are independent of the superpotential (as it may be written as an integral over just ) but depend holomorphically on the twisted superpotential, and vice versa for the B model. Dualities Dualities between TSTs A number of dualities relate the above theories. The A-model and B-model on two mirror manifolds are related by mirror symmetry, which has been described as a T-duality on a three-torus. The A-model and B-model on the same manifold are conjectured to be related by S-duality, which implies the existence of several new branes, called NS branes by analogy with the NS5-brane, which wrap the same cycles as the original branes but in the opposite theory. Also a combination of the A-model and a sum of the B-model and its conjugate are related to topological M-theory by a kind of dimensional reduction. Here the degrees of freedom of the A-model and the B-models appear to not be simultaneously observable, but rather to have a relation similar to that between position and momentum in quantum mechanics. The holomorphic anomaly The sum of the B-model and its conjugate appears in the above duality because it is the theory whose low energy effective action is expected to be described by Hitchin's formalism. This is because the B-model suffers from a holomorphic anomaly, which states that the dependence on complex quantities, while classically holomorphic, receives nonholomorphic quantum corrections. In Quantum Background Independence in String Theory, Edward Witten argued that this structure is analogous to a structure that one finds geometrically quantizing the space of complex structures. Once this space has been quantized, only half of the dimensions simultaneously commute and so the number of degrees of freedom has been halved. This halving depends on an arbitrary choice, called a polarization. The conjugate model contains the missing degrees of freedom, and so by tensoring the B-model and its conjugate one reobtains all of the missing degrees of freedom and also eliminates the dependence on the arbitrary choice of polarization. Geometric transitions There are also a number of dualities that relate configurations with D-branes, which are described by open strings, to those with branes the branes replaced by flux and with the geometry described by the near-horizon geometry of the lost branes. The latter are described by closed strings. Perhaps the first such duality is the Gopakumar-Vafa duality, which was introduced by Rajesh Gopakumar and Cumrun Vafa in On the Gauge Theory/Geometry