| We discuss three examples of nonperturbative phenomena in the topological string. First, we consider the computation of amplitudes in N = 4 super Yang-Mills theory using the B model topological string as proposed by Witten. We give an argument suggesting that the computations using connected or disconnected D-instantons of the B model are in fact equivalent. Second, we formulate a conjecture that the squared modulus of the open topological string partition function can be defined nonperturbatively as the partition function of a mixed ensemble of BPS states in d = 4. This conjecture is an extension of a recent proposal for the closed topological string. In a particular example involving a non-compact Calabi-Yau threefold, we show that the conjecture passes some basic checks, and that the square of the open topological string amplitude has a natural interpretation in terms of 2-dimensional Yang-Mills theory, again generalizing known results for the closed string case. Third, we discuss an action for an abelian two-form gauge theory introduced by Hitchin which describes variations of G 2 structures in seven dimensions. Upon reducing to six dimensions this action splits into two pieces, one related to the complex structure and one related to the symplectic structure; we argue that these two pieces are related to the A and B model topological string theories. In this sense Hitchin's gauge theory is a candidate for a "topological M-theory" in seven dimensions. We also note that upon reduction to lower dimensions this two-form gauge theory naturally reduces to gauge theory descriptions of lower-dimensional gravity theories. |
