Real topological string theory

The Thesis comprises work done at SISSA (Trieste) and UAM (Madrid) under supervision of A. Uranga during academic years 2013-2016 and published in the following works. In the first one we describe the type IIA physical realization of the unoriented topological string introduced by Walcher, describe its M-theory lift, and show that it allows to compute the open and unoriented topological amplitude in terms of one-loop diagram of BPS M2-brane states. This confirms and allows to generalize the conjectured BPS integer expansion of the topological amplitude. The M-theory lift of the orientifold is freely acting on the M-theory circle, so that integer multiplicities are a weighted version of the (equivariant subsector of the) original closed oriented Gopakumar-Vafa invariants. The M-theory lift also provides new perspective on the topological tadpole cancellation conditions. We finally comment on the M-theory version of other unoriented topological strings, and clarify certain misidentifications in earlier discussions in the literature. In the second we consider the real topological string on certain non-compact toric Calabi-Yau three-folds X, in its physical realization describing an orientifold of type IIA on X with an O4-plane and a single D4-brane stuck on top. The orientifold can be regarded as a new kind of surface operator on the gauge theory with 8 supercharges arising from the singular geometry. We use the M-theory lift of this system to compute the real Gopakumar-Vafa invariants (describing wrapped M2-brane BPS states) for diverse geometries. We show that the real topological string amplitudes pick up certain signs across flop transitions, in a well-defined pattern consistent with continuity of the real BPS invariants. We further give some preliminary proposals of an intrinsically gauge theoretical description of the effect of the surface operator in the gauge theory partition function. In the third, which is in preparation, we focus on target space physics related to real topological strings, namely we discuss the physical superstring correlation functions in type I theory (or equivalently type II with orientifold) that compute real topological string amplitudes. As it turns out that direct computation presents a problem, which also affects the standard case, we consider the correlator corresponding to holomorphic derivative of the real topological amplitude $G_\chi$, at fixed worldsheet Euler character $\chi$. This corresponds in the low-energy effective action to N=2 Weyl tensor, appropriately reduced to the orientifold invariant part, and raised to power $g'=-\chi+1$. In this case, we are able to perform computation, and show that appropriate insertions in the physical string correlator give precisely the holomorphic derivative of topological amplitude. Finally, we apply this method to the standard closed oriented case as well, and prove a similar statement for the topological amplitude $F_g$, which solves a small issue affecting that computation.