I will present three studies of string theory on twisted geometries.In the first calculation included in this dissertation we use gauge/gravity dualityto study the Coulomb branch of an unusual type of nonlocal field theory, called PuffField Theory. On the gravity side, this theory is given in terms of D3-branes intype IIB string theory with a geometric twist. While the field theory description,available in the IR limit, is a deformation of Yang-Mills gauge theory by an orderseven operator which we here compute.In the rest of this disertation we explore N = 4 super Yang-Mills (SYM) theorycompactied on a circle with S-duality and R-symmetry twists that preserve N = 6supersymmetry in 2 + 1D. It was shown that abelian theory on a flat manifold givesChern-Simons theory in the low-energy limit and here we are interested in the non-abelian counterpart. To that end, we introduce external static supersymmetric quarkand anti-quark sources into the theory and calculate the Witten Index of the resultingHilbert space of ground states on a two-torus. Using these results we compute the ac-tion of simple Wilson loops on the Hilbert space of ground states without sources. Insome cases we find disagreement between our results for the Wilson loop eigenvaluesand previous conjectures about a connection with Chern-Simons theory.The last result discussed in this dissertation demonstrates a connection betweengravitational Chern-Simons theory and N = 4 four-dimensional SYM theory com-pactified on a circle twisted by S-duality where the remaining three-manifold is notflat starting with the explicit geometric realization of S-duality in terms of (2, 0)theory
