Remodeling the B-Model

Abstract

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C 3 /Z 3 .

Torus Knots and Mirror Symmetry

“…The simplest non-trivial knot, the trefoil knot, is the (2, 3) torus knot. It is depicted, together with the more complicated (3,8) torus knot, in Fig. 2.…”

“…The framed unknot can be also studied in the B-model [3,41]. As usual in local mirror symmetry, the mirror is an algebraic curve in C * × C * , and the invariants of the framed unknot can be computed as open topological string amplitudes in this geometry using the formalism of [39,8]. The Hopf link can be also understood in the framework of topological strings and Gromov-Witten theory (see for example [24]).…”

“…Their colored U (N ) invariants can then be computed systematically by applying the topological recursion of [19] to the spectral curve, exactly as in [8]. In this description, the (P, Q) torus knot comes naturally equipped with a fixed framing of QP units, just as in Chern-Simons theory [31].…”

Topological Strings on Local Curves

“…Therefore, topological strings on X p can be described by a matrix model formalism. The matrix model/topological string correspondence was first found by [12] in some special affine backgrounds and later generalized to toric manifolds [24,5]. In the case of X p the existence of a matrix model description was proved by Eynard in [14].…”

“…The study of topological string theory on X p has led to many insights. For example, the conjecture relating toric backgrounds to matrix models stated in [24] and further refined in [5] was motivated to a large extent by the genus zero solution on X p presented in [8]. These backgrounds have been an important testing ground for recent techniques and ideas, but one has to keep in mind that they are rather unconventional in many respects.…”