlands Abstract In the present talk I review the construction of a new supersymmetric 3–brane solution of type IIB supergravity on an ALE transverse space in presence of a non trivial flux of the complex 2–form across the ALE cycles. After introducing the ansatz for the solution I will show how the supergravity field equations reduce to a differential equation of Heun type in the non–conformal warp factor H of the metric. Although our analysis is worked out purely at a macroscopic level comments will be made on the general features of the (non–conformal) gauge theory dual to this solution 1 Introduction So far considerable effort has been directed towards extending Maldacena’s AdS/CFT con- jecture to non–conformal gauge theories in order be able to describe using supergravity tools non–perturbative phenomena occurring in gauge theories like QCD. Along this line of re- search fractional branes [3]- [16] have been playing a fundamental role. However all these attempts of extending the holographic correspondence to non–conformal field theories have been hampered on the supergravity side by the presence of singularities of repulson type [17] in the space–time configurations. In the present talk I shall review the main results of a recent work [1] in which a new class of BPS 3–brane solutions was constructed in the framework of Type IIB supergravity. These extended objects are transverse to a space with geometry of R 2 × ALE, where ALE denotes a resolved Asymptotically Locally Euclidean 4–manifold, and couple to a non trivial flux of the complex 2–form potential across the ALE cycles. It was shown that the field equations reduce to a second order inhomogeneous differential equation in the warp factor H characterizing the ansatz for the metric. The solution to this equation will be analyzed in detail in the particular ∗ e-mail: m.trigiante@phys.uu.nl c  WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2002 0015-8208/02/8-909-0802 $ 17.50+.50/0