Geometrical transitions and strong coupling effects in supersymmetric field theories

Geometrical transitions are powerful tools designed to study strong coupling e ect in a general class of N = 1 theories realized on type IIB D5 branes wrapping 2-cycles of local Calabi-Yau threefolds or as e ective eld theories on D4 branes in type IIA brane con gurations. In this notes we discuss the issue of the classical moduli space for N = 2; Qn i=1 U(Ni) theories deformed by a general superpotential for the adjoint and bifundamental elds. We investigate the geometric transitions in the ten dimensional theories as well as in M-theory. Strong coupling e ects in eld theory are analyzed in the deformed geometry with uxes. The connection between the geometrical transition and strong coupling results constitutes a background for the methods of the Dijkgraaf-Vafa approach, where nonperturbative results in eld theories can be studied by using perturbative results in matrix models. 1400 Parallel Sessions 1 Geometric Transitions One of the most exciting directions developed in string theory and eld theory in of the last few years was to considerer the duality between open strings ( eld theory) and the closed string (supergravity). The by-now classical example of such dualities is the Maldacena conjecture [1] between closed strings propagating on AdS spaces and conformal eld theories (see [2] for some more recent developments). The same idea appeared in the context of topological theories and was pursued by Gopakumar-Vafa in studying the duality between Chern-Simons theories on S cycles and closed topological strings on the resolved conifold [3]. The topological transition was lifted to a type IIA transition by Vafa, in the seminal paper [4] where it becomes a geometric transition between D6 branes on the S of the deformed conifold and type IIA strings on the resolved conifold, with uxes. After a geometric transition of the non-singular Calabi-Yau threefold to another non-singular Calabi-Yau threefold, the Dbranes disappear and they are replaced by RR and NS uxes through the dual cycles, the eld theory being described in the large 't Hooft parameter region. It is diÆcult and cumbersome to generalize the type IIA picture, this is because not much it is known about the Chern-Simons theory for more complicated geometries (nevertheless see [5] for a worldsheet proof of the transition, which is based on the type IIA picture or [11, 12, 13] for developments of the type IIA picture). The type IIB picture can be obtained as a mirror symmetric picture. The transition becomes one between D5 branes wrapped on the P 1 cycle of the resolved conifold and type IIB on the deformed conifold with uxes. Even though it was on less rm ground as compared with the type IIA transition, the type IIB picture has been extensively extended to a large class of geometric transition dualities, for geometries which are more complicated than the conifold [14, 15, 16, 17, 19, 20, 21, 22, 23]. More recently, the type IIB geometric transition has become the backbone for a very powerful development, which relates the eld theories and matrix models. This is the Dijkgraaf-Vafa procedure [6, 7, 8] which uses eld theory, geometry and matrix model as a triangle, where the most important side of the triangle is the one connecting matrix models and eld theories. The statement becomes that perturbative results in the matrix models (in the large N limit, when only planar diagrams contribute in matrix model) give information about the non-=perturbative e ects in the eld theory. The subject is now extensively studied and the list of references is becoming very large (some other important papers are [9, 10]). In the next section I shall discuss our contribution to the study of the geometric transition, by using an approach based on brane con gurations and their lift to M theory [19, 20, 21, 22]. In the last section I shall outline possible connections between our approach and other recent lines of research, and point out some possible future developments. 6: Supersymmetry and Superstrings 1401 2 Geometric Transition from MQCD Dynamics In this section we present our method on dealing with the geometrical transition, in the type IIB approach. The main procedure is to consider N = 1 geometries