About the duality when embedding Gopakumar-Vafa into superstring theory

Question

Vafa proposed a duality when embedding the Gopakumar-Vafa duality into superstring theory. Vafa's duality is about a correspondence N=1 supersymmetric gauge theory and superstring propagating on noncompact CY manifolds with flux turned on. I am puzzled about this relation.

a) On superstring theory side, is the total dimension six or ten? In the other words, it is the duality about ordinary string theory or topological string theory?

b) Where does the ${\cal N}=1$ gauge theory come from? IIA superstring theory compactification on conifold internal space with $D_6$ branes wrapped around 3-cycles, its geometric transition counterpart, or $D_6$ branes world volume theory?

Answer 1

a) Full Superstring Theory lives in 10D space of the form $\mathbb{R}^4 \times CY_3$, where $CY_3$ is a non-compact Calabi-Yau space, namely, deformed conifold. The duality proposed in the first paper you mentioned is about full String Theory. See, this classical paper by Vafa and friends in which a relation between Topological String Theory and certain F-terms in the effective field theory obtained by compactifying the original String Theory is discussed.

b) ${\cal N}=1$ gauge theory is the effective field theory describing the physics of non-compact part of 7-dimensional worldvolume of D6 brane filling $\mathbb{R}^4 \times S^3$, where $S^3$ is the base of $CY_3=T^*S^3$ bundle.