We study open B-model representing D-branes on 2-cycles of local Calabi-Yau geometries. To this end we work out a reduction technique linking D-branes partition functions and multimatrix models in the case of conifold geometries so that the matrix potential is related to the complex moduli of the conifold. We study the geometric engineering of the multi-matrix models and focus on two-matrix models with bilinear couplings. We show how to solve this models in an exact way, without resorting to the customary saddle point/large N approximation. The method consists of solving the quantum equations of motion and using the flow equations of the underlying integrable hierarchy to derive explicit expressions for correlators. Finally we show how to incorporate in this formalism the description of several group of D-branes wrapped around different cycles.
We determine the gluino condensate Trλ 2 in the pure N = 1 super Yang-Mills theory (SYM) for the classical gauge groups SU (r+1), SO(2r+1), U Sp(2r) and SO(2r), by deforming the pure N = 2 SYM theory with the adjoint scalar multiplet mass, following the work by Finnell and Pouliot, and Ritz and Vainshtein. The value of the gluino condensate agrees in all cases with what was found in the weak coupling istanton calculation.The value of the gluino condensate in the pure N = 1 super Yang-Mills theory with gauge group SU (r + 1) has been calculated in the eighties by using two different methods. In the first one [1], called the strong coupling instanton calculation, one evaluates the (r+1) point function Trλ 2 (x 1 ) . . . Trλ 2 (x r+1 ) which is saturated by one instanton zero modes at short distances, and being constant due to supersymmetric Ward-Takahashi identities, it is set equal to the product of Trλ 2 by clustering.The second method [2], the so-called weak coupling istanton calculation, makes use of a deformation of the original theory by addition of some matter fields. Classically, the *
We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.
MSC: 14D15, 14H45, 83E30
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