Worldsheet Realization of the Refined Topological String

A worldsheet realization of the refined topological string is proposed in terms of physical string amplitudes that compute generalized N=2 F-terms of the form F_{g,n} W^{2g}Y^{2n} in the effective supergravity action. These terms involve the chiral Weyl superfield W and a superfield Y defined as an N=2 chiral projection of a particular anti-chiral T-bar vector multiplet. In Heterotic and Type I theories, obtained upon compactification on the six-dimensional manifold K3xT2, T is the usual K\"ahler modulus of the T2 torus. These amplitudes are computed exactly at the one-loop level in string theory. They are shown to reproduce the correct perturbative part of the Nekrasov partition function in the field theory limit when expanded around an SU(2) enhancement point of the string moduli space. The two deformation parameters epsilon_- and epsilon_+ of the Omega-supergravity background are then identified with the constant field-strength backgrounds for the anti-self-dual graviphoton and self-dual gauge field of the T-bar vector multiplet, respectively.Comment: 35 pages, typos corrected, published in NP

The decoupling of Ω¯ in string theory

In this note, we study the deformation of the topological string by Ω¯. Namely, adopting the perturbative string amplitudes approach, we identify the Ω¯-deformation in terms of a physical state in the sting spectrum. We calculate the topological amplitudes Fg in heterotic string theory in the presence of the latter. In particular, we show that it is crucial to include quadratic terms in the effective action in order for Ω¯ to decouple. It turns out that this decoupling happens at the full string level, suggesting that this holds non-perturbatively